Estimating fractions

Estimating fractions

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about estimating a sum of fractions.
 

These questions are about estimating not solving to find the exact answer. For the each of the following three problems:

  1. Estimate what you think the sum of all the fractions is; and
  2. Explain how you worked out the estimate.
a)
\(8 \over 9\) + \(14 \over 15\) + 2\(1 \over 6\)
 

Estimate:

How I estimated:
 
 
 
 
 
b)
 
\(7 \over 15\) + \(11 \over 12\) + \(19 \over 18\) + 3
 

Estimate:

How I estimated:
 
 
 
 
 
c)
 
2\(3 \over 4\) + \(1 \over 2\) + 2\(5 \over 8\) + \(3 \over 11\)
 

Estimate:

How I estimated:
  
 
 
 
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, estimation, addition, key competencies
Description of task: 
Students estimate the sum of a number of fractions.
Curriculum Links: 
Key competencies
This resource involves explaining how they performed estimates of several addition of fraction problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, sets 7-8
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y8 (05/2008)
a) About 4 (or 4 and a bit) and
students used an estimation strategy:

  • drawing an approximate diagram and estimating visually
  • combining only the fractions of significant size and not small fractions (like front-end estimation for whole numbers without compensation)
  • combining the fractions of significant size and compensating the remaining fraction (like front-end estimation for whole numbers and compensation)
  • rounding fractions to the nearest whole (or half) and then mentally adding
  • combining fractions to 1 or 0 (like compatible numbers); or
  • any other estimation strategy that does not involve calculating.
 difficult
b) About 5 and a half and students used an estimation strategy (see above) difficult
c) About 6 and students used an estimation strategy (see above) very difficult

Based on a representative sample of 224 students.

Teaching and learning: 
This resource involved an understanding about the size of fractions and how to estimate rather than the addition of fractions.  The fractions were selected to be difficult to add to ensure that calculation was a barrier, and encourage the use of estimation.  However, about a third of students did not write down a strategy for solving and a quarter did not identify an answer.
Diagnostic and formative information: 

Although the fractions in the questions made calculation difficult, students indicated some sophisticated strategies and understandings about fraction size, and about how they can estimate the addition of fractions. Essentially, the most successful students either rounded the fractions to a benchmark of 0 or 1 or a fraction they were comfortable with, e.g., 25/8 becomes 21/2 .  Students who could round to other benchmarks (such as a half) than 0 or 1 tended to have even more success with the accuracy of their estimation.  Another strategy that students used to was to combine fractions, e.g., 1/2 + 25/8 becomes 3 (that 5/8 is essentially the same as 1/2 is implicit).

  Common error Likely misconception
a)
b)
c)		 223/40 or 25/40
337/45 or 40/45
412/25		 Students attempt to add (calculate) the top and bottom numbers of the fractions.  This error is twofold: calculating rather than estimating, and adding incorrectly.
Next steps: 
No strategies or solution
Students who did not write down any strategy or did not identify an answer may need to develop the two understandings of

  • estimation being a mental approximation strategy, and
  • that fractions have a size and can be compared before trying to combine the two.

Guessing as a strategy
Students who indicated they were guessing need to develop their understanding about explaining/sharing their strategies for solving maths problems.  Although a number of these "guesses" of the right magnitude, explaining strategies is an important maths skill to develop. Estimation problems involving wholes numbers could be explored and students may find it easier to explain the how they solved the problems.  Additionally, students could draw diagrams to visually estimate fraction addition problems and then be asked if they can further describe without the use of diagrams (i.e., indicate and understanding about what the top and bottom number represent in terms of fraction size). 

Incorrect calculation
A number of students Calculated the top and bottom numbers as separate whole numbers. This resource is not about knowing how to add fractions.  It is about recognising the approximate size of fractions and being able to estimate the combination of these.  Students could find out about estimation by starting with whole numbers and showing/explaining their working, move onto developing an understanding about the part-whole relationship a fraction represents, and then combine these two ideas without the need to know how to "add fractions with different denominators".  Students would simply need to know about how big (or how much of a whole) a fraction is. To support students to learn about addition of fractions, refer to the Fractional thinking concept map: addition of fractions.

Classroom discussion
To help students expand their own repertoire of strategies and develop their understanding about partitioning, have them explain, compare, and justify their strategies as a class or in groups. They could look at similarities and differences between the strategies and identify which are more sophisticated or efficient.  See Mathematical classroom discourse for more information on this assessment strategy.