Why it happens: You made one number twice as big and the other half as big. Twice as many rows, but each row only half as long. The extra and the missing cancel each other out exactly.
Q2.
(b) 22 × 5: fill in ___ × ___ = ____
Answer
11 × 10 = 110.
Look at the tall block. It is 22 packets tall and 5 packets wide.
Cut it across the middle. Each half is 11 tall and 5 wide.
Put the two halves side by side. Now the block is 11 tall and 10 wide.
Check it yourself: Count the other way. 22 × 5 = 22 × 5. Twenty 5s is 100 and two more 5s is 10, so 100 + 10 = 110 ✓
Q3.
(c) Solve the following problems like the previous ones (fill in the Half and Double boxes): 14 × 3 = , 38 × 5 = , 16 × 4 = , 35 × 14 =
Answer
Halve the first number, double the second, then multiply.
Problem
Half
Double
New problem
Answer
14 × 3
7
6
7 × 6
42
38 × 5
19
10
19 × 10
190
16 × 4
8
8
8 × 8
64
35 × 14
7 (half of 14)
70 (double of 35)
70 × 7
490
35 × 14 Step 1. This time halve the second number: half of 14 = 7 Step 2. Double the first: 35 + 35 = 70 Step 3. 70 × 7 = 490
Choose which number to halve: Halve the even number. In 35 × 14, only 14 is even, so 14 must be the one you halve. In 16 × 4 both are even, so you may choose either.
This halving and doubling strategy works well when we have to multiply with numbers like 5 and 25. Discuss why?
Answer
Because doubling 5 gives 10, and doubling 25 twice gives 100 — and multiplying by 10 or 100 is the easiest job in arithmetic.
Start with 5. Double it once and you get 10. Multiplying by 10 just puts a zero on the end.
Start with 25. Double it once and you get 50. Double it again and you get 100. Multiplying by 100 puts two zeros on the end.
Each time you double one number, halve the other. The answer never changes, but the sum becomes far easier.
Example with 5 18 × 5 → 9 × 10 = 90
Example with 25 16 × 25 → 8 × 50 = 4 × 100 = 400
Why 10 and 100 are so friendly: Our numbers are built in tens. When you multiply by 10 every digit simply shifts one house to the left. There is nothing to carry and nothing to remember.
Sample answer for the class discussion: We said 5 is half of 10 and 25 is a quarter of 100. So whenever 5 or 25 turns up, we can climb up to 10 or 100 by doubling, as long as we halve the other number the same number of times.
Q5.
(d) Find the product by halving and doubling either the multiplier or the multiplicand. 1) 5 × 18 2) 50 × 28 3) 15 × 22 4) 25 × 12 5) 12 × 45 6) 16 × 45
Check one by a second route: 16 × 45 = 16 × 45. Ten 45s is 450 and six 45s is 270. 450 + 270 = 720 ✓
Q6.
(e) Give 5 examples of multiplication problems where halving and doubling will help in finding the product easily. Find the products as well.
Answer
Sample answer: Pick problems where one number is even and the other is 5, 25 or 50 — those are the ones that climb to 10 or 100.
My example
Halve and double
Becomes
Product
5 × 24
24 → 12, 5 → 10
10 × 12
120
25 × 16
16 → 8, 25 → 50; again 8 → 4, 50 → 100
100 × 4
400
15 × 14
14 → 7, 15 → 30
30 × 7
210
5 × 36
36 → 18, 5 → 10
10 × 18
180
50 × 18
18 → 9, 50 → 100
100 × 9
900
What makes a good example: One number must be even, or you cannot halve it neatly. The other should be 5, 15, 25 or 50, because doubling those gives 10, 30, 50 or 100.
Try this: A bad example is 7 × 9, because neither number is even and neither is close to 10 or 100. Halving and doubling would only make it messier. Choosing the right tool for the numbers is part of the skill.