1.8: Problem Solving Strategies
Introduction
The Four Thousand Footers
While at Galehead Hut, Kelly found a book on the different mountains in the Presidential Range. She was fascinated to learn that so many of them were above 4,000 feet in elevation. Laurel, one of the leaders, saw her reading it and came over to her.
“Pretty interesting, huh?” she asked Kelly.
“Yes. I had no idea.”
“Well if you continue hiking maybe you’ll become part of the Four Thousand Footers Club,” Laurel said.
“What is that?” Kelly asked.
“That is a group that climbs all of the peaks above 4,000 feet. There are 48 of them.”
Wow! Kelly couldn’t believe it. If each peak was at least 4,000 feet, that would be quite a collection. Kelly looked in the book again and found a whole chapter on the Four Thousand Footers Club. She was fascinated. She wrote down the following mountains in her journal.
Washington 6,288 ft.
Adams 5,774 ft
Jefferson 5,712 ft.
Monroe 5,384 ft.
Madison 5,367 ft.
Lafayette 5,260 ft.
Lincoln 5,089 ft.
If she climbed each of these peaks, how many feet would that be in all? If each peak took two days average to climb, how many days would it take her to climb them all?
Use your problem solving to help Kelly figure this out. You will solve this problem by the end of the lesson.
What You Will Learn
In this lesson you will learn 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
Taking the time to read and understand a problem is the key to finding a solution. Unfortunately, it is the part that students often rush through and then end up making mistakes or becoming confused. You can avoid that problem yourself by using the questions that you learned in the last section. Remember those?
Also, keep in mind that sometimes there will be extra information in a problem deliberately put there to throw you off. Pay close attention as you read and don’t be fooled!!
Let’s look at an example that has extra information in it.
Example
Ron arranged his herb garden in rows, with the following number of plants in each row:
2 plants, 5 plants, 11 plants, 23 plants
The garden has an area of \begin{align*}25 \ yd^2\end{align*}.
How many plants will be in the fifth row?
Whenever you see a series of numbers, this should alert you to use the "find a pattern" strategy.
This problem is asking us to identify a pattern rule and use that pattern rule to find how many plants will be in the fifth row.
The pattern rule is \begin{align*}2x + 1\end{align*} where "x" is the previous number, so there will be 47 plants in the fifth row.
The question has nothing to do with the area of the garden, so we can ignore that information.
Make a note to in your notebook that when you see a series of numbers, you need to look for a pattern.
Example
Melissa has 72 cookies she wants to put as evenly as possible into 7 gift bags.
There are half as many chocolate chip cookies as peanut butter cookies.
How many cookies will be left over after she puts them in the bags?
When we see the phrase “how many will be left over, we can guess the problem might be asking about remainders and that to solve this problem we will need to divide.
We don’t care about the numbers of different types of cookies, so we can ignore that information.
\begin{align*}72 \div 7 = 10r2\end{align*}
There will be two cookies left over.
II. Develop and Use a Variety of Strategies
There are a variety of strategies that you could select from when working on a problem, such as: find a pattern, guess and check, work backward, draw a picture, write an equation, and use a formula. The more you practice solving problems, the quicker you will become at identifying the most appropriate strategies to use when solving specific types of problems.
Here are some hints to aid in selecting the best strategy:
- Find a Pattern - best used when there is a series of numbers and/or when you are being asked for a later quantity. For example, find the number in the tenth step.
- Guess and Check - best used when you are looking for one or two numbers and you think one of them might work. You can take a guess, try out a number and then adjust your answer from there.
- Work Backwards - think about the problems that you had earlier in this chapter when you were given the area or perimeter and you needed to find a side length. Working backwards is very helpful for problems like these where an answer of some sort is given right away.
- Draw a Picture - look for examples that have some kind of visual in them; problems with geometric shapes work best with the drawing a picture strategy.
- Write an Equation - writing an equation is great when there is a missing quantity that needs to be figured out.
- Use a Formula - Formulas are helpful for area and perimeter problems. You will also encounter other formulas as you work through this book and those can be applied when problem solving as well.
Take a few notes before continuing in this lesson. Be sure that you understand all of the different strategies and when to use each one.
IV. Plan and Compare Alternative Approaches to Solving Problems
There are many different ways to solve a problem and still get the correct answer. Strategies are designed to help make problem-solving faster. Working through lots of problems, you will find the approaches and strategies that work best for you. Look back at the notes you wrote in the last section. When selecting a strategy, those will be helpful. Also, remember that sometimes more than one can be used. For example, a problem with an unknown could be solved by an equation, but if it has a series of numbers, it could also be solved by looking for a pattern.
Let’s practice.
Example
Ms. Powell wants to hang a large tapestry lengthwise on her living room wall. The tapestry has a perimeter of 42 ft and a width of 9 ft. Ms. Powell’s wall is 10 ft high. Will the length of the tapestry fit the height of Ms. Powell’s wall without hitting the ceiling?
This is a multi-step problem requiring a variety of different strategies to solve. To start off, it might help to draw a picture to get a feel for what the problem is asking. The problem asks if the tapestry will fit. We are going to have to find the length of the tapestry and compare it to the height of Ms. Powell’s wall.
Because the question involves perimeter, we know we are going to have to use the formula for perimeter. \begin{align*}P = 2l + 2w\end{align*}. We know the perimeter of the tapestry and we know the width, so we can write an equation to solve for the length.
\begin{align*}P &= 2l + 2w.\\ 42 &= 2l + 2(9)\\ 42 &= 2l + 18\\ 24 &= 2l\\ l &= 12 \ feet\end{align*}
Now we need to return to the problem and compare the length of the tapestry with the height of the wall. The tapestry is 12 feet long; the ceiling is only 10 feet high. The tapestry won’t fit!
An alternative approach to solving this problem would have been to only look at a drawing. We could have drawn each piece of the problem and then compared. Looking at the dimensions, you would have been able to see that the tapestry would not fit. Let's look at another example:
Example
The band leader has arranged 56 musicians that will be participating in a parade. The number of players he placed in each row was 10 more than the number of rows. How many rows were there?
To approach this problem, we need to assume that the musicians are arranged in equal rows. Then we know we are going to have to look for factors of 56 which fit these qualifications. We can work to “guess and check” for numbers that work.
The guessing and checking strategy uses \begin{align*}7 \times 8 = 56\end{align*}, but that doesn’t work. \begin{align*}28 \times 2 = 56\end{align*}, but that doesn’t work either. \begin{align*}14 \times 4 = 56\end{align*} and that works! If there are 14 musicians in each row, there will be 10 more than the number of rows.
Guessing and checking isn’t the only way to solve this problem. You might also choose to draw a picture or make an organized list.
Real Life Example Completed
The Four Thousand Footers
Here is the original problem once again. Reread it and underline any important information.
While at Galehead Hut, Kelly found a book on the different mountains in the Presidential Range. She was fascinated to learn that so many of them were above 4,000 feet in elevation. Laurel, one of the leaders, saw her reading it and came over to her.
“Pretty interesting, huh?” she asked Kelly.
“Yes. I had no idea.”
“Well if you continue hiking maybe you’ll become part of the Four Thousand Footers Club,” Laurel said.
“What is that?” Kelly asked.
“That is a group that climbs all of the peaks above 4,000 feet. There are 48 of them.”
Wow! Kelly couldn’t believe it. If each peak was at least 4,000 feet, that would be quite a collection. Kelly looked in the book again and found a whole chapter on the Four Thousand Footers Club. She was fascinated. She wrote down the following mountains in her journal.
Washington 6,288 ft.
Adams 5,774 ft
Jefferson 5,712 ft.
Monroe 5,384 ft.
Madison 5,367 ft.
Lafayette 5,260 ft.
Lincoln 5089 ft.
If she climbed each of these peaks, how many feet would that be in all? If each peak took two days average to climb, how many days would it take her to climb them all?
First, let’s find the sum of all of the elevations in Kelly’s list.
\begin{align*}6288 + 5774 + 5712 + 5384 + 5367 + 5260 + 5089 = 38874 \ feet\end{align*}
Keep in mind that this is only one way. Kelly still has to hike back down. You will need to double your number to accurately represent the total feet hiked for these mountains.
How many miles is that?
77,748 feet \begin{align*}\div\end{align*} 5280 feet (the distance in one mile) = 14.72 miles
Now if it were to take 2 days for each peak, how many days would it take Kelly to climb these mountains on her list?
We can write an equation.
7 mountains
2 days
\begin{align*}7 \times 2 = 14 \ days\end{align*}
At Galehead Hut, Kelly looked out at the views of the mountains and was glad to be a part of the summer teen adventure program. While she was tired, she was also satisfied and proud of what she had accomplished so far. She was excited to think about what the next adventure would be!
Time to Practice
Directions: Use what you have learned to solve the following problems.
- Mary went to the music store with her babysitting money. She bought two CDs for $12.50 each and two magazines for $4.25 each. She left the store with $10.25. How much money did she start with?
- Since he began his fitness routine, Mr. Trigg has measured his weight every week. His weights for the first six weeks are as follows: 236, 230, 232, 226, 228, 222. If the pattern continues, how much will he weigh in the tenth week?
- The area of City Park is \begin{align*}75 \ km^2\end{align*}. The length of the park is 3 times the width. What is the perimeter of the park?
- A farmer planted corn, wheat, and cotton in a total of 88 fields. He planted twice as many fields in corn than in wheat and half as many in cotton than in corn. How many fields did he plant of each?
- Mrs. Whitaker is mailing a pair of shoes to her daughter. She wants to fit the rectangular shoebox inside a larger square box. The area of the base of the shoebox is \begin{align*}84 \ in.^2\end{align*}; the length of one side is 12 inches. One side of the larger square box measures 14 inches. Will the shoebox fit in the larger box? How do you know?
- After a pinball game, the score board showed that the combined points of Peter, Ella, and Ned is 728. Ella scored half the points of Ned, and Peter scored one-fourth the points of Ned. How many points did each player score?
- Tami made a total of $47 babysitting on New Year’s Eve. She made her hourly rate plus a $7 tip. If she worked 5 hours, what is her hourly rate?
- A weightlifter lifts weights in the following order: 0.5lb, 1.5lb, 4.5lb, 13.5lb. How many pounds will he lift next?
- Figure A is a square with a side that measures 9 cm. Figure B is a square with a side that measures 6 cm. Which figure has the greater area, Figure A or Figure B?
- Mr. and Mrs. Rowe are driving 959 miles to have a vacation on the beach. They want to split the driving distance over 4 days, driving the exact same amount on the first three days and the remainder on the fourth day. If they drive 119 miles on the fourth day, how many miles will they drive on the first day?
- Cedric spent $27.75 on pizza for his friends. Each cheese pizza cost $8 and each extra topping cost $0.75. If Cedric bought 3 cheese pizzas, how many extra toppings did he get?
- At the trading fair, Chi Wong arranged 72 baseball cards in rows on the trading table. Each row had 14 more cards than the number of rows. How many cards were in each row?
Take a few minutes to check the strategies you used with a peer. Did you use the same strategies or different strategies in problem solving?
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