# 1.16: Problem-Solving Models

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

Remember the hikers? Take a look at this dilemma.

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 to the Galehead Hut. Then 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 estimated time is 3:30 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 being one of the most famous of the mountains. However, there are definitely exciting other ones to see and climb.

The group set off at 7 am. It was a challenging hike and took longer than expected. Instead of 3:30 minutes, the group arrived in 5 hours. It was an extra 1:30 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 5:20 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 Concept is about problem solving. Use this Concept to help the group make some decisions.**

### Guidance

**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.

**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 a problem 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.

Look at this situation.

On Monday, Jake spent a total of 180 minutes on her 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 question that needs answering?

**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 plus 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.**

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

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

\begin{align*}75=x\end{align*}

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

**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.

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 has run in the fifth week.

**What information have we been given?**

We have been given his mileage for week’s 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 week’s 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 week’s 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.**

\begin{align*}& 1 && 2\\ & 2 && 2.5\\ & 3 && 3\\ & 4 && 3.5\\ & 5 && 4\end{align*}

**During week’s 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.**

\begin{align*}2 + 2.5 + 3 + 3.5 + 4 = 15 \ miles\end{align*}

**Our answer is 15 miles.**

**Thinking backwards, we can check our work and see that our answer is accurate and makes sense.**

Answer the following questions about the problem solving plan.

#### Example A

When approaching a problem, what is the first thing that you should do?

**Solution: Read the problem carefully.**

#### Example B

True or false. You can solve a problem without knowing what the problem is asking.

**Solution: False. You must know what the problem is asking.**

#### Example C

Once you have an answer your work is finished.

**Solution: False. You must check your answer to be sure that it makes sense.**

Here is the original problem once again.

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 to the Galehead Hut. Then 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 estimated time is 3:30 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 being one of the most famous of the mountains. However, there are definitely exciting other ones to see and climb.

The group set off at 7 am. It was a challenging hike and took longer than expected. Instead of 3:30 minutes, the group arrived in 5 hours. It was an extra 1:30 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, Raul 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 5:20 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 is an estimated 5:20 minutes, but the group is taking 1:30 minutes longer.**

**5:20 + 1:30 = 6:50 or 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.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Product
- the answer in a multiplication problem

- Difference
- the answer in a subtraction problem

- Sum
- the answer in an addition problem

- Quotient
- the answer in a division problem

- Problem Solving
- solving a problem mathematically that is written in verbal language

### Guided Practice

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?

**Answer**

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. \begin{align*}36 \times 3 = 108\end{align*}.**

**The shop sold 108 turkey sandwiches.**

### Video Review

Here is a video for review.

- This is a Khan Academy video on how to solve word problems using a plan.

### Practice

Directions: Use each of the key 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 his 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 on 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 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 the 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 their total combined points 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 \begin{align*}40.5 \ m^2\end{align*}. 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.

### Image Attributions

Here you'll learn to solve real-world problems involving a plan.