H3 Mathematics Problem Solving Lecture 3 (Looking Back, Expanding and Thinking About Thinking)

H3 Mathematics Problem Solving Lecture 3: Looking Back, Expanding, and Thinking About Thinking Dr. Ho Weng Kin 1 / 53

Reconnecting to Lecture 2 Last lesson, we studied: mathematical resources; heuristics; structure recognition; the importance of checking and extending. Today we move fromsolving problems tounderstanding how we solve problems. Book link:Chapter 3; Chapter 4; Chapter 5 2 / 53

Reconnecting to Lecture 2 Last lesson, we studied: mathematical resources; heuristics; structure recognition; the importance of checking and extending. Today we move fromsolving problems tounderstanding how we solve problems. Book link:Chapter 3; Chapter 4; Chapter 5 2 / 53

Lecture 3 Roadmap 1 Discuss Homework (2) 2 Extract the key counting structures 3 Look back, check and expand 4 Introduce Schoenfeld’s framework 5 Apply Schoenfeld’s framework to a problem Big Theme Problem solving is not just about getting an answer. It is about developing control over mathematical thinking. 3 / 53

Lecture 3 Roadmap 1 Discuss Homework (2) 2 Extract the key counting structures 3 Look back, check and expand 4 Introduce Schoenfeld’s framework 5 Apply Schoenfeld’s framework to a problem Big Theme Problem solving is not just about getting an answer. It is about developing control over mathematical thinking. 3 / 53

Homework (2) Discussion Homework (2): Problem 1 H3 2018 Question 4 A clothes shop sells a particular make of T-shirt in four different colours. The shopkeeper has a large number of T-shirts of each colour. A customer wishes to buy seven T-shirts. (i)(a) In how many ways can he do this? (i)(b) In how many ways can he do this if he buys at least one of each colour? Book link:Chapter 3, Exercises 4 / 53

Homework (2) Discussion T-shirt Problem: Part (i)(a) Let x1 +x 2 +x 3 +x 4 = 7, wherex i is the number of T-shirts bought in colouri. We need the number of non-negative integer solutions. By stars and bars: ( 7 + 4−1 4−1 ) = ( 10 3 ) = 120. Heuristic Use a suitable representation: turn colours into variables. 5 / 53

Homework (2) Discussion T-shirt Problem: Part (i)(a) Let x1 +x 2 +x 3 +x 4 = 7, wherex i is the number of T-shirts bought in colouri. We need the number of non-negative integer solutions. By stars and bars: ( 7 + 4−1 4−1 ) = ( 10 3 ) = 120. Heuristic Use a suitable representation: turn colours into variables. 5 / 53

Homework (2) Discussion T-shirt Problem: Part (i)(a) Let x1 +x 2 +x 3 +x 4 = 7, wherex i is the number of T-shirts bought in colouri. We need the number of non-negative integer solutions. By stars and bars: ( 7 + 4−1 4−1 ) = ( 10 3 ) = 120. Heuristic Use a suitable representation: turn colours into variables. 5 / 53

Homework (2) Discussion T-shirt Problem: Part (i)(a) Let x1 +x 2 +x 3 +x 4 = 7, wherex i is the number of T-shirts bought in colouri. We need the number of non-negative integer solutions. By stars and bars: ( 7 + 4−1 4−1 ) = ( 10 3 ) = 120. Heuristic Use a suitable representation: turn colours into variables. 5 / 53