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6.18: Problem Solving Plan, Diagrams

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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 Concept will show you how to effectively use a diagram to solve a problem.

Guidance

You have been learning about fractions and mixed numbers and about how to add and subtract them. Many of the examples in these Concepts 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.

John ate \frac{1}{5} 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.

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.

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.

Shannon jogged 1 \frac{3}{20} miles yesterday. Today, she jogged \frac{1}{2} 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 1 \frac{3}{20} of the diagram.

This represents the 1 \frac{3}{20} miles that Shannon jogged yesterday.

Shannon also jogged \frac{1}{2} mile today.

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

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

Method two –– Set up an addition problem:

To find out how many miles she jogged all together, add 1 \frac{3}{20} + \frac{1}{2} . The fractional part of the mixed number has a different denominator than \frac{1}{2} . Find the least common multiple (LCM) of both denominators. The least common multiple of 20 and 2 is 20. Next, we rename the fractions.

\frac{1}{2} = \frac{10}{20}

Now we can add the two together.

1 \frac{3}{20} + \frac{10}{20} = 1 \frac{13}{20}

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.

Now it's time for you to try a few. Draw a diagram and solve each problem.

Example A

2 \frac{1}{3} + 4

Solution: 6 \frac{1}{3}

Example B

\frac{4}{5} - \frac{1}{5}

Solution: \frac{3}{5}

Example C

\frac{3}{4} + \frac{2}{4}

Solution: 1 \frac{1}{4}

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

\frac{1}{3} = \frac{4}{12}

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 \frac{8}{12} 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.

Vocabulary

Here are the vocabulary words in this Concept.

Problem Solving
using key words and operations to solve mathematical dilemmas written in verbal language
Diagram
a drawing used to represent a mathematical problem.

Guided Practice

Here is one for you to try on your own.

Teri ran 1 \frac{1}{2} miles yesterday, and she ran 2 \frac{1}{2} miles today. How many miles did she run in all?

If John ran 7 miles, what is the difference between his total miles and Teri’s total miles? How many miles have they run altogether?

Answer

We can solve this problem a couple of different ways. First, we could draw a diagram of the path of both runners.

Let's start with Terri.

1 \frac{1}{2} + 2 \frac{1}{2} = 4 miles

Terri ran 4 miles.

John ran 7 miles.

There is a difference of 3 miles.

Together, they ran 11 miles.

These are our answers.

Video Review

Here is a video for review.

Khan Academy: Word Problem Solving Strategies

Practice

Directions: Solve each of the following problems by using a problem solving strategy.

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. John and Terri each ran 18 miles. If Kyle ran half the distance that both John and Teri ran, how many miles did he run?

10. If Jeff ran 3 \frac{1}{2} miles, how much did he and Kyle run altogether?

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

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

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

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

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

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Date Created:

Oct 29, 2012

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Jun 25, 2014
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