lesson 8 problem solving solve a simpler problem

Curriculum  /  Math  /  8th Grade  /  Unit 6: Systems of Linear Equations  /  Lesson 8

Systems of Linear Equations

Lesson 8 of 11

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Solve systems of linear equations using elimination (linear combinations) when there is already a zero pair. 

Common Core Standards

Core standards.

The core standards covered in this lesson

Expressions and Equations

8.EE.C.8.B — Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Foundational Standards

The foundational standards covered in this lesson

8.EE.C.7 — Solve linear equations in one variable.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Understand that adding equivalent expressions will maintain the equivalence; equations can be added together. 
• Understand elimination as an algebraic approach to solving a system of equations.
• Define and identify zero pairs in systems of equations. 
• Solve systems of equations using elimination. 

Suggestions for teachers to help them teach this lesson

This is the first of two lessons on solving systems using elimination or linear combinations. In Lesson 8, students focus on the concepts behind elimination and why it works. Students look at problems that already contain a zero pair and do not require multiplication. In Lesson 9, students will learn how to multiply one or both equations in order to use this method. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Investigate if you can add equations to each other and still maintain balance and equality. 

a.   Given:

$$2 + 5 = 7$$ , and

$$1 + 9 = 10$$

Does  $$(2 + 5) + (1 + 9) = 7 + 10$$ ?

Does  $$(2+1) + (5+9) = 7 + 10$$ ?

b.   Given

$$-3 + 11 = 8$$ , and

$$7 - 2 = 5$$

Does  $$ (-3 + 11) + (7 - 2) = 8 + 5$$ ?

Does  $$(-3 + 7) + (11 - 2) = 8 + 5$$ ?

c.   Based on your conclusion from parts (a) and (b), write a single equation that accounts for both equations below.

$$x + 3 = 8 $$

$$x - 4 = 1$$

Guiding Questions

Grade 8 Mathematics > Module 4 > Topic D > Lesson 28 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

Solve the system of equations below by using the elimination method. Write your answer as a coordinate point. 

$${6x-5y=21}$$

$${2x+5y=-5}$$

Solve the three systems using the elimination method. Compare and contrast each solution.

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Tamar writes two equations:

$$2y=4x+14$$

$$-2y=-4x-14$$

Tamar says that together, the two equations create a system with no solution because both equations have the same slope. 

Do you agree with Tamar? Explain your reasoning. 

Solve the system.

$${9x+2y=9}$$ $${6x-2y=-4}$$

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include problems where students solve systems using elimination to reinforce the procedure and process; ensure problems all have a zero pair and do not yet require multiplication; include problems that are no solution or infinite solution.
• Kuta Software Free Algebra 1 Worksheets Solving Systems of Equations by Elimination — Only use #1–4, or change the equations in the other examples to include a zero pair.

Topic A: Analyze & Solve Systems of Equations Graphically

Define a system of linear equations and its solution.

Solve systems of linear equations by graphing. 

8.EE.C.8.A 8.EE.C.8.B

Classify systems of linear equations as having a unique solution, no solutions, or infinite solutions. 

Solve real-world and mathematical problems by graphing systems of linear equations. 

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Topic B: Analyze & Solve Systems of Equations Algebraically

Solve systems of linear equations using substitution when one equation is already solved for a variable.

Solve systems of linear equations using substitution by first solving an equation for a variable. 

 Solve real-world and mathematical problems using linear systems and substitution.

Solve systems of linear equations using elimination (linear combinations) by first creating a zero pair.

Solve real-world and mathematical problems using systems and any method of solution.

8.EE.C.8.B 8.EE.C.8.C

Model and solve real-world problems using systems of equations.

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Problem Solving Strategies: Solve a Simpler Problem

Jan 19, 2017

Welcome to Rithm’s series on problem-solving strategies. If you’re just joining us, you may want to start at the beginning. Here’s a list of the articles we’ve written:

• Understand the Problem
• Explore Concrete Examples
• Break It Down
• Solve a Simpler Problem
• Use Tools Strategically
• Look Back and Refactor

We’ve talked about how to initially approach a problem by understanding what it’s asking and exploring concrete examples. We’ve also discussed developing a plan to solve the problem, by breaking it down into smaller and more manageable pieces.

But planning is sometimes easier said than done. After you’ve broken a problem down, you may find that one of the components is quite difficult. For particularly challenging problems, you may not even know how to break down the problem at all. In these situations, it may help to turn to our next problem solving strategy: solve a simpler problem.

Strategy #4: Solve a Simpler Problem

In How To Solve It , Polya writes that “If there is a problem you can’t solve, then there is an easier problem you can solve: find it.”

The idea here is that you want to relax the assumptions of the original problem, without sacrificing too much of the nature of the solution. In this way, the hope is that you can simplify the problem sufficiently that it becomes solvable, and that the solution will provide insight into the original problem.

To put it another way, solving a simpler problem is about finding the core difficulty in what you’re trying to do, trying to temporarily ignore that difficulty, and then incorporate that difficulty back in to a solution you derive for the simplified problem.

Let’s return to our coding example from before:

Write a function which takes in a string and returns counts of each character in the string.

Someone who can’t solve this problem could be struggling for a variety of reasons. Here are a few:

• They have trouble with looping and iteration.
• They aren’t comfortable manipulating keys and values in objects,
• They aren’t comfortable with string manipulations (for instance, not differentiating between upper- and lowercase letters in the count),

Depending on where the difficulty lies, there are a number of simpler problems that could try to address the underlying problem:

• Rather than returning an object of character counts, return an object with the first character as a key and a value of 1. In other words, only count the first character. (This problem requires no iteration, it focuses more on understanding key-value pairs in objects.)
• Rather than counting all characters, return an object whose keys are the characters in the input string, and whose values are all 1. This requires looping, but less manipulation of values in an object.
• Make some simplifying assumptions on the input. For example, assume the input string consists of only lowercase letters. This simplification removes the need to worry about the distinction between uppercase and lowercase characters, as well as the need to worry about characters that aren’t letters.

Solving a simpler problem is all about trying to find the root of the difficulty you’re experiencing, and modifying the problem so that the difficulty can be safely ignored.

One caveat with this approach is that it’s possible to oversimplify a problem. When this happens, you may wind up with an easier problem that you can solve, but whose solution provides little insight into the original problem.

Here’s one oversimplification of the current example: rather than returning a count for each character, just return a count of the total number of characters. Here are two solutions to this simplified problem:

While the first solution may be helpful for someone still struggling with for loops in JavaScript, the second solution provides no insight into the nature of the original problem.

But not to fear; solving a simpler problem is a process. If you find that your simplification makes the problem too simple, try to find a problem that is harder than your simplification, but still easier than the original problem.

And if you’re having trouble solving your simpler problems too, there are a couple more problem solving strategies that may help. We’ll discuss another strategy next time.

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1.6: Problem Solving Strategies

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• Page ID 132869

• Michelle Manes
• University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solve them), you learn strategies and techniques that can be useful. But no single strategy works every time.

How to Solve It

George Pólya was a great champion in the field of teaching  effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, How to Solve it . Pólya died at the age 98 in 1985. [1]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• First, you have to understand the problem.
• After understanding, then make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

A Problem Solving Strategy: Try Something!

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

Note that being "good at mathematics" is not about doing things right the first time. It is about figuring things out. Practice being okay with having done something incorrectly. Try to avoid using an eraser and just lightly cross out incorrect work (do not black out the entire thing). This way if it turns out that you did something useful, you still have that work to reference! If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what was left after paying Brianna. Finally, Alex saw David and gave him 1/2 of the remaining money. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

A Problem Solving Strategy: Draw a Picture

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

A Problem Solving Strategy: Make Up Numbers

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

Try this: Assume (that is, pretend) Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person.

Or try working backward: suppose Alex has some specific amount left at the end, say $10. Since he gave David half of what he had before seeing David, that means he had $20 before running into David. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

Most people want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. Instead of asking the teacher, “Is this right?”, you should be ready to justify it and say, “Here’s my answer, and here is how I got it.”

A Problem Solving Strategy: Try a Simpler Problem

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said, “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

The ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

A Problem Solving Strategy: Work Systematically

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

A Problem Solving Strategy: Use Manipulatives to Help You Investigate

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

A Problem Solving Strategy: Look for and Explain Patterns

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table. If possible, actually describe these to a friend.
• Explain and justify any of the patterns you see (if possible, actually do this with a friend). If you don't have a partner to work with, imagine they asked you, "How can you be sure the patterns will continue?"
• Expand this to find what calculation(s) you would perform to find the total number of squares on a 100 × 100 chess board.

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What progress have you made?

A Problem Solving Strategy: Find the Math, Remove the Context

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

A Problem Solving Strategy: Check Your Assumptions

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

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McGraw Hill My Math Grade 3 Chapter 12 Lesson 8 Answer Key Problem-Solving Investigation: Solve a Simpler Problem

All the solutions provided in  McGraw Hill My Math Grade 3 Answer Key PDF Chapter 12 Lesson 8 Problem-Solving Investigation: Solve a Simpler Problem will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 3 Answer Key Chapter 12 Lesson 8 Problem-Solving Investigation: Solve a Simpler Problem

Learn the Strategy

2. Plan I will collect and organize the data in a line plot. Then I will decide if Shane’s estimate is ________________. Answer: I will collect and organize the data in a line plot. Then I will decide if Shane’s estimate is reasonable.

4. Check Does your answer make sense? Explain. Answer: Yes,because he only rolled 15 twice.

Practice the Strategy

1. Understand What facts do you know? Answer: From the given data, we know that Julina needs to make 100 favors for the family reunion and also about the estimate of relatives coming on friday and saturday. What do you need to find? Answer: We need to find the total number of relatives coming.

2. Plan Answer: Determine the total number of relatives coming.Add the relatives coming on friday and saturday and compare that too 100.

3. Solve Answer: Count of relatives coming on friday=62 Count of relatives coming on saturday = half the count of friday=62÷2=31 Therefore, the total count of relatives will be = 62+31=93

4. Check Does your answer make sense? Explain. Answer: Total 93 relatives will be attending, 93 is less than 100.So, Julina’s estimation of making 100 favors for the family reunion is resonable.

Apply the Strategy

Solve each problem by first solving a simpler problem.

Review the Strategies

Use any strategy to solve each problem.

• Solve a simpler problem.
• Determine reasonable answers.
• Make a table.

From the above data, How many fewer people chose mango salsa than pineapple or tomato salsa combined? Answer: The total number of people who like pineapple or tomato salsa combined will be 7+10=17 The fewer people who choose mango salsa than pineapple or tomato salsa combined will be 17-5=12

How many more cars had either 1 or 2 people rather than 4 people? Answer: Number of cars who had either 1 or 2 people will be 7+9=16 Number of cars who had either 4 people will be 8 Therefore, the number of cars who had either 1 or 2 people rather than 4 people will be 16-8=8

Question 5. Mathematical PRACTICE Reason Is it reasonable to say that about twice as many cars had 1 passenger than 3 passengers? Explain. Answer: Yes, that is reasonable to say because the number of cars with 1 passenger is 7 and twice of 3 passengers is 6. As 7 is greater than 6, the given statement is reasonable.

McGraw Hill My Math Grade 3 Chapter 12 Lesson 8 My Homework Answer Key

Problem Solving

Question 3. Mathematical PRACTICE Make Sense of Problems Look at Exercise 2. What is the simpler problem you solved first? Answer: The first simpler problem solved is to find out if the estimate is reasonable.I had to find the actual number of friends Elizabeth survyed.

Question 4. How many more students chose football and baseball together than basketball? Write an equation. Answer: The number of students who chose football and baseball together are 8+4=12 The number of students who chose basketball are 10 Therefore, the number of students chose football and baseball together than basketball will be 12-10=2 Let ‘a’ be the number of students chose football and baseball together than basketball. So, the equation will be 12-10=a=2

Question 5. Mathematical PRACTICE Model Math Write a problem about the data above that would take two steps to solve. Then solve. Answer: Is it reasonable to say that the number of Elizabeth’s friends who like football are twice that of  baseball? From the above data, we know that the number of Elizabeth’s friends who like football are 8. The number of Elizabeth’s friends who like baseball are 4. As twice of 4 (2×4=8) is 8, the estimate is reasonable.

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Problem Solving: Simplify the Problem

What Is It?

Simplifying a mathematics problem is a strategy that often is used along with other problem-solving strategies. When a problem is too complex to solve in one step, it often helps to divide it into simpler problems and solve each one separately. Creating a simpler problem from a more complex one may involve rewording the problem; using smaller, simpler numbers; or using a more familiar scenario to understand the problem and find the solution. For example, consider the problem:

A soccer team won 24 of 36 games in the first season. If the team had the same ratio of wins to games in the second season, and they won 16 games, how many games did they play in the second season?

The answer can be found by simplifying the ratio of 24:36 to 2:3, and then cross-multiplying to find the total number of games in the second season, 24 games.

Why Is It Important?

This is a problem-solving strategy that can be used with difficult concepts such as manipulating ratios or fractions. If a problem is confusing, the numbers can be rounded, or simpler numbers can be used to help make a plan to solve it.

How Can You Make It Happen?

Introduce a problem to students that is complex and might be easier to solve if it were simplified. For example:

On your way to visit a friend, you leave your house at 2:45 P.M. and travel 1 3/4 miles to the train, 12 1/2 miles on the train, and 3/4 mile to your friend's house from the train station. If you get there at 4:15 P.M., how many miles per hour did you travel?

Understand the Problem

Demonstrate that the first step is understanding the problem. This involves identifying the key pieces of information needed to find the answer. Students may need to read the problem several times and/or put the problem into their own words.

I know I left at a certain time, arrived at a certain time, and traveled a certain distance. I need to find how many miles per hour I traveled.

Choose a Strategy

For this problem, it might be helpful for students to use simpler numbers to learn the steps they need to follow to solve it. Have students change the problem to:

I left the house 1:00, traveled 12 miles, and arrived at 4:00. How many miles per hour did I travel?

Solve the Problem

First, have students solve the problem using the simpler numbers.

I left the house 1:00, traveled 12 miles, and arrived at 4:00. How many miles per hour did I travel? I traveled 12 miles. It took 3 hours. To find the miles per hour, I divide 12 by 3 to get 4 miles per hour.

Next, have them write down the steps they used to solve the problem.

Find the distance traveled.

Find the time spent.

Divide to find the miles per hour.

Then, have them use the actual numbers from the problem and follow the same steps.

Find the distance traveled. 1 3/4 + 12 1/2 + 3/4 = 15 miles

Find the time spent. The time from 2:45 to 4:15 is 1 hour and 30 minutes, or 1 1/2 hours.

Divide to find the miles per hour. 15 divided by 1 1/2 = 10 miles per hour

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State standards, national science education standards - science.

Scientists, engineers, and ordinary people use problem solving each day to work out solutions to various problems. Using a systematic and iterative procedure to solve a problem is efficient and provides a logical flow of knowledge and progress.

In this unit, we use what is called "The Technological Method of Problem Solving." This is a seven-step procedure that is highly iterative—you may go back and forth among the listed steps, and may not always follow them in order. Remember that in most engineering projects, more than one good answer exists. The goal is to get to the best solution for a given problem. Following the lesson conduct the associated activities Egg Drop and Solving Energy Problems for students to employ problem solving methods and techniques. 

Lesson Background and Concepts for Teachers

The overall concept that is important in this lesson is: Using a standard method or procedure to solve problems makes the process easier and more effective.

The specific process of problem solving used in this unit was adapted from an eighth-grade technology textbook written for New York State standard technology curriculum. The process is shown in Figure 1, with details included below. The spiral shape shows that this is an iterative, not linear, process. The process can skip ahead (for example, build a model early in the process to test a proof of concept) and go backwards (learn more about the problem or potential solutions if early ideas do not work well).

This process provides a reference that can be reiterated throughout the unit as students learn new material or ideas that are relevant to the completion of their unit projects.

Brainstorming about what we know about a problem or project and what we need to find out to move forward in a project is often a good starting point when faced with a new problem. This type of questioning provides a basis and relevance that is useful in other energy science and technology units. In this unit, the general problem that is addressed is the fact that Americans use a lot of energy, with the consequences that we have a dwindling supply of fossil fuels, and we are emitting a lot of carbon dioxide and other air pollutants. The specific project that students are assigned to address is an aspect of this problem that requires them to identify an action they can take in their own live to reduce their overall energy (or fossil fuel) consumption.

The Seven Steps of Problem Solving

1.  Identify the problem

Clearly state the problem. (Short, sweet and to the point. This is the "big picture" problem, not the specific project you have been assigned.)

2.  Establish what you want to achieve

• Completion of a specific project that will help to solve the overall problem.
• In one sentence answer the following question: How will I know I've completed this project?
• List criteria and constraints: Criteria are things you want the solution to have. Constraints are limitations, sometimes called specifications, or restrictions that should be part of the solution. They could be the type of materials, the size or weight the solution must meet, the specific tools or machines you have available, time you have to complete the task and cost of construction or materials.

3.  Gather information and research

• Research is sometimes needed both to better understand the problem itself as well as possible solutions.
• Don't reinvent the wheel – looking at other solutions can lead to better solutions.
• Use past experiences.

4.  Brainstorm possible solutions

List and/or sketch (as appropriate) as many solutions as you can think of.

5.  Choose the best solution

Evaluate solution by: 1) Comparing possible solution against constraints and criteria 2) Making trade-offs to identify "best."

6.  Implement the solution

• Develop plans that include (as required): drawings with measurements, details of construction, construction procedure.
• Define tasks and resources necessary for implementation.
• Implement actual plan as appropriate for your particular project.

7.  Test and evaluate the solution

• Compare the solution against the criteria and constraints.
• Define how you might modify the solution for different or better results.
• Egg Drop - Use this demonstration or activity to introduce and use the problem solving method. Encourages creative design.
• Solving Energy Problems - Unit project is assigned and students begin with problem solving techniques to begin to address project. Mostly they learn that they do not know enough yet to solve the problem.
• Energy Projects - Students use what they learned about energy systems to create a project related to identifying and carrying out a personal change to reduce energy consumption.

The results of the problem solving activity provide a basis for the entire semester project. Collect and review the worksheets to make sure that students are started on the right track.

Learn the basics of the analysis of forces engineers perform at the truss joints to calculate the strength of a truss bridge known as the “method of joints.” Find the tensions and compressions to solve systems of linear equations where the size depends on the number of elements and nodes in the trus...

Through role playing and problem solving, this lesson sets the stage for a friendly competition between groups to design and build a shielding device to protect humans traveling in space. The instructor asks students—how might we design radiation shielding for space travel?

A process for technical problem solving is introduced and applied to a fun demonstration. Given the success with the demo, the iterative nature of the process can be illustrated.

The culminating energy project is introduced and the technical problem solving process is applied to get students started on the project. By the end of the class, students should have a good perspective on what they have already learned and what they still need to learn to complete the project.

Hacker, M, Barden B., Living with Technology , 2nd edition. Albany NY: Delmar Publishers, 1993.

Other Related Information

This lesson was originally published by the Clarkson University K-12 Project Based Learning Partnership Program and may be accessed at http://internal.clarkson.edu/highschool/k12/project/energysystems.html.

Contributors

Supporting program, acknowledgements.

This lesson was developed under National Science Foundation grants no. DUE 0428127 and DGE 0338216. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: August 16, 2023

Chapter 5, Lesson 3: Problem-Solving Strategy: Solve a Simpler Problem

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Problem Solving: Use a Simpler Problem

In this problem solving worksheet, 5th graders focus on using a simpler problem to solve 6 word problems. They work with decimals, fractions, times, and money.

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