Solve multiplication and division problems by using proportional adjustment.
Number Framework Stage 7
Containers
Post-its (optional).
Using Materials
Set up a container with twenty-four unifix cubes joined in packets of four. Tell the students that there are twenty-four cubes in total in the container. Take out a packet of four cubes. Ask how many packets of four will be in the container. Discuss the mental strategies used by students. At this stage students should be using reversibility, that is, “Six fours are twenty-four, so there are six packets.”
Ask, How could this problem be written as an equation? (24 ÷ 4 = 6) Ask, How could this answer be used to work out how many packets of eight could be made using all the cubes in the container?
Let the students discuss their strategies. Rather than solving 24 ÷ 8 = 3 as a separate problem look for students to use proportional adjustment:
Six fours are in twenty-four so are three eights.
Ask, How can you use this idea to work out how many packets of two could be made with the cubes?
Students will know the solution using doubling of twelve but look for them to justify it using proportional adjustment:
Six fours are in twenty-four so are twelve twos.
Provide other examples in which a division answer is found and proportionality could be used to work out other results. The power of this strategy will become more apparent to students as the number size increases. Examples might be:
32 ÷ 4 = ? so what are 32 ÷ 2 = ? and 32 ÷ 8 = ?
40 ÷ 5 = ? so what are 40 ÷ 10 = ? and 40 ÷ 20= ?
18 ÷ 3 = ? so what are 18 ÷ 6 = ? and 18 ÷ 9 = ?
Using Imaging
Role playing: Suppose that Emily (student’s name) is working at the biscuit factory. She has a jar of 36 biscuits. How many packets of two can she make?
Ask the students to image what the packets look like laid flat. There will be 18 packets of two. Draw a diagram to assist the imaging.
Emily thinks that packets of two are small. How many packets of four can she make with the jar of biscuits?
Look for students to realise that two packets of two make a packet of four so the number of packets will be halved.
Provide further examples in the biscuit story context, using diagrams to assist the imaging process.
120 ÷ 10 = ?so what are 120 ÷ 5 = ? and 120 ÷ 20 = ?
48 ÷ 6 = ? so what are 48 ÷ 12 = ? and 48 ÷ 3 = ?
72 ÷ 2 = ? so what are 72 ÷ 4 = ? and 72 ÷ 8 = ?
Using Number Properties
Give the students division problems to solve, in biscuit packet context, where use of proportion is a sensible strategy. Encourage them to decide which factor they willdivide by to make the problem easier. Examples might be:
170 ÷ 5 = ? as 170 ÷ 10 = 17 so 170 ÷ 5 = 34
56 ÷ 4 = ? as 56 ÷ 2 = 28 so 56 ÷ 4 = 14
96 ÷ 6 = ? as 96 ÷ 3 = 32 so 32 ÷ 2 = 16 (or vice versa)
81 ÷ 3 = ? as 81 ÷ Ù± = 9 so 72 ÷ 3 = 27
64 ÷ 4 = ? as 64 ÷ 8 = 8 so 64 ÷ 4 = 16 or 64 ÷ 2 = 32 so 64 ÷ 4 = 16.