INTEGERS

CLASS ASSIGNMENT 1

1. What does it mean if you say a person is “negative”? Explain this in mathematical context.

2. What do you think is a “negative number”? Use an illustration to substantiate your explanation.

3. Give two examples of where you would use “negative” numbers on earth.

4. Give a definition of integers:

5. What symbol represents the set of integers?

6. How would you represent the following on a number line (graphically)?

x ≥ -3 ; x size 12{ in } {}z.

(how would you express the above in words? – all integers greater than -3)

Figure 1

[shaded dots – indicate number is included -- therefore also equal toa circle (not coloured dot) – indicates that the specific number is not included]

Different types of notations:

(read as follows: set x in which x ∈ zand x is greater than and equal to -3)

(Numbers greater than -3 up to infinity on the positive side)

6.1 Now represent the following graphically (by means of a number line):

Draw your number line:

6.1.1 x < 2 , x ∈ Z

6.1.2 x ≥ -2 , x ∈ Z

6.1.3 2 ≤ x < 5

6.2 Write the following in set builder notation:

Figure 2

CLASS ASSIGNMENT 2

Can you still remember the following from Module 1?

(+) × or ÷ (+) →

(+) × or ÷ ( - ) →

( - ) × or ÷ ( - ) →

(You will need the above even when adding and subtracting integers, because you have to remember: you may never have two signs next to each other, you must always multiply the two signs with each other)

Can you still remember the properties of 0 (zero)? Look at this....

b × 0 =

b + 0 =

b- 0 =

b0b0 size 12{ { {b} over {0} } } {} = and 0b0b size 12{ { {0} over {b} } } {} =

1. Can you carry out the following instructions with regard to a number line?

1.1 3 + 4

1.2 8 - 12

2. The temperature in Bloemfontein is 4 °C. It drops by 8 °C.What is the temperature now?

3. Calculate the following:

3.1: -5 - 18

3.2: 15 - 8 - 17 + 5

3.3: - 30 + 7 - 4

3.4: - 8 + (-5) + (+7)

4. Can you think of a way to do 3.2; 3.3 and 3.4?

(A short cut?)

How would you do the following?

Decide which number has to come first: 3 - (-5)

remember the rule – multiply the two signs next to each other.

( - ) × ( - ) → ( + )

5. Now calculate the following:

5.1: - 9 - ( -6)

5.2: -18 + (-13) - (-7)

5.3: 20 - (25 + 50)

5.4: 10 - (16 - 18)

6. Calculate the difference between -31 and -17

7. Replace ___ by a ( + ) or ( - ) to make the following statements true:

7.1: - 6 ___ (-3) = -9

7.2: 5 ___ (-5) = 10

HOMEWORK ASSIGNMENT 1

1. Calculate each of the following:

1.1: 13 - 18 + 4 - 17

1.2: - 9 - ( -8 ) + ( - 16 )

1.3: - ( -16 )² + ( -3)²

1.4: ( - 13 )² - ( - 13 )

1.5: [a + (-b) ] + b

1.6: [a + (-b)] + (-a)

1.7: (-b) + [(-b) + a ]

1.8: (-y)² - (-x)² - (-x ²)

2. By doing a calculation in each case, say whether the following is true or false.

2.1 - (-x) = x

2.2 - (x + y) = - x - (-y)

2.3 y+ z = z- (-y)

2.4 -( x - y) = - x + y

3. Calculate the value of a to make each of the following true.

3.1: -5 + a = -7

3.2: a + (-5) = 7

3.3: -6 + a = -9

3.4: 18 + a = 10

4. Your financial transactions for the past two months are as follows:

Holiday work: R 615 Expenses: Stationery: R 46

Petrol consumption: R 480 Personal expenses: R 199

Will you have a profit or a loss for the past period?

Show how you calculated this.

Assessment

 good  average  not so good

CLASS ASSIGNMENT 3

(You must always do these four in sequence in any sum)

Look at: +4 × ( -3) = -12

What about -12 ÷ (+4) = -3

1. Calculate the following:

1.1: -7 x (-3) x (-2)

1.2: -18 x (-2) + (-17) x (-2)

1.3: -5 x (-7)

1.4: 3 x (8 - 19) + 6

1.5: 3 x (-8) x (19 + 6)

1.6: (-2)³

1.7: (-4)³ - (-2)²

1.8: (15 - 9)²

1.9: (9 - 15)²

1.10: -2 (-3)²

1.11: −5−63−5−63 size 12{ { { - 5` - `6} over {3} } } {}

1.12: −6(−4)-12−(-2)−6(−4)-12−(-2) size 12{ { { - 6` \( - `4 \) } over {"-12"`-` \( "-2" \) } } } {}

1.13: −6×(−5)-7−6×(−5)-7 size 12{ - 6`` times `` { { \( - 5` \) } over {"-7"} } } {}

1.14: 53-2553-25 size 12{ { {"53"} over {"-25"} } } {}

1.15: -50 ÷ ? = -10

2. Calculate p if a = -2 and b = 3

2.1 p = a x b ÷ a²

2.2 p = 4ab ÷ ab

HOMEWORK ASSIGNMENT 2(Mixed examples)

1. Simplify:

1.1 (13)² - (-13)² - 13²

1.2 (7 - 8)² - (8 - 7)² - 8² - 7²

1.3 (3 + 2)3 - 33 - 22

2. Divide -147 by -21 and then subtract -55 from the quotient.

3. Divide the product of 17 and -15 by -7

4. Subtract - 58 from the sum of -88 and 7.

5. Subtract the product of -5 and 17 from -7

6. Calculate p in each case:

6.1: 20 + p = -40

6.2: -8 + (-p) = 0

6.3: -10 + (-17) + p = -20

6.4: 2p - (-6) = -4

7. If -a = -4, then a= …

8. If x = 3 , then -(- x) = …

9. x ∈ {-3; -2; -1; 0; 1; 2; 3; 4; 5} ; Select from the set of integers and tabulate all the possible answers.

9.1: -2 < x < 4

9.2: x > 1

9.3: x < 0

Assessment

 good  average  not so good

Tutorial 1: (Integers)

Total: 40

1. Complete:

[4]

2. Select from the set of integers:

2.1 4n+ 3 > 30 n∈ { } [2]

2.2 n2n2 size 12{ { {n} over {2} } } {} - 1 < 2 n ∈ { } [2]

3. Represent the following graphically:

3.1 { x / x ∈ z , -3 < x < 5 } [2]

3.2 { x / x ∈ z, x < 1 } [2]

4. Calculate each of the following:

4.1: [- (-2)²]³ [2]

4.2: - 8 + (-9) - (-8) + 9 [2]

4.3: 15 + 8 x (-5) + 3 x (-4) [3]

4.4: 611611 size 12{ { {6} over {"11"} } } {} ÷ (-24) [2]

4.5: (-0,3)² x (-0,4) [2]

4.6: - (-1)² [2]

4.7: What should be added to -17 to give + 70? [2]

²4.8: -0,75a² x 0,3a³ [3]

4.9: −126b44a5b−126b44a5b size 12{ left ( { { - "12" rSup { size 8{6} } b rSup { size 8{4} } } over {4a rSup { size 8{5} } b} } right )} {} [3]

5. If a= -2 and b= -1 , calculate:

5.1: (3b - 3a)² [2]

5.2: -3a³ + 3b² [3]

5.3: 3a² [2]

IntegersTutorial

Parent signature: Date:

Test 1: (Integers)

Total: 40

1. Simplify:

1.1: 834 n⁴ x 0 [1]

1.2: (-1)¹⁰ [1]

1.3: -8m⁶ ÷ 2m³ [2]

1.4: (-2c⁴d³)³ [2]

1.5: 2p³qx (-3pq³) x (-5pq²) [3]

1.6: -6a⁸ ÷ (-2a²) + 4a² x 3a⁴ [3]

1.7: (-2) + (+3) - (-4) - (-1) [2]

1.8: -6a³ + (-2a²b) + (-4a³) - (+5b²a) [3]

1.9: - 3k6m3-9k2m12- 3k6m3-9k2m12 size 12{ { {"- 3"k rSup { size 8{6} } m rSup { size 8{3} } } over { size 11{"-9"}k rSup { size 8{2} } m rSup { size 8{"12"} } } } } {} [3]

1.10: -3ab(ab- 2b) - (-4ab) [3]

1.11: 12a6 - 4a²- 4a²12a6 - 4a²- 4a² size 12{ { {"12"a rSup { size 8{6} } " - 4"a²} over {"- 4"a²} } } {} [3]

2. If A = 2p - 3q- 4rand B = -2p + 3r - 4q

Determine: -2A - 3B

[4]

3. Subtract the product of -3a + 12ab and -6(ab)² from 5a³b- 10a³b³

[4]

4. Calculate the quotient of -2(a + b) and -3a

[3]

5. Write in set builder notation:

Figure 3

[2]

6. Bonus question

Prove that the product of three consecutive integers plus one will always be a perfect square.

[2]

Enrichment exercise for the quick worker

(Learning unit 1)

1. If 1x−13=121x−13=12 size 12{ { {1} over {x` - ` { { size 8{1} } over { size 8{3} } } } } `=` { {1} over {2} } } {} , then x is equal to ....

2. The figure shows a magic square in which the sum of the numbers in any row, column of diagonal is equal. The value of n is...

3. A train passes completely through a tunnel in 5 minutes. A second train, twice as long, passes through the tunnel in six minutes. If both trains were travelling at 24 km/h determine the length of the tunnel.

4. A clock loses exactly 4 minutes every hour. At 06:00 it is set correctly. What will the correct time be when the clock shows 15:48 for the first time?

5. The last digit of the number 3 ¹⁹⁹³ is ...

6. You are travelling along a road at a constant speed of 105 km per hour, and you notice that you pass telephone pylons at the side of the road at regular intervals. If it takes 72 seconds to travel from the first pylon to the fifteenth, then the distance in metres between tow successive pylons is …

CLASS ASSIGNMENT 1

2.

Figure 4

4. Numbers with no fractions or decimals added to it e.g. 2 not 2½ or 2,5

5. Z

6.

6.1.1

6.1.2

6.1.3

CLASS ASSIGNMENT 2

1.1

2. 4⁰ – 8⁰ = –4⁰C

4. Add all (+) numbers; Add all (–) numbers; Subtract them from each other.

7.1 :–6 + (–3) = –9

7.2 :5 – (–5) = 10

HOMEWORK ASSIGNMENT 2

–9 + 8 – 16 = –17

= –256 + 9

= –247

= 169 + 13

= 179

1.5 :a – b + b = a

1.6 :a – b – a = –b

1.7 :–b – b + a = –2b

1.8 : y2−x2−x2=y2y2−x2−x2=y2 size 12{y rSup { size 8{2} } - x rSup { size 8{2} } - x rSup { size 8{2} } =y rSup { size 8{2} } } {}

3.1 : a = –2

3.2 : a = 12

3.3 : a = –3

3.4 :–8 = a

4. R615 – R(46 + 480 + 199)

= R615 – R725

= R110 (–) Loss

CLASS ASSIGNMENT 3

1. ( )

2. of

3. x or ÷ : from left to right

4. + or – : from left to right

= –66

2. p = (–2) x (3) ÷ (–2)²

= –6 ÷ 4

= −64−64 size 12{ { { - 6} over {4} } } {} = –1 1212 size 12{ { {1} over {2} } } {} / –1,5

= –24 ÷ (–6)

= 4

HOMEWORK ASSIGNMENT 2

= 169 – 169 – 169 = –169

= (–1)² – (1)² – 64 – 49

= +1 – 1 – 64 – 49

= –113

= 15 – 55

= –40

2. −147−21−147−21 size 12{ { { - "147"} over { - "21"} } } {} – (–55)

= 7 + 55

= 62

3. 17 x (–15) ÷ (–7)

= –255 ÷ (–7)

= 36,4

4. (–88 + 7) – (–58)

= –81 + 58

= –23

5. –7 – (–5 x 17)

= –7 + 85

= 78

p = –5

7. a = 4

8. –(–3) = 3