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Level Four > Number and Algebra

Doubling and halving

Achievement Objectives:

Achievement Objective: NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
AO elaboration and other teaching resources
Achievement Objective: NA4-8: Generalise properties of multiplication and division with whole numbers.
AO elaboration and other teaching resources

Purpose: 

These activities are designed to teach doubling and halving strategies.

Specific Learning Outcomes: 
solve problems using doubling and halving
identify problems that could be solved using doubling and halving
solve problems using trebling and thirding
generalise the doubling and halving strategy using algebraic notation
list factors of numbers
 
Description of mathematics: 

Multiplication and division, AM (Stage 7) 

Required Resource Materials: 
Practice exercises with answers (PDF or Word)
Activity: 

Prior knowledge

Recall multiplication facts
Double numbers to 100
Halve numbers to 100

Background

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.

Algebraic notation
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

Exercise 1
Asks students to solve problems using doubling and halving. 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.
 
Exercise 2
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…
 
Exercise 3
Asks students to solve problems using doubling and halving, and trebling and thirding strategy.   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.
 
Exercise 4
Asks students to solve problems using doubling and halving. 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.
 
Exercise 5
Asks students to solve problems involving decimals uisng proportional adjustment.
 
Exercise 6
Asks students to solve word problems.  These are all straightforward and can use the doubling and halving strategy.
 
Exercises 7 and 8
Asks students to generalise the doubling and halving strategy by introducing algebraic notation.
 
Exercise 9
Asks students to list factors of numbers.
AttachmentSize
DoublingAndHalvingSheet.pdf132.76 KB
DoublingAndHalvingSheet.doc53.5 KB