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Problem-Solving Plan & Strategies

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Problem-Solving Plan & Strategies

Grade 8 TAKS Objective 6 8.14 (B) & (C)

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Step 1:

Understand the Problem

- What are you asked to find?
- Make sure that you understand exactly what the problem is asking.
- Restate the problem in your own words.

- What information is given in the problem?
- List every piece of information the problem gives you.

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Step 1:

Understand the Problem

- Is all the information relevant?
- Sometimes problems have extra information that is not needed to solve the problem.
- Try to determine what is and is not needed.
- This helps you stay organized when you are making a plan.

- Were you given enough information to solve the problem?
- Sometimes there simply is not enough information to solve the problem.
- List what else you need to know to solve the problem.

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Step 2:

Make a Plan

- What problem-solving strategy or strategies can you use to help you solve the problem?
- Think about strategies you have used in the past to solve the problems.
- Would any of them be helpful in solving this problem?

- Create a step-by-step plan of how you will solve the problem.
- Write out your plan in words to help you get a clearer idea of how to solve the problem mathematically.

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Step 3:

Solve

- Use your plan to solve the problem.
- Translate your plan from words to math.
- Show each step in your solution and write your answer in a complete sentence.

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Step 4:

Look Back

- Did you completely answer the question that was asked?
- Be sure answered the question that was asked and that your answer is complete.

- Is your answer reasonable?
- Your answer should make sense.

- Could you have used a different strategy to solve the problem?
- Solving the problem again with a different strategy is a good way to check your answer.

- Did you learn anything that could help you solve similar problems in the future?
- You may want to take notes about this kind of problem and the strategy you used to solve it.

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- To be a good problem solver you need a good problem-solving plan.
- Using a problem-solving plan along with a problem-solving strategy helps you organize your work and correctly solve the problem.

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- A problem-solving strategy is a plan for solving a problem.
- Different strategies work better for different types of problems.
- Sometimes you can use more than one strategy to solve a problem.
- As you practice solving problems, you will discover which strategies you prefer and which work best in various situations.

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- Some problem-solving strategies include
- drawing a picture;
- looking for a pattern;
- guessing and checking;
- acting it out;
- making a table;
- working a simpler problem; and
- working backwards.

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- You can solve problems by drawing pictures.
- Step 1: Draw a picture to represent the situation.
- Step 2: Finish the picture to show the action in the story.
- Step 3: Interpret the picture to find the answer.

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- Example: Suzanne was stacking boxes in a display window. The boxes were shaped in the shape of a triangle. How many boxes are needed altogether if there are 7 boxes on the bottom row?

Bottom row: 3 boxes

Total: 6 boxes

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- Read and understand
- What do you know?
- Boxes are being stacked.
- 3 on the bottom gives 6 altogether.

- What are you trying to find?
- The total number of boxes needed if there are 7 boxes in the bottom row.

- What do you know?

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- Plan and Solve
- What strategy will you use?
- Strategy: Draw a picture.

- What strategy will you use?

Bottom row: 5 boxes

Total: 15 boxes

Bottom row: 3 boxes

Total: 6 boxes

Bottom row: 4 boxes

Total: 10 boxes

Bottom row: 6 boxes

Total: 21 boxes

Bottom row: 7 boxes

Total: 28 boxes

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- Look Back and Check
- Is your answer reasonable?
- Yes, the pattern continues in the triangle shapes.
- The bottom row has one more box than the bottom row of the previous stack.

- Is your answer reasonable?

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- A pilot is 3,800 ft below the clouds. A plane at a lower altitude is 5,500 ft above the ground. The clouds are at 12,000 ft altitude. Find the difference in altitude between the planes.

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- Looking at how numbers or figures in a pattern compare can help you find the rule that creates the pattern.
- Example: The president of Best Idea has a 5-phase emergency plan.
- Phase 1: The president emails her 2 vice presidents.
- Phase 2: Each vice president forwards the email to 2 employees.
- Phase 3: Each employee who receives the email sends it to 2 employees.
- How many employees work for the company if everyone knows about the emergency in 5 phases?

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- Read and understand
- What do you know?
- Three employees know in Phase 1.
- In Phase 2, seven employees know.
- Fifteen employees know in Phase 3.

- What are you trying to find?
- Find the number of employees in the company.

- What do you know?

Phase 1

Phase 2

Phase 3

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- Plan and Solve
- What strategy will you use?
- Strategy: Look for a pattern

- What strategy will you use?

Phase

1

Phase

2

Phase

3

Phase

4

Phase

5

3

7

15

31

?

22

23

24

25

Answer: There are 63 employees in the company.

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- Look Back and Check
- Is your answer reasonable?
- Yes, the difference from one phase to the next doubles.

- Is your answer reasonable?

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- Name the missing numbers: 1, 3, 9, 27, __, __, __

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- Predicting and verifying can help solve problems.
- Step 1: Think to make a reasonable first try.
- Step 2: Check using information given in the problem.
- Step 3: Revise. Use your first try to make a reasonable second try. Check.
- Step 4: Use previous tries to continue trying and checking until you get the answer.

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- Example: Kyle needs an average score of 85 (without rounding) on three exams. His scores on the first two exams were 82 and 86. What score must he get on the last exam?

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- Read and Understand
- What do you know?
- Kyle needs an average score of 85.
- He scored 82 and 86 on two of the exams.

- What are you trying to find?
- Find the score on the last exam that will give him an average greater than or equal to 85.

- What do you know?

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- Plan and Solve
- Try 90: 82 + 86 + 90 = 258; 258 ÷ 3 = 86; 90 works. Since any score greater than or equal to 90 works, s ≥ 90
- Try 85: 82 + 86 + 85 = 253; 253 ÷ 3 ≈ 84.3; 84 is too low.
- Try 86: 82 + 86 + 86 = 254; 254 ÷ 3 ≈ 84.7; This rounds to 85 but Kyle’s teacher does not round grades.
- Try 87: 82 + 86 + 87 = 255; 255 ÷ 3 = 85; s ≥ 87

- Answer: A score of 87 or greater on the last exam will give Kyle an average of 85 or greater.

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- Look Back and Check
- Is your work correct?
- Yes. It makes sense that the third score must be greater than 85.

- Is your work correct?

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- Carla needs an average score of 75 (without rounding) on 3 tests. Her first two scores were 73 and 70. What score must she get on the third test?

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- To act out a problem means to perform or model the actions in the problem.
- Example: Marco tosses two coins, a quarter and a nickel. Write a proportion that can be used to find p, the percent of times Marco can expect both coins to land heads up.

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- You can act out this problem by using two actual coins. Take a quarter and a nickel and turn them over one at a time until you have listed all the possible outcomes of tossing both coins.
- There are four possible outcomes to this experiment: both coins land heads up; the quarter lands heads up and the nickel lands tails up; both coins land tails up; or the quarter lands tails up and the nickel lands heads up.

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H

H

H

T

Quarter

Nickel

Quarter

Nickel

T

T

T

H

Quarter

Nickel

Quarter

Nickel

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- Only one of these outcomes—both coins land heads up—is a favorable outcome. The probability of both coins landing heads up is the ratio of the number of favorable outcomes to the number of possible outcomes: ¼ .
- A percent is a ratio that compares a number to 100. Find the number of favorable outcomes for every 100 possible outcomes.
- Represent p, the percent of favorable outcomes, in a ratio. p/100
- Write a proportion using these two ratios. p/100 = ¼
- The proportion p/100 = ¼ can be used to find p, the percent of times Marco can expect both coins to land heads up.

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- Organizing data in a table can help you find the information you are looking for.
- Look for patterns in the data.
- Step 1: Set up the table with the correct labels.
- Step 2: Enter known data into the table.
- Step 3: Look for a pattern. Extend the table.
- Step 4: Find the answer in the table.

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- Example: A scientist fills a balloon with a gas. As the gas is heated, the balloon expands in proportion to the temperature of the gas.
- The table below shows the measurements the scientist recorded.
- Write a proportion that can be used to find V, the volume of the balloon when the temperature is 35ºC.

Balloon Measurements

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- Write a ratio that compares a known volume to a known temperature.
Volume = 125

Temperature 25

- Write a ratio that compares the unknown volume to the temperature 35ºC.
Volume = V

Temperature 35

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- The volume of the balloon is proportional to the temperature of the gas. Write a proportion setting these two ratios as equal.
- The following proportion can be used to find the value of V, the volume of the balloon when the temperature is 35ºC:
125=V

2535

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- For every 3 items you buy at the school store you earn 25 points toward the purchase of other items. How many items would you need to buy in order to earn enough points for a backpack worth 175 points? Make a table to solve the problem.

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- Sometimes solving a simpler problem can help you solve a more difficult problem.
- Step 1: Break apart or change the problem into one or more problems that are simpler to solve.
- Step 2: Solve the simpler problem(s).
- Step 3: Use the answer(s) to the simpler problem to help you solve the original problem.

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- Example: The diagram shows a swimming pool in the shape of a rectangle with semicircular ends.
- Write an equation that can be used to find A, the approximate area of the pool.

34 feet

16 feet

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- By looking closely at the pool, you can see that it consists of a rectangle and two semicircles of equal size.
- One way to find the area of the pool is to solve two simpler problems.
- Find the area of the rectangle.
- Find the area of the circle equal to the sum of the areas of the two semicircles.
- Add the area of the rectangle and the area of the circle to represent the total area of the pool.

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34 feet

- Area of rectangle = lw = 34 x 16
- Area of circle = πr2
- The radius is equal to the diameter divided by 2.
- r =16 ÷ 2 = 8
- Use 3 to approximate the value of π.
- Area of circle ≈ 3 x 82
- The equation A ≈ (34 x 16) + (3 x 82) can be used to find the approximate area of the pool.

16 feet

16 feet

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- The Werner’s dinner bill is $75 and they want to leave a 15% tip. Explain how you can solve a simpler problem to calculate the tip amount. Then find the tip amount.

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- Start with what you know and work backwards to find the answer.
- Step 1: Identify what you are trying to find.
- Step 2: Draw a diagram to show each change, starting from the unknown.
- Step 3: Start at the end. Work backward using the inverse of each change.

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- Start with what you know and work backwards to find the answer.
- Example: Melissa wants to see a movie with her friend Gillian. The movie starts at 3:15 P.M., and they want to get to the theater 10 minutes early to buy tickets. It will take Melissa 10 minutes to ride her bike to Gillian’s house and twice that time for the girls to ride from Gillian’s house to the movie theater. What time should Melissa leave her house?

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- Identify the steps you would take to solve this problem logically. Begin with the starting time of the movie, 3:15 P.M., and work backwards.
- Find the time the girls need to arrive at the theater.
- Ten minutes before 3:15 P.M. is 3:05 P.M.

- Find the time they need to leave Gillian’s house. Multiply 10 minutes by 2 to find the number of minutes needed to ride from Gillian’s house to the theater.
- Twenty minutes before 3:05 P.M. is 2:45 P.M.

- Find the time Melissa needs to leave her house.
- Ten minutes before 2:45 P.M. is 2:35 P.M.

- Find the time the girls need to arrive at the theater.
- Melissa needs to leave her house at 2:35 P.M.

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- After completing repairs on an elevator, a maintenance engineer tested his work by sending the elevator up 12 floors, down 18 floors, and back up 20 floors. If the elevator ended up on the 30th floor, on which floor did the engineer begin his test?

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- 1) A tree branch divides into 2 branches. Each smaller branch divides into 3 branches. Each of these divides into 5 branches. How many branches of all sizes are there?

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- 2) Find the sum of the first 100 even positive numbers.

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- 3) Which of the numbers 4, 5, or 6 is a solution to (n + 3)(n - 2) = 36?

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- 4) Apply the act it out strategy to solve this problem. A baseball card manufacturer is holding a contest. Each package of cards contains a puzzle piece. If you collect all 6 different pieces, you win two tickets to a major league game. There is an equally-likely chance of getting a different puzzle piece each time. How many packages of cards would you need to buy to win the contest?

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- 5) Ramir makes his own pottery vases. It costs him $2.25 in materials for each vase. He sells them for $10 each. How many vases does he need to sell in order to make a profit of $93?

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- 6) Emma wants to buy a car that is priced at $8,000. She put a deposit on the car that is 15% of the price. How much was the deposit? Explain how you can solve a simpler problem to calculate the deposit. Then find the deposit.

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- 7) One January night, the temperature fell 13° F between midnight and 6am. By 9am, the temperature had doubled from what it had been at 6am. By noon, it had risen another 8 degrees to 32° F. What was the temperature at midnight?

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