Multiplication
The signs used for multiplication are x and ·
2 x 3 means to add 2 to itself 3 times:
2 + 2 + 2 = 6
-3 · 2 means to add -3 to itself 2 times:
-3 + -3 = -6
4 x -2 means to add -2 to itself 4 times: -2 + -2 +
-2 + -2 = -8
-2 · -3 has no meaning except,
possibly, the opposite of 2 · -3,
that is, add -3 to itself 2 times then use the opposite of the result:
-3 + -3 = -6, the inverse of -6 is 6. So, -2 · -3 = 6.
It is probably best to learn and use the rules:
a. 2 same sign integers multiplied yield a
positive product.
b. 2 mixed sign integers
multiplied yield a negative product.
But what if you have more than 2 integers
multiplied together?
Example: -2 · 3 · -4
well, simplify by taking 2 integers at a time from left to right:
-2 · 3 is -6, so we have -6 · -4.
Now
-6 · -4 = 24, the answer.
Example: 2 · 5 · -6 · -2 · 3
10 · -6 · -2 · 3
-60 · -2 · 3
120 · 3
360
Division
Diviosion is the opposite operation for multiplication. Instead of adding an integer to
itself a number of times, we substract an integer from another repeatedly until we reach either zero or a smaller integer
than the one we are using to do the subtraction.
The operation sign we use is ÷.
For example:
12 ÷ 3
12 - 3 = 9 (once)
9 - 3 = 6 (twice)
6 - 3 = 3 (3 times)
3 - 3 = 0 (4 times)
We subtracted 3 starting from 12 four times until we reached 0 so 12 ÷ 3 = 4.
For example:
8 ÷ 4
8 - 4 = 4 (once)
4 - 4 = 0 (twice)
We subtracted 4 starting from 8 two times until we reached 0 so 8 ÷ 4 = 2.
We say "12 divided by 3 is 4." And we say "3 divides into 12, 4 times."
Likewise, we say "8 divided by 4 is 2." And we say "4 divides into 8, 2 times."
When teaching division of integers start with divisions that do not create remainders.
Reinforce the multiplication after each division.
In the previous examples after finding the result of the division, write both operations down and
spaced as shown. Doing this will reinforce both operations and how each number is related to the other.
Be careful to maintain the order of the numbers in the multiplication. Doing so will reinforce
the meanings of a dividend, a divisor and a quotient.
12 ÷ 3 = 4
4 · 3 = 12
8 ÷ 4 = 2
2 · 4 = 8
Before we continue, the multiplication table for the integers from 1 to 10
must be memorized. Yes, I said memorized.
How can you expect your child to have confidence in any further mathematics if your child cannot
handle routine multiplications and divisions without having to resort to the calculator? Sigh, when
my daughter had to stop to determine 7 x 9 trying to answer a statistics question, the sadness that came across me
made me regret trusting the school system in her mathematics education. Let's just say, that changed
in a hurry.
Here's the table.
| x |
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 |
Added to this table are the following squares:
11 x 11 = 121
12 x 12 = 144
13 x 13 = 169
14 x 14 = 196
15 x 15 = 225
Once these multiplication facts are known, then we can proceed.
Trust me, the following will be a struggle until these facts are committed to memory.
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