Dividing by four is halving twice, because four is two twos
Halve, then halve again
Dividing by 4 shares a number into 4 equal parts. Because 4 = 2 × 2, you can do it by halving twice: cut the number in half, then cut each half in half. That makes 4 equal parts.
For 84 ÷ 4: half of 84 is 42, and half of 42 is 21. So 84 ÷ 4 = 21. Written as one number sentence, 84 ÷ 4 = (84 ÷ 2) ÷ 2 = 42 ÷ 2 = 21.
The same bar of 84, cut in half, then with each half cut in half again. Each of the 4 equal parts is 21.
Each hop halves: 84 to 42, then 42 to 21. Two halvings make ÷ 4.
Halving a two-digit number
To halve a number, halve the tens and halve the ones, then add. Half of 84 is half of 80 plus half of 4: 40 + 2 = 42.
When the tens digit is odd, the half of the tens has a 5 in it. Half of 90 is 45, because half of 80 is 40 and half of 10 is 5. So half of 92 is 45 + 1 = 46.
Eight is three twos
8 = 2 × 2 × 2, so dividing by 8 is halving three times. For 96 ÷ 8: 96 halves to 48, 48 halves to 24, and 24 halves to 12. So 96 ÷ 8 = 12.
Count the halvings. Stopping one halving early gives twice the right answer, and one halving too many gives half of it. To check, double back the same number of times: 12, 24, 48, 96.
Three hops, each one ÷ 2: 96, 48, 24, 12. So 96 ÷ 8 = 12.
Halve 96 until it is 96 ÷ 8
Each halving cuts every part in two: 1 part of 96, then 2 of 48, then 4 of 24, then 8 of 12. Three halvings make ÷ 8.
Multiplying and dividing in your head · A Ribbon Halved Twice into 4 Equal Pieces
QuestionMrs Lim has a ribbon that is 92 cm long. She cuts it in half. Then she cuts each half in half again, so she has 4 equal pieces. (a) How long is each half? (b) How long is each of the 4 pieces?
1.Dividing by 4 is halving twice, because 2 × 2 = 4.
Dividing by 4 is halving twice. 2.(a) Halve 92. Half of 90 is 45 and half of 2 is 1, so each half is 45 + 1 = 46 cm long.
(a) Half of 92 is 45 + 1 = 46 cm. 3.Halve 46. Half of 40 is 20 and half of 6 is 3.
Halve 46: half of 40 is 20 and half of 6 is 3. 4.(b) 20 + 3 = 23, so each of the 4 pieces is 23 cm long. Check: double 23 is 46 and double 46 is 92.
(b) Each of the 4 pieces is 20 + 3 = 23 cm long.
Answer: (a) 46 cm; (b) 23 cm
Watch out for
- Halving only once and giving 46 cm for each of the 4 pieces. One halving makes 2 pieces. A second halving is needed to make 4.
- Writing half of 90 as 40. Half of 80 is 40 and half of 10 is 5, so half of 90 is 45.
Multiplying and dividing in your head · Marbles of 25 g Grouped in Fours
QuestionEach marble has a mass of 25 g. A bag holds 32 marbles. (a) How many groups of 4 marbles can be made from the bag? (b) What is the mass of all the marbles in the bag?
1.Four 25s make 100: 4 × 25 = 100. So a group of 4 marbles has a mass of 100 g.
Four 25s make 100, so 4 marbles have a mass of 100 g. 2.Find the number of groups of 4 in 32 by halving twice. Half of 32 is 16, and half of 16 is 8.
Put the 32 marbles in groups of 4. Half of 32 is 16, and half of 16 is 8. 3.(a) 32 ÷ 4 = 8, so 8 groups of 4 marbles can be made.
(a) 32 ÷ 4 = 8 groups of 4 marbles. 4.Each group has a mass of 100 g, so the total mass is 8 × 100 g.
Each group of 4 marbles has a mass of 100 g. 5.(b) 8 × 100 = 800, so the marbles have a mass of 800 g.
(b) 8 × 100 = 800 g.
Answer: (a) 8 groups; (b) 800 g
Watch out for
- Multiplying 32 by 100 and writing 3200 g. One marble is 25 g. It takes 4 marbles to make 100 g, so the number of marbles is divided by 4 first.
- Halving 32 only once and using 16 groups. Halving once divides by 2. The marbles are in groups of 4, so halve twice.
Your turn
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