Multiplication: An Adventure in Number Sense
Times Five Pattern
Mentor: Now is my turn to color the table. Here are the facts that are left uncolored, 28 of them:
|
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 |
Mentor: I am going to color times five facts:
|
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 |
Student: They do seem easy for some reason. Maybe because I "practice" them a lot when I am counting money, using nickels and dimes and quarters?
 |
If you use only nickels, dimes and quarters (5 cent, 10 cent and 25 cent coins), but as many of them as you want, what amounts of money can you pay? What if you only use nickels (5 cent coins)? Only dimes (10 cent coins)? Only quarters (25 cent coins)? |
Mentor: I want to share my trick for multiplying by five. As many of my tricks, it involves multiplying by ten. Say, I want to do 6*5. The total of six 5s is half as many as six 10s, right?
Student: Right, because five is half as many as ten.
Mentor: But half as many as six 10s is three 10s, so I get my answer 6*5=30 (three 10s is thirty, of course). Want to try it with 8*5?
Student: Eight 5s is a half of eight 10s, but a half of eight 10s is four 10s, so 8*5=40. Easy!
Student: Now let me try 7*5. Seven 5s is a half of seven 10s... Wait, now I need to find a half of seven, but that is three and a half. What do I do with three and a half 10s?
Mentor: You know what a half of ten is, right?
Student: A half of ten is five, so three and a half 10s will be three 10s and 5, or 35. That is right: 7*5=35. I will try with the rest of the facts about multiplication by five, just to practice.
 |
To multiply a number by five, multiply a half of that number by ten. |
Mentor: I will color times five facts in the big table, because now that you know how to compute them fast you do not have to memorize them:
|
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: We had 28 uncolored facts, so now we only have 28-7=21 facts. And it is my turn to color!
|
Write down times five facts (or look at the fifth row of the table). Can you find any patterns in answers? Look at the first and the second digit of the answer, as well as the number that you multiply by five. |
| © 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. |