These activities are designed to teach doubling and halving strategies.
Multiplication and division, AM (Stage 7)
This activty involves using proportional adjustment to solve multiplication problems. Doubling and halving, and trebling and thirding can be used to make multiplication problems easier to solve. For example, 3 x 16 is the same as 6 x 8. The activity asks students to solve problems, fill in missing numbers in equations using proportional adjustment and solve word problems.
The essential notation of 2 x (and 2) for doubling and /2 or ÷ 2 or ½ as notation for halving. This is then repeated with letters. This not only bridges the gap between “fill in the box” type problems and the x as an unknown number but also introduces students to such notational forms before they are expected to use them. This is essential introductory algebra to build understanding of the language of mathematics. Good discussion is warranted as a follow-up.
Students are also introduced to the concept of proof. It is likely that when students are asked to “explain why doubling and halving always gives the same answer as the original problem” many are likely to write a story. However, the concept of proof and the power of algebra can be followed up in discussions. A teacher led explanation of “what is going on when we play with the symbols using the rules of mathematics we know” should help decode the answer sheet for the problem. An explanation along the lines of “as we don’t know what numbers we actually started with – and just ended up with the same numbers, what we have shown must work for every pair of numbers we can think of…regardless of whether or not the process is actually useful!” should help explain what manipulating the symbols has shown (or proved).
Doubling and halving to find factors
Doubling and halving (tripling and thirding etc) is a very useful strategy for finding a full set of factors. However, it does require some idea of prime numbers and how these operate. Start with 1 x n, and double and halve from there. For example
1 x 60
2 x 30
4 x 15 ← look for what goes into 15
20 x 3 ← 3 is a prime so this thread stops, work on the 20
10 x 6
5 x 12 ← other side is now a prime – so stop
Comments on the Exercises
Asks students to identify what problems are suitable to solve using the doubling and halving strategy. For question 10 students should identify that doubling and halving is a useful strategy when both numbers are even – to start with, but this question deserves more discussion, as simply doubling and halving may not necessarily produce a question that is easier to do…