Introduction Multiplication is an important component in primary mathematics learning.
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Transcript of Introduction Multiplication is an important component in primary mathematics learning.
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Mathematical Thinking
through investigation in multiplication
For Primary Teacher Education programme in Hong Kong
CHENG Chun Chor-Litwin
The Hong Kong Institute of Education
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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.
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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..
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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.
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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
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Topic A: Finishing the multiplication table Filling in table, so that multiplication established.
2 6 4
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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
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Question:Fill in integers so that the calculation is correct
7
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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 :
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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
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Using the numbers 1 、 2 、 3 、 4 、 5 、 6. Fill in the boxes so that multiplication established.
5 4
3
1 6 2
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The number 1or 5 could not be placed on the unit
digit (marked by a cross )
5 4
3
1 6 2
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Posing Question : Using the numbers 2 、 3 、 4 、 5 、 6 、 7
5 7 5 2
6 7
3 4 2 3 6 4
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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
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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
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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
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Topic B:Fill in suitable integers
A B C D 2 1 7 8
4 4
D C B A 8 1 7 2
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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
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Topic B
Examples used
A B C 4 1 7 2 4
3 3
5 A B C 5 1 7 2
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Topic B
Examples used
A B C D 1 0 8 9
9 9
D C B A 9 8 0 1
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Topic C: Finding maximum / minimum Using 1, 2, 3, 4 to fill in the boxes to make the greatest product
4 1
3 2
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Topic C: Finding maximum / minimum Using 1 , 2 , 3 , 4 , 5 , 6 , 7
6 5 4 3 2 1 7
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Topic C: Finding maximum / minimum Using 1 , 2 , 3 , 4 , 5 , 6 , 7
7 5 3 2 1 6 4
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Topic C: Finding maximum / minimum Using 1 , 2 , 3 , 4 , 5 , 6 , 7 .
7 5 3 6 4
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Exercise:
identify the greatest product
6 4 2 1 6 5 3 1
7 5 3 7 4 2
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Exercise:
identify the greatest product
7 4 2 1 7 5 3 1
6 5 3 6 4 2
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Topic C: Finding maximum / minimum Using 1 , 2 , 3 , 4 .
2 1
3 4
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Topic C: Finding maximum / minimum Using 1 , 2 , 3 , 4 , 5.
2 1
3
4 5
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Topic C: Finding maximum / minimum Using 1 , 2 , 3 , 4 , 5 , 6.
3 2
4 1
5 6
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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 ,
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Topic D: Using factor / common factorStep (1), doing multiplication
8 15
18 12
8 15 120
18 12 216
144 180
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Topic D: Using factor / common factorStep (2), reverse the thinking
35
12
21 20
7 5 35
3 4 12
73 5 4
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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
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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.
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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
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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.
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Topic E: Finding Equal product Using the numbers 2 、 3 、 4 、 6 、 8
4
2
6
3
8
X
A
C
Y
B
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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
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Topic E: Finding Equal product
Using numbers 1 、 2 、 3 、 4 、 6 、 8
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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
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Topic E: Finding Equal product Using numbers
1 、 2 、 3 、 4 、 6 、 8
3
2
64
19
8