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

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.

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..

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.

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

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

There are many answers find the answers and the number of answers : 6644  4  6     8 

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

 7969    3  3   7  7  There are many answers find the answers and the number of answers :

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

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

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

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

Topic B: Filling in numbers represented by English Letters ABC  4  4  4 CDA832872

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

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

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

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

Topic B Examples used ABC41724  3  3 5ABC5172

ABCD1089  9  9 DCBA9801

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

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

    64

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

Exercise: identify the greatest product  753  742

 653  642

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

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

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

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  ( ) = 30 ， 3  ( ) = 42 ， 5  ( ) = 60 ， 7  ( ) = 70 ，

Topic D: Using factor / common factor Step (1), doing multiplication

Topic D: Using factor / common factor Step (2), reverse the thinking 3735  4

Topic D: Using factor / common factor Step (2), reverse the thinking

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  = 5  21 = 7  = 7  = 11  15 a < b < c < d. answer are 3 、 5 、 7 、 11.

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

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.

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

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

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

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