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# 6.8: Problem – Solving Strategy-Draw a Diagram

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Tiled Floor

On his last day with Uncle Larry, Travis worked with Mr. Wilson on laying tile on the kitchen floor. Travis worked hard all morning and he was a bit discouraged when he reached his first break and realized that he had only finished about one-third of the floor.

It had taken Travis two hours to tile one-third of the floor. He thought about this as he drank from his water bottle and ate an apple.

“If it took me this long to tile one-third, how long will it take me to finish?” Travis wondered.

The floor is divided into 12 sections. If he has finished one-third of them, how many sections has he completed? This is the number that he completed in the two hours.

How many sections does he have left to complete? About how long will it take him to finish the rest?

There are many different strategies you could use to help Travis solve this problem, but drawing a diagram is probably the most useful. This lesson will show you how to effectively use a diagram to solve a problem.

What You Will Learn

By the end of this lesson you will be able to demonstrate the following skills:

• Read and understand given problem situations.
• Develop and use the strategy: Draw a diagram.
• Plan and complete 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

In this chapter, you have been learning about fractions and mixed numbers and about how to add and subtract them. Many of the examples in this chapter have used pictures to help you learn to solve them.

Drawing a diagram or a picture is a strategy to help you solve many different problems. The first thing that you have to do when approaching a problem is to read and understand the problem and how to solve it.

Example

John ate 15\begin{align*}\frac{1}{5}\end{align*} of the cake. What fraction is left?

First, you can see that we have the amount of cake the John ate and we need to know how much he has left. We are going to be subtracting. Let’s draw a diagram to show what we know about John and his cake.

II. Develop and Use the Strategy: Draw a Diagram

Now that we have looked at what we know and what we need to know, we can draw the diagram. This is a diagram of fraction bars to represent John’s cake. The blue section shows how much of the cake John has eaten. The white bars represent the amount of cake that is left.

Here is the one-fifth that John ate. You can see that there are four-fifths left.

The answer to the problem is four-fifths.

III. Plan and Compare Alternative Approaches to Solving Problems

Sometimes, we can set up a problem as addition and sometimes we can set it up as subtraction. Often times both ways will work but one will make more sense than the other.

Let’s look at an example.

Example

Shannon jogged 1320\begin{align*}1 \frac{3}{20}\end{align*} miles yesterday. Today, she jogged 12\begin{align*}\frac{1}{2}\end{align*} mile.

How many total miles did Shannon jog?

Method one –– Draw a diagram:

One way to solve this problem is to draw a diagram. Let’s start by looking at the first distance that Shannon jogged. Draw two same-sized rectangles. Divide one rectangle into 20 equal-sized sections. Then shade 1320\begin{align*}1 \frac{3}{20}\end{align*} of the diagram.

This represents the 1320\begin{align*}1 \frac{3}{20}\end{align*} miles that Shannon jogged yesterday.

Shannon also jogged 12\begin{align*}\frac{1}{2}\end{align*} mile today.

So, shade 12\begin{align*}\frac{1}{2}\end{align*} of the partially filled rectangle to represent the distance she jogged today.

The diagram is 11320\begin{align*}1 \frac{13}{20}\end{align*} shaded. So, Shannon jogged a total of 11320\begin{align*}1 \frac{13}{20}\end{align*} miles on those two days.

Method two –– Set up an addition problem:

To find out how many miles she jogged all together, add 1320+12\begin{align*}1 \frac{3}{20} + \frac{1}{2}\end{align*}.

The fractional part of the mixed number has a different denominator than 12\begin{align*}\frac{1}{2}\end{align*}.

Find the least common multiple (LCM) of both denominators. The least common multiple of 20 and 2 is 20.

Next, we rename the problems.

12=1020\begin{align*}\frac{1}{2} = \frac{10}{20}\end{align*}

Now we can add the two together.

1320+1020=11320\begin{align*}1 \frac{3}{20} + \frac{10}{20} = 1 \frac{13}{20}\end{align*}

Notice that our answer is the same. Both methods will produce the same result. You can choose the method that you find easiest when working on problems like this.

## Real Life Example Completed

The Tiled Floor

Let’s use a diagram to help Travis with his tiling project. Here is the problem once again.

On his last day with Uncle Larry, Travis worked with Mr. Wilson on laying tile on the kitchen floor. Travis worked hard all morning and he was a bit discouraged when he reached his first break and realized that he had only finished about one-third of the floor.

It had taken Travis two hours to tile one-third of the floor. He thought about this as he drank from his water bottle and ate an apple.

“If it took me this long to tile one-third, how long will it take me to finish?” Travis wondered.

The floor is divided into 12 sections. If he has finished one-third of them, how many sections has he completed? This is the number that he completed in the two hours.

How many sections does he have left to complete? About how long will it take him to finish the rest?

First, let’s underline all of the important information to help us read and understand the problem.

Let’s figure out how much of the floor Travis has finished. First, let’s find an equivalent fraction for one-third with a denominator of 12.

13=412\begin{align*}\frac{1}{3} = \frac{4}{12}\end{align*}

Next, we can draw a diagram of the finished part of the floor.

Here is a picture of what Travis has finished.

How much does he have left?

We can count the units and see that he has 812\begin{align*}\frac{8}{12}\end{align*} of the floor left to tile. This is double what he did in two hours.

Travis has about four hours of work left.

Travis finishes his break and gets back to work. If he continues working at the same pace, he will finish working around 2 pm just in time for some pizza for lunch.

## Time to Practice

Directions: Solve each of the following problems by drawing a diagram. Show your answer and your diagram.

1. Tyler has eaten one-fifth of the pizza. If he eats another two-fifths of the pizza, what part of the pizza does he have left?

2. What part has he eaten in all?

3. How many parts of this pizza make a whole?

4. Maria decides to join Tyler in eating pizza. She orders a vegetarian pizza with six slices. If she eats two slices of pizza, what fraction has she eaten?

5. What fraction does she have left?

6. If Tyler was to eat half of Maria’s pizza, how many pieces would that be?

7. If Maria eats one-third, and Tyler eats half, what fraction of the pizza is left?

8. How much of the pizza have they eaten altogether?

9. Teri ran 112\begin{align*}1 \frac{1}{2}\end{align*} miles yesterday, and she ran 212\begin{align*}2 \frac{1}{2}\end{align*} miles today. How many miles did she run in all?

10. If John ran 7 miles, what is the difference between his total miles and Teri’s total miles?

11. How many miles have they run altogether?

12. If Kyle ran half the distance that both John and Teri ran, how many miles did he run?

13. If Jeff ran 312\begin{align*}3 \frac{1}{2}\end{align*} miles, how much did he and Kyle run altogether?

14. What is the distance between Jeff and Kyle’s combined mileage and John and Teri’s combined mileage?

15. Sarah gave Joey one-third of the pie. Kara gave him one-fourth of another pie. How much pie did Joey receive altogether?

16. Is this less than or more than one-half of a pie?

17. Who gave Joey a larger part of the pie, Kara or Sarah?

18. What is the difference between the two fractions of pie?

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