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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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Task The focus of this unit is on developing and using good number sense. Good number sense includes knowledge about how numbers are structured in our.

Task The focus of this unit is on developing and using good number sense. Good number sense includes knowledge about how numbers are structured in our.

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