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3.2 Divided by 2

Remainders 1

............ ............ ............ ............ ............ ............

4. Let us see how well you do in the following mental test. Complete it as quickly and accurately as you can:

Table 2.19.

4.1 35 – 5 – 5 – 5 = ............ 4.11 108 – 12 – 12 – 12 = ............

4.2 64 – 8 – 8 = ............

4.12 72 – 9 – 9 = ............

4.3 42 ÷ 7 = ............

4.13 Halve: 612: ............

4.4 54 ÷ 6 = ............

4.14 Halve: 487: ............

4.5 ............ ÷ 7 = 8

4.15 Halve: 1 036: ............

4.6 ............ ÷ 12 = 6

4.16 ............ ÷ 9 = 4

4.7 28 ÷ 4 = ............

4.17 ............ ÷ 8 = 6

4.8 280 ÷ 4 = ............

4.18 60 ÷ ............ = 5

4.9 280 ÷ 40 = ............

4.19 36 ÷ ............ = 4

4.10 2 800 ÷ 40 = ............

4.20 3 600 ÷ 90 = ............

DID YOU KNOW?

Division is the inverse of multiplication.

Thus: 5 × 3 = 1515 ÷ 3 = 5 en15 ÷ 5 = 3

When I want to check a division sum, I multiply. When I check a multiplication sum, I divide

Activity 2:

To recognise, describe and use the reciprocal relation between multiplication and

division

[LO 1.12.1]

1. Now use your knowledge of the "inverse" to complete the following:

1.1 If 26 × 17 = 442, then 442 ÷ 17 = ................ and 442 ÷ ................ = 17

If 24 × 30 = 720, then 720 ÷ ................ = 24 and 720 ÷ 24 = ................

DID YOU ALSO KNOW?

When we divide, we do the following:

REPEATED SUBTRACTION

12 ÷ 3 = 12 – 3 – 3 – 3 – 3 = 4

GROUPING

12 ÷ 3: We must divide 12 into 3 equal groups

DIVISION

12 ÷ 3: We must divide 12 equally between 3

123

Activity 3:

To recognise and represent numbers and to describe and compare them [LO 1.3]

1. Let us do revision. Do you remember what happens when we divide any number by 1? Work

with a friend and find the answers to the following:

1.1 5 ÷ 1 = .................................. 1.2 86 ÷ 1 = ............................

1.3 359 ÷ 1 = .............................. 1.4 4 625 ÷ 1 = .......................

1.5 32 174 ÷ 1 = .........................

2. Write down a rule for dividing by 1.

BRAIN-TEASER!

If 5 ÷ 1 = 5 and 86 ÷ 1 = 86, then:

1 ÷ 5 = .........................

1 ÷ 86 = .........................

1 ÷ 359 = .........................

1 ÷ 4 625 = .........................

Thus, if 1 is the dividend, then the answer is always a...........................................

3. Now let us see what happens when we divide any number by 0. Work with a friend once more

and do the following with the aid of your calculator:

3.1 6 ÷ 0 = ................................... 3.2 38 ÷ 0 = ..............................

3.3 438 ÷ 0 = ............................... 3.4 1 679 ÷ 0 = .........................

63 827 ÷ 0 = .........................

4. Why does the calculator give these answers?

DO YOU STILL REMEMBER?

Division by 0 is not allowed. We say it is undefined.

BRAIN-TEASER!

I think of a number. If I subtract 3 and then divide the number by 6, the quotient will be 7. What is the number? ..................

I think of another number. If I halve it and then divide it by 12, the quotient will be 9. What is the number? ..................

Activity 4:

To describe and illustrate other cultures' ways of writing [LO 1.2]

DID YOU KNOW?

The Chinese write either from top to bottom or from left to right. Some of their numbers look like these:

Figure 2.5.

Figure 2.6.

Figure 2.7.

Figure 2.8.

Figure 2.9.

Figure 2.10.

Figure 2.11.

Figure 2.12.

Figure 2.13.

10.

Figure 2.14.

100.

Figure 2.15.

1.1 Can you write the following in Chinese? Then give the answer in our number system.

Table 2.20.

a) 10 ÷ 1 =

b) 100 ÷ 10 =

c) 9 ÷ 3 =

d) 100 ÷ 4 =

e) 100 ÷ 5 =

f) 100 ÷ 7 =

Division by 10, 100 and 1 000

Activity 5:

To describe observed patterns and rules in own terms [LO 2.2]

1. In Mathematics certain rules enable us to calculate answers quickly. However, we must be able to "spot" or "deduce" the rule first before we can apply it. Work with a friend and complete the table.

Table 2.21.

Number 20

120

360

1 470

2 560

14 380 26 520

÷ 10

............ ............ ............ ............ ............ ............ ............ ............ ............

What do you notice when you look at the answer?

2. Complete the following also:

Table 2.22.

Number 300

800

1 200

2 600

14 700

32 500

÷ 100

................. ................. ................. ................. ................. .................

Can you also write down the answers to the following?

3. Can you also write down the answers to the following?

Table 2.23.

Number 5 000

7 000

13 000

45 000

126 000

382 000

÷ 1 000 .................. .................. .................. .................. .................. ..................

What happens when we divide by 1 000?

Division by multiples of 10 and 100:

Activity 6:

To calculate by using selected computations [LO 1.8.4]

1. You have now discovered rules for division by 10, 100 and 1 000. Now let us look at division by multiples of 10 and 100. Work through the following calculations with a friend:

Table 2.24.

1.1 180 ÷ 20 = (180 ÷ 10) ÷ 2 = 18 ÷ 2 = 9

1.2 4 200 ÷ 60 = (4 200 ÷ 10) ÷ 6 = 420 ÷ 6 = 70

1.3 3 600 ÷ 400 = (3 600 ÷ 100) ÷ 4 = 36 ÷ 4 1.4 54 000 ÷ 900 = (54 000 ÷ 100) ÷ 9 = 540 ÷ 9

= 9

= 60

2. Complete the following on your own:

2.1 1 380 ÷ 60 = (1 380 ÷ 10) ÷ ..................

= .................. ÷ ..................

= ..................

2.2 32 840 ÷ 40 = (32 840 ÷ ..................) ÷ ..................

= ......................... ÷ ..................

= ..................

2.3 15 500 ÷ 500 = (15 500 ÷ 100) ÷ ..................

= .................. ÷ ..................

= ..................

2.4 312 300 ÷ 300 = (312 300 ÷ ..................) ÷ ..................

= ......................... ÷ ..................

= ..................

DID YOU KNOW?

We can also break up the divisor into factors:

E.g. 108 ÷ 12 = 108 ÷ 4 ÷ 3 = 27 ÷ 3 = 9

108 ÷ 12 = 108 ÷ 2 ÷ 6 = 54 ÷ 6 = 9

3. Calculate the following by breaking up the divisor into factors:

3.1 105 ÷ 21 = ...................................... 3.2 216 ÷ 24 = ......................................

= ...................................... = ......................................

3.3 432 ÷ 24 = ...................................... 3.4 126 ÷ 14 = ......................................

= ...................................... = ......................................

4. Work together with a friend. Look carefully at the following and then explain it to two other friends.

4.1 184 ÷ 4 = (180 + 4) ÷ 4

= (180 ÷ 4) + (4 ÷ 4)

= 45 + 1

= 46

4.2 2 515 ÷ 5 = (2 000 + 500 + 15) ÷ 5

= (2 000 ÷ 5) + (500 ÷ 5) + (15 ÷ 5)

= 400 + 100 + 3

= 503

4.3 3 672 ÷ 12 = (3 600 ÷ 12) + (72 ÷ 12)

= 300 + 6

= 306

5. Complete the following:

5.1 3 045 ÷ 15 = (3 000 + ....................) ÷ 15

= (.................... ÷ 15) + (.................... ÷ 15)

= 200 + ....................

= ....................

5.2 2 575 ÷ 25 = (2 000 + .................... + ....................) ÷ 25

= (2 000 ÷ 25) + (.................... ÷ .............) + (............. ÷ ............)

= .................... + .................... + ....................

= ....................

5.3 Can you write down your own explanation?

Activity 7:

To be able to do mental arithmetic [LO 1.9.2]

1. Let us test your mental skills again. Find the correct answer and encircle it with your pencil.

Find out who or what is hidden away.

1.1 96 ÷ 12

1.2 108 ÷ 9

1.3 72 ÷ 9

1.4 42 ÷ 6

1.5 54 ÷ 6

1.6 32 ÷ 8

1.7 27 ÷ 9

1.8 66 ÷ 11

1.9 81 ÷ 9

1.10 35 ÷ 7

1.11 21 ÷ 3

1.12 ____ ÷ 6 = 9

1.13 ____ ÷ 8 = 6

1.14 ____ ÷ 6 = 12

1.15 ___ ÷ 11 = 12

1.16 ____ ÷ 9 = 5

1.17 ____ ÷ 7 = 8

1.18 Halve: 96

1.19 Halve: 134

1.20 Halve: 576

Colour the pictures in neatly so that your answers are clearly visible.

Figure 2.16.

Activity 8:

To estimate and calculate by using selected computations [LO 1.8.5]

1. We have already discussed the value of estimation. Let us see again how well you can estimate.

Encircle the best answer.

Table 2.25.

I ESTIMATE

a) How many 16’s are there in 35?

b) How many 15’s are there in 93?

c) How many 13’s are there in 271?

200

d) How many 25’s are there in 4 130? 27 175 750

e) How many 21’s are there in 8 565? 4

400

2. Now use your calculator to see whether you were spot on.

Activity 9:

To be able to solve problems in context [LO 1.6.1]

The following activity is a group project that you can add to your portfolio. Your educator will provide you with the necessary paper and will explain exactly what you are expected to do.

Divide into groups of four or five and then read the assignment closely.

Your group must visit at least three florists or flower vendors in your area. Your job is to

determine which florist or vendor is the cheapest.

Remember to use the following as guidelines:

1. Look only at mixed bouquets.

2. Consider the number of flowers in every bunch.

3. State in your report which of the vendors or florists is the cheapest and motivate your answer.

Remember to show all your calculations.

4. Work neatly and hand in your project on time.

Activity 10:

To solve problems in context [LO 1.6]

1. Divide into groups of three. Ask your educator for the necessary paper and solve the following problems without the use of a calculator:

2 924 spectators have to be transported by bus to a soccer stadium. If every bus can seat 68

passengers, how many buses are needed to take everybody to the stadium?

55 educators were taken by taxi to the same stadium. If they had to pay R2 475 in total for the trip, how much money did each educator have to pay the driver?

A group of rugby players decided to support their soccer-playing friends. If there were 2 010

rugby players in total, how many rugby teams attended the soccer matches? (There are 15

players in a rugby team).

2. Check your answers with a calculator.

3. Compare your solutions to those of the other groups in class.

4. Discuss the differences and / or similarities between the different methods used.

Activity 11:

To determine the equivalence and validity of different representations of the

same problem by means of comparison and discussion [LO 2.6]

In the previous activity you used your own strategies and methods to solve the problems. In the next activity the answers have already been calculated. Work with a friend and explain to each

other the different solutions to the problem:

There are 7 310 books in the school library. The librarian wants to put 34 books on each shelf.

How many shelves does she need?

1. I calculate my answer by doing repeated subtraction.

I know 34 × 100 = 3 400.

Thus: 7 310

- 3 400 (100)

3 910

- 3 400 (100)

510

- 340 (10)

170

- 170 (5)

- - -

The answer is thus 100 + 100 + 10 + 5 = 215

2. I must calculate 7 310 ÷ 34. I do this by multiplication.

34 × 100 = 3 400, so 34 × 200 = 6 800

7 310 – 6 800 = 510

34 × 10 = 340

510 – 340 = 170

34 × 5 = 170

170 – 170 = 0

The quotient is thus 200 + 10 + 5 = 215

1. 7 310 = 3 400 + 3 400 + 340 + 170

Thus: (3 400 ÷ 34) + (3 400 ÷ 34) + (340 ÷ 34) + (170 ÷ 34)

= 100 + 100 + 10 + 5

= 215

6. Can you think of another method to calculate 7 310 ÷ 34?

7 Explain the method used at no. 5 to your class. (Imagine YOU are the teacher!)

8. Which of the above methods do you prefer?

Why?

Activity 12:

To calculate by using selected computations [LO 1.8.5]

1. You have now looked at a variety of methods that we can use for division. Now let us

concentrate on only one of them. Find the quotient of the following by using the method as

applied to no. 5 on the previous page.

Table 2.26.

1.1 4 650 ÷ 25

1.2 5 250 ÷ 21

.................................................. ..................................................

Table 2.27.

1.3 2 648 ÷ 42

1.4 9 850 ÷ 50

.................................................. ..................................................

2. Now use any method of your choice and calculate:

Table 2.28.

2.1 15 350 ÷ 45

2.2 32 648 ÷ 28

.................................................. ..................................................

.................................................. .................................................

Assessment

Table 2.29.

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:

1.2 describes and illustrates various ways of writing numbers in different cultures (including

local) throughout history;

1.3 recognises and represents numbers in order to describe and compare them:

1.3.6 multiples of single-digit numbers to at least 100;

1.3.7 factors of at least any 2-digit whole number;

1.6 solves problems in context including contexts that may be used to build awareness of other

Learning Areas, as well as human rights, social, economic and environmental issues;

1.6.1 financial (including buying and selling, profit and loss, and simple budgets);

1.8 estimates and calculates by selecting and using operations appropriate to solving problems: 1.8.1 rounding off to the nearest 5, 10, 100 or 1 000;

1.8.4 multiplication of at least whole 3-digit by 2-digit numbers;

1.8.5 division of at least whole 3-digit by 2-digit numbers;

1.9 performs mental calculations:

1.9.2 multiplication of whole numbers to at least 10 × 10;

1.10 uses a range of techniques to perform written and mental calculations with whole numbers:

1.10.1 adding and subtracting in columns;

1.10.4 doubling and halving;

1.10.5 using a calculator;

1.11 uses a range of strategies to check solutions and judge the reasonableness of solutions;

1.12 recognises, describes and uses:1.12.1 the reciprocal relationship between multiplication and division (e.g. if 5 × 3 = 15 then 15 3 = 5 and 15 5 = 3);1.12.3 The commutative, associative and distributive properties with whole numbers (the expectation is that learners should be able to use the properties and not necessarily know the names).

LU 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.

We know this when the learner:

2.2 describes observed relationships or rules in own words;

2.6 determines, through discussion and comparison, the equivalence of different descriptions of the same relationship or rule presented:

verbally;

2.6.3 by number sentences.

Memorandum

ACTIVITY 1

3.1

Table 2.30.

2 4 4 6 2 7

3.2

Table 2.31.

Res

1 1 5 5 6 1

4. 4.1 20 4.11 72

4.2 48 4.12 54

4.3 6 4.13 306

4.4 9 4.14 243

4.5 56 4.15 518

4.6 72 4.16 36

4.7 7 4.17 48

4.8 70 4.18 12

4.8 7 4.19 9

4.10 70 4.20 40

ACTIVITY 2

1.1 26 26

1.2 30 30

ACTIVITY 3

1.1 5

1.2 86

1.3 359

1.4 4 625

1.5 32 174

2. When any number is divided by 1, the answer always remains the number

BRAIN TEASERS

(2.1)

(2.2)

(2.3)

Fractions

3. 3.1 O E 3.2 O E

3.3 O E 3.4 O E

3.5 O E

4. Shows “Error” (Answer is incorrect)

BRAIN TEASERS

216

ACTIVITY 4

1.1 a) 10

1. 10

2. 3

3. 25

4. 20

5. 14 rem 2

ACTIVITY 5

Table 2.32.

2 5 9 12 147 256 1 438 2 652

O falls away. Each number moves one place to the right.

Table 2.33.

3 8 12 26 147 325

2 Noughts fall away. Each number moves two places to the right.

Table 2.34.