Multiplication: An Adventure in Number Sense
Times Nine "Tricks"
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Hold both your hands in front of you. Count as many fingers as the number by which you multiply nine, starting from the left. Bend that finger. The number of fingers to the left of the bent finger will be the first digit of the answer, and the number of fingers to the right of it will be the second digit of the answer. |
Student: Let me try. I am multiplying nine by seven. So I count the seventh finger from the left and bend it:
There are six fingers to the left of the bent finger, and three fingers to the right. So we have 9*7=63. It works!
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Why do you think it works? |
Student: I see why the number of fingers to the left of the bent finger is the first digit of the answer. It is because the first digit of the answer is one less than the number by which you multiply nine. We talked about it before! So, if you bend the seventh finger, there are six fingers to the left of it, which is one less than seven. What about the second digit of the answer and the fingers to the right?
Mentor: Remember our easy way of multiplying by nine?
Student: Yeah, you just multiply the number by ten and then subtract the extra number.
Mentor: Let us look at your example of 9*7 closely. First, we do 10*7. Let us use seven groups of ten marbles each to do that:
You will see in a moment why I used a different color for the last group of marbles. Now we are subtracting seven. I will remove seven green marbles:
Student: Oh, I see why the three fingers to the right of the bent finger represent the last digit of the answer. Because I have ten fingers, they are like the last pile of marbles. When I bend the seventh finger, there are 10-7=3 fingers to the right of it:
It is the same as removing the seven extra marbles from the last pile of ten marbles, so that we have three marbles left in the pile. Wow, everything is so connected! Anyway, I can color the ninth row in the table, because there is no need to memorize the facts there:
|
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: Last time we counted, there were 36 uncolored facts. I have just colored eight, so there are only 36-8=28 left.
| © 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. |