Multiplication

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

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!

¹ 2 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:

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.