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Integers and their organisation

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 8

INTEGERS, EQUASIONS AND GEOMETRY

Module 7

INTEGERS AND THEIR ORGANISATION

INTEGERS

CLASS ASSIGNMENT 1

  • Step by step, discover more about… what integers are, their organisation and how you can write them down....

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
Figure 1 (Picture 27.png)

[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:

  • Graphically: i.e. using a number line
  • Set builder notation: { x / xz, x ≥ -3 }

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

  • Interval notation: [-3; ∞) , only real numbers can be indicated in this way.

(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 , xZ

6.1.2 x ≥ -2 , xZ

6.1.3 2 ≤ x < 5

6.2 Write the following in set builder notation:

Figure 2
Figure 2 (Picture 28.png)

CLASS ASSIGNMENT 2

  • Step by step, discover more about... adding and subtracting integers

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?

  • Subtract - 5 from 3

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

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

( - ) × ( - ) → ( + )

  • Thus: 3 + 5 = 8 (You can see how easy it is)

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

Table 1
Assessment of myself:   by myself:   Assessment by Teacher:
I can…     1 2 3 4   Critical Outcome 1 2 3 4
define an integer; (Lo 1.2.1);                     Critical and creative thinking        
order integers; (Lo 1.2.1);                     Collaborating        
represent integers graphically; (Lo 1.2.1);                     Organising en managing        
use set builder notation correctly; (Lo 1.2.1);                     Processing of information        
use interval notation correctly; (Lo 1.2.1);                     Communication        
use the properties of 0 and 1; and (Lo 1.2.1);                     Problem solving        
add and subtract integers. (Lo 17).                     Independence        

goodaveragenot so good

Table 2
Comments by the learner:     My plan of action:     My marks:
I am very satisfied with the standard of my work.     <   Date:      
I am satisfied with the steady progress I have made.         Out of:      
I have worked hard, but my achievement is not satisfactory.         Learner:      
I did not give my best.     >          
Table 3
Comments by parents:   Comments by teacher:
     
     
     
Signature: Date:   Signature: Date:

CLASS ASSIGNMENT 3

  • Step by step, discover more about … multiplying and dividing integers.
  • Do you still remember the sequence of operations? Write them down:

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

Look at: +4 × ( -3) = -12

  • Step 1: first multiply the signs: (+) × (-) → (-)
  • Step 2: now multiply the numbers: 4 × 3 = 12

What about -12 ÷ (+4) = -3

  • Step 1: first divide (same as for multiplying) the two signs (-) ÷ (+) → (-)
  • Step 2: now do 12 ÷ 4 = 3 OR 12/4

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: 563563 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)-76×(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

Table 4
Assessment of myself:   by myself:   Assessment by Teacher:
I can…     1 2 3 4   Critical Outcomes 1 2 3 4
multiply integers; (Lo 1.2.5);                     Critical and creative thinking        
divide integers by integers; (Lo 1.2.5);                     Collaborating        
do a mixture of examples ( +; - ; × and  ); and (Lo 1.2.1; 1.2.5);                     Organising en managing        
calculate the value of unknown ones. (Lo 2.5).                     Processing of information        
                      Communication        
                      Problem solving        
                      Independence        

goodaveragenot so good

Table 5
Comments by the learner:     My plan of action:     My marks:
I am very satisfied with the standard of my work.     <   Date:      
I am satisfied with the steady progress I have made.         Out of:      
I have worked hard, but my achievement is not satisfactory.         Learner:      
I did not give my best.     >          
Table 6
Comments by parents:   Comments by teacher:
     
     
     
Signature: Date:   Signature: Date:

Tutorial 1: (Integers)

Total: 40

1. Complete:

Table 7
n 2 5     -20
7n - 5 9   -58 65  

[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 / xz , -3 < x < 5 } [2]

3.2 { x / xz, 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: 126b44a5b126b44a5b 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

Table 8
I demonstrate knowledge and understanding of: Learning outcomes   0000 000 00 0
1. the ordering of integers; 1.2.1 ; 1.2.2          
2. graphic representation of integers; 1.2.1; 1.2.2          
3. numbers in set builder notation; 1.2.1 ; 1.2.2          
4. representing numbers in interval notation; 1.2.1 ; 1.2.2          
5. calculating and subtracting integers; 1.2.1; 1.2.2 ; 1.7          
6. multiply integers with one another; 1.2.1; 1.2.2 ; 1.7          
7. dividing integers with one another. 1.2.1; 1.2.2; 1.2.5; 1.7          
8.              
9.              
10.              
11.              
12.              
13.              
14.              
               
               
Table 9
The learner’s … 1 2 3 4
work is… Not done.. Partially done. Mostly complete. Complete.
layout of the work is… Not understandable. Difficult to follow. Sometimes easy to follow. Easy to follow.
accuracy of calculations… Are mathematically incorrect. Contain major errors. Contain minor errors. Are correct.
Table 10
  My BEST marks:   Comments by teacher:
Date:          
Out of:          
Learner:          
           
          Signature: Date:

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 - 4- 412a6 - 4- 4 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
Figure 3 (Picture 29.png)

[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 1x13=121x13=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...

Table 11
8    
9    
4 n  

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 …

Memorandum

CLASS ASSIGNMENT 1

  1. “Less than 0” not positive

2.

Figure 4
Figure 4 (Picture 128.png)
  1. temperatures; bank balances; etc.

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

  • –1 ≤ x < 2; x size 12{ in } {} Z
  • x ≥ 3; x size 12{ in } {}Z
  • –2 < x < 2; x size 12{ in } {} Z

CLASS ASSIGNMENT 2

1.1

2. 40 – 80 = –40C

  • :–13
  • :–5
  • :–27
  • :–8 – 5 + 7 = –6

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

  • :–9 + 6 = –3
  • :–18 – 13 + 7 = –14
  • :20 – 75 = –55
  • :10 – (–2) = 10 + 2 = 12
  1. :–31 – (–17) = –31 + 17 = –14

7.1 :–6 + (–3) = –9

7.2 :5 – (–5) = 10

HOMEWORK ASSIGNMENT 2

  • :13 – 18 + 4 – 17 = –18
  • :–9 – (–8) + (–16)

–9 + 8 – 16 = –17

  • :– (–16)2 + (–3)2

= –256 + 9

= –247

  • :(–13)2 – (–13)

= 169 + 13

= 179

1.5 :a – b + b = a

1.6 :a – b – a = –b

1.7 :–b – b + a = –2b

1.8 : y2x2x2=y2y2x2x2=y2 size 12{y rSup { size 8{2} } - x rSup { size 8{2} } - x rSup { size 8{2} } =y rSup { size 8{2} } } {}

  • True
  • : xyx+yxyx+y size 12{ - x - y <> - x+y} {} False
  • : y+z=z+yy+z=z+y size 12{y+z=z+y} {} True
  • : x+y=x+yx+y=x+y size 12{ - x+y= - x+y} {} True

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

  • :–42
  • :36 + 34 = 70
  • :35
  • :3 x (–1) + 6 = –3 + 6 = 3
  • :–24 x 25 = –600
  • :(–2)3 = –8
  • :(–64) – (+2) = –64 – 2

= –66

  • :(15 – 9)2 = (6)2 = 36
  • :(–6)2 = 36
  • :–2(9) = –18
  • 113113 size 12{ { { - "11"} over {3} } } {} = –3 1313 size 12{ { {1} over {3} } } {}
  • 2412+22412+2 size 12{ { {"24"} over { - "12"+2} } } {} = 24102410 size 12{ { {"24"} over { - "10"} } } {} = –2,4
  • –6 x 5757 size 12{ { {5} over {7} } } {} = 307307 size 12{ { { - "30"} over {7} } } {} = –4 2727 size 12{ { {2} over {7} } } {}
  • 53255325 size 12{ { {"53"} over { - "25"} } } {} = –2 325325 size 12{ { {3} over {"25"} } } {} or –2,12

  • :–50 ÷ 5 = –10

2. p = (–2) x (3) ÷ (–2)2

= –6 ÷ 4

= 6464 size 12{ { { - 6} over {4} } } {} = –1 1212 size 12{ { {1} over {2} } } {} / –1,5

  • p = 4(–2)(3) ÷ (–2)(3)

= –24 ÷ (–6)

= 4

HOMEWORK ASSIGNMENT 2

  • (13)2 – (–13)2 – 132

= 169 – 169 – 169 = –169

  • (7 – 8)2 – (8 – 7)2 – 82 – 72

= (–1)2 – (1)2 – 64 – 49

= +1 – 1 – 64 – 49

= –113

  • (5)3 – 33 – 22

= 15 – 55

= –40

2. 1472114721 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 = –60
  • :p = –8
  • :p = 7
  • :2p + 6 = –4

p = –5

7. a = 4

8. –(–3) = 3

  • {–1; 0; 1; 2; 3}
  • {2; 3; 4; 5}
  • {–1; –2; –3}

TUTORIAL 1

1. :30; 537537 size 12{ { { - "53"} over {7} } } {} = –7,6 √; 10; –145 √

2.1 : nn size 12{n} {} = 274274 size 12{ { {"27"} over {4} } } {} = 6 3434 size 12{ { {3} over {4} } } {}nn size 12{n} {} > 6 3434 size 12{ { {3} over {4} } } {}nn size 12{n} {} size 12{ in } {} {7; 8; 9; . . . } √√

  • :{5; 4; 3; 2; 1} √
  • √√
  • √√
  • :[–(4)]3 √ = –64 √
  • :–8 – 9 + 8 + 9 √ = 0 √
  • :15 + (–40) √ + (–12) √ = –37 √
  • : 61116111 size 12{ { { { {6}} rSup { size 8{1} } } over {"11"} } } {} x 12441244 size 12{ { {1} over { - { {2}} { {4}} rSub { size 8{4} } } } } {}

= – 144144 size 12{ { {1} over {"44"} } } {}

  • :(0,09) √ x (–0,04) = –0,0036 √
  • :–1 √√
  • :87 √√

√ √ √

  • :–0,225a5
  • 1 2 3 a a b b 4 b 3 4 a 5 b 1 2 3 a a b b 4 b 3 4 a 5 b size 12{ left ( { { - { {1}} { {2}} rSup { size 8{3} } { {a}} rSup { size 8{a { {b}}} } { {b}} rSup { size 8{ { {4}}b rSup { size 6{3} } } } } over { { {4}} { {a}} rSup { { {5}}} { { size 12{b}}}} } right )} {}

√ √ √

= 9a2b

  • :[3(–1) – 3{–2)]2

= [–3 + 6]2

= 9 √

  • :–3(–2)3 + 3(–1)2

= –3(–8) + 3(1) √

= 24 + 3

= 27√

5.3 :3(–2)2

= 3(4)

= 12 √

TEST (INTEGER)

  • :0 √
  • :1 √
  • : 84M632M384M632M3 size 12{ { { - { {8}} rSup { size 8{4} } { {M}} rSup { size 8{ { {6}} rSup { size 6{3} } } } } over { { {2}} { {M}} rSup { { {3}}} } } } {} - –4M3 √√

√ √

  • :–8c12d9

√ √ √

  • :30p5q6
  • : 63a862a263a862a2 size 12{ { { - { {6}} rSup { size 8{3} } a rSup { size 8{ { {8}} rSup { size 6{6} } } } } over { - { {2}}a rSup { { {2}}} } } } {} + 12a6

√ √ √

= 3a6 + 12a6 = 15a6

  • :–2 + 3 + 4 + 1 √ = 6 √
  • :–6a3 – 2a2b – 4a3 – 5ab2

= –10a3 – 2a2b – 5ab2 √√

√√√1.9 : k43M9k43M9 size 12{ { {k rSup { size 8{4} } } over {3M rSup { size 8{9} } } } } {}

√ √ √

  • :–3a2b2 + 6ab2 + 4ab

√ √ √

  • :–3a4 + 1 [12a64a2[12a64a2 size 12{\[ { {"12"a rSup { size 8{6} } } over { - 4a rSup { size 8{2} } } } } {}4a24a2]4a24a2] size 12{ { {4a rSup { size 8{2} } } over { - 4a rSup { size 8{2} } } } \]} {}
  1. :–2(2p – 3q – 4r) – 3(–2p + 3r – 4q) √

= –4p + 6q + 8r + 6p – 9r + 12q √√

= 2p + 18qr

3. :5a3b – 10a3b3 – [–6a2b2(–3a + 12ab)] √

5a3b – 10a3b3 – [18a3b2 – 72a3b3] √

5a3b – 10a3b3 – 18a3b2 – 72a3b3

5a3b – 18a3b2 + 62a3b3

4. 2(a+b)3a2(a+b)3a size 12{ { { - 2 \( a+b \) } over { - 3a} } } {} = 2a2b3a2a2b3a size 12{ { { - 2a - 2b} over { - 3a} } } {}

= 2a3a2a3a size 12{ { { - 2 { {a}}} over { - 3 { {a}}} } } {}2b3a2b3a size 12{ { {2b} over { - 3a} } } {}

= 2323 size 12{ { {2} over {3} } } {} + 2b3a2b3a size 12{ { {2b} over {3a} } } {}

√ √

√ √

5. {x / –2 ≤ x ≤3; x size 12{ in } {} 1R}

6. x (x + 1)( x + 2) + 1

= (x 2 + x)( x + 2) + 1

= x 3 + 2 x 2 + x 2 + 2 x + 1

= x 3 + 3 x + 2 x + 1

2(3)(4) + 1 = 25

4(5)(6) + 1 = 35 False

5(6)(7) + 1 = 211 False

ENRICHMENT EXERCISE

1. 13x1313x13 size 12{ { {1} over { { {3x - 1} over {3} } } } } {}

= 1212 size 12{ { {1} over {2} } } {}2121 size 12{ { {2} over {1} } } {} = 3x133x13 size 12{ { {3x - 1} over {3} } } {}

6 = 3 x + 1

7 = 3 x

(2 1313 size 12{ { {1} over {3} } } {}) 7373 size 12{ { {7} over {3} } } {} = x

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