Using doubling and halving

Using doubling and halving

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about using doubling and halving to help solve multiplication problems.
         
To solve the problems below James needs to use doubling and halving to make the numbers easier to work with. He only knows his 2 × , 3 × , and 10 × tables.  He does not know his other tables.

 

a)

 

 Show how James could use doubling and halving to solve 6 × 4.

 
 
 
 
 
 
Answer: _____ 
 
b)

 

  
Show how James could use doubling and halving to solve 5 × 8.

 
 
 
 
 
 
Answer: _____ 
 
c)

 

  
Show how James could use doubling and halving to solve 8 × 15.

 
 
 
 
 
 
Answer: _____ 
Task administration: 
This task is completed with pencil and paper only.
Levels:
2, 3
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
doubling, halving, multiplication
Description of task: 
Students use doubling and halving to solve multiplication problems.
Curriculum Links: 
This resource can help to identify students' ability to use basic facts and knowledge of place value and partitioning whole numbers to solve multiplication problems.
  • Skip counting are is Stage 4: Advanced counting (in the multiplication and division domain). 
  • Repeated addition and the additive doubling are Stage 5 (Early additive part-whole) strategies. 
  • Doubling and halving is Stage 6: Early multiplicative part-whole. 
  • More sophisticated part-whole multiplication strategies are Stage 7: Advanced multiplicative part-whole or above.
Key competencies
This resource involves recording the strategies they used to solve multiplication problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 3-4
Multiplicative thinking, sets 4-5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y4 (11/2006)
a) 24
Working using doubling and halving (answer need not be correct):

  • halving 6 and doubling 4, 3 × 8  or doubling 6 and halving 4, 12 × 2 [doubling one number & halving the other];
  • 6 × 2 × 2 = 12 × 2 [breaking down 4 to 2 × 2];
  • 6 × 2 = 12 [halving one number], 12 × 2 = 24 [doubling the solution];
  • using an additive doubling, e.g., 6+6 = 12, and 12+12 = 24; or
  • any other strategy that involves doubling and halving.
 moderate
difficult
b) 40
Working using doubling and halving (answer need not be correct):

  • double 5 and halve 8, 10 × 4 [doubling one number & halving the other];
  • 5 × 4 = 20 [halving one number], 20 × 2 = 40 [doubling the solution];
  • 5 × 8 = 10 × 4 = 20 × 2 = 40 [multiple doubling and halving]; or
  • any other strategy that involves doubling and halving.
 easy
difficult
c) 120
Working using doubling and halving (answer need not be correct):

  • halve 8, and double 15, 4 × 30 [doubling one number & halving the other];
  • 4 × 15 = 60 [halving one number], 60 × 2 = 120 [doubling the solution];
  • 8 × 15 = 4 × 30 = 2 × 60 = 120 [multiple doubling and halving]; or
  • any other strategy that involves doubling and halving.
 difficult
very difficult

Based on a representative sample of 183 Y4 students.

Teaching and learning: 

This resource is about being able to use doubling and halving as a strategy to solve multiplication problems, and the numbers were chosen to support the use of doubling and halving.  Doubling and halving is a Stage 6 strategy of the Number Framework. Being able to use a variety of strategies is what indicates a students understanding to move from early to advanced understanding of either additive or multiplicative stages.

Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a)
b)		 16
30		 6 × 4 = (6+2) × (4–2) =8 × 2
5 × 8 = (5–2) × (8+2) =  3 × 10		 Rather than multiplying and dividing by 2 to double and halve, students add and subtract 2, and then solve, i.e., a × b = (a–2) × (b+2).
c) 85 8 × 15 = 8 × 10+5 Distributive error
Students misapply the distributive law
[a × (b+c) = ab +ac] by not expanding one of the components, i.e.,  a × (b+c) = ab+c.
a) 40 6 × 4 = 6+4 = 10,
and then 10×4 = 40		 Some students made a procedural error and misapplied addition and multiplication rules.

Averaged over the three questions about 14% of students used a repeated addition or skip counting strategy to solve the problem. A small number of students solved the problem using a part-whole strategy.

Next steps: 

Misapplication of rules for addition and multiplication
Students who misapplied rules for adding and multiplying need to explain their reasoning for strategies and if required use drawings or diagrams to help show how their strategy works.

Not using a doubling and halving strategy
Students who used another strategy to solve the multiplication problem (correct or incorrect solution) should be encouraged to show how they could use doubling and halving to solve find their solution.  These students could either be using a sophisticated advanced multiplicative strategy (stage 7), or a visual or additive strategy.  The numbers have been selected to make the doubling and halving strategy an effective way to solve the problems.

Visual and additive strategies
Students who used visual strategies (e.g., drawing, tallying, and grouping) need to begin looking at developing more efficient additive strategies (skip counting and repeated addition).  Then encourage students to connect additive strategies to multiplicative strategies (see below).
Students who used visual or additive strategies (e.g., skip counting and repeated addition) need to begin looking at developing multiplicative strategies.  Some students doubled by adding.  Doubling could also be regarded as one of the most basic multiplication operations.  Students could be encouraged to recognise that adding a number to itself is "2 lots of that number", and that this is true for any number, i.e., the count of repeated additions is how many lots (factor) of that number.  Exploring the link between repeated addition and multiplication, and basic doubling of numbers should support them starting to use multiplication.

Effectively using a doubling and halving strategy
Students who could apply doubling and halving efficiently could look at the other multiplicative inverse relationships and see if they can apply and generalise those, e.g., tripling and thirding (3 × 27 = 9 × 9); quadrupling and quartering (multiple halving and doubling); or pentupling and fifthing (6 × 45 = 30 × 9).  Students should also look at a wider range of multiplicative strategies and discuss which are more efficient and explain their reasoning.