Math Homework

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Section 1-3

Strategies for Problem Solving

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Strategies for Problem Solving

• A General Problem-Solving Method

• Using a Table or Chart

• Working Backward

• Using Trial and Error

• Guessing and Checking

• Considering a Similar Simpler Problem

• Drawing a Sketch

• Using Common Sense

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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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Example: Working Backward

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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Example: Solution

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 60.

Divide by 4 to undo quadruple = 15. Add 47 to get 62, then divide by 2 = 31.

Add 5 to get 36 and divide by 3 = 12.

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Example: Solution

Solution

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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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?

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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 1849. Subtract 43 from 1849 to get that he was born in 1806.

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Example: Guessing and Checking

Find a positive natural number that satisfies the

equation below.

2

4 8

x x x 

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Example: Solution

Try this by guess and check: 2

4 If 4 : 4 4 4

8 x   

16 4(2) 4

8   

2 9

If 9 : 4 9 9 8

x    81

4(3) 9 8

  

2 16

If 16 : 4 16 16 8

x    256

4(4) 16 8

  

Solution

x = 16 satisfies the equation.

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What is the ones (or units) digit in 3200?

Example: Considering a Simpler

Problem

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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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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.