Multiplication: An Adventure in Number Sense
Twin Facts and Commutative Law
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
1 |
1*1=1 |
1*2=2 |
1*3=3 |
1*4=4 |
1*5=5 |
1*6=6 |
1*7=7 |
1*8=8 |
1*9=9 |
1*10=10 |
2 |
2*1=2 |
2*2=4 |
2*3=6 |
2*4=8 |
2*5=10 |
2*6=12 |
2*7=14 |
2*8=16 |
2*9=18 |
2*10=20 |
3 |
3*1=3 |
3*2=6 |
3*3=9 |
3*4=12 |
3*5=15 |
3*6=18 |
3*7=21 |
3*8=24 |
3*9=27 |
3*10=30 |
4 |
4*1=4 |
4*2=8 |
4*3=12 |
4*4=16 |
4*5=20 |
4*6=24 |
4*7=28 |
4*8=32 |
4*9=36 |
4*10=40 |
5 |
5*1=5 |
5*2=10 |
5*3=15 |
5*4=20 |
5*5=25 |
5*6=30 |
5*7=35 |
5*8=40 |
5*9=45 |
5*10=50 |
6 |
6*1=6 |
6*2=12 |
6*3=18 |
6*4=24 |
6*5=30 |
6*6=36 |
6*7=42 |
6*8=48 |
6*9=54 |
6*10=60 |
7 |
7*1=7 |
7*2=14 |
7*3=21 |
7*4=28 |
7*5=35 |
7*6=42 |
7*7=49 |
7*8=56 |
7*9=63 |
7*10=70 |
8 |
8*1=8 |
8*2=16 |
8*3=24 |
8*4=32 |
8*5=40 |
8*6=48 |
8*7=56 |
8*8=64 |
8*9=72 |
8*10=80 |
9 |
9*1=9 |
9*2=18 |
9*3=27 |
9*4=36 |
9*5=45 |
9*6=54 |
9*7=63 |
9*8=72 |
9*9=81 |
9*10=90 |
10 |
10*1=10 |
10*2=20 |
10*3=30 |
10*4=40 |
10*5=50 |
10*6=60 |
10*7=70 |
10*8=80 |
10*9=90 |
10*10=100 |
Student: Well, but how will I know the part of the table you have just colored blue?
Mentor: Look, every fact in this part has a twin in the other part of the table. For example, for 2*8=16, there is 8*2=16. And so on. Can you guess why I call them "twins"?
Student: Because they give the same answer.
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Can you find a reason why "twin facts," such as 3*8 and 8*3, give the same results? Try to find several different explanations. |
Student: Well, I can draw why they give the same answer, probably. Take 3*5 and 5*3, for example:
Mentor: This law is called commutative law of multiplication. A very useful law indeed!
Student: Oh yes, now I do not have to memorize the whole table! If I remember 5*3, then I will know 3*5. So, we have 55 facts instead of 100. Not bad!
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Commutative Law of Multiplication: a*b = b*a for any numbers a and b. |
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How many pairs of "twin facts" (such as 5*7 and 7*5) are there in the multiplication table for numbers from 1 to 10? What if the table covered numbers from 1 to 20? From 1 to 30? From 1 to 5? From 7 to 11? From 200 to 300? Can you find a pattern? What is it? |
Mentor: We have just started to color the table. Now it is your turn.
| © Copyright 1998 by Maria Droujkova and Dmitri Droujkov. All rights reserved. No part of these materials should ever be used in any situation that involves compulsory teaching. See also copyright notes and student rights. |