Doubling and halving
Number Framework domain and stage:
Multiplication and division. Advanced additive to advanced multiplicative
Number, levels 3 to 4
Numeracy Project book reference
These activities can be used to follow the teaching episodes based on cut and paste, Book 6, pages 25-26 and are for those students who are able to use the associated number properties.
Prior knowledge. Students should be able to:
- Recall all their multiplication facts to 100 (at least)
- Double numbers to 100
- Halve numbers to 100
During these activities, students will meet:
- Doubling and halving, tripling and thirding etc (ex 3)
- Box equations, including the use of different shapes to represent different numbers (ex 4)
- Doubling and halving with decimals (ex 5)
- Word problems that may lead to doubling and halving (ex 6)
- Simple algebraic notation for doubling and halving (ex 7)
- Simple proof (ex 8)
- Finding a set of factors for a number using doubling and halving, tripling and thirding etc.
In the first exercise, the size of the numbers being used progressively increases. As students should already be at stage 6 before they attempt such an exercise, this should not cause problems for students, though work with large numbers should be forming part of the teaching.
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…
The last 2 problems require some thinking. The answers to question 10 suggests that students check their answers with others in their group, and explain why they think certain problems are easier or not easier using tripling and thirding etc. Question 11 requires students use a mindmap to show their answer. They may need to be taught what a mind map is and what it looks like.
In exercise 4, the doubling and halving (etc) met in earlier exercises is now met as box equations, that is written equations with an unknown. The first few problems have the box on the right hand side of the equation, (and all involve halving to find the number in the box). Teaching should also cover boxes where the missing number is double one of the other numbers, and tripling and thirding too. Problems 4 – 6 have the box on the left hand side of the equation, and involve students working in reverse. This could prove tricky for some students.
The final question have 2 unknowns – and the instruction identifies that they involve doubling and halving. (Students who do not read this instruction could find they cannot do them, unless they assume that they have been doubled and halved.
Overall this exercise has inadequate practice for the number of different skills students need to practice. It is a good revision exercise, but will not really help students convert their strategies to knowledge, for this work is different conceptually to that above. Additional work needs to be set. For example. Students can be challenged to make up 10 (or 15 ) of their own problems that look like this. By putting their problems on one side of a piece of paper, and the answers on another, they can swap papers and check their problems and answers are right before handing them in form marking.
These are all straightforward and can use the doubling and halving strategy
Exercise 7 gives teachers the opportunity to visit 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.
Exercise 8 informally introduces students 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
Practice exercises with answers:
Doubling and halving bingo (23KB)