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Illustrative Mathematics (6-8 OUR Mathematics)

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U1 , U2 , U3 , U4 , U5 , U6 , U7 , U8

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Preparation for High School Algebra or Integrated Math 1

University High School Summer Bridge Course

High School Illustrative M athematics - McGra w Hill, Kendall Hunt, Imagine Learning, Learn Zillion)

Algebra 1 Resources

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OPEN UP RESOURCES Mathematics - OUR

OUR - Algebra 1 Resources

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Home > CCA > Chapter 7 > Lesson 7.1.2 > Problem 7-27

Write an equation or system of equations to solve this problem.

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Master Math with Morgan: Homework Help for Math 7

Updated on Dec 06,2023

Table of Contents

• Introduction

Understanding Circumference

Exploring diameter and circumference, measuring circumference and diameter, proportional relationship between diameter and circumference, using constant proportionality to estimate circumference, homework and practice problems.

• Identifying Inaccurate Measurements
• Using Approximate Values of Pi

The Equation and Proportional Relationship

In this lesson, we will be exploring the concept of circumference and how it relates to circular objects. We will start by understanding what circumference actually represents - the distance around a circular object. To measure the circumference of an object, we typically use a piece of STRING or a measuring tape to wrap around the object and then measure the length of that string. However, measuring the circumference directly can be quite challenging due to the circular Shape of the object.

To better understand the relationship between circumference and other measurements of a circle, such as diameter or radius, we will be conducting measurements and calculations. By analyzing the measurements of different objects and plotting them on a coordinate plane, we can identify Patterns and establish a proportional relationship between the diameter and circumference.

To begin our exploration, we will measure the diameter and circumference of several circular objects. By recording these measurements in a table, we can analyze the relationship between the two.

For example, I measured the diameter of my coffee mug and found it to be 9.9 centimeters. Using a string, I measured the circumference of the mug to be approximately 32 centimeters. Similarly, I measured the diameter of a quarter to be 2.3 centimeters and its circumference to be around 7.2 centimeters. Finally, I measured the diameter of a megaphone to be 19.5 centimeters, with a corresponding circumference of approximately 60 centimeters.

In activity number two, we will be practicing the measurement of circumference and diameter using various objects. Your teacher will provide you with objects such as a mug, a quarter, or even a megaphone for you to measure. Using a string or a measuring tape, wrap it around the object to measure the circumference, and use a ruler to measure the diameter. By recording these measurements in a table, we can further analyze their relationship and observe any patterns that may emerge.

For instance, I measured the circumference of a mug to be approximately 32 centimeters, while its diameter was 9.9 centimeters. The quarter, on the other HAND , had a circumference of around 7.2 centimeters and a diameter of 2.3 centimeters. Lastly, I measured the circumference of a megaphone to be about 60 centimeters, with a diameter of 19.5 centimeters.

Upon analyzing the measurements and plotting them on a coordinate plane, I noticed a proportional relationship between the diameter and circumference of these objects. Although not perfect, the data points appear to be relatively close to a straight line. This indicates that as the diameter increases, the circumference also increases, suggesting a proportional relationship between the two.

When we analyze the values, we find that the ratios of circumference to diameter are consistent. For example, the mug has a ratio of approximately 3.09 (32/9.9), the quarter has a ratio of around 3.13 (7.2/2.3), and the megaphone has a ratio of about 3.07 (60/19.5). These ratios are all quite close, indicating a proportional relationship.

Based on the consistent ratios we observed, we can establish a constant proportionality to estimate the circumference of a circle based on its diameter. We will refer to this constant proportionality as "k." In our case, the value of "k" is approximately 3.1, derived from the ratios we calculated earlier.

Now, let's consider another circular object with a diameter that is half as long as the diameter of our largest circle. If the largest circle has a diameter of 20, then its circumference is 60. Therefore, if the diameter is halved to 10, the new circumference will also be halved to 30. This demonstrates the proportional relationship between diameter and circumference, where halving the diameter results in halving the circumference.

For the homework assignment, Diego measured the diameter and circumference of several circular objects and recorded his measurements in a table. However, one of his measurements is inaccurate. By analyzing the values and the relationships between diameter and circumference, we can identify the inaccurate measurement.

Upon examination, we find that the measurement for the flying disc seems to be off. While the other values Align closely with the proportional relationship between diameter and circumference, the flying disc measurement deviates significantly. This suggests that the measurement for the flying disc is inaccurate.

Moreover, the homework also includes a task to complete a table using the constant proportionality of pi, represented by 3.14 or 22/7. By multiplying or dividing using the given measurements, we can estimate the missing values for the circumference or diameter. This allows us to Apply the constant proportionality of pi and demonstrate its usage in real-world calculations.

To further Delve into the subject, we will explore the equation y = 1.5x + 2. This equation represents a linear relationship between two variables, where y is dependent on x. By finding four pairs of x and y values that satisfy the equation, we can plot these points on a coordinate plane.

For instance, when x is 0, y equals 2 (0 × 1.5 + 2 = 2). When x is 1, y is 3.5 (1 × 1.5 + 2 = 3.5). Similarly, for x values of 2 and 3, the corresponding y values are 5 and 6.5, respectively. Plotting these points on the coordinate plane, we observe a linear relationship between the two variables.

However, it is essential to note that although the equation represents a linear relationship, it doesn't satisfy the requirements for a proportional relationship. A proportional relationship must pass through the origin, but in this case, the equation is shifted upward by 2 units. Thus, it is not a direct proportional relationship.

In conclusion, understanding the concept of circumference and its relationship with diameter is crucial in geometry and real-world applications. By measuring and analyzing different circular objects, we can observe patterns and establish proportional relationships. The constant proportionality, pi, plays a significant role in these relationships and allows us to estimate the circumference or diameter of circles.

The above is a brief introduction to Master Math with Morgan: Homework Help for Math 7

Let's move on to the first section of Master Math with Morgan: Homework Help for Math 7

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