Solve multiplication and division problems by splitting factors.
Number Framework Stage 7
Give out sets of animal strips to the students. Tell them to take a collection of the same strips, e.g., take all the four strips. Follow this sequence of instructions for laying down cards and recording:
Lay down one card ... double the number and record 2 x ?... double this amount and record 2 x 2 x ? ... double the amount again.
Ask the students how they should record what they have. They may give you responses about their personal collections, for example, “I’ve got eight sixes”, that can in turn be generalised as 2 x 2 x 2 x ? is the same as 8 x ?.
Ask them how many times their number they will have if the eight sets are doubled (sixteen sets). Record this as 2 x 2 x 2 x 2 x ? is the same as 16 x ?. Focus what connection there is between the 2 x 2x 2 x 2 and the 16 (2 x 2 x 2 x 2 is a factorisation of 16). Tell the students to use the same collection of strips to work out what happens when a number is multiplied by two, then multiplied by three. Record their ideas as expressions, for example, 3 x (2 x ?)
Provide other examples for students to explore such as:
2 x (5 x ?) (is the same as 10x ?)
5 x (3 x ?) (is the same as 15 x ?)
Explore whether the same idea can be applied to division. Begin with a set-up array composed of animal strips, for example 8 sets of the same strip. Ask students to record a division equation for their array, for example 32 ÷ 8 = 4. Discuss what feature of the array gives them the answer in this case (the number on each strip).
Ask them to halve the number of strips and record the division equation, for example, 32 ÷ 8 = 16 ÷ 4. Ask them to halve their array again and record the equation. This can be generalised as 32 ÷ 2 ÷ 2 ÷ 2 is the same as 32 ÷ 8. Reinforce the significance of 2 x 2 x 2 being a factorisation of 8.
Provide similar division examples, such as 24 ÷ 2 ÷ 3 is the same as 24 ÷ 6, and40 ÷ 2 ÷ 5 is the same as 40 ÷ 10.
Using Imaging
Ask the students to image repeated multiplication and division examples and to represent them by recording box diagrams. Examples might be:
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or 2 x 3 x 3x ? is the same as 18 x ?
and 3 x 3 x 3 is the same as 27 x ?
Provide division examples like the steps below"
or ? ÷ 3 ÷ 5 is the same as ? ÷ 15 and ? ÷ 2 ÷ 2 ÷ 3 ÷ 3 is the same as ? ÷ 36.
Using Number Properties
Provide examples for students to solve using repeated division by factors. Good examples might be:
8 x 43 as 43 x 2 = 86, 86 x 2 = 172, 172 x 2 = 344
12 x 56 as 56 x 2 = 112, 112 x 2 = 224, 224 x 3 = 672
15 x 68 as 68 x 5 = 340, 340 x 3 = 1 020
18 x 26 as 26 x 2 = 52, 52 x 3 = 156, 156 x 3 = 468
36 x 77 as 77 x 2 = 154, 154 x 2 = 308, 308 x 3 = 924, 924 x 3 = 2 772
216 ÷ 12 as 216 ÷ 2 = 108, 108 ÷ 2 = 54, 54 ÷ 3 = 18
688 ÷ 16 as 688 ÷ 2 = 344, 344 ÷ 2 = 172, 172 ÷ 2 = 86, 86 ÷ 2 = 43
612 ÷ 18 as 612 ÷ 2 = 306, 306 ÷ 3 = 102, 102 ÷ 3 = 34
1 536 ÷ 24 as 1 536 ÷ 3 = 512, 512 ÷ 2 = 256, 256 ÷ 2 = 128, 128 ÷ 2 = 64
Practice
Students will enjoy playing the game of Factor Leapfrog (Material Master 6–10) to consolidate these strategies for multiplication.