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1.7: A Problem-Solving Plan

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Introduction

Galehead Hut

When their work at the Lonesome Lake Hut was finished, the group learned that their next destination was to be another hut. They were to hike from Lonesome Lake Hut, to Greenleaf Hut, then head to the Galehead Hut. They would stop at the Galehead hut.

“That looks pretty simple on this map,” Yalisha said looking at the map above.

“Yes, but it’s not. Look at this one,” Kelly said, pulling out a more detailed map of the White Mountains.

The group would take the Old Bridle Path from Lonesome Lake to Greenleaf Hut. It is 4.5 miles and the estimated travel time is 3 hours and 30 minutes.

“We’ll be in the Presidential range,” John said happily.

The Presidential Range of Mountains, in the White Mountains, contains peaks named after several US Presidents. Mount Washington is one of the most famous. However, there are other exciting ones to see and climb.

The group set off at 7 am. It was a challenging hike and took longer than expected. Instead of 3 hours and 30 minutes, the group arrived after 5 hours. It took them an extra 1 hour and 30 minutes.

At the hut, they tried to decide whether they should go on, or stay at the hut for the rest of the day. The group was undecided and the leaders tried to let the students work it out amongst themselves. While all the arguing was going on, Raoul pulled out the map and made these notes on a piece of paper.

Greenleaf to Galehead by Garfield Ridge Trail is 7.7 miles estimated 5 hours and 20 minutes.

If the group decides to move on, what is their estimated arrival time at the next hut given today’s pace? If they want to be at the next hut by dinner time, will they make it?

This lesson is about problem solving. Problem solving is what the group needs to work on together, in order to make some decisions.

What You Will Learn

In this lesson you will learn 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 solve the problem.
  • Solve real-world problems using a plan.

Teaching Time

I. Read and Understand a Given Problem Situation

One of the most important skills for any mathematics student is the ability to understand real-world problems, develop an approach to solving the problem, and carry out a problem solving plan.

There are countless ways to solve problems.

Drawing pictures, making lists, working backwards, guessing and checking, looking for patterns, and writing equations are just a few of the approaches mathematicians take to solving problems.

In this lesson we will read and examine all kinds of problems. Then we will develop a plan and solve them.

How do we start when faced with a difficult problem?

The key to developing an appropriate problem solving plan is taking the time to read and understand the problem or situation.

You can’t expect to be able to grasp the important information in a problem if you just skim it for the important numbers.

When you come across a complicated problem, don’t panic!

Slow down and carefully read the problem until you understand what it’s really asking.

You have already been doing this work in each introduction problem. Now we can learn specific methods.

Example

On Monday, Jake spent a total of 180 minutes on his math, history, and science homework.

He spent 45 minutes on his math homework and 1 hour on his science homework.

How many minutes did he spend on his history homework?

Key questions can help you to understand a problem.

Here are a few key questions.

1. What is the information that needs to be found?

The amount of time Jake spent on his history homework.

2. What do you need to know to answer this question?

The total amount of time he spent and the time he spent on his other two subjects.

3. What information have you been given?

He spent 180 minutes total. Of that, he spent 45 minutes on science and 1 hour (60 minutes) on math.

4. How do we solve this problem?

Think about what skills you have already learned. We could write an equation to figure out this problem.

\text{Total time} &= \text{math time} + \text{history time} + \text{science time}\\180 &= 60+x+45\\180 &= 105+x

Now use mental math, “What number plus 105 is equal to 180?”

75=x

He spent 75 minutes, or 1 hour and 15 minutes, on his history homework.

Example

A sandwich shop sold 36 tuna fish sandwiches and 45 roast beef sandwiches. The shop sold three times as many turkey sandwiches as tuna sandwiches. How many turkey sandwiches did the shop sell?

Use the following questions to help you understand the problem:

1. What is the question?

How many turkey sandwiches did the shop sell?

2. What do you need to know?

How many tuna sandwiches the shop sold

3. What do you know?

The shop sold 36 tuna sandwiches and three times as many turkey sandwiches as tuna sandwiches. You don’t need to know how many roast beef sandwiches the shop sold.

4. How can you solve the problem?

By multiplying the number of tuna fish sandwiches by three. 36 \times 3 = 108.

The shop sold 108 turkey sandwiches.

Reading a problem carefully and using these key questions will help you when working through a story problem.

II. Make a Plan to Solve the Problem

Many times a problem situation will require more than one step to find a solution. This is where having a plan becomes so important. When confronted with this type of problem remember to carefully read to find out what the problem is asking. With multi-step problems, it often helps to solve a simpler problem, then return to the main problem and apply what you found in the simpler problem to solve.

Example

William began a fitness schedule. He ran 2 miles the first week, 2.5 miles the second week, and 3 miles the third week. If the pattern continues, how many miles will William have run after five weeks?

First, let’s think about what the problem is asking.

The problem is asking for the number of miles that William will run in the fifth week.

What information have we been given?

We have been given his mileage for weeks one, two and three. We have also been told that there is a pattern.

What is our plan?

First, we need to figure out the mileage for weeks four and five. Then, we need to find the sum of all of the weeks.

Think about what you have already learned. We need to use the pattern to figure out the next two weeks' mileage. What strategies did we use with patterns?

Yes! A table would be perfect! Let’s organize the data in a table. The left column is the number of the week, and the right column is the mileage.

1 \qquad 2\\2 \qquad 2.5\\3 \qquad 3\\4 \qquad 3.5\\5 \qquad 4

During weeks four and five, here is the mileage. Notice that the pattern increased by one-half mile each week.

Now we can find the sum of the total miles.

2 + 2.5 + 3 + 3.5 + 4 = 15 \ miles

Our answer is 15 miles.

III. Solve the Problem and Check the Results

Particularly when a problem requires multiple steps to solve, it is important to examine your solution and ask, “Does my solution make sense?” A good habit in problem solving is to take your solution, return to the original problem, and check to see if your solution makes sense.

Example

Ms. Hahn has three square rugs. The second rug has twice the area of the first rug. The third rug has three times the area of the second rug. If the third rug has an area of 24 \ ft^2, what is the length of 1 side of the first rug?

It is clear in this problem that we need to work backward, using the area of the third rug to find the area of the second rug and using the area of the second rug to find the area of the first rug. From the area of the first rug, we need to find the length of one of the sides.

Step 1: Find the length of one of the sides.

The problem tells us that the third rug has an area of 24 \ ft^2. We know that the third rug has 3 times the area of the second rug. If x is the area of the second rug, 3x is the area of the third rug. Now we can solve for the area of the second rug.

3x = 24

Ask yourself, “What number times 3 is 24?”

x = 8 \ ft^2

Now we know the area of the second rug is 8 \ ft^2. We can use this to find the area of the first rug because the problem tells us the area of the second rug is twice the area of the first rug. If the area of the first rug is y, the area of the second rug is 2y. Now we can solve for the area of the first rug.

2y = 8

Ask yourself, “What number times two is eight?”

y = 4 \ ft^2

Now that we know the area of the first rug, we can find the length of one of its sides. Remember the formula for area of a square? A = s^2. Now we can solve for one of the sides.

A &= s^2\\4 &= s^2

What number times itself is 4?

2 \times 2 = 4, so one of the sides of the first rug is 2 ft

To check our answer, we can return to the problem, starting with the first rug and working up to the third rug.

If one of the sides of the first rug is 2 feet, the area of the first rug is 4 \ ft^2.

The area of the second rug is twice the area of the first rug, or 8 \ ft^2.

The area of the third rug is three times the area of the second rug, or 24 \ ft^2.

Our solution is correct!

IV. Compare Alternative Approaches to Solving the Problem

There usually isn’t one right way to solve a problem. This means that logical reasoning is one of the most important skills you can bring to problem solving. Strategies and plans are in place because they make solving certain types of problems easier. As long as you take the time to understand what the problem is asking and develop a logical approach, you can find the solution, no matter what strategy you choose to use.

For example, let’s look back at the question about William and the running.

Example

William began a fitness schedule. He ran 2 miles the first week, 2.5 miles the second week, and 3 miles the third week. If the pattern continues, how many miles will William have run after five weeks?

When working on this problem, we created a table and then added up the number of miles ran per week.

We could have tackled this problem in another way. We could have written a list of the miles.

2, 2.5, 3

Then we would have seen the pattern of +.5.

We could have continued to add on .5 to each of the previous terms in the list.

2, 2.5, 3, 3.5, 4

That would have given us the correct mileage to then find the sum.

Our answer would have been the same, but our way of getting there would have differed.

V. Solve Real World Problems Using a Plan

Any time we encounter sets of real-world problems, we need to be flexible in our approach and willing to apply whatever strategy seems to best fit the situation. Remember these steps as you work through a problem.

The best approach is a careful reading of the problem.

Then ask yourself the following types of questions to help solve the problem: What is the problem asking?

What information do I have? What information do I need?

What is the best strategy to use?

After you solve, remember to ask: Does my solution make sense? Always go back to the original problem and double check!

Now we can go back to the original problem and help our hikers to solve their dilemma.

Real Life Example Completed

Galehead Hut

Here is the original problem once again. Reread this problem and underline any important information.

When their work at the Lonesome Lake Hut was finished, the group learned that their next destination was to be another hut. They were to hike from Lonesome Lake Hut, to Greenleaf Hut, then head to the Galehead Hut. They would stop at the Galehead hut.

“That looks pretty simple on this map,” Yalisha said looking at the map above.

“Yes, but it’s not. Look at this one,” Kelly said, pulling out a more detailed map of the White Mountains.

The group would take the Old Bridle Path from Lonesome Lake to Greenleaf Hut. It is 4.5 miles and the estimated travel time is 3 hours and 30 minutes.

“We’ll be in the Presidential range,” John said happily.

The Presidential Range of Mountains, in the White Mountains, contains peaks named after several US Presidents. Mount Washington is one of the most famous. However, there are other exciting ones to see and climb. The group set off at 7 am. It was a challenging hike and took longer than expected. Instead of 3 hours and 30 minutes, the group arrived after 5 hours. It took them an extra 1 hour and 30 minutes. At the hut, they tried to decide whether they should go on, or stay at the hut for the rest of the day. The group was undecided and the leaders tried to let the students work it out amongst themselves. While all the arguing was going on, Raoul pulled out the map and made these notes on a piece of paper. Greenleaf to Galehead by Garfield Ridge Trail is 7.7 miles estimated 5 hours and 20 minutes.

If the group decides to move on, what is their estimated arrival time at the next hut given today’s pace? If they want to be at the next hut by dinner time, will they make it?

First, let’s make a note that it is noon when the group arrives at the first hut. Now let’s figure out the estimated hiking time from one hut to the next one. It was an estimated 5 hours and 20 minutes, but the group required 1 hour and 30 minutes longer.

5 hours and 20 minutes + 1 hour and 30 minutes = 6 hours and 50 minutes or an approximated 7 hours

If the group left right away, they wouldn’t arrive before 7 pm. That would definitely make them late for dinner.

Raul showed this math to his friends while they were still arguing. Given the time it would take, the group decided to stay at Greenleaf hut for the night.

Resources

Here are some places you can learn more about the Presidential Range.

http://home.earthlink.net/~ellozy/greenleaf-hut.html#tocref5 – Routes to Greenleaf Hut and Galehead Hut with maps, times and mileage.

http://en.wikipedia.org/wiki/Presidential_Range – This website contains overall information about the mountains and has some pictures.

Time to Practice

Directions: Read each of the questions to develop a strategy and solve each problem. Be sure to check your answer and make sure that it makes sense.

1. Giovanni is working to improve the scores on his weekly science quizzes. The following were his scores for the first four weeks of school: 64, 70, 76, 82. If the pattern continues, in which week will Giovanni’s score be 100?

2. Lola’s Bakery uses 62 pounds of flour and 19 pounds of sugar each week. The bakery uses half as much butter as flour each week. How much butter will the bakery use in a month’s time?

3. A children’s pool holds 6 cubic meters of water. The length of the pool is three times the height and the width of the pool is twice the height. What is the height of the pool?

4. Martin went to the state fair with $30. He rode 17 rides and came home with $4.50. How much did each ride cost?

5. Lionel arranged 24 photos in an album. The number of photos in each row is 5 more than the number of rows. How many rows of photos are there?

6. An aquarium has four fish tanks it wants to arrange on a shelf. The shelf has an area of 196 square feet. The area of the second tank is twice the area of the first tank, and the area of the third tank is four times the area of the second tank. The area of the third tank is 56 square feet. If the aquarium puts all three tanks on the shelf, how much shelf area will be left over?

7. The Durands are driving 456 miles to their family reunion. If they split the drive equally over three days, how many miles will they drive each day?

8. At the end of a video game tournament, Raul and Martha had both scored twice as many points as Justice. If the total combined points of all three players is 285, how many points did each player score?

9. At the lunch counter, Ariana bought a sandwich and lemonade. The sandwich cost five times as much as the lemonade. She paid with $10 and got $2.50 in change. How much did the sandwich cost? How much did the lemonade cost?

10. The area of a playground is 40.5 \ m^2. The width of the playground is half the length. What is the length of the playground?

Directions: Write five of your own problems. Be sure that one uses addition, one uses subtraction, one uses multiplication, one uses division and one uses a pattern that requires a table.

When finished, exchange papers with a friend and solve each other’s problems. Then discuss your solutions.

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