Estimating fractions

These questions are about estimating not solving to find the exact answer. For the each of the following three problems:
- Estimate what you think the sum of all the fractions is; and
- Explain how you worked out the estimate.
a)
|
89 + 1415 + 216
|
|
b)
|
715 + 1112 + 1918 + 3
|
|
c)
|
234 + 12 + 258 + 311
|
Y8 (05/2008) | ||
a) |
About 4 (or 4 and a bit) and students used an estimation strategy:
|
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.
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. |
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.
- Estimate these
- Estimate the menu
- Estimate these II
- Estimating sums of money
- Estimating scores and crowds
- Estimating farm animals
- Estimating team scores
- Who is estimating? Addition
- Estimating in sport
- Estimating addition II
- Estimating with addition III
- Estimating addition IV
- Journalling about estimation
- Estimating with addition
- Estimation or not?