Remedial
Some learners may have difficulty with doing correct calculations with integers, especially where negative integers or large integers are involved. Revise the basic “sign rules” with them and show them how to use brackets and how to remove brackets, especially where negative integers are involved.
Exercise 5 Guidelines on how to implement this activity
Demonstrate the commutative, associative and distributive properties for the operations with integers on the board. Also, the fact that 1 is the identity for multiplication and that the only integers that have multiplicative inverses are 1 and –1. Motivate the latter statement by showing that, for example: although (−2) ×
the rational number − 1
( − 1 __
Suggested answers 1.1
the answers once they completed the exercise. Learner’s Book page 39
2 ) = 1, the multiplicative inverse of –2, it is not an integer but
__ 2 . Let the learners do Exercise 5 in the classroom and discuss
Yes, 7 × ( 5 + 3 ) = 7 × 8 = 56 and ( 7 × 5 ) + ( 7 × 3 ) = 35 + 21 = 56
1.2 Yes, ( 10 + 5 + 3 ) × 2 = 18 × 2 = 36 and ( 10 × 2 ) + ( 5 × 2 ) + ( 3 × 2 ) = 20 + 10 + 6 = 36
1.3 Yes, 5 × ( 11 − 4 ) = 5 × 7 = 35 and ( 5 × 11 ) − ( 5 × 4 ) = 55 − 20 = 35 1.4 Yes, ( 9 − 4 ) × 3 = 5 × 3 = 15 and ( 9 × 3 ) − ( 4 × 3 ) = 27 − 12 = 15 1.5
a × ( b − c ) = ( a × b ) − (a × c) Multiplication of integers is distributive over subtraction. 2.1
No, 14 − 9 = 5 but 9 − 14 = −5 Subtraction of integers is not commutative.
2.2 No, 2 − ( 5 − 6 ) = 2 − ( −1 ) = 3 but ( 2 − 5 ) − 6 = −3 − 6 = −9 Subtraction of integers is not associative.
2.3 No, 12 ÷ ( 6 ÷ 2 ) = 12 ÷ 3 = 4 but ( 12 ÷ 6 ) ÷ 2 = 2 ÷ 2 = 1 Division of integers is not associative.
2.4 3.1
No, −8 ÷ 4 = −2 but 4 ÷ ( −8 ) = −0,5 Division of integers is not commutative.
83 × 57 + 83 × 43 = 83 × ( 57 + 43 ) = 83 × 100 = 8 300
3.2 15 × ( −8 ) + 25 × ( −8 ) = −8 × ( 15 + 25 ) = −8 × 40 = −320 3.3 8 × ( −73 ) – 8 × ( −23 ) = 8 × ( −73 − ( −23 ) ) = 8 × − 50 = −400
Remedial
Some learners may have difficulty with interpreting and applying the associative and distributive properties for operations with integers. Demonstrate these with easy examples and progress to more involved cases like applying the distributive property “in reverse”.
Chapter 1: Numbers and integers
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