m30499 Numeric patterns 1.1 2009/07/25 05:10:13.482 GMT-5 2009/07/25 05:15:09.820 GMT-5 gert bezuidenhout gert bezuidenhout gertb@mweb.co.za gert bezuidenhout gert bezuidenhout gertb@mweb.co.za gert bezuidenhout gert bezuidenhout gertb@mweb.co.za Mathematics and Statistics en

Activity 1: To recognise and represent multiples in order to be able to describe, compare and represent them [LO 1.3] To investigate numeric patterns [LO 2.1] To describe numeric patterns in your own words [LO 2.2] To find output numbers [LO 2.3] 1. When we count in 6’s we are saying the multiples of 6. 1.1 Work with a friend. One of you must count in 6’s from 0 to 102. The other one must use a calculator to check you and stop you if you make a mistake. If that happens the one with the calculator must just say, ”Stop!” and show the calculator. The counting goes on from there. Then swap over. 1.2 Now fill in the missing multiples of 6 in the table below: 1 2 3 4 5 6 7 8 9 10 11 12 × 6 6 12 18 24 30 60 66 1.3 Now count backwards in 6’s from 102 to 0. Let a friend check you. Then swap over. Do you notice anything? Yes, multiples of 6 are all even numbers. 1.4 What patterns do you notice? Yes, the last digits seem to be 6;..2; ..8; ..4; ..0. They are repeated and so form a pattern. Now that you are aware of this, you can count in 6’s for ever (if you concentrate)! 1.5 Count in 6’s and complete the flow diagram:

1.6 How do we programme the calculator to count in 6’s? Press clear and 1.7 Write down the multiples of 6 from 102 to 0: 1.8 How do we programme the calculator to count backwards in 6’s from 102? Press clear and Now you can really count in 6’s easily, so let’s move on to 7’s. 2. Multiples of 7 2.1 Now use your calculator (if necessary) to count in 7’s and complete the flow diagram:

Can you spot any patterns? Two are written below. Try to spot some more and discuss with a friend. They are both odd and even numbers. They seem to be: odd, even, odd, even. 2.2 Now fill in the missing multiples of 7: 1 2 3 4 5 6 7 8 9 10 11 12 × 7 7 14 21 70 77 There seems to be some sort of repetition after the first 9 multiples. Does it continue? Are there any other patterns? Write down what you have noticed. Count backwards in 7’s from 105 to 0. Use your calculator to do this if necessary. Write the missing multiples of 6 in the table below and then the multiples of 7 in the next row. Do you notice anything? 1 2 3 4 5 6 7 8 9 10 11 12 × 6 6 12 18 24 × 7 7 14 21 28 2.5 Now compare the two rows (across) of answers. A very interesting pattern appears to be emerging. Look: 1 × 7 = 1× 6 + 1 5 × 7 = 5 × 6 + …… 9 × 7 = 9 × 6 + …… 2 × 7 = 2 × 6 + ……. 6 × 7 = 6 × 6 + …… 10 × 7 = 10 × 6 + ….. × 7 = 3 × 6 + …… 7 × 7 = 7 × 6 + …… 11 × 7 = 11 × 6 +…… 4 × 7 = 4 × 6 + …… 8 × 7 = 8 × 6 + …… 12 × 7 = 12 × 6 +…… 3. Multiples of 8 3.1 The numbers jump in 8 wholes. Use the calculator to count in 8’s and write down the missing multiples of 8. 0, 8, 16, 24, 32,……., ……., …………, ………, ………….,…………,………., …………, ……., …….., …………, ……… . 3.2 Another way of saying: 8 + 8 + 8 + 8 is 4 × 8. Now complete the flow diagram:

3.3 Spot a pattern. Write down the missing multiples of 8 in the table below and look at the last digit of each one. 1 2 3 4 5 6 7 8 9 10 11 12 × 8 8 16 80 88 Look carefully: 8; 16; 24; 32; 40; 48; 56; 64; 72; 80; 88; 96 If you are aware of the pattern in which 8; ..6; ..4; ..2; ..0; is repeated, and concentrate, you should be able to count in 8’s forever, without mistakes. Now, work with a friend. Count in 8’s from 0 to 104 while your friend checks you on a calculator. Then swap over. 3.5 Now count backwards in 8’s from 104 to 0 while your friend checks you on a calculator. Then swap over. 4. Multiples of 9

4.1 Do what Sue suggested. Write down the missing multiples of 9 in the flow diagram and look at the last digit of each one. Spot the pattern.

4.2 Counting in 9’s is the easiest of all! Now complete the table below. 1 2 3 4 5 6 7 8 9 10 11 12 × 9 9 18 27 90 99 4.3 Now count backwards in 9’s while a friend checks you on the calculator. Begin with 108. Now you should feel comfortable when you count in 6’s; 7’s; 8’s and 9’s and you already know how to count in tens. Practise this with a friend, forwards and backwards. TEST YOUR SKILLS 1. Complete the following by doing one column each day or all four columns in one day, or two columns per day, as your educator chooses: (a) 7 × 4 =………… 6 × 8 =…………….. 8 × 9 =…………….. 2 × 8 =……………. (b) 9 × 8 = 3 × 7 = 1 × 7 = 6 × 9 = (c) 7 × 6 = 8 × 7 = 6 × 5 = 8 × 7 = (d) 2 × 5 = 3 × 9 = 7 × 9 = 5 × 5 = (e) 10 × 10 = 2 × 8 = 9 × 9 = 4 × 6 = (f) 3 × 6 = 8 × 5 = 9 × 0 = 9 × 8 = (g) 4 × 2 = 5 × 8 = 7 × 3 = 7 × 10 = (h) 5 × 6 = 6 × 6 = 8 × 8 = 3 × 8 = (j) 4 × 4 = 0 × 10 = 4 × 9 = 5 × 7 = (k) 5 × 9 = 4 × 8 = 5 × 7 = 4 × 7 = TOTAL: TOTAL: TOTAL: TOTAL: 2. Now write all those that you got wrong and write down a way of finding the correct answer for each of them. Activity 2: To do mental calculations involving multiplication of whole numbers [LO 1.9] 1. Now complete this flow diagram:

2. Ways of solving 7 x 8. Example of solving 7 x 8. 7 x 4 + 7 x 4 OR double (7 x 4) 8 + 8 + 8 + 8 + 8 + 8 + 8 Counting in 8’s till you reach 7 x 8 7 x 5 + 7 x 3 i.e. breaking up the 8 as we did previously 2 x 8 + 5 x 8 i.e. breaking up the 7 to make it easier; I know 2 x 8 and 5 x 8 Double 3 x 8 + 8 7 x 10 - 2 x 7Be sure that you know what you are doing! 3. Now take each one that bothers you, and find a way of reaching the answer without a calculator and write your solution in the table below. My way of solving each problem: e.g. 8 × 8 Double 4 × 8 =32 + 32 = 64 3 × 6 6 × 3 (I know my 3 times table) = 18 TEST YOUR SKILLS Complete each of the following: 1.

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6. Explain the link between 4 and 5. 7. Use what you have learnt in 4 and 5 to complete the following: (a) 6 × 20 = (f) 40 × 20 = (b) 7 × 70 = (g) 60 × 60 = (c) 50 × 3 = (h) 50 × 80 = (d) 9 × 70 = (j) 60 × 7 = (e) 8 × 30 = (k) 90 × 50 = Activity 3: To estimate and calculate by selecting and using a range of techniques [LO 1.1, 1.8] 1. You have found ways of calculating answers of tables that you have not memorized. Now estimate the answer by rounding off. Estimations should be quick and easy, but they only give approximate answers. Then find ways of reaching the exact answer of the following sums and write down all your steps: 1.1 7 x 18 24 x 6 Discuss your methods with other friends in your group. 1.3 36 x 54 Examples of methods: 7 × 18 Estimation: 7 × 20 = 140; the answer is about 140 7 × 10 + 7 × 8 i.e. 7 times all of 18 Other ways of finding the exact answer: or: 18 + 18 + 18 + 18 + 18 +18 + 18 or: 7 × 20 － 14 There are other methods. Try to think of them. 64 × 35 i.e. all of 64 × all of 35 Estimation: 60 × 40 =2 400 60 × 30 = 1 800 60 × 5 = 300 i.e. add all answers 4 × 30 = 120 4 × 5 = 20 2 240 When you are really comfortable with this method, you can proceed to the old traditional vertical method. Don’t try to get there too quickly; more haste, less speed. 2. Word sums. 2.1 Share 54 Smarties equally amongst 9 friends. What does each one receive? 2.2 54 learners have to be transported to an Athletics Meeting. The coach wants to hire vehicles that may take 8 passengers. How many vehicles will be needed? 2.3 A shopkeeper has 106 apples. He puts them on little trays to sell them in his shop. There are 6 apples on each tray. How many trays can he fill? Some methods: 9 + 9 + 9 + 9 + 9 + 9 (How many 9’s in 54?) or: 54  9 9 × ? = 54 There are other methods 54  8 6 × 8 = 48; 6 rem. 6, but they are people who have to get to the athletics meeting. 7 × 8 = 56 7 vehicles will be needed and there will be 2 empty seats on one of them. 106  6 10 × 6 = 60 5 × 6 = 30 2 × 6 = 12 102 17 trays rem. 4 apples (The key word here is “fill”.) There are other ways. Discuss them with your friends and find the way that you understand best. 3. Calculate the answer. Write down the steps of your calculations, explaining (in numbers) how you reached your solution. Then write down the steps which you used to check that your answer is reasonable. 3.1 The mass of a van after it has been loaded is 2 500 kg. If the mass of the load is 500 kg, what is the mass of the van when it is empty? 3.2 The mass of one bag of cement is 25 kg. How many bags of cement will there be if the mass of the whole load is 500 kg? 3.3 The petrol tank of the van can hold 55 litres of petrol when it is full. The van needs one litre of petrol to travel 13 km. The driver fills his tank with petrol. How far can he travel before he needs to fill his tank again? 3.4 When the van is used for short trips around the town, it only travels 11 km per litre of petrol. The driver fills his petrol tank, which holds 55 litres of petrol. How many kilometres can it travel in the town? 3.5 Your school soccer team has to travel from Cape Town to Grahamstown to take part in a soccer tournament. Your team can go by bus along the coast, a distance of 899 km, or you can go by train via De Aar. The distance between Cape Town and De Aar is 762 km. From De Aar to Grahamstown is 444 km.

How much further will you travel if you go by train? 3.6 A special tour to the Kruger National Park is arranged for 134 tourists from overseas. At the rest camp 8 people can sleep in a rondavel. How many rondavels are needed for the tourists? 3.7 There is a tourist shop at the rest camp in the Kruger Park. In this shop special torches called Bush Baby Lanterns are sold. Seven tourists buy a torch each and together they pay R273 for them. How much does one Bush Baby Lantern cost? 3.8 On one side of the car park at the rest camp there is a straight fence consisting of wooden poles. The upright poles for this fence are 3 m apart. There are 18 upright poles. How long is the fence? (To understand the story, draw a fence with 6 poles – technique: substituting smaller numbers in order to understand the story.)

Learning outcomes(LOs) LO 1 Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems. Assessment standards(ASs) We know this when the learner: 1.1 counts forwards and backwards in a variety of intervals; 1.3 recognises and represents the following numbers in order to describe and compare them: common fractions with different denominators, common fractions in diagrammatic form, decimal fractions and multiples of single-digit numbers; 1.3.2 common fractions with different denominators, including halves, thirds, quarters, fifths, sixths, sevenths and eighths; 1.3.3 common fractions in diagrammatic form; 1.3.4 decimal fractions of the form 0,5; 1,5 and 2,5; etc., in the context of measurement; 1.3.6 multiples of single-digit numbers to at least 100; 1.5 recognises and uses equivalent forms of the numbers including common fractions and decimal fractions; 1.5.1 common fractions with denominators that are multiples of each other; 1.5.2 decimal fractions of the form 0,5; 1,5 and 2,5, etc., in the context of measurement; 1.7 solves problems that involve comparing two quantities of different kinds (rate); 1.7.1 comparing two or more quantities of the same kind (ratio); 1.8 estimates and calculates by selecting and using operations appropriate to solving problems that involve addition of common fractions, multiplication of at least whole 2-digit by 2-digit numbers, division of at least whole 3-digit by 1-digit numbers and equal sharing with remainders; 1.8.3 addition of common fractions in context; 1.8.6 equal sharing with remainders; 1.9 performs mental calculations involving: 1.9.2 multiplication of whole numbers to at least 10 x 10; 1.12 recognises, describes and uses:, and 1.12.1 the reciprocal relationship between multiplication and division (e.g. if 5 x 3 = 15 then 15 ÷ 3 = 5 and 15 ÷ 5 = 3; 1.12.2 the equivalence of division and fractions (e.g. 1 ÷ 8 = ⅛); 1.12.3 the commutative, associative and distributive properties with whole numbers. Learning outcomes(LOs) LO 2 Patterns, Functions and AlgebraThe learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills. Assessment standards(ASs) We know this when the learner: 2.1 investigates and extends numeric and geometric patterns looking for a relationship or rules; 2.1.1 represented in physical or diagrammatic form; 2.1.2 not limited to sequences involving constant difference or ratio; 2.1.3 found in natural and cultural contexts; 2.1.4 of the learner’s own creation; 2.2 describes observed relationships or rules in own words; 2.3 determines output values for given input values using verbal descriptions and flow diagrams; 2.3.1 verbal descriptions; 2.3.2 flow diagrams.

ACTIVITY 1

Multiples of 6 Oral Missing multiples of 6 1 2 3 4 5 6 7 8 9 10 11 12 x 6 36 42 48 54 72 Oral Own answers e.g. they’re all even numbers; they’re all also multiples of 3; some are multiples of 9; some are multiples of 12; they are no prime numbers etc. Flow diagram: Output numbers are: 12; 18; 24; 30; 36; 42; 48; 54; 60 Using the calculator to count in 6’s: press clear; 6 + = = = or 6 + + = = = 6; 12; 18; 24; 30; 36; 42; 48; 54; 60; 66; 72; 78; 84; 90; 96; 102 clear; 102 – 6 = = = or clear; 6 － － 102 = = = 2. Multiples of 7 2.1 Flow diagram – output numbers: 14; 21; 28; 35; 42; 49; 56; 63; 70 Missing multiples of 7 1 2 3 4 5 6 7 8 9 10 11 12 x 7 28 35 42 49 56 63 84

1. (a) 28, 48, 72, 16 (b) 72, 21, 7, 54 (c) 42, 56, 30, 56 (d) 10, 27, 63, 25 (e) 100, 16, 81, 24 (f) 18, 40, 0, 72 (g) 8, 40, 21, 70 (h) 30, 36, 64, 24 (j) 16, 0, 36, 35 (k) 45, 32, 35, 28 2. Own ACTIVITY 2 1. Flow diagram – multiples of 8, mixed up Missing output numbers: 16; 72; 32; 48; 64; 56; 40; 24; 80 2. Own 3. Own 4. Own methods of solving multiplication of single digit x single digit.

1. Flow diagram – missing output numbers: 50; 57; 64; 71; 78; 85 2. Flow diagram – missing output numbers: 6; 7; 8; …;11 – missing output numbers: …; 86; 95 3. Flow diagram – missing output numbers: …; 8 – missing output numbers: 28; 40; 46; …; 58; 76 4. Flow diagram – missing output numbers: …; 6; 7 missing output numbers: 240; 300; …; 480; 540 5. Flow diagram – missing output numbers: …; 6; 7 missing output numbers: 240; 300; …; 480; 540 6. x 60 ≈ x 6 x 10 7. Missing answers (a) 120 (b) 490 (c) 150 (d) 630 (e) 240 (f) 800 (g) 3 600 (h) 4 000 (j) 420 (k) 4 500 ACTIVITY 3 – estimations and calculations 1.1 126; 7 x 20 = 140 or 10 x 18 = 180 1.2 144 20 x 6 = 120 or 24 x 10 = 240 or 20 x 10 = 200 1.3 1 944 40 x 50 = 2 000 2. Word sums 2.1 6 Smarties 2.2 7 vehicles 2.3 17 trays and 4 apples left over 3. 3.1 2 000 kg: 2 000 + 500 3.2 20 bags: 25 x 20 = 500 3.3 715 km: rounding off: 60 x 10 = 600 (various methods of checking) 3.4 605 km; rounding off: 60 x 10 = 600 3.5 307 km 899 km + 307 km = 762 km + 444 km 3.6 17 rondavels; 17 x 8 = 80 + 56 = 136 3.7 R39: 39 x 7 = 210 + 63 = 273 3.8 I – I – I – I – I – I : 51 m