Integers and their organisation

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)

[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 / x ∈ z, x ≥ -3 } (read as follows: set x in which x ∈ zand 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 , x ∈ Z 6.1.2 x ≥ -2 , x ∈ Z 6.1.3 2 ≤ x < 5 6.2 Write the following in set builder notation:

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 = b0 size 12{ { {b} over {0} } } {} = and 0b 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 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

 good  average  not so good 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. >

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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: −5−63 size 12{ { { - 5` - `6} over {3} } } {} 1.12: −6(−4)-12−(-2) size 12{ { { - 6` \( - `4 \) } over {"-12"`-` \( "-2" \) } } } {} 1.13: −6×(−5)-7 size 12{ - 6`` times `` { { \( - 5` \) } over {"-7"} } } {} 1.14: 53-25 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 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

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Tutorial 1: (Integers) Total: 40 1. Complete: 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 n2 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: 611 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 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 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.

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.

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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 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- 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:

[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=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... 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 “Less than 0” not positive 2.

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 :–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 : y2−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} } } {} True : −x−y≠−x+y size 12{ - x - y <> - x+y} {} False : y+z=z+y size 12{y+z=z+y} {} True : −x+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 −113 size 12{ { { - "11"} over {3} } } {} = –3 13 size 12{ { {1} over {3} } } {} 24−12+2 size 12{ { {"24"} over { - "12"+2} } } {} = 24−10 size 12{ { {"24"} over { - "10"} } } {} = –2,4 –6 x 57 size 12{ { {5} over {7} } } {} = −307 size 12{ { { - "30"} over {7} } } {} = –4 27 size 12{ { {2} over {7} } } {} 53−25 size 12{ { {"53"} over { - "25"} } } {} = –2 325 size 12{ { {3} over {"25"} } } {} or –2,12 :–50 ÷ 5 = –10 2. p = (–2) x (3) ÷ (–2)² = –6 ÷ 4 = −64 size 12{ { { - 6} over {4} } } {} = –1 12 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. −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 = –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; −537 size 12{ { { - "53"} over {7} } } {} = –7,6 √; 10; –145 √ 2.1 : n size 12{n} {} = 274 size 12{ { {"27"} over {4} } } {} = 6 34 size 12{ { {3} over {4} } } {}n size 12{n} {} > 6 34 size 12{ { {3} over {4} } } {}n 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 √ : 6111 size 12{ { { { {6}} rSup { size 8{1} } } over {"11"} } } {} x 1−244 size 12{ { {1} over { - { {2}} { {4}} rSub { size 8{4} } } } } {} √ = – 144 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 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 √ : −84M632M3 size 12{ { { - { {8}} rSup { size 8{4} } { {M}} rSup { size 8{ { {6}} rSup { size 6{3} } } } } over { { {2}} { {M}} rSup { { {3}}} } } } {} - –4M³ √√ √ √ :–8c¹²d⁹ √ √ √ :30p⁵q⁶ : −63a86−2a2 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 : k43M9 size 12{ { {k rSup { size 8{4} } } over {3M rSup { size 8{9} } } } } {} √ √ √ :–3a²b² + 6ab² + 4ab √ √ √ :–3a⁴ + 1 [12a6−4a2 size 12{\[ { {"12"a rSup { size 8{6} } } over { - 4a rSup { size 8{2} } } } } {} – 4a2−4a2] size 12{ { {4a rSup { size 8{2} } } over { - 4a rSup { size 8{2} } } } \]} {} :–2(2p – 3q – 4r) – 3(–2p + 3r – 4q) √ = –4p + 6q + 8r + 6p – 9r + 12q √√ = 2p + 18q – r √ 3. :5a³b – 10a³b³ – [–6a²b²(–3a + 12ab)] √ 5a³b – 10a³b³ – [18a³b² – 72a³b³] √ 5a³b – 10a³b³ – 18a³b² – 72a³b³√ 5a³b – 18a³b² + 62a³b³ √ 4. −2(a+b)−3a size 12{ { { - 2 \( a+b \) } over { - 3a} } } {} = −2a−2b−3a size 12{ { { - 2a - 2b} over { - 3a} } } {} √ = −2a−3a size 12{ { { - 2 { {a}}} over { - 3 { {a}}} } } {} – 2b−3a size 12{ { {2b} over { - 3a} } } {} = 23 size 12{ { {2} over {3} } } {} + 2b3a size 12{ { {2b} over {3a} } } {} √ √ √ √ 5. {x / –2 ≤ x ≤3; x size 12{ in } {} 1R} 6. x (x + 1)( x + 2) + 1 = (x² + x)( x + 2) + 1 = x³ + 2 x² + x² + 2 x + 1 = x³ + 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. 13x−13 size 12{ { {1} over { { {3x - 1} over {3} } } } } {} = 12 size 12{ { {1} over {2} } } {}21 size 12{ { {2} over {1} } } {} = 3x−13 size 12{ { {3x - 1} over {3} } } {} 6 = 3 x + 1 7 = 3 x (2 13 size 12{ { {1} over {3} } } {}) 73 size 12{ { {7} over {3} } } {} = x