1. Section 1-3 Chapter 1 Strategies for Problem Solving
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2. A General Problem-Solving Method
Polya’s Four-Step Method
Step 1 Understand the problem. Read and analyze carefully. What are you to find?
Step 2 Devise a plan.
Step 3 Carry out the plan. Be persistent.
Step 4 Look back and check. Make sure that your answer is reasonable and that you’ve answered the question.
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3. Strategies for Devising a Plan
Use a Table or Chart
Look for a Pattern
Solve a Similar Simpler Problem
Draw a Sketch
Use Inductive Reasoning
Write an equation and solve it
Use a formula
Working Backward
Using Trial and Error
Guess and Check
Using Common Sense
Look for a “catch” if an answer seems too obvious
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4. Example: Working Backward or Use Equation
Start with an unknown number. Triple it and then subtract 5. Now, take the new number and double it but then subtract 47. If you take this latest total and quadruple it you have 60. What was the original unknown number?
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5. Example: Solution (working backwards)
Step 1 Understand the problem. We are looking for a number that goes through a series of changes to turn into 60.
Step 2 Devise a plan. Work backwards to undo the changes.
Step 3 Carry out the plan. The final amount was Divide by 4 to undo quadruple = 15.
Add 47 to get 62, then divide by 2 = Add 5 to get 36 and divide by 3 = 12.
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6. Example: Solution (working backwards)
The original unknown number was 12.
Step 4 Look back and check. We can take 12 and run through the computations to get 60.
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7. Example: Solution (using an equation)
Plan: Set up an equation.
Carry out the plan:
Let x = unknown number
Triple the number and subtract 5: 3x – 5
Double the new number and subtract 47: 2(3x – 5) – 47
Quadruple this number and set it equal to 60:
4[2(3x - 5) - 47] = 60
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8. Example: Solution (using an equation)
Carry out the plan:
Solve the equation: 4[2(3x - 5) - 47] = 60
x = 12
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9. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Common Sense
I’m thinking of a number between 4 and 10. If I square the number and find the difference of the digits I will end up with the cube root of the number. What is my number?
Test Question
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10. Example: Using Trial and Error
The mathematician Augustus De Morgan lived in the nineteenth century. He made the following statement: “I was x years old in the year x 2.” In what year was he born?
Test Question
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11. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Solution
He lived in the nineteenth century, which means during the 1800s. Find a perfect square that is between 1800 and 1900.
42 2 = 1764
43 2 = 1849
44 2 = 1936
43 is the only natural number that works De Morgan was 43 in Subtract 43 from 1849 to get that he was born in 1806.
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12. Example: Considering a Simpler Problem / Pattern
What is the ones (or units) digit in 3200?
Test Problem
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13. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Solution
Step 1 Understand the problem. We are looking for the last digit if 3200 is multiplied out.
Step 2 Devise a plan. Look for a pattern with multiplication by 3s.
Step 3 Carry out the plan.
31 = 3, 32 = 9, 33 = 27, 34 = 81
35 = 243, 36 = 729, 37 = 2187, 38 = 6561,…
Notice that if the power is divisible by 4 then the units digit is a 1.
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14. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Solution
Solution
The units digit in 3200 is 1 because the power, 200, is divisible by 4.
Step 4 Look back and check. We can try a few more powers of 3 to make sure that the pattern continues and also check the multiplication.
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15. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Use a table
You have brought two unmarked buckets to a stream. The buckets hold 7 gallons and 3 gallons of water, respectively. How can you obtain exactly 5 gallons of water to take home?
Solution Lg 7 4 1 5 Sm 3
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16. Example: Look for a pattern
Find the missing digit x 3 2 4 8 7 1 X 5 6 9
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17. Example: Look for a pattern
Solution 3 2 4 8
3 x 8 = 24 7 1
7 x 3 = 21 X 5
8 x 5 = 40 6 9
4 x 9 = 36
so X must be 0
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18. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Magic Square
Use each number 5 through 13 once to complete the magic square 8 6 9 11 5
Test Problem
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19. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Magic Square
Solution 8 13 6 7 9 11 12 5 10
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