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10.8: Problem Solving Strategy: Solve a Simpler Problem

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Introduction

The Windmill Star

Jillian has finished her first quilt and has decided to create another one. This quilt square has a specific pattern. Each square is made up of a pattern of parallelograms and triangles. The colors are mixed up.

Jillian is having a difficult time deciphering the pattern. Since her quilt will have twenty squares total, she wants to be sure that the same colors don’t bump up against each other. To do this, Jillian will need to simplify the pattern. She isn’t sure how to do this.

This is where you come in. This lesson is about solving a simpler problem. Often in mathematics, problems can be very complicated and need to be broken down before they can be solved. Pay close attention to this lesson and in the end you will be able to help Jillian decipher the pattern.

What You Will Learn

In this lesson you will learn the following skills:

  • Read and understand given problem situations.
  • Develop and use the strategy: Solve a Simpler Problem.
  • Plan and Compare Alternative Approaches to Solving Problems.
  • Solve real-world problems using selected strategies as part of a plan.

Teaching Time

I. Read and Understand Given Problem Situations

When working on a problem, the first thing that you need to do is to read the problem through carefully. As you have been doing in this book, you can then underline the important information. Often it makes sense to underline it in a different color too.

This lesson is about using the strategy: Solve a Simpler Problem. You can use this strategy when you problem solve any situation that seems too complicated to do in one step. In other words, it may require multiple operations, or you may have to break down the problem to be able to solve it.

Let’s look at an example where Solving a Simpler Problem would be very useful.

Example

How many cubes are in the next step? The tenth step? The twentieth?

Here is a problem that can’t be solved in one step. You can see that multiple things are being asked for in this problem. When looking at a problem that has multiple steps, we can look at the strategy: Solve a Simpler Problem. Let’s look at how to do that.

II. Develop and Use the Strategy: Solve a Simpler Problem

If we wanted to break this problem down into simpler steps, we could first create a table to look for a pattern.

1 \qquad 2\\2 \qquad 3\\3 \qquad 4\\4 \qquad 5

Now we have simplified the problem into a table. The left hand column is the step. The right hand column is the number of cubes.

Next, we look for a pattern. The number of cubes is one more than the step. Therefore, we could say that the step plus one is the number of cubes.

Now it is manageable to figure out the fifth step, the tenth, the twentieth, even the number of cubes on the 100^{th} step. We just add one.

Step 5 = 6

Step 10 = 11

Step 20 = 21

By simplifying the problem into a simpler problem, we could easily solve this one.

III. Plan and Compare Alternative Approaches to Solving Problems

Is there another way that we could have solved this problem?

Yes, definitely. We could have drawn out the pattern until we knew the number of cubes in each step. Look at what that would have looked like.

Notice how time consuming this strategy is. We could keep going. If you were to choose this strategy could certainly get an accurate answer.

The strategy of simplifying the problem into parts and then solving each part is quicker and simpler. You also have a way to check your work with numbers not just pictures.

Real Life Example Completed

The Windmill Star

Here is the original problem once again. Reread the problem and underline the important information.

Jillian has finished her first quilt and has decided to create another one. This quilt square has a specific pattern. Each square is made up of a pattern of parallelograms and triangles. The colors are mixed up.

Jillian is having a difficult time deciphering the pattern. Since her quilt will have twenty squares total, she wants to be sure that the same colors don’t bump up against each other. To do this, Jillian will need to simplify the pattern. She isn’t sure how to do this.

To solve this problem, Jillian needs to break down the pattern that she is working with. Let’s help her do this by looking at the components or parts of the pattern.

There are triangles and parallelograms in the pattern. Each square can be broken down into four smaller squares. Then each smaller square can be broken in half on the diagonal.

Now Jillian can see the pattern. Each section of the smaller square has two triangles and one parallelogram.

Here is a list of what she has discovered by breaking down the pattern.

  1. Two flowered triangles and one orange parallelogram
  2. Two orange triangles and one flowered parallelogram
  3. Two flowered triangles and one orange parallelogram
  4. Two orange triangles and one flowered parallelogram

And the pattern repeats itself.

Now that Jillian has broken down the pattern, as long as she follows it no two colors will bump up against each other. Her dilemma is solved!!

Time to Practice

Directions: Use the strategy Solve a Simpler Problem. Each problem will have multiple steps to it. Please show all of your work in your answer.

1. How many prime numbers are there between 1 and 50?

2. How many numbers are there between two and fifty that are divisible by two?

3. How many numbers between two and fifty are divisible by three?

4. How many numbers between two and fifty are divisible by four?

5. How many numbers between two and thirty are multiples of five?

6. How many multiples of three are there in 100?

7. How many different ways can you make 10 by adding the numbers in the set 1 – 10 without repeating any numbers in each sum?

8. Look at this pattern.

3, 6, 12, 24, ____

What is the next step in the pattern?

9. Look at this pattern.

5, 7, 9, 11, _____

What is the next step in the pattern?

10. Look at this pattern.

2, 5, 11, ____

What is the next step in this pattern?

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