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Comparison of Problem-Solving Models

Compare methods of organizing and solving story problems

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Comparison of Problem-Solving Models

Lisa's grandmother gave her some money for her birthday. Lisa decided to save it to help pay for a trip with her friends this summer. Since her birthday, Lisa has saved an additional $5 every week! It's been 8 weeks and Lisa has$95. How much money did Lisa's grandmother give her for her birthday?

In this concept, you will learn how to choose an appropriate strategy for solving a real world problem.

Comparing Problem-Solving Models

To start solving a real world problem it can help to ask yourself the following key questions.

Key Questions:

1. What am I trying to find out?
2. What do I know?
3. How can I solve the problem?

By reading the problem a few times you should be able to answer the first two key questions. However, sometimes answering the third key question is not that easy. It is often not obvious how to solve a problem! Here are some commonly used strategies for solving problems.

1. 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.
2. 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.
3. Work Backwards - Best used when you are given a total or final amount and you are looking for some partial amount or original amount.
4. Draw a Picture - Best used when the problem describes some sort of visual.
5. Write an Equation - Best used when there is a missing quantity that needs to be figured out. Then you can write an equation and solve for the answer.
6. Use a Formula - At this point, best used for area and perimeter problems.

The more you practice solving problems, the quicker you will become at identifying the most appropriate strategy to use.

Here is an example.

Ron arranged his herb garden rows in the following order: 2 plants, 5 plants, 11 plants, 23 plants. How many plants will be in the fifth row?

First, ask yourself “what am I trying to find out?”

You are trying to find out how many plants will be in the fifth row of Ron's herb garden.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• The first row has 2 plants.
• The second row has 5 plants.
• The third row has 11 plants.
• The fourth row has 23 plants.

Then, ask yourself “how can I solve the problem?”

Notice that in this problem you were given a list of numbers and asked about a later quantity. This is a perfect time to use the find a pattern strategy. First, find the pattern and describe the pattern rule. Then, extend the pattern to figure out how many plants are in the fifth row.

Now, implement your plan. So far the pattern is 2, 5, 11, 23, . . .

The pattern rule is multiply by 2 and add 1.

\begin{align*}\begin{array}{rcl} 2×2+1 & = & 5\\ 5×2+1 & = & 11\\ 11×2+1 & = & 23 \end{array}\end{align*}

Next, extend the pattern.

\begin{align*}23 \times 2+1=47\end{align*}

The answer is that there will be 47 plants in the fifth row.

Remember that once you have an answer, you need to make sure you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how many plants are in the fifth row and that's what you did. 47 plants is a lot, but it fits the pattern so your answer makes sense.

Examples

Example 1

Earlier, you were given a problem about Lisa and her saved money.

After her grandmother gave her some money for her birthday, Lisa has been saving $5 a week for 8 weeks. She now has$95! You wonder how much money Lisa got from her grandmother for her birthday.

First, ask yourself “what am I trying to find out?”

You are trying to find out how much money Lisa got from her grandmother for her birthday.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• Currently Lisa has $95 dollars. • Lisa is saving$5 each week.
• It has been 8 weeks.
• Lisa started out with just the money from her grandmother.

Then, ask yourself “how can I solve the problem?”

Notice that in this problem you were given a final amount of $95 and asked about an initial amount. This is a good time to use the work backwards strategy. First, figure out how much money Lisa has saved over the 8 weeks since she got the money from her grandmother. Then, work backwards from the$95 to figure out how much money she must have started with from her grandmother.

Now, implement your plan. Lisa has saved 5 a week for 8 weeks. \begin{align*}\5×8=\40\end{align*} So over the 8 weeks Lisa has saved an additional40.

Next, work backwards. Lisa has $95 now. She added$40 to the money from her grandmother to get to the $95. That means you can subtract$40 form 95 to figure out how much money Lisa started with from her grandmother. \begin{align*}\95-\40=\55\end{align*} The answer is Lisa got55 from her grandmother.

Make sure you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how much money Lisa's grandmother gave her for her birthday and you found out that Lisa got $55 from her grandmother.$55 is a realistic amount of money for a birthday present so your answer makes sense.

Example 2

Ms. Powell wants to hang a large rectangular tapestry lengthwise on her living room wall. The tapestry has a perimeter of 42 feet and a width of 9 feet. Ms. Powell’s wall is 10 feet high. Will the length of the tapestry fit against the height of Ms. Powell’s ceiling?

First, ask yourself “what am I trying to find out?”

You are trying to find out if the length of the tapestry is less than the height of Ms. Powell's ceiling which is 10 feet.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• The tapestry is a rectangle.
• The tapestry has a perimeter of 42 feet.
• The tapestry has a width of 9 feet.
• Ms. Powell's ceilings are 10 feet high.
• The tapestry will be hung lengthwise.

Then, ask yourself “how can I solve the problem?”

Notice that this problem referenced perimeter and a rectangle, so this is a great time to use the use a formula strategy. First, use the rectangle perimeter formula to find the length of the tapestry. Then, see if the length of the tapestry is less than 10 feet.

Now, implement your plan. The rectangle perimeter formula is \begin{align*}P=2l+2w\end{align*}. You know \begin{align*}P=42\end{align*} and \begin{align*}w=9\end{align*}. You want to figure out the length, \begin{align*}l\end{align*}.

Substitute the values for \begin{align*}P\end{align*} and \begin{align*}w\end{align*} into the rectangle perimeter formula.

\begin{align*}\begin{array}{rcl} P & = & 2l+2w\\ 42 & = & 2l+2(9)\\ 42 & = & 2l+18 \end{array}\end{align*}

Now, solve for \begin{align*}2l\end{align*}. “What number plus 18 is equal to 42?” You know that 24 plus 18 is equal to 42, so \begin{align*}2l\end{align*} must be equal to 24.

\begin{align*}2l=24\end{align*}

Next you can solve for the length \begin{align*}l\end{align*}. “2 times what number is equal to 24?” You know that 2 times 12 is equal to 24 so \begin{align*}l\end{align*} must be equal to 12.

\begin{align*}l=12\end{align*}

Ms. Powell's tapestry is 12 feet long. Since her ceilings are 10 feet high, the tapestry is too long to fit on the wall.

The answer is that Ms. Powell's tapestry is too long to fit on the wall.

Now make sure you answered the question. The question asked if the length of the tapestry was less than the height of Ms. Powell's ceiling. You determined that no, the length of the tapestry is not less than the height of the ceiling. 12 feet for the length of a tapestry that you would put on the wall is realistic, so your answer makes sense.

Example 3

Melissa has 144 cookies she wants to put evenly into 8 gift bags. How many cookies will go into each bag?

First, ask yourself “what am I trying to find out?”.

You are trying to find out how many cookies will go into each gift bag.

Next, ask yourself “what do I know?”.

You know the following pieces of information:

• There are 144 cookies total.
• There are 8 gift bags.
• Each gift bag needs to have the same number of cookies.

Then, ask yourself “how can I solve the problem?”

Notice that we have a missing quantity to figure out, the number of cookies in each bag. This is a good time to use the write an equation strategy. First, write an equation to show the relationship between the different numbers in the problem. Then, solve the equation.

Now, implement your plan. You know the number of cookies in each bag times the number of bags will equal the total number of cookies. The unknown quantity is the number of cookies in each bag so that will be your variable.

Let \begin{align*}x=\end{align*}  the number of cookies in each bag.

\begin{align*}x \cdot 8=144\end{align*}

Now, solve the equation. “What number times 8 is equal to 144?” If you need to, you can use your calculator to divide 144 by 8 to get the answer. 18 times 8 is equal to 144 so \begin{align*}x\end{align*} is equal to 18.

\begin{align*}x=18\end{align*}

Next, make sure you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how many cookies were in each bag and you did. 18 cookies in a bag is realistic so your answer makes sense.

Example 4

A list of numbers is found in the problem. Which strategy might you use?

When there is a list of numbers, the find a pattern strategy is a useful one to try.

The answer is the find a pattern strategy.

Example 5

A final quantity is given, but the initial amount is missing. Which strategy might you use?

When you know the final amount but not the starting amount, the work backwards strategy is a good one to try.

The answer is the work backwards strategy.

Review

Solve the following problems.

1. 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? 2. 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? 3. The area of City Park is \begin{align*}75 \ km^2\end{align*}. The length of the park is 15 feet. What is the perimeter of the park? 4. A farmer planted corn, wheat, and cotton in a total of 88 fields. He planted 10 rows of corn. If he planted even numbers of rows for the other crops, how many rows of wheat did he plant? 5. 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 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? 6. After a pin-ball 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? 7. Tami made a total of47 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? 8. A weightlifter lifts weights in the following order: 0.5 lb, 1.5 lb, 4.5 lb, 13.5 lb. How many pounds will he lift next? 9. 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? 10. Mr. Rowe and Mrs. Rowe are driving 959 miles to a beach vacation. They want to split the 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? 11. 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?
12. What was the total cost without the toppings?
13. If 29 people went to the zoo the first day and double went the second, how many went in both days combined?
14. If on the third day, double the people from the second day went, how many people went to the zoo?
15. If ten less went to the zoo on the fourth day verses the third day, how many people went to the zoo on the fourth day?

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Color Highlighted Text Notes

Vocabulary Language: English

Difference

The result of a subtraction operation is called a difference.

Problem Solving

Problem solving is using key words and operations to solve mathematical dilemmas written in verbal language.

Product

The product is the result after two amounts have been multiplied.

Quotient

The quotient is the result after two amounts have been divided.

Sum

The sum is the result after two or more amounts have been added together.