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An introduction to Problem Solving Math 110 Iris Yang

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Instructional programs from prekindergarten through grade 12 should enable all students to: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving.

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Polya’s Problem-Solving Principles

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Understand the Problem Restate the problem Highlight or identify important facts Determine the question or problem to be solved

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Devise a Plan Guess and check Look for a pattern Make an orderly list Draw a picture Eliminate possibilities Solve a simpler problem Use symmetry Use a model Consider special cases Work backwards Use direct reasoning Use a formula Solve an equation Be ingenious

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Carry Out the Plan Identify Needed Facts Choose an Appropriate Strategy Pencil and Paper Addition, Subtraction, Multiplication, Division Show All Work

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Look Back Restate the Question Check the Answer Does the Answer Make Sense? Record the Answer Add Necessary Units or Labels

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Example A rectangle has an area 120 cm 2. Its length and width are whole numbers. (a) What are the possibilities for the two numbers. (b) Which possibility gives the smallest perimeter?

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Example List the three-digit numbers that can be written using each of the digits 2,5, and 8 once and only once. (a) What is the greatest number in your list? (b) What is the smallest number in your list?

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Example Suppose today is Wednesday. What day of the week will it be 100 days from now?

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Example I have 30 coins consisting of nickels and quarters. The total value of the coins is $4.10. How many of each kind do I have?

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Example If we count by 3s starting with 1, the following sequence is obtained: 1 4, 7, 10, …… (a)What is the 100 th number in the sequence? (b) What is the nth number in the sequence?

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Example (a)Compute the sum 1+2+3+…+25 (b)Find the formula of 1+2+3+…+n

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Example Look at these corresponding geometrical and numerical sequences. (a) How many dots are there for a 5-layers triangular tower? (b) Find a formula of number of dots for the nth- triangular triangular tower?

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Example A set of marbles can be divided in equal shares among 2,3 or 5 children with now marbles left over. What is the least number if marbles that the set could have?

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Example

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