☰ Year plan · Mathematics

Operations on integers

Week 4 Β· Operations on integers: addition, subtraction, multiplication and division Β· 3 lessons
3

Multiplying and dividing integers: sign rules and properties

Mathematics Prep 1 Term 1, Unit 1, Lesson 4
GoalMultiply and divide integers with the sign rules and use the properties to calculate cleverly.
New words
sign rule Β· same signs give +, different signs give βˆ’undefined Β· has no answer, like 7 Γ· 0closure Β· the answer is always in the same setcommutative Β· the order does not change the answer
Explanation

Multiplying integers: the product of two numbers with the same sign is positive, and with different signs is negative. Why is (βˆ’2) Γ— (βˆ’2) positive? Look at the pattern: 2 Γ— (βˆ’2) = βˆ’4, 1 Γ— (βˆ’2) = βˆ’2, 0 Γ— (βˆ’2) = 0, and each time the answer goes up by 2, so the next answers are 2 and then 4. Dividing follows the same signs: βˆ’12 Γ· 4 = βˆ’3 and βˆ’12 Γ· (βˆ’3) = 4. Zero divided by any number except 0 is 0, but dividing by 0 is undefined. With three or more factors, count the negatives: an odd number gives a negative product. Integers are closed under +, βˆ’ and Γ—, but not Γ·, since 1 Γ· 2 is not an integer. The order and grouping of a product do not change it.

ProductResult
3 Γ— (βˆ’2)βˆ’6
2 Γ— (βˆ’2)βˆ’4
1 Γ— (βˆ’2)βˆ’2
0 Γ— (βˆ’2)0
(βˆ’1) Γ— (βˆ’2)2
(βˆ’2) Γ— (βˆ’2)4
Worked examples
Compare (βˆ’3) Γ— (βˆ’5) = 15, (βˆ’3) Γ— 5 = βˆ’15 and βˆ’3 βˆ’ 5 = βˆ’8. The first two are products, so we use the sign rule. The last is a subtraction: βˆ’3 + (βˆ’5) = βˆ’8. Do not mix them up.
Division: βˆ’36 Γ· (βˆ’9) = 4 (same signs) and 36 Γ· (βˆ’9) = βˆ’4 (different signs). Also 0 Γ· (βˆ’7) = 0, but βˆ’7 Γ· 0 is undefined, because no number times 0 gives βˆ’7.
Three factors: (βˆ’2) Γ— (βˆ’3) Γ— (βˆ’4). First (βˆ’2) Γ— (βˆ’3) = 6, then 6 Γ— (βˆ’4) = βˆ’24. Three negatives, so the answer is negative. Calculate cleverly: (βˆ’25) Γ— 7 Γ— (βˆ’4) = ((βˆ’25) Γ— (βˆ’4)) Γ— 7 = 100 Γ— 7 = 700.
Class activity

Sign sort. Write ten multiplications on cards, such as (βˆ’4) Γ— 3 or (βˆ’5) Γ— (βˆ’2). With a grown-up, sort them into piles: answer positive, answer negative, answer zero. Say the sign rule aloud for each card.

Practice
1
Find (βˆ’6) Γ— 7.
2
Find (βˆ’8) Γ— (βˆ’5).
3
Find βˆ’36 Γ· (βˆ’9) and 0 Γ· (βˆ’7). What about βˆ’7 Γ· 0?
4
Find (βˆ’2) Γ— (βˆ’5) Γ— (βˆ’3).
5
Salma says (βˆ’3) Γ— (βˆ’5) = βˆ’15. What is wrong?