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Multiplication

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 5

MULTIPLICATION AND DIVISION

Module 32

MULTIPLICATION

Activity 1:

To recognise and represent the multiples of single-digit numbers [LO 1.3.6]

I think you already know how important it is to know your tables well. In this learning unit it will help you to multiply quickly and correctly. Let us see how well you know your tables.

1. Take the fruit-shaped sweets containing the correct answers out of the jar. Colour them in neatly and then write down the letter of the alphabet written on it. Use these letters to complete the sentence:

_____________ is a shorter way of doing repeated addition.

1.1 5 × 4

1.2 2 × 3

1.3 7 × 7

1.4 7 × 4

1.5 5 × 5

1.6 7 × 3

1.7 6 × 4

1.8 9 × 2

1.9 9 × 7

1.10 9 × 9

1.11 7 × 6

1.12 3 × 5

1.13 8 × 8

1.14 12 × 4

1.15 6 × 12

1.16 9 × 12

1.17 6 × 6

Figure 1
Figure 1 (graphics1.png)

Activity 2:

To recognise and represent the factors of at least any two-digit whole number [LO 1.3.7]

1. Now that you know exactly what factors are, you are probably able to see their important function in multiplication. Factors also help us to test our answers when we multiply. Yes, you are right! We use them when we divide in order to test whether we have multiplied correctly. Now see how many pairs of factors you can write down for the following numbers.

Table 1
a) 24 = ............ × ............ b) 36 = ............ × ............
= ............ × ............ = ............ × ............
= ............ × ............ = ............ × ............
= ............ × ............ = ............ × ............
  = ............ × ............

2. Let us do some more exercises, using factors. Look at the products (numbers) and then write down suitable factors for them. The first one has been done for you.

Table 2
42 45 63 54 64
7 × 6 ...................... ...................... ...................... ....................................
72 108 48 88 96
...................... ...................... ...................... ...................... ....................................

DO YOU STILL REMEMBER?

When you multiply any number by 0, the answer is always 0. (0 × 6 = 0)

Any number multiplied by 1, is that number.(9 × 1 = 9)

Activity 3:

To be able to do mental arithmetic [LO 1.9.2]

Now the factors are supplied to you. All you have to do in your first mental arithmetic test is to complete the multiplication block by writing down the missing answers only. Easy, isn’t it?

Table 3
X 4 5 6 7 8 9 12
5 20 ............ 30 ............ 40 ............ 60
6 ............ 30 36 42 ............ 54 ............
7 28 ............ 42 ............ 56 ............ 84
8 ............ 40 ............ 56 ............ 72 ............
9 ............ ............ 54 ............ 72 ............ 108
2 48 60 ............ 84 ............ 108 ............

Activity 4:

To describe and illustrate different ways of writing in different cultures [LO 1.2]

By doubling numbers we can find the answer to a multiplication sum much more easily. Here we can learn from the Egyptians. Work with a friend and see whether you are able to find out how this calculation was done.

28 × 324

Table 4
1 × 324: 324 (1 × 324)
Double 324: 648 (2 × 324)
Double l 648: 1 296 (4 × 324)
Double 1 296: 2 592 (8 × 324)
Double 2 592: 5 184 (16 × 324)
Thus: 28 × 324 = 9 072  

BRAIN-TEASER!

Calculate the following in the same way as the Egyptians did thousands of years ago:

15 × 241

  • Is there an easier way of doing it? Discuss this with your friend. Check your answer with a calculator.

DID YOU KNOW?

Thus we can also make use of DOUBLING when we want to multiply:

2 × 280 = (200 + 200) + (80 + 80)

= 400 + 160

= 560

Activity 5:

To use a series of techniques to do computations with whole numbers [LO 1.10.4]

1. If you looked carefully at the example above, you would have seen that this method of doubling differs from that of the Egyptians. Can you calculate the following by doubling it in exactly the same way?

1.1 2 × 1 430 =

1.2 2 × 2 315 =

2. We would very much like to use the shortest possible method in Mathematics, because it saves a lot of time, trouble and paper. Calculate the product again by doubling, but this time use a shorter method if you can.

2.1 2 × 14 820 =

2.2 2 × 36 947 =

3. There is another technique that you can use to calculate the product. You can double and halve! This makes it easier to multiply with the “big” numbers. Look carefully at the following:

126 × 5= 63 × 10 (halve 126 ; double 5)= 630

Can you find the following answers by doubling and halving?

Table 5
3.1 50 × 24 3.2 5 × 346
   
   
   

BRAIN-TEASER!

Can you fill in the missing numbers/factors WITHOUT using your calculator?

Table 6
48 × 12 = 576 64 × 10 = 640
........... × 24 = 576 32 × ........... = 640
12 × ........... = 576 ........... × 40 = 640
6 × ........... = 576 8 × ........... = 640
........... × ........... = 576 ........... × ........... = 640

Did you discover a pattern?

Activity 6:

To calculate by using selected computations [LO 1.9.2]

1. Now let us look at multiplication with multiples of 10 and 100. As we work through the examples, you will see that there are certain rules that you must follow in order to calculate the product. If you apply these rules, you will be able to calculate the answers without doing long computations on paper. Work with a friend and write down the product of the following:

1.1 10 × 6 = ......................... ; 10 × 60 = ......................... ;

10 × 600 = ......................... and 10 × 6 000 = .........................

1.2 10 × 9 = ......................... ; 10 × 90 = ......................... ;

10 × 900 = ......................... and 10 × 9 000 = .........................

1.3 10 × 15 = ......................... ; 10 × 150 = ......................... ;

10 × 1 500 = .........................

1.4 10 × 26 = ......................... ; 10 × 260 = ......................... ;

10 × 2 600 = .........................

Can you now write down a rule for multiplying any number by 10?

Does this “work” for 0,5 × 10?

2. Let us look at multiplication with multiples of 100. Work with the same friend and write down the product of:

2.1 100 × 8 = ......................... ; 100 × 80 = ......................... ;

100 × 800 = .........................

2.2 100 × 13 = ......................... ; 100 × 130 = ......................... ;

100 × 1 300 = .........................

2.3 100 × 27 = ......................... ; 100 × 270 = ......................... ;

100 × 2 700 = .........................

  • Write a rule for multiplying any number by 100:

Does your rule “work” for 0,2 × 100?

Activity 7:

To recognise and use the characteristics of multiplication with whole numbers [LO 1.12.3]

1. In our previous module we looked at the associative and commutative characteristics of addition – do you remember this? There are similar characteristics of multiplication. Do you want to know how it works? Fill in the missing factors to balance the equasions:

8 x 6 = 6 x ____

23 x ____ = 13 x 23

124 x 85 = ____ x 124

2. Is the following true or false?

2.1 6 × 3 × 4 = 3 × 4 × 6 = 4 × 6 × 3 ..................................................

2.2 (2 × 4) × 5 = (5 × 4) × 2 = 4 × (2 × 5) ..................................................

3. What do you realise from the above examples?

Activity 8:

To estimate and calculate by means of rounding off [LO 1.8.1]

1. To be able to estimate answers is an important skill. If you are good at estimating, it will be easier for you to realise that you may have made a mistake in your calculations. Round off the following numbers first and in this way you will be able to estimate what the product of each of the following will be:

Table 7
SUM ESTIMATE CALCULATOR DIFFERENCE
E.g. 19 × 21 400 399 1
a) 38 × 27 ......................... ......................... .........................
b) 99 × 146 ......................... ......................... .........................
c) 45 × 69 ......................... ......................... .........................
d) 998 × 78 ......................... ......................... .........................
e) 409 × 18 ......................... ......................... .........................

BRAIN-TEASERS!

a)Which two numbers, smaller than 10, have 3 factors each?

......................... and .........................

b) Colour in the correct block:

Table 8
The product of 2 uneven numbers is always an EVEN UNEVEN number.

c) Which two numbers have a product of 48 and a sum of 16?

......................... and .........................

Activity 9 :

  • To be able to solve problems in context [LO 1.6.1]
  • To be able to use a series of techniques to do calculations [LO 1.10]

Let’s do some brainstorming now. Your teacher will give you the necessary paper.

1. Split up into groups of three and, as quickly as possible, make a list of all the means of transportation that exist in our country.

2. Which of the above-mentioned would you choose if you had to travel? Why?

3. Where is your favourite seat in:

  1. a) a car
  2. b) a school bus
  3. c) a boat
  4. d) an aeroplane

4. Now see whether you are able to solve the following problems:

Messrs Slobo, Mugathle and Sisula have each bought a brand new taxi with which to transport passengers. For their first trip their plans are the following:

Table 9
Owner Amount per person Number of passengers km travelled
Slobo R15,80 13 76
Mugathle R14,60 14 84
Sisula R16,25 12 59
  • Who made the most money with the first trip?
  • If the petrol cost R1,05 for every kilometre that was travelled, how much did Mr Mugathle’s petrol account come to?
  • If all the passengers in Mr Slobo’s taxi were athletes and they had to run back, how far did they run altogether?
  • Explain to the rest of the class how your group calculated the answers.
  • Compare your methods. How do your methods differ?
  • Check the answers, using a pocket calculator.

Activity 10:

  • To determine the equivalence and validity of different representations [LO 2.6.3]
  • To use strategies to check solutions [LO 1.11]

1. In Activity 1.9 you used your own techniques and strategies to solve the problems. In your feedback to the class you probably noticed that there are many ways in which numbers can be multiplied. Split up into groups of three. Read the following problem and then see whether you can understand all the different solutions.

Chairs must be arranged in rows in a hall. There must be 23 rows with 16 chairs in each row. How many chairs are needed?

1.1 I calculate 23 × 16 like this:

23 = 10 + 10 + 10 + 1 + 1 + 1

Table 10
23 × 16
10 160
10 160
2 32
1 16
23 368

1.2 I calculate 23 × 16 in this way:

23 = 20 + 3

16 = 10 + 6

Thus: (20 + 3) × (10 + 6)

20 × 10 = 200

20 × 6 = 120

3 × 10 = 30

3 × 6 = 18

368

1.3 I divide 16 into its factors:

23 × 16 = 23 × 2 × 8

= 23 × 2 × 2 × 4

= 46 × 2 × 4

= 92 × 4

= 368

1.4 I calculate it like this:

23 × 16 = (23 × 20) – (23 × 4) (I took 4 × 23 too much, so I had to subtract it)

= (23 × 10 × 2) – (23 × 4)

= 460 – 92

= 368

1.5 My method is shorter!

23

× 16

18 (6 × 3)

120 (6 × 20)

30 (10 × 3)

200 (10 × 20)

368

1.6 My method is shorter still!

12 3

× 1 6

1 3 8

2 3 0

3 6 8

1.7 Which method do YOU choose?

Why?

Activity 11:

To use a series of techniques to do calculations [LO 1.10.5]

1. It is better to use a technique that you understand perfectly when you have to do a calculation. Use any method and calculate the following:

Table 11
1.1 58 × 29 1.2 76 × 54

2. In the following activity we are going to see how certain factors form patterns when we multiply them by each other. We are also going to use the pocket calculator to help us with our calculations.

2.1 Using the pocket calculator, complete the following:

77 × 13 = ....................

77 × 26 = ....................

77 × 39 = ....................

77 × ............. = 4 004

77 × ............. = 5 005

77 × 78 = ....................

77 × 91 = ....................

2.2 Predict the answer without using your calculator: 77 × ............. = 8 008

77 × 117 = .............

2.3 What is the pattern in the above example?

2.4 Calculate the following, first without using the pocket calculator, and then by using it.

1 × 9 + 2 = .........................

12 × 9 + 3 = .........................

123 × 9 + 4 = .........................

................. × 9 + 5 = 11 111

12 345 × 9 + ........ = 111 111

................. × 9 + ........ = .........................

................. × 9 + ........ = .........................

2.5 Predict now without your calculator:

12 345 678 × 9 + 9 = .

123 456 789 × 9 + 10 =

2.6 Can you explain to a friend the pattern in the above example?

Activity 12:

To be able to do mental arithmetic [LO 1.9.2]

Let us see whether you can improve on your previous mental arithmetic test. This exercise will serve as preparation for the activities to follow. Fill in the missing answers of the MARKED BOXES ONLY.

Table 12
X 2 6 7 8 9 10 12 100
5 ......... ........     .......      
6       ......     .......  
7   ........     .......   .......  
8     ......       ........  
9       ......     ........  
36 .......         ........   ......
48 .......         ........   .......
124 .......         ........   .......

Activity 13:

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

To be able to use a series of techniques to do calculations [LO 1.10.5]

By now you know how to multiply by smaller numbers. Now we are going to see whether you are also able to do calculations with bigger numbers. Split up into groups of 3 and ask your teacher to give you the paper you need. Then read the instructions very carefully.

Michaela and her friends visited the zoo and heard the following interesting facts: a hippopotamus eats 45 kg of fodder per day, while a large bull elephant needs 225 kg of fodder per day!

1. How many kilograms of fodder will 329 hippopotamuses eat per day?

2. How many kilograms of fodder must the zoo buy per day to feed 76 bull elephants?

3. If the entrance fee was R15 per person and 475 people visited the zoo that day, how much money was taken at the entrance gate?

4. Explain to the rest of the class how your group calculated the answers.

5. Compare your methods. How do the various methods differ from each other?

6. Use a pocket calculator to check the answers.

Activity 14:

To determine the equivalence and the validity of different representations [LO 2.6.1]

To use strategies to check solutions [LO 1.11]

1. In the previous activity you probably noticed that there is more than one method to do multiplication. Split up into groups of 3 again. Look carefully at the solutions to the following problem and explain the methods to one another.

Mrs Cele sells clothes. There are 46 different pairs of jeans and 238 different shirts. How many combinations can she make up?

1.1 I calculate 46 × 238 in this way

238 × 46 = (200 + 30 + 8) × (40 + 6)

200 × 40 = 8 000

30 × 40 = 1 200

8 × 40 = 320

200 × 6 = 1 200

30 × 6 = 180

8 × 6 = 48

10 948

1.2 I write it in this way:

238

× 46

48 (8 × 6)

180 (30 × 6)

1 200 (200 × 6)

320 (40 × 8)

1 200 (40 × 30)

8 000 (40 × 200)

10 948

1.3 My method looks like this:

22 43 8

× 4 6

1 4 2 8

9 5 2 0

1 0 9 4 8

2. Now use any method that you understand and with which you feel comfortable, and calculate the product of:

2.1 576 × 47

2.1 576 × 47

CHALLENGE!

Can you calculate the product of the following by using the method shown in 1.3?

1. 347 × 251

2. 526 × 438

ANOTHER BRAIN-TEASER

Study the following sum carefully: 24 × 13 = 377

One of the digits is incorrect. Can you find out which one it is? ..............

What must it be? .........................

Activity 15:

To solve problems in context [LO 1.6.1]

Let’s go out to dinner.

This assignment is for your portfolio. Your teacher will go through it with you and explain exactly what you are expected to do. Make sure that you understand perfectly before you start. Remember to work neatly.

Read the following menu carefully:

Figure 2
Figure 2 (graphics2.png)

Form groups of three. Your group decides to eat something from each section of the menu. To make it easier for the waiter, each person in your group chooses the same soup, starter, main course and dessert.

Decide what you are going to order and then calculate your account. Use the block below. You may use a calculator. First estimate what the meal will cost the group. ..............................

Please work in the given blocks in order to answer the following questions:

If each of you has R85, will your group have enough money to pay the bill? .........…

If this is so, how much change will your group receive?

How much will you tip the waiter? ......................... Why?

Let us do a bit of research.

At which restaurant do you like eating the most?

Why?

According to you, what is a ‘MUST’ for a restaurant to be successful?

Can you think of any ‘MUST NOT’S’?

Visit any two restaurants of your choice and compare their menus and prices. Write down what you will report back to the class.

Assessment

Table 13
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 2

1. (a) 24 x 1 (b) 36 x 1

  1. x 2 18 x 2

6 x 4 9 x 4

3 x 8 6 x 6

  1. x 3

2.

Table 14
7 x 6 9 x 5 7 x 9 9 x 6 8 x 8; 32 x 2
9 x 8 12 x 9 6 x 8 8 x 11 12 x 8; 32 x 3

ACTIVITY 3

Table 15
               
  20 25 30 35 40 45 60
  24 30 36 42 48 54 72
  28 35 42 49 56 63 84
  32 40 48 56 64 72 96
  36 45 54 63 72 81 108
  48 60 72 84 96 108 144

BRAIN TEASERS

9. 1 x 241 : 241 (1 x 241)

Double 241 : 482 (2 x 241)

Double 482 : 964 (4 x 241

Double 964 : 1 928 (8 x 241)

615

ACTIVITY 5

1.1 (1 000 + 1 000) + (400 + 400) + (30 + 30)

2 000 + 800 + 60

= 2 860

1.2 (2 000 + 2 000) + (300 + 300) + (10 + 10) + (5 + 5)

4 000 + 600 + 20 + 10

= 4 630

2.1 (14 000 + 14 000) + (800 + 800) + (20 + 20)

28 000 + 1 600 + 40

= 29 640

2.2 (36 000 + 36 000) + (900 + 900) + (47 + 47)

72 000 + 1 800 + 94

= 73 894

3.1 = 100 x 12 3.2 = 10 x 173

= 1 200 = 1 730

BRAIN TEASERS

24 20

  1. 16

96 80

3 x 192 4 x 160

Halve multiplicand

Double multiplier

ACTIVITY 6

1.1 60; 600; 6 000; 60 000

1.2 90; 900; 9 000; 90 000

1.3 150; 1 500; 15 000

1.4 260; 2 600; 26 000

Add a naught only to multiplier.

2.1 800; 8 000; 80 000

2.2 1 300; 13 000; 130 000

2.3 2 700; 27 000; 270 000

Add two noughts to multiplier.

ACTIVITY 7

1. 8 x 6 6 x 8 23 x 13 13 x 23 124 x 85 85 x 124

2. (a) true

  1. (a) true

3. It does not matter in which order you multiply.

ACTIVITY 8

1. (a) 1 026

  1. (a) 14 454
  2. (b) 3 105
  3. (c) 77 844
  4. (d) 7 362

BRAIN TEASERS

(a) 4 9

  1. (a) uneven
  2. (b) 12 4

own answer

own answer

ACTIVITY 11

1.1 1 628

1.2 4 104

2.1 1 001

2 002

3 003

52

65

6 006

7 007

2.2 104

9 009

2.3 Multiplier becomes 13 times more each time.

2.4 11

111

1 111

1 234

6

123 456 7 1 111 111

1 234 567 8 11 111 111

2.5 111 111 111

1 111 111 111

ACTIVITY 12

Table 16
x 2 6 7 8 9 10 12 100
5   30     45      
6       48     72  
7   42     63   84  
8     56       96  
9       72     108  
36 72         360   3 600
48 96         480   4 800
124 248         1 240   12 400

ACTIVITY 14

2.1 27 072

2.2 26 358

1. 347

x 251

___

347

17 350

69 400

______

87 097

2. 526

x 438

_____

4 208

15 780

210 400

_______

230 388

ANOTHER BRAIN-TEASER

4

9

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