Finding fractions

Finding fractions

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about how to find fractional amounts of a total.
a)
A piece of timber was 36 cm long. 
If \(1 \over 4\) was cut off, what would the new length be?

Show your working.

 

 

Answer: __________cm
b)
James started with 32 marbles and lost \(3 \over 8\) of them. 
How many did he lose?  

Show your working.

 

 

Answer: __________
c)
Aroha won \(7 \over 9\) of the 72 tennis matches that she played. 
How many matches did she win?  

Show your working.

 

 

Answer: __________
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
fractions, fractions as operators, key competencies
Description of task: 
Students work out fractional values of whole numbers in a range of contexts and show their working.
Curriculum Links: 
This resource can help to identify students' ability to apply additive or multiplicative strategies flexibly to find fractions quantities.
 
Key competencies
This resource involves recording the strategies students use to find fractions of quantities. 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 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y9 (03/2004)
a) 27
Working that uses the information in the question to show how to calculate the amount cut off and adjustment of the length accordingly. Any 1 of:

  • 36 ÷ 4 = 9 and 36 - 9
  • 3/4 of 36 = 3 × (1/4 of 36) = 3 × 9 (find the unit fraction and adjust)
  • Other working showing how to find 3/4 of 36.
 moderate
easy
b) 12
Working that uses the information in the question to show how to calculate the amount of marbles lost. Any 1 of:

  • 1/8 of 32 = 4 -> 3/8 = 4 × 3 (find the unit fraction and adjust)
  • 3/8 × 32 and simplify (formal method)
  • Other working showing how to find 3/8 of 32.
 moderate
moderate
c) 56
Working that uses the information in the question to show how to calculate the amount of tennis matches lost. Any 1 of:

  • 72 ÷ 9 = 8 and 8 × 7 (find the unit fraction and adjust)
  • 7/9 × 72 and simplify (formal method)
  • 72 ÷ 9 = 8, 8 × 2 = 16 and 72 - 16
  • (find the unit fraction × 2 and subtract from 72.)
  • Other working showing how to find 7/9 of 72.
 difficult
moderate

NOTE: For all working, the final answer does not have to be correct to attain a mark. This mark should be allocated for the communication of the attempted calculation as well as a stepwise sequence that could lead to a correct answer.

Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a) 9   Gives the amount cut off rather than the new length.
a)
b)		 35.75 or 353/4
31.625 or 31 5/8		 36 - 1/4
32 - 1/8		 Manipulates fractions and whole numbers incorrectly (treats fractions as numbers not proportions).
b) 24 Solves for 3/4 of 32  
c) 63 7/9 of 72 is 9 -> 7 × 9 Initial miscalculation of unit fraction, then correct calculations.

A common, successful way used by students to answer all 3 questions was to:

  • find the unit fraction (using division, reverse multiplication or successive halving); then
  • adjust the original amount (by subtracting the appropriate amount or multiplying the unit fraction) to find the given fraction.