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A Step-by-Step Guide to Structuring a Math Lesson Plan

$A Step-by-Step Guide to Structuring a Math Lesson Plan$

As a math teacher, I discovered that teaching math could be a challenging task, as it requires a structured approach to ensure that students understand the concepts taught. The truth is math is considered to be one of the most challenging subjects by many students, but I believe that mathematics shouldn’t be as complicated as many students find it to be.

As a teacher, we have the power to make math fun and accessible for all learners; and it all starts with how you structure your math lesson plan. With a well-organized lesson plan, you can create an engaging and effective learning experience for your students.

In this blog post, I will guide you through how I structure my math lesson plan, from setting your learning objectives to selecting instructional strategies and assessing student progress. You might also enjoy reading: 7 Every day online math resources for teachers .

Table of Contents

What Is a Lesson Plan?

A lesson plan is a teacher’s road map of what students must learn and how it will be accomplished effectively during class. Generally, a lesson plan includes the learning objectives (what students need to know), how the learning objectives will be effectively achieved (the method of delivery and procedure), and a way to assess how well the learning objectives were reached (typically through homework assignments, exit ticket, or testing).

A successful lesson plan should include three fundamental components:

Learning Objectives.
Learning activities.
Assessment to assess for student understanding.

And I believe that math lesson plans are an essential tool for us teachers because they provide us with a blueprint for how we will teach, what materials we will use, and how we will assess our students’ understanding.

Why Is a Lesson Plan Necessary For Maths?

A lesson plan is essential because it helps students and teachers understand the learning objectives. A well-designed lesson plan keeps us teachers focused and helps students know what they need to learn, how it will be delivered, and how the learning objectives will be measured.

In a study where students were given a challenging task to activate their thinking around equivalence and patterns, the teachers reported that the lesson succeeded in student learning and contributed to the class discussion.

What Makes an Effective Maths Lesson Plan?

Opening : Where you introduce your learning objectives
Body : delivering the main activities
Closing the lesson : Summary and check for your students’ understanding of the learning objectives
9 Exciting Smartboard Activities for Kindergarten in Math .
Get Ready to Challenge Your Brain With Math Puzzles .
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Keep your lesson plans relevant : Even when a teacher explains maths, some children may still don’t understand. I generally use examples that are relevant to my students. In your maths lesson, I encourage you to develop maths activities that relate to real-life problems and their day-to-day lives.

$A Step-by-Step Guide to Structuring a Math Lesson Plan$

How Do You Write a Structure For a Lesson Plan?

Structuring a math lesson plan is critical in creating successful learning experiences for your students. By setting your learning objectives, planning your maths activities, preparing your materials, assessing student understanding, and reflecting and revising your lesson plan, you can create a structured approach that ensures students learn the required math concepts effectively.

Remember that effective math lesson plans evolve over time through continued reflection and revision. Here is a step-by-step guide to structuring a math lesson plan:

Step 1: Identify the learning objective

I believe that the first step in structuring a math lesson plan is to determine what you want your students to learn. Setting learning objectives means determining what skills and concepts your students will be learning or practicing during the lesson. Generally, learning objectives provide the framework for the lesson, guiding what instructional strategies to use and how to assess student understanding.

As you’re deciding on your objective, keep in mind the level of your students and any prior knowledge they may have in the subject.

I encourage you to ensure that your learning objectives are specific, measurable, achievable, relevant, and time-bound using SMART goal setting.

For instance, instead of stating that the students will be able to ‘read a graph,’ a more specific objective could be, ‘students will be able to interpret data from a line graph and use it to make predictions.’

The table below includes the main characteristics of effective learning objectives:

Step 2: Plan your lesson activities

Once you have identified the learning objective for your students, it is time to plan out the activities for the lesson.

I encourage you to vary the types of activities to keep your students engaged and interested, including using visual aids, manipulatives, interactive technology, or group work.

You can also use other strategies, including direct instruction, problem-based learning, collaborative learning, and visual learning. I suggest you consider factors such as your students’ age and learning level, the materials at your disposal, and the time available.

For instance, I found that problem-based learning may work well for high school students and more extended lesson times, while direct instruction may work best for younger students and shorter lesson blocks.

The table below includes examples of class learning activities for a mathematics class:

Step 3: Anticipate and address student misconceptions

Misconceptions are a common challenge in math lessons, but they don’t have to derail the learning experience. To prepare for potential misconceptions, I suggest you think about where students might get stuck or misunderstand the concepts. In your lesson plan, plan out how you will address these misconceptions and ensure that all students understand the content.

Before the lesson begins, I also encourage you to ensure you have all the materials you need to conduct the lesson effectively. These materials may include textbooks, workbooks, calculators, models, and other manipulatives.

I recommend you look at the purpose of each material and ensure that it aligns with your learning objectives and instructional strategies. Additionally, ensure that your materials are accessible to all students, including those with disabilities, if possible.

Step 4: Assess student understanding

Assessment is an integral part of any lesson, as it ensures that students have understood the concepts taught.

I encourage you to plan out how you will assess your students’ understanding, be it through formative assessments (quizzes, exit tickets, etc.) or summative assessments (tests), as it will help you adjust your lesson plan as necessary and to provide helpful feedback to your students.

I believe that assessing your students’ learning is crucial in measuring the effectiveness of your math lesson plan. Consider using various assessment methods, such as formative and summative assessments, performance assessments, and diagnostics.

Typically, formative assessments include quizzes, problem sets, and verbal feedback, while summative assessments may include tests or exams.

I suggest you make sure that your assessments align with your learning objectives and instructional strategies and communicate the assessment criteria to your students.

Step 5: Reflect And Revise The Math Lesson Plan

Finally, the last step is to reflect on your lesson plan and assess any changes that need to be made. At this step, I suggest you consider the feedback you receive from your students provide and adjust your lesson plan accordingly to make your teaching experience better in the future.

I encourage you to look at your assessments, instructional strategies, and the materials used to teach. Additionally, I suggest you ask yourself what went well and what needs improvement and revise your lesson plan appropriately.

Continual reflection and revision are essential for developing successful math lesson plans that work best for your students. Check out these math lesson planning guides shared by the Colorado Department of Education .

Closing thoughts

As a teacher of math, I believe that it is important to have a structured, well-organized lesson plan to keep your students engaged and getting the most out of their lessons.

By keeping in mind the steps I have discussed, you can create an effective lesson plan that not only helps your students learn but makes math more accessible and fun.

With these tips in your back pocket, it is time to show your students just how exciting and approachable math can be.

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

5.1: Quadratic Functions

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Learning Objectives

Recognize characteristics of parabolas.
Understand how the graph of a parabola is related to its quadratic function.
Determine a quadratic function’s minimum or maximum value.
Solve problems involving a quadratic function’s minimum or maximum value.

Curved antennas, such as the ones shown in Figure $\PageIndex{1}$, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

$Satellite dishes.$

In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.

Recognizing Characteristics of Parabolas

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex . If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value . In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry . These features are illustrated in Figure $\PageIndex{2}$.

$alt$

The y-intercept is the point at which the parabola crosses the $y$-axis. The x-intercepts are the points at which the parabola crosses the $x$-axis. If they exist, the x-intercepts represent the zeros , or roots , of the quadratic function, the values of $x$ at which $y=0$.

Example $\PageIndex{1}$: Identifying the Characteristics of a Parabola

Determine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown in Figure $\PageIndex{3}$.

$Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).$

The vertex is the turning point of the graph. We can see that the vertex is at $(3,1)$. Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is $x=3$. This parabola does not cross the x-axis, so it has no zeros. It crosses the $y$-axis at $(0,7)$ so this is the y-intercept.

Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions

The general form of a quadratic function presents the function in the form

\[f(x)=ax^2+bx+c\]

where $a$, $b$, and $c$ are real numbers and $a{\neq}0$. If $a>0$, the parabola opens upward. If $a<0$, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.

The axis of symmetry is defined by $x=−\frac{b}{2a}$. If we use the quadratic formula, $x=\frac{−b{\pm}\sqrt{b^2−4ac}}{2a}$, to solve $ax^2+bx+c=0$ for the x-intercepts, or zeros, we find the value of $x$ halfway between them is always $x=−\frac{b}{2a}$, the equation for the axis of symmetry.

Figure $\PageIndex{4}$ represents the graph of the quadratic function written in general form as $y=x^2+4x+3$. In this form, $a=1$, $b=4$, and $c=3$. Because $a>0$, the parabola opens upward. The axis of symmetry is $x=−\frac{4}{2(1)}=−2$. This also makes sense because we can see from the graph that the vertical line $x=−2$ divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, $(−2,−1)$. The x-intercepts, those points where the parabola crosses the x-axis, occur at $(−3,0)$ and $(−1,0)$.

$Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.$

The standard form of a quadratic function presents the function in the form

\[f(x)=a(x−h)^2+k\]

where $(h, k)$ is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function .

As with the general form, if $a>0$, the parabola opens upward and the vertex is a minimum. If $a<0$, the parabola opens downward, and the vertex is a maximum. Figure $\PageIndex{5}$ represents the graph of the quadratic function written in standard form as $y=−3(x+2)^2+4$. Since $x–h=x+2$ in this example, $h=–2$. In this form, $a=−3$, $h=−2$, and $k=4$. Because $a<0$, the parabola opens downward. The vertex is at $(−2, 4)$.

$Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.$

The standard form is useful for determining how the graph is transformed from the graph of $y=x^2$. Figure $\PageIndex{6}$ is the graph of this basic function.

$Graph of y=x^2.$

If $k>0$, the graph shifts upward, whereas if $k<0$, the graph shifts downward. In Figure $\PageIndex{5}$, $k>0$, so the graph is shifted 4 units upward. If $h>0$, the graph shifts toward the right and if $h<0$, the graph shifts to the left. In Figure $\PageIndex{5}$, $h<0$, so the graph is shifted 2 units to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a|>1$, the point associated with a particular x-value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a|<1$, the point associated with a particular x-value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure $\PageIndex{5}$, $|a|>1$, so the graph becomes narrower.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.

\[\begin{align*} a(x−h)^2+k &= ax^2+bx+c \\[4pt] ax^2−2ahx+(ah^2+k)&=ax^2+bx+c \end{align*} \]

For the linear terms to be equal, the coefficients must be equal.

\[–2ah=b \text{, so } h=−\dfrac{b}{2a}. \nonumber\]

This is the axis of symmetry we defined earlier. Setting the constant terms equal:

\[\begin{align*} ah^2+k&=c \\ k&=c−ah^2 \\ &=c−a\cdot\Big(-\dfrac{b}{2a}\Big)^2 \\ &=c−\dfrac{b^2}{4a} \end{align*}\]

In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h)=k$.

Definitions: Forms of Quadratic Functions

A quadratic function is a function of degree two. The graph of a quadratic function is a parabola.

The general form of a quadratic function is $f(x)=ax^2+bx+c$ where $a$, $b$, and $c$ are real numbers and $a{\neq}0$.
The standard form of a quadratic function is $f(x)=a(x−h)^2+k$.
The vertex $(h,k)$ is located at \[h=–\dfrac{b}{2a},\;k=f(h)=f(\dfrac{−b}{2a}).\]

HOWTO: Write a quadratic function in a general form

Given a graph of a quadratic function, write the equation of the function in general form.

Identify the horizontal shift of the parabola; this value is $h$. Identify the vertical shift of the parabola; this value is $k$.
Substitute the values of the horizontal and vertical shift for $h$ and $k$. in the function $f(x)=a(x–h)^2+k$.
Substitute the values of any point, other than the vertex, on the graph of the parabola for $x$ and $f(x)$.
Solve for the stretch factor, $|a|$.
If the parabola opens up, $a>0$. If the parabola opens down, $a<0$ since this means the graph was reflected about the x-axis.
Expand and simplify to write in general form.

Example $\PageIndex{2}$: Writing the Equation of a Quadratic Function from the Graph

Write an equation for the quadratic function $g$ in Figure $\PageIndex{7}$ as a transformation of $f(x)=x^2$, and then expand the formula, and simplify terms to write the equation in general form.

$alt$

We can see the graph of $g$ is the graph of $f(x)=x^2$ shifted to the left 2 and down 3, giving a formula in the form $g(x)=a(x+2)^2–3$.

Substituting the coordinates of a point on the curve, such as $(0,−1)$, we can solve for the stretch factor.

\[\begin{align} −1&=a(0+2)^2−3 \\ 2&=4a \\ a&=\dfrac{1}{2} \end{align}\]

In standard form, the algebraic model for this graph is $g(x)=\dfrac{1}{2}(x+2)^2–3$.

To write this in general polynomial form, we can expand the formula and simplify terms.

\[\begin{align} g(x)&=\dfrac{1}{2}(x+2)^2−3 \\ &=\dfrac{1}{2}(x+2)(x+2)−3 \\ &=\dfrac{1}{2}(x^2+4x+4)−3 \\ &=\dfrac{1}{2}x^2+2x+2−3 \\ &=\dfrac{1}{2}x^2+2x−1 \end{align}\]

Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.

We can check our work using the table feature on a graphing utility. First enter $\mathrm{Y1=\dfrac{1}{2}(x+2)^2−3}$. Next, select $\mathrm{TBLSET}$, then use $\mathrm{TblStart=–6}$ and $\mathrm{ΔTbl = 2}$, and select $\mathrm{TABLE}$. See Table $\PageIndex{1}$

The ordered pairs in the table correspond to points on the graph.

Exercise $\PageIndex{2}$

A coordinate grid has been superimposed over the quadratic path of a basketball in Figure $\PageIndex{8}$. Find an equation for the path of the ball. Does the shooter make the basket?

$alt$

Figure $\PageIndex{8}$: Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes. (credit: modification of work by Dan Meyer)

The path passes through the origin and has vertex at $(−4, 7)$, so $h(x)=–\frac{7}{16}(x+4)^2+7$. To make the shot, $h(−7.5)$ would need to be about 4 but $h(–7.5){\approx}1.64$; he doesn’t make it.

$alt$

Identify $a$, $b$, and $c$.
Find $h$, the x-coordinate of the vertex, by substituting $a$ and $b$ into $h=–\frac{b}{2a}$.
Find $k$, the y-coordinate of the vertex, by evaluating $k=f(h)=f\Big(−\frac{b}{2a}\Big)$.

Example $\PageIndex{3}$: Finding the Vertex of a Quadratic Function

Find the vertex of the quadratic function $f(x)=2x^2–6x+7$. Rewrite the quadratic in standard form (vertex form).

The horizontal coordinate of the vertex will be at

\[\begin{align} h&=–\dfrac{b}{2a} \\ &=-\dfrac{-6}{2(2)} \\ &=\dfrac{6}{4} \\ &=\dfrac{3}{2}\end{align}\]

The vertical coordinate of the vertex will be at

\[\begin{align} k&=f(h) \\ &=f\Big(\dfrac{3}{2}\Big) \\ &=2\Big(\dfrac{3}{2}\Big)^2−6\Big(\dfrac{3}{2}\Big)+7 \\ &=\dfrac{5}{2} \end{align}\]

Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic.

\[f(x)=ax^2+bx+c \\ f(x)=2x^2−6x+7\]

Using the vertex to determine the shifts,

\[f(x)=2\Big(x–\dfrac{3}{2}\Big)^2+\dfrac{5}{2}\]

One reason we may want to identify the vertex of the parabola is that this point will inform us what the maximum or minimum value of the function is, $(k)$,and where it occurs, $(h)$.

Exercise $\PageIndex{3}$

Given the equation $g(x)=13+x^2−6x$, write the equation in general form and then in standard form.

$g(x)=x^2−6x+13$ in general form; $g(x)=(x−3)^2+4$ in standard form.

Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.

Definition: Domain and Range of a Quadratic Function

The domain of any quadratic function is all real numbers.

The range of a quadratic function written in general form $f(x)=ax^2+bx+c$ with a positive $a$ value is $f(x){\geq}f ( −\frac{b}{2a}\Big)$, or $[ f(−\frac{b}{2a}),∞ ) $; the range of a quadratic function written in general form with a negative a value is $f(x) \leq f(−\frac{b}{2a})$, or $(−∞,f(−\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x)=a(x−h)^2+k$ with a positive $a$ value is $f(x) \geq k;$ the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \leq k$.

Identify the domain of any quadratic function as all real numbers.
Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum.
Determine the maximum or minimum value of the parabola, $k$.
If the parabola has a minimum, the range is given by $f(x){\geq}k$, or $\left[k,\infty\right)$. If the parabola has a maximum, the range is given by $f(x){\leq}k$, or $\left(−\infty,k\right]$.

Example $\PageIndex{4}$: Finding the Domain and Range of a Quadratic Function

Find the domain and range of $f(x)=−5x^2+9x−1$.

As with any quadratic function, the domain is all real numbers.

Because $a$ is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x-value of the vertex.

\[\begin{align} h&=−\dfrac{b}{2a} \\ &=−\dfrac{9}{2(-5)} \\ &=\dfrac{9}{10} \end{align}\]

The maximum value is given by $f(h)$.

\[\begin{align} f(\dfrac{9}{10})&=5(\dfrac{9}{10})^2+9(\dfrac{9}{10})-1 \\&= \dfrac{61}{20}\end{align}\]

The range is $f(x){\leq}\frac{61}{20}$, or $\left(−\infty,\frac{61}{20}\right]$.

Exercise $\PageIndex{4}$

Find the domain and range of $f(x)=2\Big(x−\frac{4}{7}\Big)^2+\frac{8}{11}$.

The domain is all real numbers. The range is $f(x){\geq}\frac{8}{11}$, or $\left[\frac{8}{11},\infty\right)$.

Determining the Maximum and Minimum Values of Quadratic Functions

The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola . We can see the maximum and minimum values in Figure $\PageIndex{9}$.

$Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).$

There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.

Example $\PageIndex{5}$: Finding the Maximum Value of a Quadratic Function

A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.

Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length $L$.
What dimensions should she make her garden to maximize the enclosed area?

Let’s use a diagram such as Figure $\PageIndex{10}$ to record the given information. It is also helpful to introduce a temporary variable, $W$, to represent the width of the garden and the length of the fence section parallel to the backyard fence.

$Diagram of the garden and the backyard.$

a. We know we have only 80 feet of fence available, and $L+W+L=80$, or more simply, $2L+W=80$. This allows us to represent the width, $W$, in terms of $L$.

\[W=80−2L\]

Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so

\[\begin{align} A&=LW=L(80−2L) \\ A(L)&=80L−2L^2 \end{align}\]

This formula represents the area of the fence in terms of the variable length $L$. The function, written in general form, is

\[A(L)=−2L^2+80L\].

The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since $a$ is the coefficient of the squared term, $a=−2$, $b=80$, and $c=0$.

To find the vertex:

\[\begin{align} h& =−\dfrac{80}{2(−2)} &k&=A(20) \\ &=20 & \text{and} \;\;\;\; &=80(20)−2(20)^2 \\ &&&=800 \end{align}\]

The maximum value of the function is an area of 800 square feet, which occurs when $L=20$ feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.

This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function in Figure $\PageIndex{11}$.

$Graph of the parabolic function A(L)=-2L^2+80L, which the x-axis is labeled Length (L) and the y-axis is labeled Area (A). The vertex is at (20, 800).$

Write a quadratic equation for revenue.
Find the vertex of the quadratic equation.
Determine the y-value of the vertex.

Example $\PageIndex{6}$: Finding Maximum Revenue

The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, $p$ for price per subscription and $Q$ for quantity, giving us the equation $\text{Revenue}=pQ$.

Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently $p=30$ and $Q=84,000$. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, $p=32$ and $Q=79,000$. From this we can find a linear equation relating the two quantities. The slope will be

\[\begin{align} m&=\dfrac{79,000−84,000}{32−30} \\ &=−\dfrac{5,000}{2} \\ &=−2,500 \end{align}\]

This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the y-intercept.

\[\begin{align} Q&=−2500p+b &\text{Substitute in the point $Q=84,000$ and $p=30$} \\ 84,000&=−2500(30)+b &\text{Solve for $b$} \\ b&=159,000 \end{align}\]

This gives us the linear equation $Q=−2,500p+159,000$ relating cost and subscribers. We now return to our revenue equation.

\[\begin{align} \text{Revenue}&=pQ \\ \text{Revenue}&=p(−2,500p+159,000) \\ \text{Revenue}&=−2,500p^2+159,000p \end{align}\]

We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex.

\[\begin{align} h&=−\dfrac{159,000}{2(−2,500)} \\ &=31.8 \end{align}\]

The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.

\[\begin{align} \text{maximum revenue}&=−2,500(31.8)^2+159,000(31.8) \\ &=2,528,100 \end{align}\]

This could also be solved by graphing the quadratic as in Figure $\PageIndex{12}$. We can see the maximum revenue on a graph of the quadratic function.

$Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue ($). The vertex is at (31.80, 258100).$

Finding the x- and y-Intercepts of a Quadratic Function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the y-intercept of a quadratic by evaluating the function at an input of zero, and we find the x-intercepts at locations where the output is zero. Notice in Figure $\PageIndex{13}$ that the number of x-intercepts can vary depending upon the location of the graph.

$<div data-mt-source="1"><p class=$

Evaluate $f(0)$ to find the y-intercept.
Solve the quadratic equation $f(x)=0$ to find the x-intercepts.

Example $\PageIndex{7}$: Finding the y- and x-Intercepts of a Parabola

Find the y- and x-intercepts of the quadratic $f(x)=3x^2+5x−2$.

We find the y-intercept by evaluating $f(0)$.

\[\begin{align} f(0)&=3(0)^2+5(0)−2 \\ &=−2 \end{align}\]

So the y-intercept is at $(0,−2)$.

For the x-intercepts, we find all solutions of $f(x)=0$.

\[0=3x^2+5x−2\]

In this case, the quadratic can be factored easily, providing the simplest method for solution.

\[0=(3x−1)(x+2)\]

\[\begin{align} 0&=3x−1 & 0&=x+2 \\ x&= \frac{1}{3} &\text{or} \;\;\;\;\;\;\;\; x&=−2 \end{align}\]

So the x-intercepts are at $(\frac{1}{3},0)$ and $(−2,0)$.

By graphing the function, we can confirm that the graph crosses the $y$-axis at $(0,−2)$. We can also confirm that the graph crosses the x-axis at $\Big(\frac{1}{3},0\Big)$ and $(−2,0)$. See Figure $\PageIndex{14}$.

$Graph of a parabola which has the following intercepts (-2, 0), (1/3, 0), and (0, -2).$

Rewriting Quadratics in Standard Form

In Example $\PageIndex{7}$, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

Substitute a and $b$ into $h=−\frac{b}{2a}$.
Substitute $x=h$ into the general form of the quadratic function to find $k$.
Rewrite the quadratic in standard form using $h$ and $k$.
Solve for when the output of the function will be zero to find the x-intercepts.

Example $\PageIndex{8}$: Finding the x-Intercepts of a Parabola

Find the x-intercepts of the quadratic function $f(x)=2x^2+4x−4$.

We begin by solving for when the output will be zero.

\[0=2x^2+4x−4 \nonumber\]

Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.

\[f(x)=a(x−h)^2+k\nonumber\]

We know that $a=2$. Then we solve for $h$ and $k$.

\[\begin{align*} h&=−\dfrac{b}{2a} & k&=f(−1) \\ &=−\dfrac{4}{2(2)} & &=2(−1)^2+4(−1)−4 \\ &=−1 & &=−6 \end{align*}\]

So now we can rewrite in standard form.

\[f(x)=2(x+1)^2−6\nonumber\]

We can now solve for when the output will be zero.

\[\begin{align*} 0&=2(x+1)^2−6 \\ 6&=2(x+1)^2 \\ 3&=(x+1)^2 \\ x+1&={\pm}\sqrt{3} \\ x&=−1{\pm}\sqrt{3} \end{align*}\]

The graph has x-intercepts at $(−1−\sqrt{3},0)$ and $(−1+\sqrt{3},0)$.

We can check our work by graphing the given function on a graphing utility and observing the x-intercepts. See Figure $\PageIndex{15}$.

$Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).$

Exercise $\PageIndex{1}$

In Try It $\PageIndex{1}$, we found the standard and general form for the function $g(x)=13+x^2−6x$. Now find the y- and x-intercepts (if any).

y-intercept at $(0, 13)$, No x-intercepts

Example $\PageIndex{9}$: Solving a Quadratic Equation with the Quadratic Formula

Solve $x^2+x+2=0$.

Let’s begin by writing the quadratic formula: $x=\frac{−b{\pm}\sqrt{b^2−4ac}}{2a}$.

When applying the quadratic formula, we identify the coefficients $a$, $b$ and $c$. For the equation $x^2+x+2=0$, we have $a=1$, $b=1$, and $c=2$. Substituting these values into the formula we have:

\[\begin{align*} x&=\dfrac{−b{\pm}\sqrt{b^2−4ac}}{2a} \\ &=\dfrac{−1{\pm}\sqrt{1^2−4⋅1⋅(2)}}{2⋅1} \\ &=\dfrac{−1{\pm}\sqrt{1−8}}{2} \\ &=\dfrac{−1{\pm}\sqrt{−7}}{2} \\ &=\dfrac{−1{\pm}i\sqrt{7}}{2} \end{align*}\]

The solutions to the equation are $x=\frac{−1+i\sqrt{7}}{2}$ and $x=\frac{−1-i\sqrt{7}}{2}$ or $x=−\frac{1}{2}+\frac{i\sqrt{7}}{2}$ and $x=\frac{-1}{2}−\frac{i\sqrt{7}}{2}$.

Example $\PageIndex{10}$: Applying the Vertex and x-Intercepts of a Parabola

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation $H(t)=−16t^2+80t+40$.

When does the ball reach the maximum height? What is the maximum height of the ball? When does the ball hit the ground?

The ball reaches the maximum height at the vertex of the parabola. \[\begin{align} h &= −\dfrac{80}{2(−16)} \\ &=\dfrac{80}{32} \\ &=\dfrac{5}{2} \\ & =2.5 \end{align}\]

The ball reaches a maximum height after 2.5 seconds.

To find the maximum height, find the y-coordinate of the vertex of the parabola. \[\begin{align} k &=H(−\dfrac{b}{2a}) \\ &=H(2.5) \\ &=−16(2.5)^2+80(2.5)+40 \\ &=140 \end{align}\]

The ball reaches a maximum height of 140 feet.

To find when the ball hits the ground, we need to determine when the height is zero, $H(t)=0$.

We use the quadratic formula.

\[\begin{align} t & =\dfrac{−80±\sqrt{80^2−4(−16)(40)}}{2(−16)} \\ & = \dfrac{−80±\sqrt{8960}}{−32} \end{align} \]

Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.

\[t=\dfrac{−80-\sqrt{8960}}{−32} ≈5.458 \text{ or }t=\dfrac{−80+\sqrt{8960}}{−32} ≈−0.458 \]

The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See Figure $\PageIndex{16}$.

$CNX_Precalc_Figure_03_02_016.jpg$

When does the rock reach the maximum height?
What is the maximum height of the rock?
When does the rock hit the ocean?

a. 3 seconds b. 256 feet c. 7 seconds

Key Equations

general form of a quadratic function: $f(x)=ax^2+bx+c$
the quadratic formula: $x=\dfrac{−b{\pm}\sqrt{b^2−4ac}}{2a}$
standard form of a quadratic function: $f(x)=a(x−h)^2+k$

Key Concepts

A polynomial function of degree two is called a quadratic function.
The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
The axis of symmetry is the vertical line passing through the vertex. The zeros, or x-intercepts, are the points at which the parabola crosses the x-axis. The y-intercept is the point at which the parabola crosses the $y$-axis.
Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
The vertex can be found from an equation representing a quadratic function. .
The domain of a quadratic function is all real numbers. The range varies with the function.
A quadratic function’s minimum or maximum value is given by the y-value of the vertex.
The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.
Some quadratic equations must be solved by using the quadratic formula.
The vertex and the intercepts can be identified and interpreted to solve real-world problems.

axis of symmetry a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by $x=−\frac{b}{2a}$.

general form of a quadratic function the function that describes a parabola, written in the form $f(x)=ax^2+bx+c$, where $a,b,$ and $c$ are real numbers and a≠0.

standard form of a quadratic function the function that describes a parabola, written in the form $f(x)=a(x−h)^2+k$, where $(h, k)$ is the vertex.

vertex the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

vertex form of a quadratic function another name for the standard form of a quadratic function

zeros in a given function, the values of $x$ at which $y=0$, also called roots

Characteristics of Modern Mathematics

What are the characteristics of mathematics, especially contemporary mathematics?

I’ll consider five groups of characteristics:

Applicability and Effectiveness,
Abstraction and Generality,
Simplicity,
Logical Derivation, Axiomatic Arrangement,
Precision, Correctness, Evolution through Dialectic…

In the article What is Mathematics? , I have posited that Mathematics arises from Man’s attempt to summarize the variety of empirical phenomena that he experiences, and that Mathematics advances through the expansion and generalization of these concepts, and the improvement of these models.

But what are the characteristics of mathematics, especially contemporary mathematics?

Applicability and Effectiveness ,
Abstraction and Generality ,
Simplicity ,
Logical Derivation, Axiomatic Arrangement ,
Precision, Correctness, Evolution through Dialectic .

Though each of these characteristics presents unique pedagogical challenges and opportunities, here I’ll focus on the characterisics themselves and leave the pedagogical discussion to ( Ebr05d ). (Pedagogical matters are discussed in the article Teaching Mathematics “in Tunic” .)

1. Wide Applicability and the Effectiveness of Mathematics

General applicability is a recurring characteristic of mathematics: mathematical truth turns out to be applicable in very distinct areas of application in phenomena from across the universe to across the street. Why is this? What is it about mathematics and the concepts that it captures that causes this?

Mathematics is widely useful because the five phenomena that it studies are ubiquitous in nature and in the natural instincts of man to seek explanation, to generalize, and to attempt to improve the organization of his knowledge. As Mathematics has progressively advanced and abstracted its natural concepts, it has increased the host of subjects to which these concepts can be fruitfully applied.

2. Abstraction and Generality

Abstraction is the generalization of myriad particularities. It is the identification of the essence of the subject, together with a systematic organization around this essence. By appropriate generalizations, the many and varied details are organized into a more manageable framework. Work within particular areas of detail then becomes the area of specialists.

Put another way, the drive to abstraction is the desire to unify diverse instances under a single conceptual framework. Beginning with the abstraction of the number concept from the specific things being counted, mathematical advancement has repeatedly been achieved through insightful abstraction. These abstractions have simplified its topics, made the otherwise often overwhelming number of details more easily accessible, established foundations for orderly organization, allowed easier penetration of the subject and the development of more powerful methods.

3. Simplicity (Search for a Single Exposition), Complexity (Dense Exposition)

For the outsider looking in, it is hard to believe that simplicity is a characteristic of mathematics. Yet, for the practitioner of mathematics, simplicity is a strong part of the culture. Simplicity in what respect? The mathematician desires the simplest possible single exposition. Through greater abstraction, a single exposition is possible at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the higher level.

This is significant: although the mathematician may indeed have found his desired single exposition (for which reason he claims also that simplicity has been achieved), the reader often bears the burden of correctly and conscientiously exploring the quite significant terrain that lies beneath the abstract language of the higher-level exposition.

Thus, I believe it is the mathematician’s desire for a single exposition that leads to the attendant complexity of mathematics, especially in contemporary mathematics.

4. Logical Derivation, Axiomatic Arrangement

The modern characteristics of logical derivability and axiomatic arrangement are inherited from the ancient Greek tradition of Thales and Pythagoras and are epitomized in the presentation of Geometry by Euclid (The Elements).

It has not always been this way. The earliest mathematics was firmly empirical, rooted in man’s perception of number (quantity), space (configuration), time, and change (transformation). But by a gradual process of experience, abstraction, and generalization, concepts developed that finally separated mathematics from an empirical science to an abstract science, culminating in the axiomatic science that it is today.

It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics.

This does not mean that there is no connection with empirical reality. Quite the contrary. But it does mean that mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. These abstractions mean that mathematical fact is now established without reference to empirical reality. It may certainly be influenced by this reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics.

Why the contemporary bias for axiomatic Mathematics? Why is axiomatic mathematics so heavily favored by modern mathematics? For the same reason it was favored in the time of Euclid: in the presence of empirical difficulties, linguistic paradox, or conceptual subtlety, it is an anchor that clarifies more precisely the foundations and the manner of reasoning that underlie a mathematical subject area. Once the difficulties of establishing an axiomatic framework have been met, such a framework is favored because it helps ease the burden of many, complicated, inter-related results, justified in various ways, and inter-mixed with paradoxes, pitfalls, and impossible problems. It is favored when new results cannot be relied upon without complicated inquiries into the chains of reasoning that justify each one.

The value of axiomatic mathematics What the axiomatic approach offers is a way to bring order to a subject area, but one which requires deciding what is fundamental and what is not, what will be set up higher as a “first principle” and what will be derived from it. When it is done, however, it sets a body of knowledge into a form that can readily be presented and expanded. Appealing and effective axiom systems are then developed and refined. Their existence is a mark of the maturity of a mathematical subfield. Proof within the axiomatic framework becomes the hygiene that the community of working mathematicians adopts in order to make it easier to jointly share in the work of advancing the field. 1

Axiomatic Mathematics as Boundaries in the Wilderness In all cases of real mathematical significance, the selection of axioms is a culminating result of intensive investigations into an entire mathematical area teeming with phenomena, and the gaining of a deep understanding that results finally in identifying a good way to separate the various phenomena that have been discovered. So, though the axioms may sound trivial, in reality, the key axioms delineate substantially different structures. In this sense, axioms are boundaries that separate structurally distinct areas from each other, and, together, from the rest of “wild” mathematics.

For example: the triangle inequality is a theorem of Euclidean geometry. But it is taken as an axiom for the study of metric spaces. By doing so, this one axiom forces much of the Euclidean isometric structure. As such, it becomes a code or litmus test for the “Euclidean-ness” of a space.

Thus, from this point of view, non-axiomatic mathematics is the mathematics of discovery. Axiomatic mathematics has been tamed and made easy to learn, present, and work within. One might regard it is a fenced off area within the otherwise unmarked wilderness of other mathematical and non-mathematical phenomena. One might think of the progress of axiomatic mathematics as paralleling the way in which mankind slowly but inexorably tamed the wilderness, chopping down the trees and pushing the truly wild animals further away, while domesticating and harnessing the desirable easier ones, and setting up buildings, and walkways, farmlands, granaries, and a functioning and productive economy.

The same holds for mathematical definitions: they are attempts to tame certain phenomena and identify them as the subjects for further domestication and as able to be safely put to use, shutting out the untamed disorder of the rest of phenomena, mathematical and non-mathematical.

The down-side of axiomatics One down-side of the axiomatic presentation of mathematics is that although deep understanding is typically hidden within the axioms, the definitions of the mathematical systems have been designed precisely to make the axioms seem trivial. Which means that it is all too easy to simply state them and move on to the “meat” of the matter.

But this would be a mistake. Time spent understanding why the axioms are there, seeing them as theorems in historically prior investigations, and understanding in what phenomena they arise and where they don’t – time spent this way leads to a much deeper understanding of the significance of taking the axioms on in the first place and understanding the boundaries of the subject that the axioms establish.

Axiomatic mathematics and density of presentation For those who are interested in learning mathematics efficiently, the axiomatic presentation is most definitely the most efficient both in presentation and in “coverage density” (you get the most amount of reach and the greatest applicability in the fewest steps).

But along with “coverage density” comes conceptual density. The abstract language of axiomatic mathematics can subsume vastly different specific examples within a single abstract statement, examples which may spread across a host of historic sub-disciplines and mathematical objects of interest. You may follow the proof, and be able to turn out your own (within the framework of the axiomatics), but do you really understand the results deeply? Have you actually rubbed shoulders with the individual mathematical animals? Would you be able to recognize the right way to subdue a specific animal if it came across your path in unfamiliar circumstances (i.e. not presented in the efficient abstract language)?

5. Precision, Correctness, Evolution Through Dialectic

The Language of Mathematics. Over the course of the past three thousand years, mankind has developed sophisticated spoken and written natural languages that are highly effective for expressing a variety of moods, motives, and meanings. The language in which Mathematics is done has developed no less, and, when mastered, provides a highly efficient and powerful tool for mathematical expression, exploration, reconstruction after exploration, and communication. Its power (when used well) comes from simultaneously being precise (unambiguous) and yet concise (no superfluities, nothing unnecessary). But the language of mathematics is no exception to being used poorly. Just as any language, it can be used well or poorly.

Once correctness in mathematics is separated from empirical evidence and moved into a model-based or axiomatic framework, the touchstone for correctness becomes other, carefully selected, statements that capture the essential elements of the underlying reality: definitions, axioms, previously established theorems. The language of mathematics, and logical reasoning using that language, form the everyday working experience of mathematics.

Symbolical mathematics.

In earlier times, mathematics was in fact, fully verbal. Now, after the dramatic advances in symbolism that occurred in the Mercantile period (1500s), mathematics can be practised in an apparent symbolic shorthand, without really the need for very many words. This, however, is only a shorthand. The symbols themselves require very careful and precise definition and characterization in order for them to be used, computed with, and allow the results to be correct.

The modern language of working mathematics, as opposed to expository or pedagogical mathematics, is symbolic, and is built squarely upon the propositional logic, the first order predicate logic, and the language of sets and functions. 2 The symbolical mode is one which should be learned by the student and used by the practitioner of mathematics. It is the clearest, most unambiguous, and so most precise and therefore demanding language. But, one might say it is a “write only” language: you don’t want to read it. So, once one has written out ones ideas carefully this way, then one typically switches to one of the other two styles: direct or expository, these being the usual methods of communicating with others. 3

Evolution Through Dialectic Mathematical definitions, mathematical notions of correctness, the search for First Principles (Foundations) in Mathematics and the elaboration of areas within Mathematics have all proceeded in a dialectic fashion, alternating between periods of philosophical/foundational contentment coupled with active productive work on the one hand, and the discovery of paradoxes coupled with periods of critical review, reform, and revision on the other. This dialectical process through its history has progressively raised the level of rigor of the Mathematics of each era. 4

The level of precision in mathematics increased dramatically during the time of Cauchy, as those demanding rigor dominated mathematics. There were simply too many monsters, too many pitfalls and paradoxes from the monsters of functions in the function theory to the paradoxes and strangenesses in the Fourier analysis and infinite series, to the paradoxes of set theory and modern logic. The way out was through subtle concepts, subtle distinctions, requiring careful delineations, all of which required precision.

A Culture of Precision

Mathematical culture is that what you say should be correct. What you say should have a definition. You should know the definition and limits of what you are saying, stating, or claiming. The distinction is between mathematics being developed informally and mathematics being done more formally, with necessary and sufficient conditions stated up front and restricting the discussion to a particular class of objects.

Thus, I would argue that the modern mathematical culture of precision arises because:

mathematics has developed a precise, highly symbolic language,
mathematical concepts have developed in a dialectic manner that allows for the adaptation, adjustment and cumulative refinement of concepts based on experiences, and
mathematical reasoning is expected to be correct.

2 comments to Characteristics of Modern Mathematics

Excellent blog and you explained it well. Life is a mathematical problem. To pick up the most, you need to realize how to change over negatives into positives. Thanks!

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Module 5: Function Basics

Characteristics of functions and their graphs, learning outcomes.

Determine whether a relation represents a function.
Find function values.
Determine whether a function is one-to-one.
Use the vertical line test to identify functions.
Graph the functions in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Characteristics of Functions

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain of the relation and the set of the second components of each ordered pair is called the range of the relation. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice the first.

[latex]\left\{\left(1,2\right),\left(2,4\right),\left(3,6\right),\left(4,8\right),\left(5,10\right)\right\}[/latex]

The domain is [latex]\left\{1,2,3,4,5\right\}[/latex]. The range is [latex]\left\{2,4,6,8,10\right\}[/latex].

Note the values in the domain are also known as an input values, or values of the independent variable , and are often labeled with the lowercase letter [latex]x[/latex]. Values in the range are also known as an output values, or values of the dependent variable , and are often labeled with the lowercase letter [latex]y[/latex].

A function [latex]f[/latex] is a relation that assigns a single value in the range to each value in the domain . In other words, no [latex]x[/latex]-values are used more than once. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, [latex]\left\{1,2,3,4,5\right\}[/latex], is paired with exactly one element in the range, [latex]\left\{2,4,6,8,10\right\}[/latex].

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

[latex]\left\{\left(\text{odd},1\right),\left(\text{even},2\right),\left(\text{odd},3\right),\left(\text{even},4\right),\left(\text{odd},5\right)\right\}[/latex]

Notice that each element in the domain, [latex]\left\{\text{even,}\text{odd}\right\}[/latex] is not paired with exactly one element in the range, [latex]\left\{1,2,3,4,5\right\}[/latex]. For example, the term “odd” corresponds to three values from the domain, [latex]\left\{1,3,5\right\}[/latex] and the term “even” corresponds to two values from the range, [latex]\left\{2,4\right\}[/latex]. This violates the definition of a function, so this relation is not a function.

This image compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.

(a) This relationship is a function because each input is associated with a single output. Note that input [latex]q[/latex] and [latex]r[/latex] both give output [latex]n[/latex]. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input [latex]q[/latex] is associated with two different outputs.

A General Note: FunctionS

A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain , and the output values make up the range .

How To: Given a relationship between two quantities, determine whether the relationship is a function.

Identify the input values.
Identify the output values.
If each input value leads to only one output value, the relationship is a function. If any input value leads to two or more outputs, the relationship is not a function.

Example: Determining If Menu Price Lists Are Functions

The coffee shop menu consists of items and their prices.

Is price a function of the item?
Is the item a function of the price?

A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.

Example: Determining If Class Grade Rules Are Functions

In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.

For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.

In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

The table below lists the five greatest baseball players of all time in order of rank.

Is the rank a function of the player name?
Is the player name a function of the rank?
yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that makes it easier to work with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables [latex]h[/latex] for height and [latex]a[/latex] for age. The letters [latex]f,g[/latex], and [latex]h[/latex] are often used to represent functions just as we use [latex]x,y[/latex], and [latex]z[/latex] to represent numbers and [latex]A,B[/latex], and [latex]C[/latex] to represent sets.

[latex]\begin{align}&h\text{ is }f\text{ of }a &&\text{We name the function }f;\text{ height is a function of age}. \\ &h=f\left(a\right) &&\text{We use parentheses to indicate the function input}\text{. } \\ &f\left(a\right) &&\text{We name the function }f;\text{ the expression is read as }''f\text{ of }a''. \end{align}[/latex]

Remember, we can use any letter to name the function; we can use the notation [latex]h\left(a\right)[/latex] to show that [latex]h[/latex] depends on [latex]a[/latex]. The input value [latex]a[/latex] must be put into the function [latex]h[/latex] to get an output value. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example [latex]f\left(a+b\right)[/latex] means “first add [latex]a[/latex] and [latex]b[/latex], and the result is the input for the function [latex]f[/latex].” We must perform the operations in this order to obtain the correct result.

A General Note: Function Notation

The notation [latex]y=f\left(x\right)[/latex] defines a function named [latex]f[/latex]. This is read as [latex]``y[/latex] is a function of [latex]x.''[/latex] The letter [latex]x[/latex] represents the input value, or independent variable. The letter [latex]y[/latex], or [latex]f\left(x\right)[/latex], represents the output value, or dependent variable.

Example: Using Function Notation for Days in a Month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month in a non-leap year.

The number of days in a month is a function of the name of the month, so if we name the function [latex]f[/latex], we write [latex]\text{days}=f\left(\text{month}\right)[/latex] or [latex]d=f\left(m\right)[/latex]. The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.

For example, [latex]f\left(\text{April}\right)=30[/latex], because April has 30 days. The notation [latex]d=f\left(m\right)[/latex] reminds us that the number of days, [latex]d[/latex] (the output), is dependent on the name of the month, [latex]m[/latex] (the input).

Analysis of the Solution

We must restrict the function to non-leap years. Otherwise February would have 2 outputs and this would not be a function. Also note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

Example: Interpreting Function Notation

A function [latex]N=f\left(y\right)[/latex] gives the number of police officers, [latex]N[/latex], in a town in year [latex]y[/latex]. What does [latex]f\left(2005\right)=300[/latex] represent?

When we read [latex]f\left(2005\right)=300[/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]N[/latex], is 300. Remember, [latex]N=f\left(y\right)[/latex]. The statement [latex]f\left(2005\right)=300[/latex] tells us that in the year 2005 there were 300 police officers in the town.

Instead of a notation such as [latex]y=f\left(x\right)[/latex], could we use the same symbol for the output as for the function, such as [latex]y=y\left(x\right)[/latex], meaning “ y is a function of x ?”

Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as [latex]f[/latex], which is a rule or procedure, and the output [latex]y[/latex] we get by applying [latex]f[/latex] to a particular input [latex]x[/latex]. This is why we usually use notation such as [latex]y=f\left(x\right),P=W\left(d\right)[/latex], and so on.

Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases these values represent all we know about the relationship; other times the table provides a few select examples from a more complete relationship.

The table below lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[/latex], where [latex]D=f\left(m\right)[/latex] identifies months by an integer rather than by name.

The table below defines a function [latex]Q=g\left(n\right)[/latex]. Remember, this notation tells us that [latex]g[/latex] is the name of the function that takes the input [latex]n[/latex] and gives the output [latex]Q[/latex].

The table below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

How To: Given a table of input and output values, determine whether the table represents a function.

Identify the input and output values.
Check to see if each input value is paired with only one output value. If so, the table represents a function.

Example: Identifying Tables that Represent Functions

Which table, A, B, or C, represents a function (if any)?

a) and b) define functions. In both, each input value corresponds to exactly one output value. c) does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by a) can be represented by writing

[latex]f\left(2\right)=1,f\left(5\right)=3,\text{and }f\left(8\right)=6[/latex]

Similarly, the statements [latex]g\left(-3\right)=5,g\left(0\right)=1,\text{and }g\left(4\right)=5[/latex] represent the function in b).

c) cannot be expressed in a similar way because it does not represent a function.

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Determine whether a function is one-to-one

Figure of a bull and a graph of market prices.

Some functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b) below.

The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[/latex] and [latex]r[/latex] both give output [latex]n[/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

A General Note: One-to-One Function

A one-to-one function is a function in which each output value corresponds to exactly one input value.

Example: Determining Whether a Relationship Is a One-to-One Function

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

A circle of radius [latex]r[/latex] has a unique area measure given by [latex]A=\pi {r}^{2}[/latex], so for any input, [latex]r[/latex], there is only one output, [latex]A[/latex]. The area is a function of radius [latex]r[/latex].

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[/latex] is given by the formula [latex]A=\pi {r}^{2}[/latex]. Because areas and radii are positive numbers, there is exactly one solution: [latex]r=\sqrt{\frac{A}{\pi }}[/latex]. So the area of a circle is a one-to-one function of the circle’s radius.

Is a balance a function of the bank account number?
Is a bank account number a function of the balance?
Is a balance a one-to-one function of the bank account number?
yes, because each bank account (input) has a single balance (output) at any given time.
no, because several bank accounts (inputs) may have the same balance (output).
no, because the more than one bank account (input) can have the same balance (output).

Evaluating and Solving Functions

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\left(x\right)=5 - 3{x}^{2}[/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

How To: EVALUATE A FUNCTION Given ITS FORMula.

Replace the input variable in the formula with the value provided.
Calculate the result.

Example: Evaluating Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], evaluate [latex]h\left(4\right)[/latex].

To evaluate [latex]h\left(4\right)[/latex], we substitute the value 4 for the input variable [latex]p[/latex] in the given function.

[latex]\begin{align}h\left(p\right)&={p}^{2}+2p \\ h\left(4\right)&={\left(4\right)}^{2}+2\left(4\right) \\ &=16+8 \\ &=24 \end{align}[/latex]

Therefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[/latex].

Example: Evaluating Functions at Specific Values

For the function, [latex]f\left(x\right)={x}^{2}+3x - 4[/latex], evaluate each of the following.

[latex]f\left(2\right)[/latex]
[latex]f(a)[/latex]
[latex]f(a+h)[/latex]
[latex]\dfrac{f\left(a+h\right)-f\left(a\right)}{h}[/latex]

Replace the [latex]x[/latex] in the function with each specified value.

Because the input value is a number, 2, we can use algebra to simplify. [latex]\begin{align}f\left(2\right)&={2}^{2}+3\left(2\right)-4 \\ &=4+6 - 4 \\ &=6\hfill \end{align}[/latex]
In this case, the input value is a letter so we cannot simplify the answer any further. [latex]f\left(a\right)={a}^{2}+3a - 4[/latex]
With an input value of [latex]a+h[/latex], we must use the distributive property. [latex]\begin{align}f\left(a+h\right)&={\left(a+h\right)}^{2}+3\left(a+h\right)-4 \\[2mm] &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \end{align}[/latex]

and we know that

Now we combine the results and simplify.

[latex]\begin{align}\dfrac{f\left(a+h\right)-f\left(a\right)}{h}&=\dfrac{\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\right)-\left({a}^{2}+3a - 4\right)}{h} \\[2mm] &=\dfrac{2ah+{h}^{2}+3h}{h}\\[2mm] &=\frac{h\left(2a+h+3\right)}{h}&&\text{Factor out }h. \\[2mm] &=2a+h+3&&\text{Simplify}.\end{align}[/latex]

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], evaluate [latex]g\left(5\right)[/latex].

[latex]g\left(5\right)=\sqrt{5- 4}=1[/latex]

Example: Solving Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], solve for [latex]h\left(p\right)=3[/latex].

[latex]\begin{align}&h\left(p\right)=3\\ &{p}^{2}+2p=3 &&\text{Substitute the original function }h\left(p\right)={p}^{2}+2p. \\ &{p}^{2}+2p - 3=0 &&\text{Subtract 3 from each side}. \\ &\left(p+3\text{)(}p - 1\right)=0 &&\text{Factor}. \end{align}[/latex]

If [latex]\left(p+3\right)\left(p - 1\right)=0[/latex], either [latex]\left(p+3\right)=0[/latex] or [latex]\left(p - 1\right)=0[/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[/latex] in each case.

[latex]\begin{align}&p+3=0, &&p=-3 \\ &p - 1=0, &&p=1\hfill \end{align}[/latex]

This gives us two solutions. The output [latex]h\left(p\right)=3[/latex] when the input is either [latex]p=1[/latex] or [latex]p=-3[/latex].

Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).

We can also verify by graphing as in Figure 5. The graph verifies that [latex]h\left(1\right)=h\left(-3\right)=3[/latex] and [latex]h\left(4\right)=24[/latex].

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], solve [latex]g\left(m\right)=2[/latex].

[latex]m=8[/latex]

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[/latex] expresses a functional relationship between [latex]n[/latex] and [latex]p[/latex]. We can rewrite it to decide if [latex]p[/latex] is a function of [latex]n[/latex].

How To: Given a function in equation form, write its algebraic formula.

Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example: Finding an Equation of a Function

Express the relationship [latex]2n+6p=12[/latex] as a function [latex]p=f\left(n\right)[/latex], if possible.

To express the relationship in this form, we need to be able to write the relationship where [latex]p[/latex] is a function of [latex]n[/latex], which means writing it as [latex]p=[/latex] expression involving [latex]n[/latex].

[latex]\begin{align}&2n+6p=12\\[1mm] &6p=12 - 2n &&\text{Subtract }2n\text{ from both sides}. \\[1mm] &p=\frac{12 - 2n}{6} &&\text{Divide both sides by 6 and simplify}. \\[1mm] &p=\frac{12}{6}-\frac{2n}{6} \\[1mm] &p=2-\frac{1}{3}n \end{align}[/latex]

Therefore, [latex]p[/latex] as a function of [latex]n[/latex] is written as

[latex]p=f\left(n\right)=2-\frac{1}{3}n[/latex]

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Example: Expressing the Equation of a Circle as a Function

Does the equation [latex]{x}^{2}+{y}^{2}=1[/latex] represent a function with [latex]x[/latex] as input and [latex]y[/latex] as output? If so, express the relationship as a function [latex]y=f\left(x\right)[/latex].

First we subtract [latex]{x}^{2}[/latex] from both sides.

[latex]{y}^{2}=1-{x}^{2}[/latex]

We now try to solve for [latex]y[/latex] in this equation.

[latex]\begin{align}y&=\pm \sqrt{1-{x}^{2}} \\[1mm] &=\sqrt{1-{x}^{2}}\hspace{3mm}\text{and}\hspace{3mm}-\sqrt{1-{x}^{2}} \end{align}[/latex]

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\left(x\right)[/latex].

If [latex]x - 8{y}^{3}=0[/latex], express [latex]y[/latex] as a function of [latex]x[/latex].

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y}[/latex], if we want to express [latex]y[/latex] as a function of [latex]x[/latex], there is no simple algebraic formula involving only [latex]x[/latex] that equals [latex]y[/latex]. However, each [latex]x[/latex] does determine a unique value for [latex]y[/latex], and there are mathematical procedures by which [latex]y[/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[/latex] as a function of [latex]x[/latex], even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[/latex].

The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function [latex]P[/latex] at the input value of “goldfish.” We would write [latex]P\left(\text{goldfish}\right)=2160[/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.

How To: Given a function represented by a table, identify specific output and input values.

Find the given input in the row (or column) of input values.
Identify the corresponding output value paired with that input value.
Find the given output values in the row (or column) of output values, noting every time that output value appears.
Identify the input value(s) corresponding to the given output value.

Example: Evaluating and Solving a Tabular Function

Using the table below,

Evaluate [latex]g\left(3\right)[/latex].
Solve [latex]g\left(n\right)=6[/latex].
Evaluating [latex]g\left(3\right)[/latex] means determining the output value of the function [latex]g[/latex] for the input value of [latex]n=3[/latex]. The table output value corresponding to [latex]n=3[/latex] is 7, so [latex]g\left(3\right)=7[/latex].
Solving [latex]g\left(n\right)=6[/latex] means identifying the input values, [latex]n[/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[/latex] and [latex]n=4[/latex].

When we input 2 into the function [latex]g[/latex], our output is 6. When we input 4 into the function [latex]g[/latex], our output is also 6.

Using the table from the previous example, evaluate [latex]g\left(1\right)[/latex] .

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example: Reading Function Values from a Graph

Given the graph below,

Evaluate [latex]f\left(2\right)[/latex].
Solve [latex]f\left(x\right)=4[/latex].

Graph of a positive parabola centered at (1, 0).

Using the graph, solve [latex]f\left(x\right)=1[/latex].

[latex]x=0[/latex] or [latex]x=2[/latex]

Identify Functions Using Graphs

As we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value [latex]x[/latex] and the output value [latex]y[/latex], and we say [latex]y[/latex] is a function of [latex]x[/latex], or [latex]y=f\left(x\right)[/latex] when the function is named [latex]f[/latex]. The graph of the function is the set of all points [latex]\left(x,y\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right)[/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x -coordinate of each point is an input value and the y -coordinate of each point is the corresponding output value. For example, the black dots on the graph in the graph below tell us that [latex]f\left(0\right)=2[/latex] and [latex]f\left(6\right)=1[/latex]. However, the set of all points [latex]\left(x,y\right)[/latex] satisfying [latex]y=f\left(x\right)[/latex] is a curve. The curve shown includes [latex]\left(0,2\right)[/latex] and [latex]\left(6,1\right)[/latex] because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]x[/latex] value. The [latex]y[/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[/latex] value. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because that [latex]x[/latex] value has more than one output. A function has only one output value for each input value.

Three graphs visually showing what is and is not a function.

How To: Given a graph, use the vertical line test to determine if the graph represents a function.

Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
If there is any such line, the graph does not represent a function.
If no vertical line can intersect the curve more than once, the graph does represent a function.

Example: Applying the Vertical Line Test

Which of the graphs represent(s) a function [latex]y=f\left(x\right)?[/latex]

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x -values, a vertical line would intersect the graph at more than one point.

Does the graph below represent a function?

The Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test . Draw horizontal lines through the graph. A horizontal line includes all points with a particular [latex]y[/latex] value. The [latex]x[/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[/latex] value. If we can draw any horizontal line that intersects a graph more than once, then the graph does not represent a one-to-one function because that [latex]y[/latex] value has more than one input.

How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
If there is any such line, the function is not one-to-one.
If no horizontal line can intersect the curve more than once, the function is one-to-one.

Example: Applying the Horizontal Line Test

Consider the functions (a), and (b)shown in the graphs below.

Are either of the functions one-to-one?

The function in (a) is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Identifying Basic Toolkit Functions

In this text, we explore functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.

Key Concepts

A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\left(x\right)[/latex].
In table form, a function can be represented by rows or columns that relate to input and output values.
To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
To solve for a specific function value, we determine the input values that yield the specific output value.
An algebraic form of a function can be written from an equation.
Input and output values of a function can be identified from a table.
Relating input values to output values on a graph is another way to evaluate a function.
A function is one-to-one if each output value corresponds to only one input value.
A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.
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College Algebra. Authored by : Abramson, Jay et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
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Unit 8: Functions

About this unit, evaluating functions.

What is a function? (Opens a modal)
Worked example: Evaluating functions from equation (Opens a modal)
Worked example: Evaluating functions from graph (Opens a modal)
Evaluating discrete functions (Opens a modal)
Worked example: evaluating expressions with function notation (Opens a modal)
Evaluate functions Get 3 of 4 questions to level up!
Evaluate functions from their graph Get 3 of 4 questions to level up!
Evaluate function expressions Get 3 of 4 questions to level up!

Inputs and outputs of a function

Worked example: matching an input to a function's output (equation) (Opens a modal)
Worked example: matching an input to a function's output (graph) (Opens a modal)
Worked example: two inputs with the same output (graph) (Opens a modal)
Function inputs & outputs: equation Get 3 of 4 questions to level up!
Function inputs & outputs: graph Get 3 of 4 questions to level up!

Functions and equations

Equations vs. functions (Opens a modal)
Obtaining a function from an equation (Opens a modal)
Function rules from equations Get 3 of 4 questions to level up!

Interpreting function notation

Function notation word problem: bank (Opens a modal)
Function notation word problem: beach (Opens a modal)
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all the numbers are changing, but what doesn't change is the relationship between x and y: y is always one more than twice x. That is, y=2x+1. Finding what doesn't change "tames" the situation. So, you have tamed this problem! Yay. And if you want a fancy mathematical name for things that don’t vary, we call these things "invariants." The number of messed-up recruits is invariant, even though they are all wiggling back and forth, trying to figure out which way is right!

3) Encourage generalizations

So, of course, the next question that comes to my mind is how to generalize what you’ve already discovered: there are 15 ways that 2 mistakes can be arranged in a line of 6 recruits. What about a different number of mistakes? Or a different number of recruits? Is there some way to predict? Or, alternatively, is there some way to predict how these 15 ways of making mistakes will play out as the recruits try to settle themselves down? Which direction interests you?

4) Inquire about reasoning and rigor

The students were looking at the number of ways the recruits could line up with 2 out of n faced the wrong way: Anyway, I had a question of my own. It looks like the number of possibilities increases pretty fast, as the number of recruits increases. For example, I counted 15 possibilities in your last set (the line of six). What I wonder is this: when the numbers get that large, how you can possibly know that you've found all the possibilities? (For example, I noticed that >>>><< is missing.) The question "How do I know I've counted 'em all?" is actually quite a big deal in mathematics, as mathematicians are often called upon to find ways of counting things that nobody has ever listed (exactly like the example you are working on).

The students responded by finding a pattern for generating the lineups in a meaningful order: The way that we can prove that we have all the possibilities is that we can just add the number of places that the second wrong person could be in. For example, if 2 are wrong in a line of 6, then the first one doesn’t move and you count the space in which the second one can move in. So for the line of six, it would be 5+4+3+2+1=15. That is the way to make sure that we have all the ways. Thanks so much for giving challenges. We enjoyed thinking!

5) Work towards proof

a) The group wrote the following: When we found out that 6 recruits had 15 different starting arrangements, we needed more information. We needed to figure out how many starting positions are there for a different number of recruits.

By drawing out the arrangements for 5 recruits and 7 recruits we found out that the number of starting arrangements for the recruit number before plus that recruit number before it would equal the number of starting arrangements for that number of recruits.

We also found out that if you divide the starting arrangements by the number of recruits there is a pattern.

To which the mentor replied: Wow! I don't think (in all the years I've been hanging around mathematics) I've ever seen anyone describe this particular pattern before! Really nice! If you already knew me, you'd be able to predict what I'm about to ask, but you don't, so I have to ask it: "But why?" That is, why is this pattern (the 6, 10, 15, 21, 28…) the pattern that you find for this circumstance (two recruits wrong in lines of lengths, 4, 5, 6, 7, 8…)? Answering that—explaining why you should get those numbers and why the pattern must continue for longer lines—is doing the kind of thing that mathematics is really about.

b) Responding to students studying a circular variation of raw recruits that never settled down: This is a really interesting conclusion! How can you show that it will always continue forever and that it doesn’t matter what the original arrangement was? Have you got a reason or did you try all the cases or…? I look forward to hearing more from you.

6) Distinguish between examples and reasons

a) You have very thoroughly dealt with finding the answer to the problem you posed—it really does seem, as you put it, "safe to say" how many there will be. Is there a way that you can show that that pattern must continue? I guess I’d look for some reason why adding the new recruit adds exactly the number of additional cases that you predict. If you could say how the addition of one new recruit depends on how long the line already is, you’d have a complete proof. Want to give that a try?

b) A student, working on Amida Kuji and having provided an example, wrote the following as part of a proof: In like manner, to be given each relationship of objects in an arrangement, you can generate the arrangement itself, for no two different arrangements can have the same object relationships. The mentor response points out the gap and offers ways to structure the process of extrapolating from the specific to the general: This statement is the same as your conjecture, but this is not a proof. You repeat your claim and suggest that the example serves as a model for a proof. If that is so, it is up to you to make the connections explicit. How might you prove that a set of ordered pairs, one per pair of objects forces a unique arrangement for the entire list? Try thinking about a given object (e.g., C) and what each of its ordered pairs tells us? Try to generalize from your example. What must be true for the set of ordered pairs? Are all sets of n C2 ordered pairs legal? How many sets of n C2 ordered pairs are there? Do they all lead to a particular arrangement? Your answers to these questions should help you work toward a proof of your conjecture.

9) Encourage extensions

What you’ve done—finding the pattern, but far more important, finding the explanation (and stating it so clearly)—is really great! (Perhaps I should say "finding and stating explanations like this is real mathematics"!) Yet it almost sounded as if you put it down at the very end, when you concluded "making our project mostly an interesting coincidence." This is a truly nice piece of work!

The question, now, is "What next?" You really have completely solved the problem you set out to solve: found the answer, and proved that you’re right!

I began looking back at the examples you gave, and noticed patterns in them that I had never seen before. At first, I started coloring parts red, because they just "stuck out" as noticeable and I wanted to see them better. Then, it occurred to me that I was coloring the recruits that were back-to-back, and that maybe I should be paying attention to the ones who were facing each other, as they were "where the action was," so I started coloring them pink. (In one case, I recopied your example to do the pinks.) To be honest, I’m not sure what I’m looking for, but there was such a clear pattern of the "action spot" moving around that I thought it might tell me something new. Anything come to your minds?

10) Build a Mathematical Community

I just went back to another paper and then came back to yours to look again. There's another pattern in the table. Add the recruits and the corresponding starting arrangements (for example, add 6 and 15) and you get the next number of starting arrangements. I don't know whether this, or your 1.5, 2, 2.5, 3, 3.5… pattern will help you find out why 6, 10, 15… make sense as answers, but they might. Maybe you can work with [your classmates] who made the other observation to try to develop a complete understanding of the problem.

11) Highlight Connections

Your rule—the (n-1)+(n-2)+(n-3)+… +3+2+1 part—is interesting all by itself, as it counts the number of dots in a triangle of dots. See how?

12) Wrap Up

This is really a very nice and complete piece of work: you've stated a problem, found a solution, and given a proof (complete explanation of why that solution must be correct). To wrap it up and give it the polish of a good piece of mathematical research, I'd suggest two things.

The first thing is to extend the idea to account for all but two mistakes and the (slightly trivial) one mistake and all but one mistake. (If you felt like looking at 3 and all but 3, that'd be nice, too, but it's more work—though not a ton—and the ones that I suggested are really not more work.)

The second thing I'd suggest is to write it all up in a way that would be understandable by someone who did not know the problem or your class: clear statement of the problem, the solution, what you did to get the solution, and the proof.

I look forward to seeing your masterpiece!

Advice for Keeping a Formal Mathematics Research Logbook

As part of your mathematics research experience, you will keep a mathematics research logbook. In this logbook, keep a record of everything you do and everything you read that relates to this work. Write down questions that you have as you are reading or working on the project. Experiment. Make conjectures. Try to prove your conjectures. Your journal will become a record of your entire mathematics research experience. Don’t worry if your writing is not always perfect. Often journal pages look rough, with notes to yourself, false starts, and partial solutions. However, be sure that you can read your own notes later and try to organize your writing in ways that will facilitate your thinking. Your logbook will serve as a record of where you are in your work at any moment and will be an invaluable tool when you write reports about your research.

Ideally, your mathematics research logbook should have pre-numbered pages. You can often find numbered graph paper science logs at office supply stores. If you can not find a notebook that has the pages already numbered, then the first thing you should do is go through the entire book putting numbers on each page using pen.

• Date each entry.

• Work in pen.

• Don’t erase or white out mistakes. Instead, draw a single line through what you would like ignored. There are many reasons for using this approach:

– Your notebook will look a lot nicer if it doesn’t have scribbled messes in it.

– You can still see what you wrote at a later date if you decide that it wasn’t a mistake after all.

– It is sometimes useful to be able to go back and see where you ran into difficulties.

– You’ll be able to go back and see if you already tried something so you won’t spend time trying that same approach again if it didn’t work.

• When you do research using existing sources, be sure to list the bibliographic information at the start of each section of notes you take. It is a lot easier to write down the citation while it is in front of you than it is to try to find it at a later date.

• Never tear a page out of your notebook. The idea is to keep a record of everything you have done. One reason for pre-numbering the pages is to show that nothing has been removed.

• If you find an interesting article or picture that you would like to include in your notebook, you can staple or tape it onto a page.

Advice for Keeping a Loose-Leaf Mathematics Research Logbook

Get yourself a good loose-leaf binder, some lined paper for notes, some graph paper for graphs and some blank paper for pictures and diagrams. Be sure to keep everything that is related to your project in your binder.

– Your notebook will look a lot nicer if it does not have scribbled messes in it.

• Be sure to keep everything related to your project. The idea is to keep a record of everything you have done.

• If you find an interesting article or picture that you would like to include in your notebook, punch holes in it and insert it in an appropriate section in your binder.

ORIGINAL RESEARCH article

“homework should be…but we do not live in an ideal world”: mathematics teachers’ perspectives on quality homework and on homework assigned in elementary and middle schools.

$\r\nPedro Rosrio*$

1 Departamento de Psicologia Aplicada, Escola de Psicologia, Universidade do Minho, Braga, Portugal
2 Departamento de Psicología, Universidad de Oviedo, Oviedo, Spain

Existing literature has analyzed homework characteristics associated with academic results. Researchers and educators defend the need to provide quality homework, but there is still much to be learned about the characteristics of quality homework (e.g., purposes, type). Acknowledging that teachers play an important role in designing and assigning homework, this study explored teachers’ perspectives regarding: (i) the characteristics of quality homework and (ii) the characteristics of the homework tasks assigned. In the current study, mathematics teachers from elementary and middle schools ( N = 78) participated in focus group discussions. To enhance the trustworthiness of the findings, homework tasks assigned by 25% of the participants were analyzed for triangulation of data purposes. Data were analyzed using thematic analysis for elementary and middle school separately. Teachers discussed the various characteristics of quality homework (e.g., short assignments, adjusted to the availability of students) and shared the characteristics of the homework tasks typically assigned, highlighting a few differences (e.g., degree of individualization of homework, purposes) between these two topics. Globally, data on the homework tasks assigned were consistent with teachers’ reports about the characteristics of the homework tasks they usually assigned. Findings provide valuable insights for research and practice aimed to promote the quality of homework and consequently students’ learning and progress.

Introduction

The extensive literature on homework suggests the importance of completing homework tasks to foster students’ academic achievement (e.g., Trautwein and Lüdtke, 2009 ; Hagger et al., 2015 ; Núñez et al., 2015a ; Valle et al., 2016 ; Fernández-Alonso et al., 2017 ). However, existing research also indicate that the amount of homework assigned is not always related to high academic achievement ( Epstein and Van Voorhis, 2001 ; Epstein and Van Voorhis, 2012 ). In the words of Dettmers et al. (2010) “homework works if quality is high” (p. 467). However, further research is needed to answer the question “What is quality homework?”.

Teachers are responsible for designing and assigning homework, thus our knowledge on their perspectives about this topic and the characteristics of the homework typically assigned is expected to be a relevant contribution to the literature on the quality of homework. Moreover, data on the characteristics of homework could provide valuable information to unveil the complex network of relationships between homework and academic achievement (e.g., Cooper, 2001 ; Trautwein and Köller, 2003 ; Trautwein et al., 2009a ; Xu, 2010 ).

Thus, focusing on the perspective of mathematics teachers from elementary and middle school, the aims of the present study are twofold: to explore the characteristics of quality homework, and to identify the characteristics of the homework tasks typically assigned at these school levels. Findings may help deepen our understanding of why homework may impact differently the mathematics achievement of elementary and middle school students (see Fan et al., 2017 ).

Research Background on Homework Characteristics

Homework is a complex educational process involving a diverse set of variables that each may influence students’ academic outcomes (e.g., Corno, 2000 ; Trautwein and Köller, 2003 ; Cooper et al., 2006 ; Epstein and Van Voorhis, 2012 ). Cooper (1989 , 2001 ) presented a model outlining the factors that may potentially influence the effect of homework at the three stages of the homework process (i.e., design of the homework assignment, completion of homework and homework follow-up practices). At the first stage teachers are expected to consider class characteristics (e.g., students’ prior knowledge, grade level, number of students per class), and also variables that may influence the impact of homework on students’ outcomes, such as homework assignment characteristics. In 1989, Cooper (see also Cooper et al., 2006 ) presented a list of the characteristics of homework assignments as follows: amount (comprising homework frequency and length), purpose, skill area targeted, degree of individualization, student degree of choice, completion deadlines, and social context. Based on existing literature, Trautwein et al. (2006b) proposed a distinct organization for the assignment characteristics. The proposal included: homework frequency (i.e., how often homework assignments are prescribed to students), quality, control, and adaptivity. “Homework frequency” and “adaptivity” are similar to “amount” and “degree of individualization” in Cooper’s model, respectively. Both homework models provide a relevant theoretical framework for the present study.

Prior research has analyzed the relationship between homework variables, students’ behaviors and academic achievement, and found different results depending on the variables examined (see Trautwein et al., 2009b ; Fan et al., 2017 ). For example, while homework frequency consistently and positively predicted students’ academic achievement (e.g., Trautwein et al., 2002 ; Trautwein, 2007 ; Fernández-Alonso et al., 2015 ), findings regarding the amount of homework assigned (usually assessed by the time spent on homework) have shown mixed results (e.g., Trautwein, 2007 ; Dettmers et al., 2009 ; Núñez et al., 2015a ). Data indicated a positive association between the amount of homework and students’ academic achievement in high school (e.g., OECD, 2014a ); however, this relationship is almost null in elementary school (e.g., Cooper et al., 2006 ; Rosário et al., 2009 ). Finally, other studies reported a negative association between time spent on homework and students’ academic achievement at different school levels (e.g., Trautwein et al., 2009b ; Rosário et al., 2011 ; Núñez et al., 2015a ).

Homework purposes are among the factors that may influence the effect of homework on students’ homework behaviors and academic achievement ( Cooper, 2001 ; Trautwein et al., 2009a ; Epstein and Van Voorhis, 2012 ; Rosário et al., 2015 ). In his model Cooper (1989 , 2001 ) reported instructional purposes (i.e., practicing or reviewing, preparation, integration and extension) and non-instructional purposes (i.e., parent-child communication, fulfilling directives, punishment, and community relations). Depending on their nature, homework instructional purposes may vary throughout schooling ( Muhlenbruck et al., 2000 ; Epstein and Van Voorhis, 2001 ). For example, in elementary school, teachers are likely to use homework as an opportunity to review the content taught in class, while in secondary school (6th–12th grade), teachers are prone to use homework to prepare students for the content to be learned in subsequent classes ( Muhlenbruck et al., 2000 ). Still, studies have recently shown that practicing the content learned is the homework purpose most frequently used throughout schooling (e.g., Xu and Yuan, 2003 ; Danielson et al., 2011 ; Kaur, 2011 ; Bang, 2012 ; Kukliansky et al., 2014 ). Studies using quantitative methodologies have analyzed the role played by homework purposes in students’ effort and achievement ( Trautwein et al., 2009a ; Rosário et al., 2015 , 2018 ), and reported distinct results depending on the subject analyzed. For example, Foyle et al. (1990) found that homework assignments with the purposes of practice and preparation improved the performance of 5th-grade students’ social studies when compared with the no-homework group. However, no statistical difference was found between the two types of homework purposes analyzed (i.e., practice and preparation). When examining the homework purposes reported by 8th-grade teachers of French as a Second Language (e.g., drilling and practicing, motivating, linking school and home), Trautwein et al. (2009a) found that students in classes assigned tasks with high emphasis on motivation displayed more effort and achieved higher outcomes than their peers. On the contrary, students in classes assigned tasks with high drill and practice reported less homework effort and achievement ( Trautwein et al., 2009a ). A recent study by Rosário et al. (2015) analyzed the relationship between homework assignments with various types of purposes (i.e., practice, preparation and extension) and 6th-grade mathematics achievement. These authors reported that homework with the purpose of “extension” impacted positively on students’ academic achievement while the other two homework purposes did not.

Cooper (1989 , 2001 ) identified the “degree of individualization” as a characteristic of homework focused on the need to design homework addressing different levels of performance. For example, some students need to be assigned practice exercises with a low level of difficulty to help them reach school goals, while others need to be assigned exercises with high levels of complexity to foster their motivation for homework ( Trautwein et al., 2002 ). When there is a disparity between the level of difficulty of homework assignments and students’ skills level, students may have to spend long hours doing homework, and they may experience negative emotions or even avoid doing homework ( Corno, 2000 ). On the contrary, when homework assignments meet students’ learning needs (e.g., Bang, 2012 ; Kukliansky et al., 2014 ), both students’ homework effort and academic achievement increase (e.g., Trautwein et al., 2006a ; Zakharov et al., 2014 ). Teachers may also decide on the time given to students to complete their homework ( Cooper, 1989 ; Cooper et al., 2006 ). For example, homework may be assigned to be delivered in the following class (e.g., Kaur et al., 2004 ) or within a week (e.g., Kaur, 2011 ). However, research on the beneficial effects of each practice is still limited.

Trautwein et al. (2006b) investigated homework characteristics other than those previously reported. Their line of research analyzed students’ perception of homework quality and homework control (e.g., Trautwein et al., 2006b ; Dettmers et al., 2010 ). Findings on homework quality (e.g., level of difficulty of the mathematics exercises, Trautwein et al., 2002 ; homework “cognitively activating” and “well prepared”, Trautwein et al., 2006b , p. 448; homework selection and level of challenge, Dettmers et al., 2010 ; Rosário et al., 2018 ) varied regarding the various measures and levels of analysis considered. For example, focusing on mathematics, Trautwein et al. (2002) concluded that “demanding” exercises improved 7th-grade students’ achievement at student and class levels, while “repetitive exercises” impacted negatively on students’ achievement. Dettmers et al. (2010) found that homework assignments perceived by students as “well-prepared and interesting” (p. 471) positively predicted 9th- and 10th-grade students’ homework motivation (expectancy and value beliefs) and behavior (effort and time) at student and class level, and mathematics achievement at class level only. These authors also reported that “cognitively challenging” homework (p. 471), as perceived by students, negatively predicted students’ expectancy beliefs at both levels, and students’ homework effort at student level ( Dettmers et al., 2010 ). Moreover, this study showed that “challenging homework” significantly and positively impacted on students’ mathematics achievement at class level ( Dettmers et al., 2010 ). At elementary school, homework quality (assessed through homework selection) predicted positively 6th-grade students’ homework effort, homework performance, and mathematics achievement ( Rosário et al., 2018 ).

Finally, Trautwein and colleagues investigated the variable “homework control” perceived by middle school students and found mixed results. The works by Trautwein and Lüdtke (2007 , 2009 ) found that “homework control” predicted positively students’ homework effort in mathematics, but other studies (e.g., Trautwein et al., 2002 , 2006b ) did not predict homework effort and mathematics achievement.

The Present Study

A vast body of research indicates that homework enhances students’ academic achievement [see the meta-analysis conducted by Fan et al. (2017) ], however, maladaptive homework behaviors of students (e.g., procrastination, lack of interest in homework, failure to complete homework) may affect homework benefits ( Bembenutty, 2011a ; Hong et al., 2011 ; Rosário et al., 2019 ). These behaviors may be related to the characteristics of the homework assigned (e.g., large amount of homework, disconnect between the type and level of difficulty of homework assignments and students’ needs and abilities, see Margolis and McCabe, 2004 ; Trautwein, 2007 ).

Homework is only valuable to students’ learning when its quality is perceived by students ( Dettmers et al., 2010 ). Nevertheless, little is known about the meaning of homework quality for teachers who are responsible for assigning homework. What do teachers understand to be quality homework? To our knowledge, the previous studies exploring teachers’ perspectives on their homework practices did not relate data with quality homework (e.g., Xu and Yuan, 2003 ; Danielson et al., 2011 ; Kaur, 2011 ; Bang, 2012 ; Kukliansky et al., 2014 ). For example, Kukliansky et al. (2014) found a disconnect between middle school science teachers’ perspectives about their homework practices and their actual homework practices observed in class. However, results were not further explained.

The current study aims to explore teachers’ perspectives on the characteristics of quality homework, and on the characteristics underlying the homework tasks assigned. Findings are expected to shed some light on the role of teachers in the homework process and contribute to maximize the benefits of homework. Our results may be useful for either homework research (e.g., by informing new quantitative studies grounded on data from teachers’ perspectives) or educational practice (e.g., by identifying new avenues for teacher training and the defining of guidelines for homework practices).

This study is particularly important in mathematics for the following reasons: mathematics is among the school subjects where teachers assign the largest amount of homework (e.g., Rønning, 2011 ; Xu, 2015 ), while students continue to yield worrying school results in the subject, especially in middle and high school ( Gottfried et al., 2007 ; OECD, 2014b ). Moreover, a recent meta-analysis focused on mathematics and science homework showed that the relationship between homework and academic achievement in middle school is weaker than in elementary school ( Fan et al., 2017 ). Thus, we collected data through focus group discussions with elementary and middle school mathematics teachers in order to analyze any potential variations in their perspectives on the characteristics of quality homework, and on the characteristics of homework tasks they typically assign. Regarding the latter topic, we also collected photos of homework tasks assigned by 25% of the participating teachers in order to triangulate data and enhance the trustworthiness of our findings.

Our exploratory study was guided by the following research questions:

(1) How do elementary and middle school mathematics teachers perceive quality homework?

(2) How do elementary and middle school mathematics teachers describe the homework tasks they typically assign to students?

Materials and Methods

The study context.

Despite recommendations of the need for clear homework policies (e.g., Cooper et al., 2006 ; Bembenutty, 2011b ), Portugal has no formal guidelines for homework (e.g., concerning the frequency, length, type of tasks). Still, many teachers usually include homework as part of students’ overall grade and ask parents to monitor their children’s homework completion. Moreover, according to participants there is no specific training on homework practices for pre-service or in-service teachers.

The Portuguese educational system is organized as follows: the last two years of elementary school encompass 5th and 6th grade (10 and 11 years old), while middle school encompasses 7th, 8th, and 9th grade (12 to 14 years old). At the two school levels mentioned, mathematics is a compulsory subject and students attend three to five mathematics lessons per week depending on the duration of each class (270 min per week for Grades 5 and 6, and 225 min per week for Grades 7–9). All students are assessed by their mathematics teacher (through continuous assessment tests), and at the end of elementary and middle school levels (6th and 9th grade) students are assessed externally through a national exam that counts for 30% of the overall grade. In Portuguese schools assigning homework is a frequently used educational practice, mostly in mathematics, and usually counts toward the overall grade, ranging between 2% and 5% depending on school boards ( Rosário et al., 2018 ).

Participants

In the current study, all participants were involved in focus groups and 25% of them, randomly selected, were asked to submit photos of homework tasks assigned.

According to Morgan (1997) , to maximize the discussion among participants it is important that they share some characteristics and experiences related to the aims of the study in question. In the current study, teachers were eligible to participate when the following criteria were met: (i) they had been teaching mathematics at elementary or middle school levels for at least two years; and (ii) they would assign homework regularly, at least twice a week, in order to have enough experiences to share in the focus group.

All mathematics teachers ( N = 130) from 25 elementary and middle schools in Northern Portugal were contacted by email. The email informed teachers of the purposes and procedures of the study (e.g., inclusion criteria, duration of the session, session videotaping, selection of teachers to send photos of homework tasks assigned), and invited them to participate in the study. To facilitate recruitment, researchers scheduled focus group discussions considering participants’ availability. Of the volunteer teachers, all participants met the inclusion criteria. The research team did not allocate teachers with hierarchical relationships in the same group, as this might limit freedom of responses, affect the dynamics of the discussion, and, consequently, the outcomes ( Kitzinger, 1995 ).

Initially we conducted four focus groups with elementary school teachers (5th and 6th grade, 10 and 11 years old) and four focus groups with middle school teachers (7th, 8th, and 9th grade, 12, 13 and 14 years old). Subsequently, two additional focus group discussions (one for each school level) were conducted to ensure the saturation of data. Finally, seventy-eight mathematics teachers (61 females and 17 males; an acceptance rate of 60%) from 16 schools participated in our study (see Table 1 ). The teachers enrolled in 10 focus groups comprised of seven to nine teachers per group. Twenty teachers were randomly selected and asked to participate in the second data collection; all answered positively to our invitation (15 females and 5 males).

Table 1. Participants’ demographic information.

According to our participants, in the school context, mathematics teachers may teach one to eight classes of different grade levels. In the current research, participants were teaching one to five classes of two or three grade levels at schools in urban or near urban contexts. The participants practiced the mandatory nationwide curriculum and a continuous assessment policy.

Data Collection

We carried out this study following the recommendations of the ethics committee of the University of Minho. All teachers gave written informed consent to participate in the research in accordance with the Declaration of Helsinki. The collaboration involved participating in one focus group discussion, and, for 25% of the participants, submitting photos by email of the homework tasks assigned.

In the current study, aiming to deepen our comprehension of the research questions, focus group interviews were conducted to capture participants’ thoughts about a particular topic ( Kitzinger, 1995 ; Morgan, 1997 ). The focus groups were conducted by two members of the research team (a moderator and a field note-taker) in the first term of the school year and followed the procedure described by Krueger and Casey (2000) . To prevent mishandling the discussions and to encourage teachers to participate in the sessions, the two facilitators attended a course on qualitative research offered at their home institution specifically targeting focus group methodology.

All focus group interviews were videotaped. The sessions were held in a meeting room at the University of Minho facilities, and lasted 90 to 105 min. Before starting the discussion, teachers filled in a questionnaire with sociodemographic information, and were invited to read and sign a written informed consent form. Researchers introduced themselves, and read out the information regarding the study purpose and the focus group ground rules. Participants were ensured of the confidentiality of their responses (e.g., names and researchers’ personal notes that might link participants to their schools were deleted). Then, the investigators initiated the discussion (see Table 2 ). At the end of each focus group discussion, participants were given the opportunity to ask questions or make further contributions.

Table 2. Focus group questions.

After the focus group discussions, we randomly selected 25% of the participating teachers (i.e., 10 teachers from each school level), each asked to submit photos of the homework tasks assigned by email over the course of three weeks (period between two mathematics assessment tests). This data collection aimed to triangulate data from focus groups regarding the characteristics of homework usually assigned. To encourage participation, the research team sent teachers a friendly reminder email every evening throughout the period of data collection. In total, we received 125 photos (51% were from middle school teachers).

Data Analysis

Videotapes were used to assist the verbatim transcription of focus group data. Both focus group data and photos of the homework assignments were analyzed using thematic analysis ( Braun and Clarke, 2006 ), assisted by QSR International’s NVivo 10 software ( Richards, 2005 ). In this analysis there are no rigid guidelines on how to determine themes; to assure that the analysis is rigorous, researchers are expected to follow a consistent procedure throughout the analysis process ( Braun and Clarke, 2006 ). For the current study, to identify themes and sub-themes, we used the extensiveness of comments criterion (number of participants who express a theme, Krueger and Casey, 2000 ).

Firstly, following an inductive process one member of the research team read the first eight focus group transcriptions several times, took notes on the overall ideas of the data, and made a list of possible codes for data at a semantic level ( Braun and Clarke, 2006 ). Using a cluster analysis by word similarity procedure in Nvivo, all codes were grouped in order to identify sub-themes and themes posteriorly. All the themes and sub-themes were independently and iteratively identified and compared with the literature on homework ( Peterson and Irving, 2008 ). Then, the themes and sub-themes were compared with the homework characteristics already reported in the literature (e.g., Cooper, 1989 ; Epstein and Van Voorhis, 2001 ; Trautwein et al., 2006b ). New sub-themes emerged from participants’ discourses (i.e., “adjusted to the availability of students,” “teachers diagnose learning”), and were grouped in the themes reported in the literature. After, all themes and sub-themes were organized in a coding scheme (for an example see Table 3 ). Finally, the researcher coded the two other focus group discussions, no new information was added related to the research questions. Given that the generated patterns of data were not changed, the researcher concluded that thematic saturation was reached.

Table 3. Examples of the coding scheme.

An external auditor, trained on the coding scheme, revised all transcriptions, the coding scheme and the coding process in order to minimize researchers’ biases and increase the trustworthiness of the study ( Lincoln and Guba, 1985 ). The first author and the external auditor examined the final categorization of data and reached consensus.

Two other members of the research team coded independently the photos of the homework assignments using the same coding scheme of the focus groups. To analyze data, the researchers had to define the sub-themes “short assignments” (i.e., up to three exercises) and “long assignments” (i.e., more than three exercises). In the end, the two researchers reviewed the coding process and discussed the differences found (e.g., some exercises had several sub questions, so one of the researchers coded it as “long assignments”; see the homework sample 4 of the Supplementary Material ). However, the researchers reached consensus, deciding not to count the number of sub questions of each exercise individually, because these types of questions are related and do not require a significant amount of additional time.

Inter-rater reliability (Cohen’s Kappa) was calculated. The Cohen’s Kappa was 0.86 for the data analysis of the focus groups and 0.85 for data analysis of the photos of homework assignments, which is considered very good according to Landis and Koch (1977) . To obtain a pattern of data considering the school levels, a matrix coding query was run for each data source (i.e., focus groups and photos of homework assignments). Using the various criteria options in NVivo 10, we crossed participants’ classifications (i.e., school level attribute) and nodes and displayed the frequencies of responses for each row–column combination ( Bazeley and Jackson, 2013 ).

In the end of this process of data analysis, for establishing the trustworthiness of findings, 20 teachers (i.e., ten participants of each grade level) were randomly invited, and all agreed, to provide a member check of the findings ( Lincoln and Guba, 1985 ). Member checking involved two phases. First, teachers were asked individually to read a summary of the findings and to fill in a 5-point Likert scale (1, completely disagree; 5, completely agree) with four items: “Findings reflect my perspective regarding homework quality”; “Findings reflect my perspective regarding homework practices”; “Findings reflect what was discussed in the focus group where I participated”, and “I feel that my opinion was influenced by the other teachers during the discussion” (inverted item). Secondly, teachers were gathered by school level and asked to critically analyze and discuss whether an authentic representation was made of their perspectives regarding quality homework and homework practices ( Creswell, 2007 ).

This study explored teachers’ perspectives on the characteristics of quality homework, and on the characteristics of the homework tasks typically assigned. To report results, we used the frequency of occurrence criterion of the categories defined by Hill et al. (2005) . Each theme may be classified as “General” when all participants, or all except one, mention a particular theme; “Typical” when more than half of the cases mention a theme; “Variant” when more than 3, and less than half of the cases mention a theme; and “Rare” when the frequency is between 2 and 3 cases. In the current study, only general and typical themes were reported to discuss the most salient data.

The results section was organized by each research question. Throughout the analysis of the results, quotes from participants were presented to illustrate data. For the second research question, data from the homework assignments collected as photographs were also included.

Initial Data Screening

All participating teachers defended the importance of completing homework, arguing that homework can help students to develop their learning and to engage in school life. Furthermore, participants also agreed on the importance of delivering this message to students. Nevertheless, all teachers acknowledged that assigning homework daily present a challenge to their teaching routine because of the heavy workload faced daily (e.g., large numbers of students per class, too many classes to teach, teaching classes from different grade levels which means preparing different lessons, administrative workload).

Teachers at both school levels talked spontaneously about the nature of the tasks they usually assign, and the majority reported selecting homework tasks from a textbook. However, participants also referred to creating exercises fit to particular learning goals. Data collected from the homework assigned corroborated this information. Most of participating teachers reported that they had not received any guidance from their school board regarding homework.

How do Elementary and Middle School Teachers Perceive Quality Homework?

Three main themes were identified by elementary school teachers (i.e., instructional purposes, degree of individualization/adaptivity, and length of homework) and two were identified by middle school teachers (i.e., instructional purposes, and degree of individualization/adaptivity). Figure 1 depicts the themes and sub-themes reported by teachers in the focus groups.

Figure 1. Characteristics of quality homework reported by mathematics teachers by school level.

In all focus group discussions, all teachers from elementary and middle school mentioned “instructional purposes” as the main characteristic of quality homework. When asked to further explain the importance of this characteristic, teachers at both school levels in all focus group talked about the need for “practicing or reviewing” the content delivered in class to strengthen students’ knowledge. A teacher illustrated this idea clearly: “it is not worth teaching new content when students do not master the material previously covered” (P1 FG3). This idea was supported by participants in all focus groups; “at home they [students] have to work on the same content as those taught in class” (P1 FG7), “students have to revisit exercises and practice” (P2 FG9), “train over and over again” (P6 FG1), “practice, practice, practice” (P4 FG2).

While discussing the benefits of designing homework with the purpose of practicing the content learned, teachers at both school levels agreed on the fact that homework may be a useful tool for students to diagnose their own learning achievements while working independently. Teachers were empathetic with their peers when discussing the instrumentality of homework as a “thermometer” for students to assess their own progress. This idea was discussed in similar ways in all focus group, as the following quotation illustrates:

P2 FG1: Homework should be a bridge between class and home… students are expected to work independently, learn about their difficulties when doing homework, and check whether they understood the content.

When asked to outline other characteristics of quality homework, several elementary school teachers in all focus group mentioned that quality homework should also promote “student development” as an instructional purpose. These participants explained that homework is an instructional tool that should be designed to “foster students’ autonomy” (P9 FG4), “develop study habits and routines” (P1 FG8), and “promote organization skills and study methods” (P6 FG7). These thoughts were unanimous among participants in all focus groups. While some teachers introduced real-life examples to illustrate the ideas posited by their colleagues, others nodded their heads in agreement.

In addition, some elementary school teachers observed that homework tasks requiring transference of knowledge could help develop students’ complex thinking, a highly valued topic in the current mathematics curriculum worldwide. Teachers discussed this topic enthusiastically in two opposite directions: while some teachers defended this purpose as a characteristic of quality homework, others disagreed, as the following conversation excerpt illustrates:

P7 FG5: For me good homework would be a real challenge, like a problem-solving scenario that stimulates learning transference and develops mathematical reasoning … mathematical insight. It’s hard because it forces them [students] to think in more complex ways; still, I believe this is the type of homework with the most potential gains for them.

P3 FG5: That’s a good point, but they [students] give up easily. They just don’t do their homework. This type of homework implies competencies that the majority of students do not master…

P1 FG5: Not to mention that this type of homework takes up a lot of teaching time… explaining, checking…, and we simply don’t have time for this.

Globally, participants agreed on the potential of assigning homework with the purpose of instigating students to transfer learning to new tasks. However, participants also discussed the limitations faced daily in their teaching (e.g., number of students per class, students’ lack of prior knowledge) and concluded that homework with this purpose hinders the successful development of their lesson plans. This perspective may help explain why many participants did not perceive this purpose as a significant characteristic of quality homework. Further commenting on the characteristics of quality homework, the majority of participants at both school levels agreed that quality homework should be tailored to meet students’ learning needs. The importance of individualized homework was intensely discussed in all focus groups, and several participants suggested the need for designing homework targeted at a particular student or groups of students with common education needs. The following statements exemplifies participants’ opinions:

P3 FG3: Ideally, homework should be targeted at each student individually. For André a simple exercise, for Ana a more challenging exercise … in an ideal world homework should be tailored to students’ needs.

P6 FG6: Given the diversity of students in our classes, we may find a rainbow of levels of prior knowledge… quality homework should be as varied as our students’ needs.

As discussed in the focus groups, to foster the engagement of high-achievers in homework completion, homework tasks should be challenging enough (as reported previously by P3 FG3). However, participants at both school levels observed that their heavy daily workload prevents them from assigning individualized homework:

P1 FG1: I know it’s important to assign differentiated homework tasks, and I believe in it… but this option faces real-life barriers, such as the number of classes we have to teach, each with thirty students, tons of bureaucratic stuff we have to deal with… All this raises real-life questions, real impediments… how can we design homework tasks for individual students?

Considering this challenge, teachers from both school levels suggested that quality homework should comprise exercises with increasing levels of difficulty. This strategy would respond to the heterogeneity of students’ learning needs without assigning individualized homework tasks to each student.

While discussing individualized homework, elementary school teachers added that assignments should be designed bearing in mind students’ availability (e.g., school timetable, extracurricular activities, and exam dates). Participants noted that teachers should learn the amount of workload their students have, and should be aware about the importance of students’ well-being.

P4 FG1: If students have large amounts of homework, this could be very uncomfortable and even frustrating… They have to do homework of other subjects and add time to extracurricular activities… responding to all demands can be very stressful.

P4 FG2: I think that we have to learn about the learning context of our students, namely their limitations to complete homework in the time they have available. We all have good intentions and want them to progress, but if students do not have enough time to do their homework, this won’t work. So, quality homework would be, for example, when students have exams and the teacher gives them little or no homework at all.

The discussion about the length of homework found consensus among the elementary school teachers in all focus group in that quality homework should be “brief”. During the discussions, elementary school teachers further explained that assigning long tasks is not beneficial because “they [students] end up demotivated” (P3 FG4). Besides, “completing long homework assignments takes hours!” (P5 FG4).

How do Elementary and Middle School Teachers Describe the Homework Tasks They Typically Assign to Students?

When discussing the characteristics of the homework tasks usually assigned to their students four main themes were identified by elementary school teachers (i.e., instructional purposes, degree of individualization/adaptivity, frequency and completion deadlines), and two main themes were raised by middle school (i.e., instructional purposes, and degree of individualization/adaptivity). Figure 2 gives a general overview of the findings. Data gathered from photos added themes to findings as follows: one (i.e., length) to elementary school and two (i.e., length and completion deadlines) to middle school (see Figure 3 ).

Figure 2. Characteristics of the homework tasks usually assigned as reported by mathematics teachers.

Figure 3. Characteristics of the homework tasks assigned by mathematics teachers.

While describing the characteristics of the homework tasks usually assigned, teachers frequently felt the need to compare the quality homework characteristics previously discussed with those practices. In fact, at this stage, teachers’ discourse was often focused on the analysis of the similarities and potential discrepancies found.

The majority of teachers at both school levels in all focus group reported that they assign homework with the purpose of practicing and reviewing the materials covered earlier. Participants at both school levels highlighted the need to practice the contents covered because by the end of 6th- and 9th-grade students have to sit for a national exam for which they have to be trained. This educational context may interfere with the underlying homework purposes teachers have, as this quotation illustrates:

P3 FG3: When teaching mathematics, we set several goals, but our main focus is always the final exam they [students] have to take. I like students who think for themselves, who push themselves out of their comfort zone. However, I’m aware that they have to score high on national exams, otherwise… so, I assign homework to practice the contents covered.

Beyond assigning homework with the purpose of practicing and reviewing, middle school teachers also mentioned assigning homework with the purpose of diagnosing skills and personal development (see Figure 2 ). Many teachers reported that they use homework as a tool to diagnose students’ skills. However, several recognized that they had previously defended the importance of homework to help students to evaluate their own learning (see Figure 1 ). When discussing the latter point, participants observed the need to find out about whether students had understood the content taught in class, and to decide which changes to teaching style, homework assigned, or both may be necessary.

Participant teachers at middle school in all focus groups profusely discussed the purpose of personal development when assigning homework. In fact, not many teachers at this school level mentioned this purpose as a characteristic of quality homework (it was a variant category, so it was not reported), yet it was referred to as a cornerstone in their homework practice. Reflecting on this discrepancy, middle school teachers explained in a displeased tone that their students were expected to have developed study habits and manage their school work with autonomy and responsibility. However, this “educational scenario is rare, so I feel the need to assign homework with this aim [personal development]” (P4 FG9).

Moving further in the discussion, the majority of teachers at both school levels reported to assign whole-class homework (homework designed for the whole class with no focus on special cases). “Individualized homework requires a great amount of time to be monitored” (P1 FG6), explained several participants while recalling earlier comments. Teachers justified their position referring to the impediments already mentioned (e.g., large number of students per class, number of classes from different grade levels which means preparing different lessons). Besides, teachers discussed the challenge of coping with heterogeneous classes, as one participant noted: “the class is so diverse that it is difficult to select homework tasks to address the needs of every single student. I would like to do it…but we do not live in an ideal world” (P9 FG4).

Moreover, teachers at both school levels (see Figure 2 ) reported to assign homework according to the availability of students; still, only elementary school teachers had earlier referred to the importance of this characteristic in quality homework. When teachers were asked to elaborate on this idea, they defended the need to negotiate with students about specific homework characteristics, for example, the amount of homework and submission deadline. In some classes, matching students’ requests, teachers might assign a “weekly homework pack” (P7 FG10). This option provides students with the opportunity to complete homework according to their availability (e.g., choosing some days during the week or weekend). Teachers agreed that ‘negotiation’ fosters students’ engagement and homework compliance (e.g., “I do not agree that students do homework on weekends, but if they show their wish and actually they complete it, for me that’s okay”, P7 FG10). In addition, teachers expressed worry about their students’ often heavy workload. Many students stay in school from 8.30 am to 6.30 pm and then attend extracurricular activities (e.g., soccer training, private music lessons). These activities leave students very little free time to enjoy as they wish, as the following statement suggests:

P8 FG4: Today I talked to a group of 5th-graders which play soccer after school three times a week. They told me that sometimes they study between 10.00 and 11.00 p.m. I was astonished. How is this possible? It’s clearly too much for these kids.

Finally, elementary school teachers in all focus group referred frequency and completion deadlines as characteristics of the homework they usually assign. The majority of teachers informed that they assign homework in almost every class (i.e., teachers reported to exclude tests eves of other subjects), to be handed in the following class.

The photos of the homework assignments (see some examples in Supplementary Material ) submitted by the participating teachers served to triangulate data. The analysis showed that teachers’ discourses about the characteristics of homework assigned and the homework samples are congruent, and added information about the length of homework (elementary and middle schools) and the completion deadlines (middle school) (see Figure 3 ).

Discussion and Implications for Practice and Research

Homework research have reported teachers’ perspectives on their homework practices (e.g., Brock et al., 2007 ; Danielson et al., 2011 ; Kaur, 2011 ; Bang, 2012 ; Kukliansky et al., 2014 ), however, literature lacks research on the quality of homework. This study adds to the literature by examining the perspectives of teachers from two school levels regarding quality homework. Moreover, participants described the characteristics of the homework assignments they typically assign, which triggered the discussion about the match between the characteristics of quality homework and the tasks actually assigned. While discussing these key aspects of the homework process, the current study provides valuable information which may help deepen our understanding of the different contributions of homework to students’ learning. Furthermore, findings are expected to inform teachers and school administrators’ homework practices and, hopefully, improve the quality of students’ learning.

All teachers at both school levels valued homework as an important educational tool for their teaching practice. Consistent with the literature, participants indicated practicing or reviewing the material covered in class as the main purpose of both the homework typically assigned ( Danielson et al., 2011 ; Kaur, 2011 ) and quality homework. Despite the extended use of this homework purpose by teachers, a recent study conducted with mathematics teachers found that homework with the purpose of practicing the material covered in class did not impact significantly the academic achievement of 6th-grade students; however, homework designed with the purpose of solving problems did (extension homework) ( Rosário et al., 2015 ). Interestingly, in the current study only teachers from elementary school mentioned the homework purpose “extension” as being part of quality homework, but these teachers did not report to use it in practice (at least it was not a typical category) (see Figure 2 ). Extension homework was not referenced by middle school teachers either as quality homework or as a characteristic of homework assigned. Given that middle school students are expected to master complex math skills at this level (e.g., National Research Council and Mathematics Learning Study Committee, 2001 ), this finding may help school administrators and teachers reflect on the value and benefits of homework to students learning progress.

Moreover, teachers at both school levels stressed the use of homework as a tool to help students evaluate their own learning as a characteristic of quality homework; however, this purpose was not said to be a characteristic of the homework usually assigned. If teachers do not explicitly emphasize this homework purpose to their students, they may not perceive its importance and lose opportunities to evaluate and improve their work.

In addition, elementary school teachers identified personal development as a characteristic of quality homework. However, only middle school teachers reported assigning homework aiming to promote students’ personal development, and evaluate students’ learning (which does not imply that students evaluate their own learning). These findings are important because existing literature has highlighted the role played by homework in promoting students’ autonomy and learning throughout schooling ( Rosário et al., 2009 , 2011 ; Ramdass and Zimmerman, 2011 ; Núñez et al., 2015b ).

Globally, data show a disconnect between what teachers believe to be the characteristics of quality homework and the characteristics of the homework assigned, which should be further analyzed in depth. For example, teachers reported that middle school students lack the autonomy and responsibility expected for this school level, which translates to poor homework behaviors. In fact, contrary to what they would expect, middle school teachers reported the need to promote students’ personal development (i.e., responsibility and autonomy). This finding is consistent with the decrease of students’ engagement in academic activities found in middle school (e.g., Cleary and Chen, 2009 ; Wang and Eccles, 2012 ). This scenario may present a dilemma to middle school teachers regarding the purposes of homework. On one hand, students should have homework with more demanding purposes (e.g., extension); on another hand, students need to master work habits, responsibility and autonomy, otherwise homework may be counterproductive according to the participating teachers’ perspective.

Additionally, prior research has indicated that classes assigned challenging homework demonstrated high mathematics achievement ( Trautwein et al., 2002 ; Dettmers et al., 2010 ). Moreover, the study by Zakharov et al. (2014) found that Russian high school students from basic and advanced tracks benefited differently from two types of homework (i.e., basic short-answer questions, and open-ended questions with high level of complexity). Results showed that a high proportion of basic or complex homework exercises enhanced mathematics exam performance for students in the basic track; whereas only a high proportion of complex homework exercises enhanced mathematics exam performance for students in the advanced track. In fact, for these students, a low proportion of complex homework exercises was detrimental to their achievement. These findings, together with our own, may help explain why the relationship between homework and mathematics achievement in middle school is lower than in elementary school (see Fan et al., 2017 ). Our findings suggest the need for teachers to reflect upon the importance of assigning homework to promote students’ development in elementary school, and of assigning homework with challenging purposes as students advance in schooling to foster high academic outcomes. There is evidence that even students with poor prior knowledge need assignments with some degree of difficulty to promote their achievement (see Zakharov et al., 2014 ). It is important to note, however, the need to support the autonomy of students (e.g., providing different the types of assignments, opportunities for students to express negative feelings toward tasks, answer students’ questions) to minimize the threat that difficult homework exercises may pose to students’ sense of competence; otherwise an excessively high degree of difficulty can lead to students’ disengagement (see Patall et al., 2018 ). Moreover, teachers should consider students’ interests (e.g., which contents and types of homework tasks students like) and discuss homework purposes with their students to foster their understanding of the tasks assigned and, consequently, their engagement in homework ( Xu, 2010 , 2018 ; Epstein and Van Voorhis, 2012 ; Rosário et al., 2018 ).

We also found differences between teachers’ perspectives of quality homework and their reported homework practices concerning the degree of individualization when assigning homework. Contrary to the perspectives that quality homework stresses individual needs, teachers reported to assign homework to the whole class. In spite of the educational costs associated with assigning homework adjusted to specific students or groups of students (mentioned several times by participants), research has reported benefits for students when homework assignments match their educational needs (e.g., Cooper, 2001 ; Trautwein et al., 2006a ; Zakharov et al., 2014 ). The above-mentioned study by Zakharov et al. (2014) also shed light on this topic while supporting our participants’ suggestion to assign homework with increasing level of difficulty aiming to match the variety of students’ levels of knowledge (see also Dettmers et al., 2010 ). However, teachers did not mention this idea when discussing the characteristic of homework typically assigned. Thus, school administrators may wish to consider training teachers (e.g., using mentoring, see Núñez et al., 2013 ) to help them overcome some of the obstacles faced when designing and assigning homework targeting students’ individual characteristics and learning needs.

Another interesting finding is related to the sub-theme of homework adjusted to the availability of students. This was reported while discussing homework quality (elementary school) and characteristics of homework typically assigned (elementary and middle school). Moreover, some elementary and middle school teachers explained by email the reasons why they did not assign homework in some circumstances [e.g., eves of assessment tests of other subjects, extracurricular activities, short time between classes (last class of the day and next class in the following morning)]. These teachers’ behaviors show concern for students’ well-being, which may positively influence the relationship between students and teachers. As some participants mentioned, “students value this attitude” (P1 FG5). Thus, future research may explore how homework adjusted to the availability of students may contribute to encouraging positive behaviors, emotions and outcomes of students toward their homework.

Data gathered from the photos of the assigned homework tasks allowed a detailed analysis of the length and completion deadlines of homework. Long assignments did not match elementary school teachers’ perspectives of quality homework. However, a long homework was assigned once and aimed to help students practice the material covered for the mathematics assessment test. Here, practices diverged. Some teachers assigned this homework some weeks before and others assign it in last class before the test. For this reason, the “long term” completion deadline was not a typical category, hence not reported. Future research could consider studying the impact of this homework characteristic on students’ behaviors and academic performance.

Finally, our findings show that quality homework, according to teachers’ perspectives, requires attention to a combination of several characteristics of homework. Future studies may include measures to assess characteristics of homework other than “challenge” and “selection” already investigated ( Trautwein et al., 2006b ; Dettmers et al., 2010 ; Rosário et al., 2018 ); for example, homework adjusted to the availability of students.

Strengths and Limitations of the Study

The current study analyzed the teachers’ perspectives on the characteristics of quality homework and of the homework they typically assigned. Despite the incapability to generalize data, we believe that these findings provide important insights into the characteristics that may impact a homework assignment’s effectiveness, especially at middle school level. For example, our results showed a disconnect between teachers’ perspectives about the characteristics of quality homework and the characteristics of the homework they assign. This finding is relevant and emphasizes the need to reflect on the consistency between educational discourses and educational practices. Teachers and school administrators could consider finding opportunities to reflect on this disconnect, which may also occur in other educational practices (e.g., teacher feedback, types of questions asked in class). Present data indicate that middle school teachers reported to assign homework with the major purpose of practicing and reviewing the material, but they also aim to develop students’ responsibility and autonomy; still they neglect homework with the purpose of extension which is focused on encouraging students to display an autonomous role, solve problems and transfer the contents learned (see discussion section). Current findings also highlight the challenges and dilemmas teachers face when they assign homework, which is important to address in teachers’ training. In fact, assigning quality homework, that is, homework that works, is not an easy task for teachers and our findings provide empirical data to discuss and reflect upon its implications for research and educational practice. Although our findings cannot be generalized, still they are expected to provide important clues to enhance teachers’ homework practices in different contexts and educational settings, given that homework is among the most universal educational practices in the classroom, is a topic of public debate (e.g., some arguments against homework are related to the characteristics of the assignments, and to the malpractices in using this educational tool) and an active area of research in many countries ( Fan et al., 2017 ).

Moreover, these findings have identified some of the most common obstacles teachers struggle with; such data may be useful to school administrators when designing policies and to teacher training. The administrative obstacles (e.g., large number of students per class) reported by teachers may help understand some of the discrepancies found between teachers’ definition of quality homework and their actual homework practices (e.g., degree of individualization), and also identify which problems related to homework may require intervention. Furthermore, future research could further investigate this topic by interviewing teachers, videotaping classroom activities and discussing data in order to design new avenues of homework practices.

We share the perspective of Trautwein et al. (2006b) on the importance of mapping the characteristics of homework positively associated with students’ homework behaviors. Data from this study may inform future studies analyzing these relationships, promote adaptive homework behaviors and enhance learning.

Methodologically, this research followed rigorous procedures to increase the trustworthiness of findings, improving the validity of the study (e.g., Lincoln and Guba, 1985 ) that should be accounted for. Data from two data sources (i.e., focus groups and the homework assignments photographed) were consistent, and the member checking conducted in both phases allowed the opportunity to learn that the findings of the focus group seem to accurately reflect the overall teachers’ perspectives regarding quality homework and their homework practices.

Despite the promising contributions of this study to the body of research regarding homework practices, this specific research provides an incomplete perspective of the homework process as it has only addressed the perspectives of one of the agents involved. Future research may consider analyzing students’ perspectives about the same topic and contrast data with those of teachers. Findings are expected to help us identify the homework characteristics most highly valued by students and learn about whether they match those of teachers.

Furthermore, data from homework assignments (photos) were provided by 25% of the participating teachers and for a short period of time (i.e., three weeks in one school term). Future research may consider conducting small-scale studies by collecting data from various sources of information aiming at triangulating data (e.g., analyzing homework assignments given in class, interviewing students, conducting in-class observations) at different times of the school year. Researchers should also consider conducting similar studies in different subjects to compare data and inform teachers’ training.

Finally, our participants’ description does not include data regarding the teaching methodology followed by teachers in class. However, due to the potential interference of this variable in results, future research may consider collect and report data regarding school modality and the teaching methodology followed in class.

Homework is an instructional tool that has proved to enhance students’ learning ( Cooper et al., 2006 ; Fernández-Alonso et al., 2015 ; Valle et al., 2016 ; Fan et al., 2017 ; Rosário et al., 2018 ). Still, homework is a complex process and needs to be analyzed thoroughly. For instance, when planning and designing homework, teachers need to choose a set of homework characteristics (e.g., frequency, purposes, degree of individualization, see Cooper, 2001 ; Trautwein et al., 2006b ) considering students’ attributes (e.g., Cooper, 2001 ), which may pose a daily challenge even for experienced teachers as those of the current study. Regardless of grade level, quality homework results from the balance of a set of homework characteristics, several of which were addressed by our participants. As our data suggest, teachers need time and space to reflect on their practices and design homework tasks suited for their students. To improve the quality of homework design, school administrators may consider organizing teacher training addressing theoretical models of homework assignment and related research, discussing homework characteristics and their influence on students’ homework behaviors (e.g., amount of homework completed, homework effort), and academic achievement. We believe that this training would increase teachers’ knowledge and self-efficacy beliefs to develop homework practices best suited to their students’ needs, manage work obstacles and, hopefully, assign quality homework.

Ethics Statement

This study was reviewed and approved by the ethics committee of the University of Minho. All research participants provided written informed consent in accordance with the Declaration of Helsinki.

Author Contributions

PR and TN substantially contributed to the conception and the design of the work. TN and JC were responsible for the literature search. JC, TN, AN, and TM were responsible for the acquisition, analysis, and interpretation of data for the work. PR was also in charge of technical guidance. JN made important intellectual contribution in manuscript revision. PR, JC, and TN wrote the manuscript with valuable inputs from the remaining authors. All authors agreed for all aspects of the work and approved the version to be published.

This study was conducted at Psychology Research Centre, University of Minho, and supported by the Portuguese Foundation for Science and Technology and the Portuguese Ministry of Education and Science through national funds and when applicable co-financed by FEDER under the PT2020 Partnership Agreement (UID/PSI/01662/2013). PR was supported by the research projects EDU2013-44062-P (MINECO) and EDU2017-82984-P (MEIC). TN was supported by a Ph.D. fellowship (SFRH/BD/80405/2011) from the Portuguese Foundation for Science and Technology (FCT).

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors would like to thank Fuensanta Monroy and Connor Holmes for the English editing of the manuscript.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpsyg.2019.00224/full#supplementary-material

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Keywords : perceived quality homework, homework characteristics, math, teachers’ perspectives, elementary school, middle school, focus group, homework samples

Citation: Rosário P, Cunha J, Nunes T, Nunes AR, Moreira T and Núñez JC (2019) “Homework Should Be…but We Do Not Live in an Ideal World”: Mathematics Teachers’ Perspectives on Quality Homework and on Homework Assigned in Elementary and Middle Schools. Front. Psychol. 10:224. doi: 10.3389/fpsyg.2019.00224

Received: 12 October 2018; Accepted: 22 January 2019; Published: 19 February 2019.

Reviewed by:

Copyright © 2019 Rosário, Cunha, Nunes, Nunes, Moreira and Núñez. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Pedro Rosário, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

Characteristics of effective student talks and papers

Draft characteristics of effective communication from jmm 2013 minicourse, the math is correct., the constraints of the assignment are addressed., the audience can perceive the author’s/presenter’s content., if the target audience pays attention, they will understand the content., paying attention is easy because the content is presented in an interesting and engaging way..

One of the minicourses of the 2013 Joint Mathematics Meetings asked participants to create a categorized list of characteristics of effective communication in various contexts. To see the results, click here:

Writing of mathematics majors
Presentations of mathematics majors
Writing of future mathematics teachers
Writing (and presentations) of nonmajors
Quantitative writing and presentations

These lists have been combined with others created in the MIT Department of Mathematics to create one general list of characteristics of effective mathematics papers and talks. The result (below) reflects the communication priorities of over 30 mathematicians at a variety of institutions.

Combined list of characteristics of effective student talks and papers

The paper is mathematically correct. For example,

In a proof, the hypothesis and assumptions are clearly identified and are differentiated from the conclusion. The logic through which the latter is obtained from the former is explained and is correct.
In a mathematical model, appropriate assumptions are made and their consequences are correctly identified.
Reasoning (logical/mathematical/quantitative) is correct.
Established notation and terminology are used correctly.
Theorems are applied only when the premises are met.
The paper/talk answers the assigned question.
The paper meets the page limit or the talk ends on time. If there isn’t enough space/time to present all of the material, thoughtful choices are made about which material to omit.
The talk has been practiced or the paper has been proofread and spelling has been checked.
A research talk/paper begins by placing the research within the context of the field and concludes with suggestions for further research.
The degree of objectivity/subjectivity is appropriate to the genre (e.g., research article vs. learning log)
The way in which sources and collaborators are acknowledged is appropriate to the genre, as are other conventions such as the purpose(s) of the introduction, the formatting of important statements, and conventions of wording (e.g., “we”) and tone.
The balance of originality and use of sources is appropriate.
Nonstandard terms and notation are introduced before or as they are used. All variables are defined.
Figures are self-contained (e.g., axes are labeled, including units of measure if appropriate, and a legend is included if needed).
Nothing prevents the audience from focusing on the content (e.g., major issues of grammar, formatting, or writing/presentation style)
Type/handwriting/visuals are legible and sufficiently large, including subscripts, labels, etc.
For presentations The speaker does not block what is written on the board, either by standing in the way or by blocking boards with other boards. Slides remain visible for long enough.
For presentations Speech is sufficiently loud at all times—for example, the speaker does not become too quiet at the ends of sentences, and the speaker faces the audience when saying important points. If the speaker’s speech may be difficult for the audience to understand (due to stutter, accent, etc), strategies are used to ensure that the audience has other means for obtaining important content (e.g., more writing on the board, a handout, etc.)
The author/presenter makes clear the extent to which ideas and presentation of ideas are due to external sources.

The author/presenter is aware of the knowledge level of the target audience and anticipates what the audience will need in order to understand the material. For example,

The focus of the paper/talk is at an appropriate level of difficulty for the audience, as are the level of formality and the use of technical language. The level of the talk/paper is consistent or, if the audience is varied, the level may progress intentionally.
The author/presenter helps the audience to understand the central problem/question and to understand difficult concepts used in the solution; e.g., by anticipating and addressing possible audience confusion, by providing well-chosen examples and conceptual explanations to reinforce concepts, by using a picture iff it is worth a thousand words, and by carefully ordering information to build understanding, and by using words to explain &/or motivate that which is demonstrated with symbols.
The author/presenter provides the target audience with all the information they need, including reminders of information that should be familiar but that may not be immediately remembered. Necessary assumptions, definitions, and theorems are carefully stated, and important steps in proofs are stated explicitly. Any gaps in explanations are appropriate for the target audience.
the focus (e.g., the problem, question, or main result) is clearly stated in the title, in the introduction, and as needed throughout
the paper/talk is carefully structured, as is each part of the paper/talk
the purpose and relevance of each chunk of detail within the talk is made clear to the audience, for example by outlining or summarizing the approach before going into detail, and by using section introductions, topic and concluding sentences, and other guiding text to indicate context, relevance, and purpose.
equations are integrated with text: equations are explained rather than merely listed
explanations include “why” in addition to “how”—they are conceptual, not just procedural
The reader is guided through the logical development of arguments, which are structured to flow logically. Whenever possible, content is ordered so new content is motivated by &/or follows logically from preceding content (e.g., terms and notation are introduced in context, conjectures are motivated by example).
Writing is precise, with deliberate word choice.
Formatting of text or structure of writing on blackboards aids understanding (e.g., fractions are built up, equations are aligned, content is grouped logically into boards / slides and information is easy to find).
If non-standard notation or terminology is defined by the author/presenter, it is “good” (unambiguous, easy to use, and memorable). The choice of variables makes sense. The use of variables and notation is consistent.
For presentations The talk is robust: if an audience member misses an important point or stops to think and then starts listening again later, it is possible to figure out what’s going on (e.g., important points are written on the board, repetition is used strategically, and reminders are given as needed). Transitions are emphasized and provide sufficient context so those who are lost can start following again.
For presentations The presenter monitors audience understanding (for example by asking for questions and waiting for a response), answers questions well, and adjusts the presentation as needed.
The introduction &/or conclusion summarizes what has been done/achieved/conveyed, so the audience is not left to determine what conclusions were reached.

In other words, the presentation communicates clearly to the target audience.

The focus of the paper/talk is chosen based on an understanding of what the audience will find interesting. That focus is used to decide which content to include and which non-essential content to omit; all included content serves an important purpose.
The paper/talk begins in a way that makes the audience want to continue listening/reading, and ends in a way that provides satisfying closure.
Examples efficiently introduce basic concepts while formal treatments are reserved for the subtler or more important points that need such treatment;
The interesting aspects of a proof are presented while less-interesting details are summarized or omitted;
Structure, formatting, pacing, voice, body language, and/or eye contact are used to indicate the relative importance of presented information;
An appendix (or post-talk handout) is used for details that are needed for completeness but that are not essential to overall understanding of the topic.
The paper/talk is concise and to the point, yet has sufficient explanation and detail for the target audience.
The author/presenter elicits interest, excitement, mystery, curiosity, etc., for example by posing well-chosen questions, by commenting about what’s interesting or surprising about the content, and by giving a human dimension to the problem, for example by telling a story from personal experience or from history.
Difficult content is presented in such a way that the audience must think, but is rewarded by being able to follow the logic.
The full generality of concepts is acknowledged, and creative blending of ideas and forms connects ideas across disciplines.
The work is insightful, unique, or elegant.
Rather than merely presenting content, the author/presenter guides the audience to discovery.
For papers Formatting and guiding text enable readers to actively move around within the paper, referring back to past text and skipping forward to text of interest.
For presentations The choice of whether to use slides or blackboards is appropriate for the content and context. Slides/boardwork support delivery without distracting from it (e.g., slide design is simple and consistent with sufficient white space; writing on the board helps rather than hinders pacing of content). The presentation style is enthusiastic and interactive, and demonstrates a solid command of the content. The presenter monitors audience interest (for example by watching facial expressions), and adjusts the presentation as needed.
Nothing interferes with audience attention to and interest in the content (e.g., spelling and grammar are correct, language is straightforward, personal bearing is appropriate, use of color is purposeful, there is no gratuitous “showmanship,” and the level of formality and rigor is appropriate for the audience and context, as is the mix of generalities and specifics.)

In other words, the paper/presentation is engaging.

This list can be used to inform both assessment and teaching (e.g., by suggesting questions to guide a class discussion about giving effective presentations ).

This list was compiled by S. Ruff based on rubrics by the participants and coordinators of JMM 2013 Minicourse 7 and by H. Miller and on conversations with S. Kleiman, H. Miller, D. Roe, S. Sheffield, A. Nachmias, and J.B. Lewis.

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Characteristics And Features Of Mathematics

Characteristics And Features of Mathematics

The key features and characteristics of mathematics are:

Logical sequence
Applicability
Mathematical systems
Generalization and classification
Mathematical Language and Symbolism
Rigor and logic
Abstractness
Precision and accuracy

1. Logical Sequence

The earliest mathematics was firmly empirical (experiential), rooted in man’s perception of:

Number ( quantity )
Space ( configuration )
Change ( transformation )

Today mathematical fact can be established without reference to empirical reality. All this transpired with the gradual process of

Experience,
Abstraction, and
Generalization in the field of mathematics.

Mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. These abstractions mean that mathematical fact is certainly be influenced by reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics.

It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics and made logical sequence a main feature of mathematics.

2. Applicability

Concepts and principles become more functional and meaningful only when they are related to actual practical applications. It is the natural instinct of man to

Seek explanation,
To generalize, and
To attempt to improve the organization of his knowledge.

“Knowledge is power only when it is applied”.

Whenever knowledge is applied, especially, related to daily life situations it makes the learning of any discipline more meaningful and significant. Mathematical truth turns are applicable in very distinct areas of application from across the universe to across the street.

The study of mathematics requires the learner to apply the skills acquired to new situations.

3. Mathematical systems

A typical mathematical system has the following four parts:

Undefined terms,
Defined terms,

a. Undefined terms

In geometry or in any other mathematical system, we have to start with some terms, these terms are typically extremely simple and basic objects , so simple that they resist being described in terms of simpler objects.

Example: point, line, set, variable, plane, etc.

The choice of the undefined terms is completely arbitrary and generally facilitates the development of the structure.

b. Defined terms

For example , A triangle having 3 equal sides is an equilateral triangle. Thus to define an equilateral triangle, one should have learned the terms

c. Axioms

There can be one and only straight line joining two points.
Two lines meet at a point.
A line has one and only one mid-point.

d. Theorems

A statement that we arrive at by successive application of the rule of implication to the axioms and statements previously arrived at is called a theorem.

For instance: The rule of implication states that

the statement p implies the statement q and
If the statement p is true, then the statement q will be true.

When we apply the rule of application to the axioms we generate new statements. Again we may apply this rule to these new statements.

4. Generalization and classification

The generalization and classification of mathematics are very straightforward in contrast to others fields of thought and activity. Mathematics unites numerous findings, conclusions, assumptions, etc. under one head, and from that makes schematic arrangements and classifications.

Some of the examples of successive generalizations in mathematics are:

Number concept has itself widened from that of the whole number when it included successively negative numbers, fractional numbers, imaginary numbers, and irrational numbers.
One of the significant traits of algebra is its generalized handling of the processes of arithmetic.
In geometry, there are frequent occurrences for grouping and generating results.

When the students evolve there own concepts, theorems, definitions he/she is making generalizations.

5. Structure

If ‘S’ is a non-empty set on which one or more operations have been uniquely defined with respect to an equivalence relation, then the set S together with the operation(s) is called a mathematical system.
Using one or more of the mathematical systems like commutative, associative, or distributive properties we may create a structure.

The mathematical structure has a variety of arrangements, formations, which results in putting parts together. For instance:

A structure that comprises of a mathematical system <S; O> with one operation, in which the operation O is associative is called a semi-group.

Thus mathematics has got definite logical structures. These structures ensure the order and beauty of mathematics .

6. Mathematical Language and Symbolism

Over the course of the past 3000 years, mankind has created sophisticated spoken and written natural languages which are tremendously efficient for expressing a variety of meanings, moods, and motives.

Man has the ability to assign symbols for ideas and objects.

The language in which Mathematics is developed is no less, and, when mastered, provides a highly effective and powerful tool for

Mathematical Expression,
Exploration,
Reconstruction After Exploration, and
Communication.

Usage of symbols constructs mathematics language and makes it more elegant and precise than any other language. Mathematical language and symbols

Cut short the lengthy statements.
Help the expression of ideas or things in the exact form.
Mathematical language is free from verbosity.
Mathematical symbols help to form and clear exact expression of facts.

All mathematical operations, relations, statements are expressed using mathematical symbols. T he training that mathematics provides in the use of symbols is excellent preparation for other sciences. For example:

We can state the commutative law of addition and multiplication in a real number system in the verbal form as: ‘the addition and multiplication of two real numbers is independent of the order in which they are combined’.
In concise form as: a + b = b + a (addition), and a * b = b * a (multiplication).

“ Mathematics is the language of physical sciences and certainly no more marvelous language was ever created by the mind of man ”. - Lindsay

7. Rigor and logic

Logic is essential in mathematics; logic regulates the pattern of deductive proof through which mathematics is developed. In modern times;

The primary pedagogical objective of Mathematics is that it must be understood institutively in geometrical or physical terms.
The secondary pedagogical objective of Mathematics is its rigorous presentation.

“ Argument concludes a question, but it does not make us feel certain, or acquiesce in the contemplation of truth, except the truth also be found to be so by experience ”- AS Roger Bacon

8. Abstractness

Everything in math cannot be learned through experiences with concrete objects the same way as other disciplines. Some mathematical concepts can be learned only through their definition and they may not have a physical matching part to be extracted from.

Mathematics is abstract in the sense that mathematics does not deal with actual objects in much the same way as physics . But, in fact, mathematics questions, as a rule, cannot be settled by direct appeal to experiment.

For instance: Our whole thinking is based on the belief or assumption that there are infinitely many numbers, there are infinitely many fractions between 0 and 1and therefore, counting never stops.

Infinity is something that we can never experience and yet it is a central concept of mathematics.

Man has no way of knowing, calculating, and justifying this as a man cannot observe and count all these which makes, Infinity, abstract concept, as it is not a concept corresponding to any object that man has seen or is likely to see.

Some other examples of abstract concepts in math are:

Prime numbers,
Probability,
Limit and function,
Continuous functions etc.

These all are abstract in the sense they can be learned only through their definitions and it is not possible to provide concrete objects to correspond to such concepts.

Even some of the concepts which one argues to be concrete are also abstract. For example:

Concepts such as a line, a diagonal, a point, a circle, a ray, etc., which seen as concepts that are learned through observation of concrete instances, and as a result, they are concrete. But a figure of a circle, a dot (point), a line drawn on a board, are all mere representations of the concepts and they are not objects themselves.

9. Precision and accuracy

Mathematics is known as an ‘exact’ science because of its precision . It is perhaps the only subject that can claim certainty of results. Even when there is an emphasis on approximation, mathematical results have some degree of accuracy.

There is no midway possible in Mathematics. Mathematics is either correct or incorrect, right or wrong, it is accepted or rejected.
Mathematics can decide whether or not its conclusions are right.
Mathematicians can verify the validity of the results and convince others of their validity with consistency and objectivity.
Mathematics true or false holds for everyone who uses mathematics, at any level, not only for the expert.

Mathematical culture is that what you say should be correct. What you say should have a definition. You should know the definition and limits of what you are stating, claiming, or saying.

Thus, the modern mathematical culture of precision arises because:

Mathematics has developed a highly symbolic and precise language.
Mathematical concepts have developed in a dialectic manner that allows for the adjustment, adaptation, and cumulative refinement of concepts based on experiences.
Mathematical reasoning is expected to be correct.

Pedagogy of Mathematics

Teaching of Mathematics

Write Down Some Of The Qualities, Properties And Features Of Mathematics Notes

Characteristics And Features Of Mathematics Notes For B.Ed In English Medium

[ 9 Major ] Mathematical Features And Characteristics- Generalization And Classification | Logical Sequence | Abstractness | Applicability | Mathematical Systems | Precision And Accuracy | Mathematical Language | Symbolism | Rigor And Logic | Structure Notes And Study Material, PDF, PPT, Assignment For B.Ed 1st and 2nd Year, DELED, M.Ed, CTET, TET, Entrance Exam, All Teaching Exam Test Download Free For Pedagogy of Maths And Teaching of Mathematics Subject.

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Homework characteristics as predictors of advanced math achievement and attitude among US 12th grade students

Thomas j. smith.

1 Northern Illinois University, DeKalb, IL USA

David A. Walker

Cornelius mckenna.

2 Chana, USA

Associated Data

Upon request from the authors.

The present study examined the 2015 Trends in international mathematics and science study “Advanced” data to examine how both the type of homework assigned and how that homework was used were related to advanced mathematics achievement and attitudes toward advanced mathematics among 12th grade students in the USA. Additionally, students’ use of the internet was examined as a predictor of these outcomes. Results showed that the use of homework assignments that required students to find one or more applications of the content covered in class was a statistically significant positive predictor of both Students like learning advanced mathematics and students value advanced mathematics , while discussing homework in class was a significant, negative predictor of Students like learning advanced mathematics. Additionally, using the internet to discuss math topics with other students and to find information was significantly, positively associated with both attitudinal outcomes. Using the internet to communicate with the teacher was positively associated with Students like learning advanced mathematics scores.

Introduction

Increasing student performance in mathematics is an important goal in K-12 education, as mathematics ability is key to success and advancement in many fields. A challenge for mathematics educators is to examine the relationships among factors that may account for the performance of students so that potential improvements in instructional practices may be identified and implemented. Because of their readily malleable nature (given adequate resources and policies that permit them), two factors that are of particular interest in this realm include the use of teacher-assigned homework and the use of the internet.

Teacher-assigned homework long has been believed to affect mathematics achievement in school-aged youth. Cooper et al. ( 2006 , p. 1) define homework as “any task assigned by schoolteachers intended for students to carry out during nonschool hours.” However, the complex interactions of various factors that influence the effectiveness of homework practices has made the study of homework practices and their effect on student academic achievement difficult for researchers to address. Cooper ( 1989a ) categorizes these factors into three broad categories: (1) characteristics of the homework assignments, (2) teacher factors, and (3) home community factors, and further concludes that homework generally has a positive effect on achievement. An early meta-analysis by Walberg et al. ( 1985 ) examining the effect of the amount of homework assigned per week on achievement concluded that this effect is large in magnitude. A meta-analysis carried out by Cooper ( 1989a , and summarized in Cooper 1989b ) reviewed nearly 120 studies that examined the effects of homework on student achievement. These studies were divided into three groups. The first group of studies simply compared the achievement of students who were assigned homework with students who were not assigned homework. Of the 20 most recent of these studies, 14 showed positive effects of homework on achievement, with an average effect size of d = 0.21 for the full set of 20 studies. The second set of 14 studies (Cooper 1989a , 1989b ) compared the effects of assigned homework on achievement to in-class supervised study. The 14 studies yielded an average effect size of d = 0.09. The third set of 18 studies (Cooper 1989a , b ) found that the average correlation between time spent on homework and subsequent achievement across the 27 samples was r = 0.19. An update on this meta-analysis carried out by Cooper et al. ( 2006 ), which included studies completed after 1989, found evidence for a positive influence of homework on academic achievement, whether achievement was measured by grades or by standardized tests. Among the studies in the Cooper et al. meta-analysis that involved control procedures, effect sizes ranged from d = 0.39 to 0.97. The authors also note that the effects of homework on achievement among K-6 students are weakest, and are stronger at the 7–12 grade levels.

De Jong et al. ( 2000 ) reported that the amount of homework (defined as the number of homework tasks assigned) was the only homework variable related to achievement. However, Cooper et al. ( 2006 ) report that time spent on homework was not linearly related to achievement, and that the effectiveness of homework diminished if too much is given. A study by Trautwein et al. ( 2002 ) examined data from N = 1976 German 7 th grade students found that the frequency of homework was positively related to math achievement, but the length of homework assignments had no significant effect on achievement. The authors also found that teacher monitoring of homework completion was not significantly related to math achievement. Fyfe ( 2016 ) found that teacher feedback on homework assignments resulted in higher test performance, but only for students with a low level of prior knowledge.

Besides achievement, a construct of interest when examining the effects of homework includes student attitude. As reported in Cooper et al. ( 2006 ), some studies (e.g., Covington 1998 ; Deslandes and Cloutier 2002 ; Harris et al. 1993 ; Jackson 2003 ) suggest that girls may exert greater effort on assigned homework and may possess more positive attitudes toward homework than boys. These studies, however, focus on attitudes toward homework as an outcome, rather than examining how homework is related to attitudes toward mathematics. Singh et al. ( 2002 )—in a structural equation model they constructed examining the effects of motivation, interest, and engagement on math and science achievement—found that time spent on math homework was significantly and positively associated with attitude toward math, and a similar positive relationship was observed between time spent on science homework and attitude toward science.

Existing studies involving the effects of homework generally have examined samples of students from a broad range of abilities. Few existing studies, however, have examined the effects of homework among students who are enrolled in advanced math or science courses and, further, few studies have examined how specific characteristics of assigned homework might relate to achievement or attitudes.

Student use of the internet

Another student activity that might have some impact on student achievement and attitudes and that is closely linked to student-completed homework is student use of the internet for learning purposes. A report by the U.S. Department of Education on teachers’ use of technology for school and homework assignments among children in grades 3–12 (Gray and Lewis, 2020 ) that was completed in response to a request by the U.S. Congress found that 77% of teachers assigned technology-based homework to their students. Among these teachers, a strong majority (86%) reported that their students encountered little or no difficulty in these tasks due to unfamiliarity with technology. Clearly, technology is becoming an integral component of class and homework activities, an observation that became clearly evident during stay-at-home conditions compelled by the COVID-19 epidemic. Cheung and Slavin ( 2013 ) summarize much of the research on technology use in education in their review of 74 such studies, where they concluded that technological innovations result in positive but small effects on student test scores. Roschelle et al. ( 2016 ) found that the use of an immediate (in the moment) online mathematics intervention afforded students with documented previous lower achievement in mathematics greater benefit; though the effect was small. However, Agasisti et al.’s ( 2017 ) analyses of PISA data across numerous countries indicated that intense use of information and communication technology (ICT) is associated, in most countries, with lower test scores. They suggest that the competency of the teacher in facilitating ICT use may be key to its success.

With regard to the effects of technology on attitudes toward mathematics, some research has been carried out that examines how technology integration is related to math attitudes. A meta-analysis carried out by Higgins et al. ( 2019 ) reviewed 24 studies involving 4522 participants found that, overall, technology had a significant, positive overall impact on student motivation and attitudes. The examined studies, however, focused on specific technological interventions, rather than students’ use of technology use (and specifically use of the internet) for the purpose of learning mathematics. A study by White and Loong ( 2004 ) surveyed pre-adolescent students and found that a greater preference for finding mathematics material on the internet (compared to textbooks) was related to significantly lower perceived value of mathematics. White and Loong’s study, however, focused on preference for the internet as a learning modality rather than the extent to which the internet actually was used.

Current study

The current study extends this research into both the effects of homework activities and the effects of internet use on student achievement as well as student attitude outcomes. Specifically, the purpose of the present study was to examine how (1) characteristics of assigned homework (type of homework, and how that homework was used) and (2) use of the internet in class activities was related to mathematics achievement and attitudes toward advanced mathematics among high school students in the US enrolled in advanced mathematics courses. To this end, a select sample of students (12th graders enrolled in advanced mathematics courses) was used, and the following research questions addressed:

To what extent is the type of homework assigned by the advanced mathematics teacher and the teacher’s use of homework related to students’ advanced math achievement?

To what extent is the type of homework assigned by the advanced mathematics teacher and the teacher’s use of homework related to students’ attitude toward advanced mathematics?

To what extent is use of the internet for class activities related to students’ advanced math achievement?

To what extent is use of the internet for class activities related to students’ attitude toward advanced mathematics?

The present study used the Trends in International Mathematics and Science Study (TIMSS) 2015 “Advanced” data set (Martin et al. 2016 ). The TIMSS Advanced data contain information on 12th grade students enrolled in advanced science or mathematics courses, including data on math and science achievement, student background characteristics, school background, and teacher background. For the present study, we used data from the “Advanced Mathematics Population” (Martin et al., p. 3.4), which included US students in their final year of secondary school (12th grade) who had enrolled in an advanced mathematics course (i.e., Advanced Placement, International Baccalaureate, or another advanced mathematics course), as well as data from their associated advanced mathematics teacher. Students in the TIMSS sample were selected from the population of 12th grade advanced mathematics students using a two-stage stratified cluster sampling procedure where the first stage constituted a random sample of schools, and the second stage involved the selection of one or two intact classes of students (Martin et al.). School type (e.g., public vs. private) and location (e.g., metropolitan, non-metropolitan), and school performance on national exams served as stratification variables. Additionally, data from the teachers of the students advanced mathematics courses were collected. Although the vast majority of advanced math courses in the TIMSS data (98.2%) were courses in which the teacher assigned homework to students, because this study focused on the effects of homework characteristics (type and use), we only considered those classes in which homework was assigned. The total unweighted sample size of student–teacher pairs was N = 4050.

To address the research questions, the outcome of overall mathematics achievement as measured by the TIMSS-administered assessment was used. Reliability for the overall scores was alpha = 0.91. Additionally, two indices of attitude toward mathematics obtained from responses to the Student Questionnaire (IEA 2014 ) were considered as dependent variables: the Students Like Learning Advanced Mathematics (SLM) scale and the Students Value Advanced Mathematics (SVM) scale. Scale scores were based on student responses to either 12 items (SLM scale) or 9 items (SVM scale) pertaining to the corresponding constructs; e.g., I enjoy figuring out challenging mathematics and I like studying for my mathematics class outside of school (SLM scale); and It is important to do well my mathematics class and Doing well in mathematics will help me get into the university of my choice (SVM scale). Each item was associated with four Likert response options ( Agree a lot to Disagree a lot ). The scale developers created composite scores from the two math attitude scales using item response theory (IRT) procedures, and these composite scores are contained in the data set. Scores from each scale have demonstrated good evidence of reliability for the US sample, with reliability alpha = 0.91 (SLM scale) and alpha = 0.81 (SVM scale). The type of homework assigned by the teacher as well as the teacher’s use of homework was assessed by items the TIMSS Teacher Questionnaire (IEA 2014 ). The specific homework characteristics assessed in TIMSS are shown in Table Table1 1 .

TIMSS teacher questionnaire items assessing type and use of homework assignments

Response options for each item were coded as 1 = Never or almost never , 2 = Sometimes , and 3 = Always or almost always

To address RQ1 and RQ2, multiple regression analyses were carried out where indicators of the type of homework assigned were entered as predictors of TIMSS advanced mathematics achievement, and also as predictors of each of the two science attitude measures. A second set of regression analyses then was carried out that included indicators of the manner in which homework was used as predictors of the same achievement and attitude outcomes. Based on prior literature supporting a positive relationship between parental educational attainment and student attitudinal and achievement outcomes (e.g., Anderson 1980 ; Bakker et al. 2007 ; Bui 2002 ; Coley et al. 2007 ; Schlecter and Mislevy 2010 ; Spera 2006 ), and based on prior research suggesting that, among high achievers, males show higher math achievement than females (Zhou et al. 2017 ), both gender and parental education attainment were used as control variables. For RQ3 and RQ4, seven indicators of student use of the internet as reported by the student (Table (Table2) 2 ) were used as predictors of the two TIMSS mathematics attitude outcomes (SLM and SVM) as well as TIMSS mathematics achievement scores. For all regression analyses, supplied student-level sampling weights were used and, to account for the complex, multi-stage sampling design, standard errors were adjusted using jackknife replications. Each set of TIMSS science achievement scores consist of a set of five plausible values; therefore, regression analyses were carried out five times, and obtained parameter estimates averaged across these regressions. For the regression analyses, error degrees of freedom were computed as the number of PSUs (clusters) minus the number of strata (unless a stratum had a single PSU), minus the number of regressors. In the TIMSS data, some students were taught by more than one mathematics teacher. Thus the sample sizes for regressions involving teacher predictors were larger than the sample sizes for regressions involved student variables. Analyses were carried out using SPSS v.26 in combination with the IEA International Database (IDB) Analyzer (IEA 2020 ).

TIMSS student questionnaire items assessing the use of internet in class

Response options were coded as 0 = No and 1 = Yes

Tables Tables3 3 and and4 4 show results for multiple regression analyses predicting Advanced Mathematics Achievement, Students Like Learning Advanced Mathematics (SLM), and Students Value Advanced Mathematics (SVM) from homework type, homework use, and the remaining regressors. For each analysis, the regression assumptions of linearity, homoscedasticity, and normality of residuals were assessed and met. Excessive multicollinearity among regressors was not evident, with no VIF values exceeding 2.0. As can be seen from these results, when controlling for student gender and parental educational background, none of the homework type and homework use indicators significantly predicted advanced mathematics achievement. However, when controlling for the other predictors, increased parental education was significantly associated with higher student advanced mathematics scores ( β = 0.25, p < 0.001), and females showed significantly lower advanced mathematics ability scores than males ( β = − 0.15, p < 0.001). For the attitudinal outcomes, homework assignments that required students to find one or more applications of the content covered in class were a statistically significant positive predictor of both Students Like Learning Advanced Mathematics ( β = 0.05, p = 0.014) and Students Value Advanced Mathematics ( β = 0.08, p = 0.037), while discussing homework in class was a significant negative predictor of Students Like Learning Advanced Mathematics ( β = 0.04, p = 0.041). In each case, however, the effects were small in magnitude, accounting for 1% of the variance in each outcome.

Regression results predicting advanced mathematics ability scores from parental education, student sex, homework type, and homework use ( N = 3941)

* p < .05 , **p < .01, **p < .001. R 2 for full model = .12. Parameter estimates have been averaged over models fitted to each of five plausible values of the outcome

Regression results predicting students like learning advanced mathematics scores from parental education, student sex, homework type, and homework use ( N = 3930)

* p < .05 , **p < .01, **p < .001. R 2 for full model = .03 (Students like learning advanced mathematics), .02 (Students value learning advanced mathematics)

When student use of the internet for classroom activities was considered as a predictor of advanced mathematics achievement and attitude (Tables (Tables5 5 and and6), 6 ), none of the indicators for internet use was significantly associated with advanced mathematics achievement. However, when the two measures of attitude toward advanced mathematics were considered as outcomes, using the internet to discuss math topics with other students was significantly, positively associated with both Students Like Learning Advanced Mathematics ( β = 0.06, p = 0.029) and Students Value Advanced Mathematics ( β = 0.07, p = 0.022). Similarly, using the internet to find information to understand math concepts also was significantly and positively associated with both attitudinal outcomes ( β = 0.10, p = 0.008, and β = 0.10, p = 0.016, respectively), while using the internet to communicate with the teacher was positively associated with Students Like Learning Advanced Mathematics ( β = 0.06, p = 0.040).

Regression results predicting advanced mathematics ability scores from use of the internet for classroom activities ( N = 2903)

* p < .05 , **p < .01, **p < .001. R 2 for full model = .10

Regression results predicting students like learning advanced mathematics scores from teachers’ use of the internet for classroom activities ( N = 2902)

* p < .05 , **p < .01, **p < .001. R 2 for full model = .03 (Students like learning advanced mathematics), .02 (Students value advanced mathematics)

Although many studies have examined the effects of homework on student achievement (Dettmers et al. 2010 ), most have focused on how the quantity or frequency of homework is related to this outcome. Other research has focused on the frequency and amount of time spent on homework by students or the quality of homework selection as related to motivation (Trautwein and Lüdtke 2009 ) and homework support resources (Kitsantas et al. 2011 ). Few studies, however, have examined the specific characteristics of homework, and few have investigated effects of such characteristics on student attitudes toward mathematics—particularly among a select group of students—those approaching the conclusion of their secondary education who are taking advanced mathematics courses. The present study found that neither homework type nor homework use significantly predicted mathematics achievement. As prior literature (e.g., De Jong et al. 2000 , 2006 ; Cooper 1989a , b ) has focused primarily on amounts of homework assigned, and there is little existing literature on the effects of homework type or homework use, these findings provide preliminary insight and suggest that differences in types of homework may matter less than the quantity of homework assigned. Similarly, use of the internet was not a significant predictor of mathematics achievement. However, when considering the extent to which students both like and value mathematics, whether students discussed math content on the internet was a positive predictor of both of these attitudinal outcomes. Conversely, discussing math content in class did not predict the extent to which students value mathematics, and was a negative predictor of the extent that students like math. This was a curious result, and suggests that the medium of communication may serve as a critical factor. Perhaps communication on the internet entails a communication style that is more intuitive and enjoyable for young students than traditional face-to-face discussion. This would be consistent with recent findings, such as Pierce ( 2009 ), who found that teens feel more comfortable talking with others in an online environment than in a face-to-face setting, and that this increased comfort with online communication was more prominent among females.

Another finding was that asking students to find information related to mathematics concepts positively predicted both liking and valuing advanced mathematics. It is likely that such class activities would entail the use of the internet and, indeed, these results were paralleled when the use of the internet to find information was specifically examined as a predictor of math attitudes. It appears then, that the internet can play a key role in shaping students attitudes toward mathematics. The implications of this are that aspects of the “digital divide” may have impact on more than student skills and achievement—they potentially may affect student attitudes, particularly among high-performing students as examined in the present study. Thus, continued efforts to ensure digital “connectedness” for all students in all schools, particularly under-privileged students or those in rural or remote areas that historically have struggled in these respects, is essential. This has become particularly relevant and important in contexts where online learning by necessity becomes to the sole educational environment, as occurred across the world during the COVID-19 pandemic. Auxier and Anderson ( 2020 ), in fact, discuss the “digital homework gap” (p. 1) whereby some school-aged children lack the digital connectivity needed to successfully complete their school work while in their homes. The authors, based on Pew Research Center data, report that 17% of teenage students reported inability to complete schoolwork at home due to lack of digital connectivity, and that this problem was more highly prevalent among Black students and students from low-income households. Clearly, continued research into the digital aspects of school work—both traditionally assigned homework and schoolwork completed during synchronous online class sessions—is critical.

Conclusions

The current study examined characteristics of teacher-assigned homework and of student use of the internet on both mathematics achievement and attitudes toward mathematics among advanced math students. Although findings suggest no discernable impacts on student achievement, they do suggest that attitudes can in particular that online discussions might foster more positive attitudes toward mathematics. These findings have implications for how teachers might optimally design homework activities to increase the extent to which student both like and value the mathematics content that they teach and also to increase the likelihood that their students will continue their study of mathematics beyond the high school level and consider educational and career trajectories that involve these skills.

Not applicable.

Data Availability

Code availability, declarations.

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MATHEMATICS ASSIGNMENT AM025 (A1T4)
NPTEL Discrete Mathematics Assignment 3 Demystified: Solutions & Strategies #assignment #discreate

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A Step-by-Step Guide to Structuring a Math Lesson Plan
Step 1: Identify the learning objective. I believe that the first step in structuring a math lesson plan is to determine what you want your students to learn. Setting learning objectives means determining what skills and concepts your students will be learning or practicing during the lesson. Generally, learning objectives provide the framework ...
PDF MATH ASSIGNMENT ANALYSIS GUIDE
a) No - the assignment focuses on mathematics outside the grade-level standards or superficially reflects the grade-level cluster(s), grade-level content standard(s), or part(s) thereof b) Yes- the assignment focuses only on mathematics within the grade-level standards and fully reflects the depth of
PDF The Effective Mathematics Classroom
4 Algebra Readiness, Cycle 1 The Effective Mathematics Classroom What are some best practices for mathematics instruction? In general, a best practice is a way of doing something that is shown to generate the desired results. In terms of mathematics instruction, we typically think of a best practice as a teaching strategy or lesson structure that promotes a deep student understanding of ...
5.1: Quadratic Functions
Recognizing Characteristics of Parabolas. The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex.If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point ...
Characteristics of Modern Mathematics « Mathematical Science & Technologies
Simplicity, Logical Derivation, Axiomatic Arrangement, Precision, Correctness, Evolution through Dialectic. Though each of these characteristics presents unique pedagogical challenges and opportunities, here I'll focus on the characterisics themselves and leave the pedagogical discussion to ( Ebr05d ). (Pedagogical matters are discussed in ...
Characteristics of Functions and Their Graphs
Graph the functions in the library of functions. A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions.
Functions
Unit 7 Inequalities (systems & graphs) Unit 8 Functions. Unit 9 Sequences. Unit 10 Absolute value & piecewise functions. Unit 11 Exponents & radicals. Unit 12 Exponential growth & decay. Unit 13 Quadratics: Multiplying & factoring. Unit 14 Quadratic functions & equations. Unit 15 Irrational numbers.
Making Mathematics: Mathematics Research Teacher Handbook
Mathematics research influences student learning in a number of ways: Research provides students with an understanding of what it means to do mathematics and of mathematics as a living, growing field. Writing mathematics and problem-solving become central to student's learning. Students develop mastery of mathematics topics.
The Creation and Implementation of Effective Homework Assignments (Part
1. EFFECTIVE HOMEWORK PRACTICES. This issue of PRIMUS is the second of a two-part special issue on The Creation and Implementation of Effective Homework Assignments. Part 1 of the special issue focused on the creation of effective homework and featured papers that discussed elements of effective homework design and presented innovative homework systems targeting specific learning goals.
"Homework Should Be…but We Do Not Live in an Ideal World": Mathematics
Teachers discussed the various characteristics of quality homework (e.g., short assignments, adjusted to the availability of students) and shared the characteristics of the homework tasks typically assigned, highlighting a few differences (e.g., degree of individualization of homework, purposes) between these two topics.
PDF Characteristics of Highly Effective Teaching and Learning
B. Teacher uses student work/data, observations of instruction, assignments and interactions with colleagues to reflect on and improve teaching practice. The teacher: 1. Establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions (MTP1); 2.
PDF 3.2 Characteristics of Quadratic Functions
Section 3.2 Characteristics of Quadratic Functions 111 Graphing Quadratic Functions Using x-Intercepts When the graph of a quadratic function has at least one x-intercept, the function can be REMEMBER written in intercept form, f(x) = a(x − p)(x − q), where a ≠ 0. An x-intercept of a graph is the x-coordinate of a point where the graph
PDF Mathematics Classroom Assessment: A Framework for Designing Assessment
emphasis on important mathematical processes, and on procedural and conceptual aspects of mathematical ideas. The MATH taxonomy is a modiﬁcation of Bloom's taxonomy for structuring mathematics assessment tasks, and describes the skills that a particular task assesses [24]. Bloom et al. developed a taxonomy for the design and assessment of ...
Classroom assignments as measures of teaching quality
The current study examines the quality of assignments in middle school mathematics (math) and English language arts (ELA) classrooms as part of a larger study of measures of teaching quality. We define teaching quality as "the quality of interactions between students and teachers; while teacher quality refers to the quality of those aspects ...
Characteristics of effective student talks and papers
Draft characteristics of effective communication from JMM 2013 minicourse. The math is correct. The constraints of the assignment are addressed. The audience can perceive the author's/presenter's content. If the target audience pays attention, they will understand the content.
The Role of Assessment in Mathematics Classrooms: A Review
Abstract. This paper tries to understand the role of assessment in bringing changes in students' mathematics performance. Quality assessment is a key factor in improving the learning of ...
Characteristics And Features Of Mathematics
Usage of symbols constructs mathematics language and makes it more elegant and precise than any other language. Mathematical language and symbols. Cut short the lengthy statements. Help the expression of ideas or things in the exact form. Mathematical language is free from verbosity.
Homework characteristics as predictors of advanced math achievement and
The present study examined the 2015 Trends in international mathematics and science study "Advanced" data to examine how both the type of homework assigned and how that homework was used were related to advanced mathematics achievement and attitudes toward advanced mathematics among 12th grade students in the USA. Additionally, students' use of the internet was examined as a predictor of ...
Roles and characteristics of problem solving in the mathematics
Since problem solving became one of the foci of mathematics education, numerous investigations have been performed to improve its teaching, develop students' higher-level skills, emphasize mathe... Roles and characteristics of problem solving in the mathematics curriculum: a review: International Journal of Mathematical Education in Science ...
(PDF) Classroom assignments as measures of teaching quality
Relationships between assignments scores, classroom characteristics, and other measures of teaching quality are examined for both domains. These findings help us understand the nature of and ...
MATH 1280 Discussion Assignment Unit 7
Word count: 392 Suppose the distribution of math scores in the SAT follows a normal distribution with a mean μ=700 and a standard deviation σ=150. Calculate the Z-score for an SAT Math score of 600. Interpret it in your own word. Since the distribution follows a normal distribution, we can use the formula Z = x - μ / σ Given x = 600, μ ...
PDF CHAPTER 15 TRANSPORTATION AND ASSIGNMENT PROBLEMS
6. Describe the characteristics of assignment problems. 7. Identify the relationship between assignment problems and transportation problems. 8. Formulate a spreadsheet model for an assignment problem from a description of the problem. 9. Do the same for some variants of assignment problems. 10.
Characteristics of Students with Mathematical Difficulties:
Students who have learning difficulties in mathematics often display problems with how they process the language of math, problems with recalling important information from their working memory, problems with their executive functioning, and their overall cognitive development (Bryant et al., 2020, p. 400). Learning disabilities often include ...

A Step-by-Step Guide to Structuring a Math Lesson Plan

What Is a Lesson Plan?

Why Is a Lesson Plan Necessary For Maths?

What Makes an Effective Maths Lesson Plan?

How Do You Write a Structure For a Lesson Plan?

Step 1: Identify the learning objective

Step 2: Plan your lesson activities

Step 3: Anticipate and address student misconceptions

Step 4: Assess student understanding

Step 5: Reflect And Revise The Math Lesson Plan

What to read next:

Closing thoughts

Recent Posts

5.1: Quadratic Functions

Recognizing Characteristics of Parabolas

Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions

Finding the Domain and Range of a Quadratic Function

Determining the Maximum and Minimum Values of Quadratic Functions

Finding the x- and y-Intercepts of a Quadratic Function

Rewriting Quadratics in Standard Form

Key Equations

Key Concepts

1. Wide Applicability and the Effectiveness of Mathematics

2. Abstraction and Generality

3. Simplicity (Search for a Single Exposition), Complexity (Dense Exposition)

4. Logical Derivation, Axiomatic Arrangement

5. Precision, Correctness, Evolution Through Dialectic

Further Reading

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Module 5: Function Basics

Characteristics of Functions

A General Note: FunctionS

How To: Given a relationship between two quantities, determine whether the relationship is a function.

Example: Determining If Menu Price Lists Are Functions

Example: Determining If Class Grade Rules Are Functions

Using Function Notation

A General Note: Function Notation

Example: Using Function Notation for Days in a Month

Analysis of the Solution

Example: Interpreting Function Notation

Representing Functions Using Tables

How To: Given a table of input and output values, determine whether the table represents a function.

Example: Identifying Tables that Represent Functions

Determine whether a function is one-to-one

A General Note: One-to-One Function

Example: Determining Whether a Relationship Is a One-to-One Function

Evaluating and Solving Functions

How To: EVALUATE A FUNCTION Given ITS FORMula.

Example: Evaluating Functions

Example: Evaluating Functions at Specific Values

Example: Solving Functions

Evaluating Functions Expressed in Formulas

How To: Given a function in equation form, write its algebraic formula.

Example: Finding an Equation of a Function

Example: Expressing the Equation of a Circle as a Function

Evaluating a Function Given in Tabular Form

How To: Given a function represented by a table, identify specific output and input values.

Example: Evaluating and Solving a Tabular Function

Finding Function Values from a Graph

Example: Reading Function Values from a Graph

Identify Functions Using Graphs

How To: Given a graph, use the vertical line test to determine if the graph represents a function.

Example: Applying the Vertical Line Test

The Horizontal Line Test

How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

Example: Applying the Horizontal Line Test

Identifying Basic Toolkit Functions

Key Concepts

Unit 8: Functions

Inputs and outputs of a function

Functions and equations