Multiplication: An Adventure in Number Sense
Times Ten for Free
Mentor: Let us first color facts on a new table. I happen to see a bunch of yet uncolored facts that we can get for free, without memorization. They are in the last row:
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
2 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
3 |
3 |
6 |
9 |
12 |
15 |
18 |
21 |
24 |
27 |
30 |
4 |
4 |
8 |
12 |
16 |
20 |
24 | 28 |
32 |
36 |
40 |
5 |
5 |
10 |
15 |
20 |
25 |
30 |
35 |
40 |
45 |
50 |
6 |
6 |
12 |
18 |
24 |
30 |
36 |
42 |
48 |
54 |
60 |
7 |
7 |
14 |
21 |
28 |
35 |
42 |
49 |
56 |
63 |
70 |
8 |
8 |
16 |
24 |
32 |
40 |
48 |
56 |
64 |
72 |
80 |
9 |
9 |
18 |
27 |
36 |
45 |
54 |
63 |
72 |
81 |
90 |
10 |
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
100 |
 |
Why do we get the last row for free? Find several different explanations for the "times ten" pattern. Use graph paper or marbles if you want. |
Student: It is easy to multiply by ten, and I do not have to memorize anything there. You just add zero after the number!
Mentor: Let me color the new free facts dark red on the big table:
|
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 |
|
The fact that it is easy to multiply by ten has something to do with decimal (base ten) system. What is the connection? What would happen if we used a different base (say, base 16 as in computers, with the following digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, A for 10, B for 11, C for 12, D for 13, E for 14, F for 15)? Why? An interesting project would be to create a multiplication table for a number system with base other than 10. Click here to see our base 16 multiplication table. |
Student: We had forty five facts before your turn, but then you colored nine facts in the last row, so now we have 45-9=36 facts left. And it is my turn to color!
| © 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. |