H3 Mathematics Problem Solving Lecture 2 (Accessing Mathematical Resources)

H3 Mathematics Problem Solving Lecture 2: Accessing Mathematical Resources Dr. Ho Weng Kin 1 / 44

Reconnecting to Last Lesson Last week: ▶What is a genuine mathematical problem? ▶Pólya’s four stages: 1 Understand 2 Devise a plan 3 Carry out 4 Look back Today’s focus: How do we actually solve problems? 2 / 44

Reconnecting to Last Lesson Last week: ▶What is a genuine mathematical problem? ▶Pólya’s four stages: 1 Understand 2 Devise a plan 3 Carry out 4 Look back Today’s focus: How do we actually solve problems? 2 / 44

A Key Idea for Today Core Message Problem solving does not happen in a vacuum. ▶We rely onmathematical resources ▶We useheuristicsto activate them ▶We must learn torecognise structure Key Insight Do not wait for a problem to match a theorem. Force the problem to match the theorem. Book link:Chapter 2: Accessing mathematical resources; Chapter 3: Deploying problem solving heuristics 3 / 44

A Key Idea for Today Core Message Problem solving does not happen in a vacuum. ▶We rely onmathematical resources ▶We useheuristicsto activate them ▶We must learn torecognise structure Key Insight Do not wait for a problem to match a theorem. Force the problem to match the theorem. Book link:Chapter 2: Accessing mathematical resources; Chapter 3: Deploying problem solving heuristics 3 / 44

A Key Idea for Today Core Message Problem solving does not happen in a vacuum. ▶We rely onmathematical resources ▶We useheuristicsto activate them ▶We must learn torecognise structure Key Insight Do not wait for a problem to match a theorem. Force the problem to match the theorem. Book link:Chapter 2: Accessing mathematical resources; Chapter 3: Deploying problem solving heuristics 3 / 44

Homework (1) Discussion Problem Leta,bandcbe positive real numbers. By using the results in parts (a) 1 a + 1 b≥ 4 a+b and (b) (a+b+c) (1 a + 1 b + 1 c ) ≥9 prove that 9 a+b+c ≤2 ( 1 a+b + 1 a+c + 1 b+c ) ≤ 1 a + 1 b + 1 c . Book link:Chapter 1, Exercises; Chapter 2, “Accessing mathematical resources” 4 / 44

Homework (1) Discussion If we take each of these sums 1 a + 1 b , 1 b + 1 c , 1 a + 1 c one at a time and apply (a), we get ▶ 1 a + 1 b≥ 4 a+b; ▶ 1 b + 1 c≥ 4 b+c; ▶ 1 a + 1 c≥ 4 a+c. Summing up all three lines, we have2 ( 1 a + 1 b + 1 c ) ≥4 ( 1 a+b + 1 b+c + 1 a+c ) . 5 / 44

Homework (1) Discussion If we take each of these sums 1 a + 1 b , 1 b + 1 c , 1 a + 1 c one at a time and apply (a), we get ▶ 1 a + 1 b≥ 4 a+b; ▶ 1 b + 1 c≥ 4 b+c; ▶ 1 a + 1 c≥ 4 a+c. Summing up all three lines, we have2 ( 1 a + 1 b + 1 c ) ≥4 ( 1 a+b + 1 b+c + 1 a+c ) . 5 / 44

Homework (1) Discussion To prove that 2 ( 1 a+b + 1 a+c + 1 b+c ) ≥ 9 a+b+c , you stare hard at part (b), which is: (x+y+z) (1 x + 1 y + 1 z ) ≥9. What do you substitutex,yandzfor? 6 / 44