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.

2 6 4

There are many answers find the answers and the number of answers :

6 6 4 4

4 6

2 6 4 2 6 4

2 6 4 3 3 8 8

8 3

2 6 4 2 6 4

Question:Fill in integers so that the calculation is correct

7

7 9 6 9

3 3

7 2 3 7 2 0 7

6 1 7 1 8 1

7 7 7

4 2 7 4 9 7 5 6 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.

A B 1 3 C 4

D E 5 2

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

5 4

3

1 6 2

The number 1or 5 could not be placed on the unit

digit (marked by a cross )

5 4

3

1 6 2

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

5 7 5 2

6 7

3 4 2 3 6 4

Topic B: Filling in numbers represented by English Letters

A B C 2 0 8 2 1 8

4 4 4

C D A 8 3 2 8 7 2

A = 4 ， 8 ， 2 ， 6 ， 0 D could be 3 or 7.

Suppose D = 3, then C 3, C = 8.

2 B C 2 D 8 2 0 8

4 4 4

C 3 2 8 3 2 8 3 2

A = 4 ， 8 ， 2 ， 6 ， 0 D could be 3 or 7.

Suppose D = 7, then C = 8, B = 1.

2 B C 2 D 8 2 1 8

4 4 4

C 7 2 8 7 2 8 7 2

Topic B:Fill in suitable integers

A B C D 2 1 7 8

4 4

D C B A 8 1 7 2

A = 1 or A = 2, but A 1, and A = 2.D = 3 or D = 8, but D 3.

C = 2 ， 7. B = 1 。

2 B C D 2 1 7 8

4 4

D C B 2 8 1 7 2

Topic B

Examples used

A B C 4 1 7 2 4

3 3

5 A B C 5 1 7 2

Topic B

Examples used

A B C D 1 0 8 9

9 9

D C B A 9 8 0 1

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

4 1

3 2

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

6 5 4 3 2 1 7

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

7 5 3 2 1 6 4

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

7 5 3 6 4

Exercise:

identify the greatest product

6 4 2 1 6 5 3 1

7 5 3 7 4 2

Exercise:

identify the greatest product

7 4 2 1 7 5 3 1

6 5 3 6 4 2

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

2 1

3 4

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

2 1

3

4 5

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

3 2

4 1

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(3 + 5 + 7) = 30 ， 3(2 + 5 + 7) = 42 ， 5(2 + 3 + 7) = 60 ， 7(2 + 3 + 5) = 70 ，

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

8 15

18 12

8 15 120

18 12 216

144 180

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

35

12

21 20

7 5 35

3 4 12

73 5 4

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

15

108

224

144 8 315

3 1 5 15

6 2 9 108

8 4 7 224

144 8 315

There are 4 numbers a 、 b 、 c 、 d (2 to 9).a(b + c + d) = 69b(a + c + d)= 105c(a + b + d) = 133d(a + b + c) = 165 。Find the values of a 、 b 、 c 、 d.

69 = 323 105 = 521 = 715 133 = 719 165 = 1115 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 、 8

4

2

6

3

8

X

A

C

Y

B

Discussion

(AYB)(BC)(AXC) = TTT ABC X YAB C .= TTT (23468)AB C = TTTÞ (2^73^2)ABC = TTT

possible value of T = (2^43^1 )= 48 BC = 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 、 8

2

4

6 1

3

8

1

8

3 2

6

4

Topic E: Finding Equal product Using numbers

1 、 2 、 3 、 4 、 6 、 8

3

2

64

19

8