1.7: A Problem Solving Plan
Introduction
The Garden
Kevin finished looking at the pictures from Laila’s trip to Yellowstone National Park. He took a deep sigh.
“I didn’t do anything that exciting this summer,” he said with another sigh.
“I’m sure you did great stuff. What did you do? Tell me about it,” Laila said smiling.
“Well, the big thing that I did was to design and build a vegetable garden. It actually was quite cool, because I was working as a Junior Counselor at the Boys and Girls Club and so I had a bunch of seven year olds who helped me,” Kevin said.
“That is terrific.”
“You know, it really was. I designed the garden to fit in this corner of the play yard. We had an area of \begin{align*}6^\prime \times 12^\prime\end{align*} to work with, and then we wanted to plant broccoli, carrots, peas, squash, zucchini, peppers, eggplant and tomatoes. It actually involved a lot of math. We had to figure out the area of the land and then the kids wanted each vegetable to have an even spot in the garden,” Kevin explained.
Laila began thinking about this problem in her head.
It is time for you to do the same thing. There are several steps to this problem and to work through it, you will need to figure out a problem solving plan. After learning the information in this section, you should be all ready to begin.
What You Will Learn
By the end of this lesson you will be able to use the following skills.
- Read and understand a given problem situation.
- Make a plan to solve the problem.
- Solve the problem and check the results
- Compare alternative approaches to solving the problem.
- Solve real-world problems using a plan.
Teaching Time
I. Read and Understand a Given Problem Situation
By now, you have probably learned that there are many ways to solve a problem. Several problem solving strategies are listed on the table below. When making a plan to solve a problem, you may choose one or several strategies.
Act it Out
Make a Model
Try and Check
Look for a Pattern
Guess and check
Make a Table
Work Backward
Write an Equation
Write a Proportion
In this lesson, you will learn how to use a few of these strategies when approaching a problem. Let’s start with the first step in any problem solving plan-reading the problem for understanding.
When you read a problem, it is helpful to underline any important information. Important information can include words that identify an operation. You can also identify key words such as distance, time, speed, area or perimeter. All of these parts help you to identify a problem and what the problem is asking you to solve.
Let’s look at an example.
Example
Mollie is planning to meet a group of friends at the movies on Saturday night at 6:00 p.m. Mollie is in charge of driving a few friends to the movies. Mollie lives 15 minutes from Sara’s house. It takes 10 minutes to get from Sara’s house to Madison’s house. It is another 20 minutes to the movie theatre from Madison’s house. It takes Mollie 1 hour to get dressed and ready for the evening. At what time should Mollie begin to get ready for the evening?
Step 1: Read and Understand the Problem
Ask: What is the problem asking me to find out?
The problem is asking you to determine the time at which Mollie should begin getting ready for the evening if she is to be at the movies at 6:00 p.m. You must take in to consideration the amount of travel time to each of her friends houses and the amount of time it takes for Mollie to get dressed.
Now that you know what you are looking for, we can work on figuring out a plan.
II. Make a Plan to Solve the Problem
This problem involves time. We need to back up the time that it takes for Molly to get to the movie theater. Because of this, we can look at the strategies mentioned in the first section and see if any of these makes sense for our problem solving plan.
Act it Out
Make a Model
Try and Check
Look for a Pattern
Write a Proportion
Make a Table
Work Backward
Write an Equation
Guess and check
We don’t need to act anything out, so that eliminates option one.
Making a model could be one option, but let’s look for a better one.
Try and check doesn’t make sense because we don’t have a number to try.
We need to find the time first, then we can try one out.
Looking for a pattern doesn’t make sense because there isn’t a pattern of numbers shown.
We don’t have any values to compare, so writing a proportion doesn’t make sense.
Making a table doesn’t make sense because we don’t have a list of values.
Working backwards, hmmm. This makes sense! We know the final time that Molly needs to arrive at the movie theater, so if we work backwards, we will be able to help her figure out the time that she needs to get there.
Step 2: Make a Plan
Mollie needs to be at the movies at 6:00 p.m. Work backward to determine the time Mollie should begin getting ready.
III. Solve the Problem and Check the Results
Now that we have selected a plan, we can use “working backwards” as our strategy and solve the problem.
Step 3: Solve the Problem
Time of Movie – Time to Dress – Time to Movies – Time to Madison’s – Time to Sara’s
6:00 – 1:00 – 0:20 – 0:10 – 0:15
5:00 – 0:20 – 0:10 – 0:15
4:40 – 0:10 – 0:15
4:30 – 0.15
4:15
With all that she has to do, Mollie should begin getting ready at 4:15 p.m.
Now that we have a solution, we need to check our results to be sure that our work is accurate.
Step 4: Check the Results
If Mollie begins getting dressed at 4:15 p.m. and takes 1 hour, she will be ready to leave at 5:15 p.m. Since it takes Mollie 15 minutes to get to Sara’s house, she’ll arrive at 5:30 p.m. 10 minutes later at 5:40 Mollie will arrive at Madison’s house. Since it takes 20 minutes to drive to the movies, Mollie will arrive at the movies promptly at 6:00 p.m.
Mollie needs to begin getting ready at 4:15 p.m.
V. Compare Alternative Approaches to Solving the Problem
While we used “working backwards” successfully, there could have been another strategy for solving this problem. Let’s look at building a model.
Begin by placing the hands at six o’ clock. Move the hour hand back one to represent the time Mollie spent getting dressed. It is now, 5:00 p.m. Move the minute hand back 15 minutes to represent the time Mollie spent picking up Sara. The clock should now read 4:45. Move the minute hand back 10 minutes to represent the time Mollie spent picking up Madison. The clock should now read 4:35. Move the minute hand back 20 minutes to represent the time it for Mollie and the girls to drive to the theatre. The clock should read 4:15.
We could have used the model to solve the problem as well. When problem solving, the important thing is to find a method that you are comfortable with and one that seems to serve the problem.
VI. Solve Real – World Problems Using a Plan
We can solve real-world problems using a problem solving plan. Think about patterns and problem solving. You see examples of patterns around you each day. The leaves of a plant, the petals on a flower, and a colorful quilt all exude patterns. In math, a pattern is a repeated set of numbers. You can analyze patterns in data to make predictions and generalizations. Sometimes you may solve a problem just by determining a pattern. You may also be asked to use the pattern to determine a solution.
Example
The bowling alley charges $12 for one person to play, $10 for the second person to play, $8 for the third to person to play, and so on. What is the total cost for a family of five to bowl? How much money does the family save bowling together rather than separately?
Step 1: Read and Understand the Problem
Ask: What is the problem asking me to do?
The problem is asking you to analyze the pattern to determine the cost for a family of five to bowl. After determining the cost of a family of five to bowl, determine the difference between the amount the family is spending and the amount it would cost for five individuals to play.
Step 2: Make a Plan
To better observe the pattern, organize the information on a table.
Family Member | Cost to Play |
---|---|
1 | $12 |
2 | $10 |
3 | $8 |
You can see that the cost to play decreases by two dollars for each additional family member.
Step 3: Solve the Problem
Continue the pattern for the fourth and fifth family members. Add the cost to play for each family member to determine the total cost. You can see that the total cost for a family of five is $40.
Family Member | Cost to Play | Total Cost |
---|---|---|
1 | $12 | $12 |
2 | $10 | $22 |
3 | $8 | $30 |
4 | $6 | $36 |
5 | $4 | $40 |
To determine the amount saved playing as a family, subtract the total cost for a family of five from the total cost for five individuals. The cost to play individually is $12, therefore the cost for five individuals is \begin{align*}\$ 60 (\$12 \times 5)\end{align*}. $40 subtracted from $60 is $20.
\begin{align*}\text{Cost to Play Individually} - \text{Cost to Play as a Family} = \text{Amount Saved}\end{align*}
\begin{align*}\$ 60 - \$ 40 = \$ 20\end{align*}
The cost for a family of five to bowl is $40. A family of five saves $20 playing together rather than individually.
Now let’s look at how we can use a problem solving plan to figure out the solution to the problem in the introduction.
Real-Life Example Completed
The Garden
Here is the original problem once again. First, reread it. Then you will need to find the area of the garden using a formula. After that, you will need to show how much of the area each vegetable was given in the garden. There are two parts to your answer.
Kevin finished looking at the pictures from Laila’s trip to Yellowstone National Park. He took a deep sigh.
“I didn’t do anything that exciting this summer,” he said with another sigh.
“I’m sure you did great stuff. What did you do? Tell me about it,” Laila said smiling.
“Well, the big thing that I did was to design and build a vegetable garden. It actually was quite cool, because I was working as a Junior Counselor at the Boys and Girls Club and so I had a bunch of seven year olds who helped me,” Kevin said.
“That is terrific.”
“You know, it really was. I designed the garden to fit in this corner of the play yard. We had an area of \begin{align*}6^\prime \times 12^\prime\end{align*} to work with, and then we wanted to plant broccoli, carrots, peas, squash, zucchini, peppers, eggplant and tomatoes. It actually involved a lot of math. We had to figure out the area of the land and then the kids wanted each vegetable to have an even spot in the garden,” Kevin explained.
Laila began thinking about this problem in her head.
Now solve for the area of the garden. Then solve for the area of the garden that was given to each vegetable. Remember, there are two parts to your answer.
Solution to Real – Life Example
First, you need to find the area of the garden.
Area is found by using the formula \begin{align*}l \times w\end{align*}.
You know that the length of the garden is 12 feet and the width of the garden is 6 feet. We can substitute those values into the formula and solve for the area of the garden.
\begin{align*}A &= lw\\ A &= 12(6)\\ A &= 72 \ sq.feet\end{align*}
Notice that you need to label your answer in square feet because we are working with area.
To figure out how much area each vegetable was given, we can use mental math. Think about what we know.
The area of the garden is 72 sq. feet.
There are 8 vegetables being planted.
\begin{align*}72 \div 8 = 9\end{align*}
Each vegetable was given 9 square feet.
Time to Practice
Directions: Read each problem and then answer the questions following each problem.
At the end of a phone call home, Brad had $0.42. During the last minute of the phone call, the operator asked him to deposit $0.15. The initial cost of the phone call was $0.75 plus $0.12 per minute. If Brad spoke on the phone for 20 minutes, how much money did he have before making the phone call home?
- Should you use working backwards or writing a proportion for this problem?
- Why wouldn’t you use a proportion for this problem?
- You will need to multiply one part of this problem, which part?
- What equation could you write to solve this problem?
- How much money did Brad have before making the phone call?
Suppose you have $75 in your savings account. You plan to save an additional $25 per week. After how many weeks will you have saved $500?
- Which strategy makes the most sense look for a pattern or working backwards?
- What is the unknown quantity that you are trying to figure out?
- What equation can you use to solve this problem?
- How many weeks will it take to save $500.00?
An online music store charges $1.89 to download 2 songs. Determine the cost of downloading 13 songs.
- Which strategy would you use look for a pattern or use a proportion?
- Why would you use that method?
- What is the cost for the 13 songs?
- What would be the cost for double the songs?
- If the cost had been $2.25 per song, how much would 2 songs cost?
- What would be the cost for 4 songs?
- If six friends each downloaded four songs, how much would the total cost be?
For every day that Jesse harvested vegetables in the garden, he collected 4 pounds of vegetables. If Jesse continued this for 45 days, how many total pounds of vegetables will he have collected?
- Does it make sense to look for a pattern or work backwards on this problem?
- What equation can you write to show the total number of vegetables that Jesse gathered?
- How many total pounds did he gather after 45 days?
- If Jesse could collect vegetables for 90 days, how many pounds would he collect in all?
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