1. Mathematical Thinking through investigation in multiplication For Primary Teacher Education programme in Hong Kong CHENG Chun Chor-Litwin The Hong Kong Institute of Education

2. Introduction Multiplication is an important component in primary mathematics learning. However, most teachers taught according to the textbooks and there is little room for investigation in this topic. The set of materials introduced her wish to complement mathematical thinking for primary schools students.

3. In Hong Kong, primary school teachers can go to Institutions to do an in-service release course, ranging from 5-week to a year, to refresh their teaching. This 5-week course may have different focus, such as small class teaching, children with special needs..

4. The following is the list of material that was used in the teachers’ course, so that teachers can try them in their classroom. The materials have been tried out in schools before they are introduced to primary teachers. In total, there are 5 topics of materials introduced in this programme.

5. Topic A: Finishing the multiplication table Topic B: Filling in numbers represented by English Letters Topic C: Finding maximum / minimum product of multiplication Topic D: Using factor and common factor Topic E: Finding Equal product

6. Topic A: Finishing the multiplication table Filling in table, so that multiplication established.    264

7. There are many answers find the answers and the number of answers : 6644  4  6  264264   2643388  8  3 264264

8. Question: Fill in integers so that the calculation is correct     7

9.  7969    3  3  7 237207 617181  7  7  7 427497567 There are many answers find the answers and the number of answers :

10. Using the numbers 1 、 2 、 3 、 4 、 5. Fill in the boxes so that multiplication established.  AB13    C  4  DE52

11. Using the numbers 1 、 2 、 3 、 4 、 5 、 6. Fill in the boxes so that multiplication established.  54    3  162

12. The number 1or 5 could not be placed on the unit digit (marked by a cross  )   54    3  162

13. Posing Question ： Using the numbers 2 、 3 、 4 、 5 、 6 、 7  5752    6  7  342364

14. Topic B: Filling in numbers represented by English Letters ABC208218  4  4  4 CDA832872

15. A = 4 ， 8 ， 2 ， 6 ， 0 D could be 3 or 7. Suppose D = 3, then C  3, C = 8. 2BC2D8208  4  4  4 C32832832

16. A = 4 ， 8 ， 2 ， 6 ， 0 D could be 3 or 7. Suppose D = 7, then C = 8, B = 1. 2BC2D8218  4  4  4 C72872872

17. Topic B: Fill in suitable integers ABCD2178  4  4 DCBA8172

18. A = 1 or A = 2, but A  1, and A = 2. D = 3 or D = 8, but D  3. C = 2 ， 7. B = 1 。 2BCD2178  4  4 DCB28172

19. Topic B Examples used ABC41724  3  3 5ABC5172

20. ABCD1089  9  9 DCBA9801

21. Topic C: Finding maximum / minimum Using 1, 2, 3, 4 to fill in the boxes to make the greatest product  41    32

22. Topic C: Finding maximum / minimum Using 1 ， 2 ， 3 ， 4 ， 5 ， 6 ， 7    654321  7

23.    75321  64

24. Topic C: Finding maximum / minimum Using 1 ， 2 ， 3 ， 4 ， 5 ， 6 ， 7.    753  64

25. Exercise: identify the greatest product 64216531  753  742

26. 74217531  653  642

27. Topic C: Finding maximum / minimum Using 1 ， 2 ， 3 ， 4.  21  3    4

28. Topic C: Finding maximum / minimum Using 1 ， 2 ， 3 ， 4 ， 5.  21  3  4    5

29. Topic C: Finding maximum / minimum Using 1 ， 2 ， 3 ， 4 ， 5 ， 6.  32  41  5    6

30. Another type of finding greatest product From 2, 3, 4, 5, 7, add 3 number and multiply this sum with the fourth numbers, so that the product is greatest 。 The following is a list of possibilities. 2  (3 + 5 + 7) = 30 ， 3  (2 + 5 + 7) = 42 ， 5  (2 + 3 + 7) = 60 ， 7  (2 + 3 + 5) = 70 ，

31. Topic D: Using factor / common factor Step (1), doing multiplication 815 1812 815120 1812216 144180

32. Topic D: Using factor / common factor Step (2), reverse the thinking 35 12 2120 7535 3412 73735  4

33. Topic D: Using factor / common factor Step (2), reverse the thinking 15 108 224 1448315 31515 629108 847224 1448315

34. There are 4 numbers a 、 b 、 c 、 d (2 to 9). a  (b + c + d) = 69 b  (a + c + d)= 105 c  (a + b + d) = 133 d  (a + b + c) = 165 。 Find the values of a 、 b 、 c 、 d. 69 = 3  23 105 = 5  21 = 7  15 133 = 7  19 165 = 11  15 a < b < c < d. answer are 3 、 5 、 7 、 11.

35. Topic E: Finding Equal product Using the numbers from 1 to 9, fill in the circle so that the product on the three sides are equal

36. Topic E: Finding Equal product Using the numbers 2 、 3 、 4 、 6 、 8, fill in the circle so that the product on the three sides are equal.

37. Topic E: Finding Equal product Using the numbers 2 、 3 、 4 、 6 、 8 4 2 6 3 8 X A C Y B

38. Discussion (A  Y  B)  (B  C)  (A  X  C) = T  T  T  A  B  C  X  Y  A  B  C.= T  T  T  (2  3  4  6  8)  A  B  C = T  T  T  (2^7  3^2)  A  B  C = T  T  T possible value of T = (2^4  3^1 )= 48  B  C = 48 。 X A C Y B

39. Topic E: Finding Equal product Using numbers 1 、 2 、 3 、 4 、 6 、 8

40. Topic E: Finding Equal product Using numbers 1 、 2 、 3 、 4 、 6 、 8 2 4 61 3 8 1 8 32 6 4

41. 3 2 6 4 1 9 8