Solve multiplication and division problems by using proportional adjustment.
Number Framework Stage 7
Using Materials
Tell the students this story:
“Prince Pikelet, heir to the throne, has been taking creative cooking lessons. He has
made a huge batch of spinach-fl avoured biscuits that he wants members of the court
to try. So he arranges a morning tea. The servants need to make sure that each guest gets the same share of the biscuits. At the first table there are 6 seats, so the Prince puts a plate of 12 biscuits on it. There are three other tables. The table arrangement look like this:
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How many seats should be at the table with 24 biscuits?”
“How many biscuits should the servants put at each table so that every guest will
get an equal share of the biscuits?”
Let the students solve the problems in pairs using materials.
Ask them how their answers could be recorded using equations:
12 ÷ 6 = 2 or 6 x 2 = 12, 4 ÷ 2 = 2 or 2 x 2 = 4, 6 ÷ 3 = 2 or 3 x 2 = 6, 24 ÷ 12 = 2 or 12 x 2 = 24.
Ask them if they notice any patterns in the numbers and to explain how the patterns
work. Look for responses such as, “To get the same share means a table with half as
many seats gets half as many biscuits.”
Provide similar examples using the same biscuit-sharing scenario. Use materials
where required to check the predictions made by the students. Suitable examples
might be:
“Sixteen biscuits are at a four-seat table. How many should go to a two-seat table?
How many to a six-seat table?”
Get students to model the operations using cubes, counters, or similar materials and
record their findings as equations such as 16 ÷ 4 = 4 or 4 x 4 = 16, 8 ÷ 2 = 4 or 2 x 4 = 8, 24 ÷ 6 = 4 or 6 x 4 = 24.
Discuss patterns that the students see in the numbers and why they occur, such as,
“If you halve both numbers, the answer is still the same.”
Arranging the materials in arrays may help some students recognise the common
factor property that is involved. For example:
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“Fifteen cookies are at a fi ve-seat table. How many should go to a one-seat table?
How many should go to a 10-seat table? How many to a two-seat table?”
Using Imaging
Students should move very quickly through this stage to using number properties.
Provide the biscuit-sharing problems without materials in a diagrammatic form and
look for students to image the sharing process.
“Thirty-six biscuits to 12 people ... How many to 6 people?” “How many to four
people?”
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Other examples might be:
30 biscuits at a 10-seat table.
How many to a ... 5-seat table? ... 2-seat table? ... 20-seat table?
18 biscuits at a 9-seat table.
How many to a ... 3-seat table? ... 6-seat table? ... 18-seat table?
Using Number Properties
Increase the size of numbers involved in the problems so that the students need to
use number properties rather than imaging. First examples should involve taking a
diffi cult division and gradually working through progressively simpler problems
that have the same answer.
For example:
208 ÷ 8 = ? as 104 ÷ 4 = ? as 52 ÷ 2 = ?
216 ÷ 12 = ? as 108 ÷ 6 = ? as 54 ÷ 3 = ?
Other problems might be:
484 ÷ 22 = ? as 242 ÷ 11 = ? (? = 22)
140 ÷ 28 = ? as 70 ÷ 14 = ? as 35 ÷ 7 = ?
368 ÷ 16 = ? as 184 ÷ 8 = ? as 92 ÷ 4 = ? as 46 ÷ 2 = ?
390 ÷ 15 = ? as 130 ÷ 5 = ? as 260 ÷ 10 = ?
408 ÷ 24 = ? as 204 ÷ 12 = ? as 102 ÷ 6 = ? as 51 ÷ 3 = ?
324 ÷ 27 = ? as 108 ÷ 9 = ? as 36 ÷ 3 = ?
Independent Activity
To reinforce using a variety of mental division strategies, students should play
Divided Loyalties (Material Master 6-8).