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10.19: Problem Solving Plan, Solve a Simpler Problem

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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 Concept 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 Concept and in the end you will be able to help Jillian decipher the pattern.

Guidance

Sometimes, there are problems that can't be solved in one step. We have to break them down from one big step into smaller steps. This is the only way to solve the problem.

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

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.

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 you 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.

Now it's time for you to try a few. Answer these questions about problem solving.

Example A

True or false. Solving a simpler problem means that we can break a problem down into its smaller parts.

Solution: True

Example B

True or false. A visual pattern can be solved using this strategy.

Solution: True

Example C

True or false. It would make sense to solve a problem with one operation using this strategy.

Solution: False

Remember the windmill star? Here is the original problem once again.

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!!

Vocabulary

There aren't any new vocabulary words in this Concept.

Guided Practice

Here is one for you to try on your own.

Which problem solving strategy should be used for this problem?

At the farmer’s market, Josh bought 230 grams of oranges, 150 grams of grapes, and 800 grams of apples. How many grams of fruit did he buy in all?

Answer

To solve this problem, it makes the most sense to use the strategy: Choose an operation. Here you can read that the key word "in all" is used. This means that addition would be the best operation for this problem.

Since the units are all the same, you can simply add up the quantities.

Josh bought 1180 grams of fruit.

Video Review

Here is a video for review.

Khan Academy: Word Problem Solving Strategies

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. Describe what is happening in the pattern.

10. Look at this pattern.

5, 7, 9, 11, _____

What is the next step in the pattern?

11. Describe what is happening in this pattern.

12. Look at this pattern.

2, 5, 11, ____

What is the next step in this pattern?

13. Describe what is happening in this pattern.

14. Look at this pattern.

4, 9, 19, _____

What is the next step in this pattern?

15. Describe what is happening in this pattern.

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Difficulty Level:

Basic

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Date Created:

Oct 29, 2012

Last Modified:

Apr 11, 2014
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