# 1.8: Problem Solving Strategies

**At Grade**Created by: CK-12

## Introduction

*Summer Flights*

“What are you guys talking about?” Carmen asked coming over to see Kevin and Laila.

“We were talking about what we did this summer. I went to Yellowstone camping and Kevin did this really great project with a group of kids making a garden. What about you? What did you do with your summer?” Laila asked Carmen as she took a drink of milk.

“I went to see my Grandparents. It was a great time, but I barely made it on the day of my flight,” Carmen said munching a carrot.

“What happened?” Kevin asked.

“Well, it started out fine. I had a 9 pm flight. I knew that I had to be at the airport 2 hours before the flight and that we live one hour from the airport. I needed \begin{align*}1 \frac{1}{2}\end{align*}

Kevin looked at Laila.

“You should have had plenty of time,” Kevin said.

**How does Kevin know this? Can you follow Kevin’s thinking? Why does Kevin make this statement? To figure this out, you will need to apply your problem solving skills. This problem will appear again at the end of the lesson.**

*What You Will Learn*

By the end of this lesson you will be able to use the following skills.

- Read and understand given problem situations.
- Develop and use a variety of strategies.
- 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**

In this chapter we have talked about and worked with many different strategies. Each of the introduction problems required you to use different strategies to think about and solve problems. You have learned to use the strategies Write an Equation, Make a Model, Work Backward, and Look for a Pattern. In this lesson, you will use all of the strategies you learned. For each situation, you will be asked to read and understand a given problem. You will make a plan to solve by choosing an appropriate strategy. Recall that there are multiple ways to solve a word problem. Therefore, you will be asked to consider and compare different approaches for each problem given.

**First, let’s think about reading a problem.**

When you read to understand a problem, you are working to determine what the problem is asking you to do. It helps to highlight the question in the problem. You may want to also underline clue words that may help you with planning and strategy.

Example

A lizard ate five hundred flies on five consecutive nights. Each night he ate twenty-five more than the night before. How many flies did the lizard eat each night?

**In this section, you are working to determine what the problem is asking you to do.**

For this problem, you are to determine the number of flies the lizard ate each night. Here is what you are told:

- You are told that the lizard ate a total of five hundred flies over the course of five nights.
- You are told that the lizard eats twenty-five more flies each night than the night before.
- You should know the word consecutive means a logical sequence or succession. In this case, it means one night after the other.

Example

A train’s caboose is 12 feet long. Each of the train’s eight cars are twice the length of the caboose. Determine the length of the entire train.

**Ask: What is the problem asking you to find out?**

You are to determine the entire length of the train. You were told some information.

- The caboose is 12 feet long.
- There area of an additional eight cars.
- Each car is twice the length of the caboose.

**Once you have had some practice reading for understanding, it is time to work with some strategies for problem solving.**

II. **Develop and Use a Variety of Strategies**

The next step after reading the problem is to make a plan to solve. When you make a plan to solve, you decide which strategy to use. In this section, you will see the problems from the first section.

Example

A lizard ate five hundred flies on five consecutive nights. Each night he ate twenty-five more than the night before. How many flies did the lizard eat each night?

**Recall what you are to find out.**

For this problem, you are to determine the number of flies the lizard ate each night. Here is what you are told:

- You are told that the lizard ate a total of five hundred flies over the course of five nights.
- You are told that the lizard eats twenty-five more flies each night than the night before.
- You should know the word consecutive means a logical sequence or succession. In this case, it means one night after the other.

**Make plan to solve the problem.**

Use a verbal model to write and solve an equation to determine the unknown number of flies eaten each day.

You are told that the lizard ate a total of five hundred flies in five days. You are also told that each night he eats twenty-five more than the night before. To determine the number of flies consumed each night, you must first determine the number of flies the lizard ate the first night. After determining the number of flies consumed the first night, add twenty-five more each day to get the daily total.

**The next step is to solve the problem.**

*Verbal Model:*

number of flies eaten on night one + (number of flies eaten on night one + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five + twenty-five) = total number of flies eaten over five nights (500)

Let “\begin{align*}x\end{align*}

*Equation:*

\begin{align*}x + (x + 25) + (x + 25 + 25) + (x + 25 + 25 + 25) + (x + 25 + 25 + 25 + 25) = 500\end{align*}

*Solution:*

\begin{align*}x + (x + 25) + (x + 25 + 25) + (x + 25 + 25 + 25) + (x + 25 + 25 + 25 + 25) &= 500\\ x + (x + 25) + (x + 50) + (x + 75) + (x + 100) &= 500\\ 5x + 250 &= 500\end{align*}

**Next, we solve the equation. Subtract 250 from 500 and divide by 5.**

\begin{align*}x=50\end{align*}

**The lizard consumed 50 flies the first night. To determine the number of flies the lizard ate on nights two, three, four, and five, substitute 50 for “\begin{align*}x\end{align*}” in the equation.**

\begin{align*}x + (x + 25) + (x + 50) + (x + 75) + (x + 100) &= 500\\ 50 + (50 + 25) + (50 + 50) + (50 + 75) + (50 + 100) &= 500\\ 50 + 75 + 100 + 125 + 150 &= 500\end{align*}

**You can see that on night one, the lizard ate 50 flies.**

**On night two, the lizard consumed 75 flies.**

**On night three, the lizard ate 100 flies.**

**On night four, the lizard ate 125 flies.**

**On the last night, the lizard ate 150 flies.**

**Check the Results**

You can check your work, by adding the number of flies consumed each night. The sum should be equal to five hundred.

\begin{align*}50 + 75 + 100 + 125 + 150 = 500\end{align*}

**Answer:**

**Night One: 50**

**Night Two: 75**

**Night Three: 100**

**Night Four: 125**

**Night Five: 150**

Example

A train’s caboose is 12 feet long. Each of the train’s eight cars are twice the length of the caboose. Determine the length entire train.

**Ask: What is the problem asking you to find out?**

You are to determine the entire length of the train. You were told:

- The caboose is 12 feet long.
- There area an additional eight cars.
- Each car is twice the length of the caboose.

**Make a plan to solve the problem.**

You can draw a diagram and use a verbal model to visualize the information given in the problem. Then, write an equation to determine the length of the entire train.

*Verbal Model:*

Eight trains twice the length of the caboose + the length of the caboose = entire length of the entire train

Let “\begin{align*}x\end{align*}” represent the unknown length of the train.

*Equation:*

\begin{align*}8(2 \cdot 12) + 12 = x\end{align*}

*Solution:*

\begin{align*}8(2 \cdot 12) + 12 &= x\\ 8(24) + 12 &= x\\ 192 + 12 &= x\\ 204 &= x\end{align*}

**The entire train is 204 feet.**

III. **Plan and Compare Alternative Approaches to Solving Problems**

When working with a problem, you have many different approaches to choose from. While some approaches may be better than others, you can select different strategies and still end up with the correct answer.

**The key here is to check your work.**

**Always make sure that your answer makes sense.**

**If you aren’t sure that your work is correct, then use a different strategy to test your results.**

IV. **Solve Real – World Problems Using Selected Strategies as Part of a Plan**

In this section, you will use all you have learned about problem solving to solve real-world problems. Remember to read and understand the problems, make plans to solve, and then check your answers.

Example

A ten year olds’ heart beats approximately 85 times per minute. How many times does the heart beat in 24 seconds?

**What is the problem asking you to do?**

Knowing that the heart beats 85 times per minute, determine the number of heart beats in 24 seconds.

**What is the plan?**

Since you are being asked to determine an unknown rate, write a proportion to solve.

**Solve the problem.**

When you look at the problem, you should notice that the number of heart beats is given in minutes. The question asks for you to give the answer in number of heart beats per second. Therefore when writing the proportion, express one minute as sixty seconds. Recall that to solve a proportion, cross multiply then use inverse operations to determine the value of the unknown quantity.

Let “\begin{align*}x\end{align*}” = the number of heart beats in 24 seconds

\begin{align*}& \ \underline{\;\; 85 \ beats\;\;} = \underline{\;\;\;\;\;\;\; x \;\;\;\;\;\;\;}\\ & \ 60 \ seconds \quad \ 24 \ seconds\\ & \qquad 85(24) = 60(x)\\ & \ \quad \underline{\;\; 2,040 \;\;} = \underline{\;\; 60x \;\;}\\ & \ \qquad \ 60 \qquad \quad \ 60\\ & \qquad \qquad \ x=34 \ beats\end{align*}

**Check the results.**

Since you were told that the heart beats 85 times per minute, you know that your answer should be less than that amount since you were determining how many times the heart beats in 24 seconds. You can see that your answer is reasonable because it is less than 85.

You can also check your results by substituting 34 for the variable “\begin{align*}x\end{align*}” into the equation \begin{align*}85(24) = 60x\end{align*}.

\begin{align*}85(24) &= 60x\\ 85(24) &= 60(34)\\ 2,040 &= 2,040\end{align*}

**The answer is accurate and our work checks out.**

**Now let’s go back to the problem from the introduction and work on solving that problem.**

## Real-Life Example Completed

*Summer Flights*

**Here is the original problem once again. Reread it and then explain why Kevin made the statement he made. What time did Carmen need to leave her home to be on time for the flight? There are two parts to your answer.**

“What are you guys talking about?” Carmen asked coming over to see Kevin and Laila.

“We were talking about what we did this summer. I went to Yellowstone camping and Kevin did this really great project with a group of kids making a garden. What about you? What did you do with your summer?” Laila asked Carmen as she took a drink of milk.

“I went to see my Grandparents. It was a great time, but I barely made it on the day of my flight,” Carmen said munching a carrot.

“What happened?” Kevin asked.

“Well, it started out fine. I had a 9 pm flight. I knew that I had to be at the airport 2 hours before the flight and that we live one hour from the airport. I needed \begin{align*}1 \ \frac{1}{2}\end{align*} to pack and take a shower and stuff like that. It would have been fine except I had a plan to play soccer at the park first. I got home at 4:00 and barely made it to the airport,” Carmen explained.

Kevin looked at Laila.

“You should have had plenty of time,” Kevin said.

*First, explain Kevin’s statement. Then figure out what time Carmen should have left her home for the airport. There are two parts to your answer.*

*Solution to Real – Life Example*

**First, why did Kevin make that statement?**

Kevin figure out that Carmen should have left for the airport at 6 pm. If she only needed \begin{align*}1 \frac{1}{2}\end{align*} hours to get ready, then she should have had plenty of time because she got home at 4 pm leaving her 2 hours to get ready.

**Here is the breakdown of her time.**

9 pm flight – 2 hours check in = 7 pm

7 pm – 1 hour drive time = 6 pm

\begin{align*}6 - 1 \frac{1}{2}\end{align*} hours to get ready = 4:30 pm

**Carmen should have left her home at 4:30 pm to be on time for the flight.**

## Time to Practice

Directions: Read each problem and then solve each problem.

**Ted has a collection of rare coins. He already had 34 coins in his collection. The first week, Ted purchases 1 new coin. During the second week, Ted purchases 4 coins. During the third week, Ted adds 9 new coins to his collection. At this rate how many weeks will it take Ted to collect 125 coins?**

- Which strategy should Ted use to solve this problem?
- What could Ted draw to help him with his solution?
- How long will it take Ted to collect 125 coins?

**Savannah purchases a pair of jeans on sale for $59.00. The price is 25% less than the original price. Determine the original price of the jeans.**

- Which strategy could Savannah use to calculate the price?
- What is the amount of the discount?
- What is the original price?

**Extension: Have the students work in pairs to write three problems. One solved through a pattern and table, one solved through a diagram and one solved with an equation then have the students solve them and explain their thinking.**