Sign of the Times

These activities extend the activities on pages 14–15 and 16 of the students’ book from the addition and subtraction of integers to the multiplication of integers (or signed numbers).
Using the model of “negative buttons” may help the students to see that multiplying a negative by a whole number can be thought of in much the same way as ordinary multiplication of rational numbers (whole numbers and fractions). It involves taking a repeated addition approach. For example, 4 x -3 can be shown as four lots of negative 3 buttons:

The diagram shows that the result is -12.
However, when it comes to equations such as -4 x -3 = 12, it is impossible to model this because there is no way that the second negative number can be shown. The students could draw the patterns they found in Activity One, question 1, to find the product of a negative number multiplied by a negative number. For example, the products in 1c increase by 2, that is:
3 x -2 = -6
2 x -2 = -4
1 x -2 = -2
0 x -2 = 0
-1 x -2 = 2
-2 x -2 = 4
-3 x -2 = 6
-4 x -2 = 8
-5 x -2 = 10
-6 x -2 = 12
This shows that a negative number multiplied by a negative number gives a positive product. The students can use this rule or the strategy of continuing patterns to complete the table in Activity Two. They could also draw a table that summarises the four possibilities:

This would provide a full answer to the question posed in Activity Two, question 2 and would perhaps enable the students to work on Activity Two, question 1 with greater understanding.
The students could also use their calculators to investigate what they get when they multiply a negative by a negative.

Answers to Activities

Activity One
1. The patterns continued are:
a. 3 x 0 = 0
3 x -1 = -3
3 x -2 = -6
3 x -3 = -9
3 x -4 = -12
Each result is 3 less than the previous one. The second half of the pattern is the 3 times table extended backwards.
b. -1 x 7 = -7
-2 x 7 = -14
-3 x 7 = -21
-4 x 7 = -28
-5 x 7 = -35
Each result is 7 less than the previous one. The second half of the pattern is the 7 times table extended backwards.
c. -2 x -2 = 4
-3 x -2 = 6
-4 x -2 = 8
-5 x -2 = 10
-6 x -2 = 12
Each result is 2 more than the previous one. The answers in the second half of the pattern look exactly like the 2 times table multiplied by positive factors.
2 . a.

b. -1 x -2 = 2 can only be imagined because the second negative quantity cannot be shown.

Activity Two

1.

2. Answers may vary. Multiplying a positive integer by a negative integer gives a negative product. Multiplying a negative integer by a negative integer gives a positive product.