Eating fractions of cake
| a) |
Patrick and Sarah each had the same sized cake. Patrick ate \(1 \over 4\) of his and Sarah ate \(1 \over 2\) of hers. Show how to work out how much cake they ate altogether.
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b) |
Sam and Nik each had the same sized cake. Sam ate \(1 \over 3\) of his cake and Nik ate \(1 \over 6\) of his cake. Show how to work out how much cake they ate altogether.
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c) |
Jake ate 2\(1 \over 2\) cakes and Maraea ate 1\(3 \over 4\) of a cake. If the cakes were the same size, show how to work out how much cake they ate.
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d) |
There were 4 cakes on the bench. Mary came and took 1\(1 \over 7\) cakes. Show how to work out much how cake is left.
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e) |
There was 1\(1 \over 5\) of a cake in the pantry. Andrew ate \(4 \over 5\) of a whole cake. Show how to work out much how cake is left.
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| Y8 (05/08) | ||
| a) |
Working involving adding 1/4 and 1/2 :
3/4 |
easy
easy |
| b) |
Working involving adding a 1/3 and a 1/6 :
1/2 |
moderate
moderate |
| c) |
Working involving adding a 21/2 and 13/4 :
41/4 |
easy
moderate |
| d) |
Working involving subtracting 1 and 1/7 from 4 wholes:
26/7 |
moderate
moderate |
| e) |
Working involving 11/5 (or 6/5) – 4/5 =
2/5 |
moderate
moderate |
Based on a representative sample of 228 students.
NOTE:
- Other shapes can also be used: many students used circles, which are harder to show equal parts and can actually be a barrier to accurate representation.
- For questions a) and b) a small number of students answered 3/8 and 3/12 (or 1/4) respectively. Students who used diagrams to work out these answers are may have interpreted the whole as all the cake rather than a cake (see note below about finding the fraction of a different referent whole).
For questions a) and b) a small number of students answered 3/8 for question a) and 3/12 (or 1/4) for question b). Students who used diagrams to work out these answers are may have interpreted the whole as all of the (two) cakes rather than a cake,
e.g., is 3 shaded out of 8 parts (3/8).
This is 3/8 of the two cakes, so it is equivalent to 3/4 of one cake.
This can be more of a comprehension issue, and an important lead in to a conversation or class discussion about what the whole is when working with this kind of question. This resource is about encouraging students to explore a single cake as the referent whole tying in more closely with the convention of addition and subtraction of fractions (maintaining the same referent whole throughout the problem (unless otherwise specified).
However, other students who gave these answers used understanding of equivalence to rename the fractions to have common denominators and then wrote 1/4 + 2/4 = 3/8 or 2/6 + 1/6 = 3/12 . Students who wrote these equations are indicating an incorrect understanding about addition of fractions.
| Common error | Likely misconception | |
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a) b) |
1/6 1/9 |
Whole number misconception about fractions Students treat the fraction as two separate numbers – and only add the bottom numbers. This error tended to be associated with adding unit fractions. |
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a) b) c) |
2/6 (or 3/8*= 2/4 + 1/4) 2/9 (or 3/12* = 2/6 + 1/6) 34/6 (or 32/3 or 35/8*= 3 + 2/4 + 3/4) |
Whole number misconception about fractions Students treat the fraction as two separate numbers – adding the top and bottom separately and then putting the sums of each top and bottom. This same error can happen if students draw a diagram and shade the parts without understanding how to combine fractions, e.g., *some students first renamed 1/2 to 2/4 and then added both top and bottom numbers, or errors equivalent to this. |
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a) d) |
31/4 36/7 |
Not recombining the whole Students' used a number of different strategies to develop this solution. |
Students who have any of the whole number misconceptions identified above need to develop a part-whole understanding of fractions before trying to devise a system to add or subtract fractions. If required, students could go back to partitioning and explore constructing the parts (unit fractions), combining these parts to make non-unit fractions that are between 0 and 1 (also called proper fractions), and naming these new fractions (part-whole fractions).
Using diagrams to show how they added/subtracted the fractions
For students who constructed incomplete or inaccurate diagrams to show how to work out these fraction problems, ask students to explain how they know the diagram they have drawn accurately shows the fraction they need to use. Then get them to explain or justify how the parts can be added or subtracted (because they are the same size, we can add/subtract fractions).
Students can be encouraged to use easier shapes to represent the fractions, e.g., rectangles can be easier to show (and compare) a range of fractions:
These fractions are then all fairly easily manipulated to solve the addition and subtraction problems using basic counting skills and an understanding of the unit fraction.
For students who showed their working using an accurate diagram, encourage them to look at other students' strategies to show or explain how they got their answer. For example in question c) students could say
"I have a whole and want to take away 1/5 , I know that there are 5 fifths in a whole which means there is one left"
and for question d)
"I know that 3/4 is 1/2 and 1/4 , so to take 3/4 away from 11/2, I take the half away which equals 1 and then a quarter from 1 and the answer is 3/4 because there are 4 quarters in 1."
Further exploration
These questions asked students to add and subtract fractions and mixed fractions with similar denominators. It is also important to encourage students to explore other less common fractions using the strategies they have come up with. This should help them develop more robust addition and subtraction strategies that may work for all fractions.
Strategies
The most common strategies that students employed were:
- Using common denominators and fractional notation. This was employed most often in parts a), b) and c), with about a quarter of the students using it for parts a) – c), but less than 10% for parts d) and e). It was typically used by students with high mathematical competency. It had a success rate of about 90%.
- Drawing an accurate diagram. This was the most common strategy for all parts, with about one-third of students using it. It was typically used by students with average mathematical competency. It had a success rate of about 70-80%.
- Part-whole partitioning the numbers. This was used mainly for parts c), d), and e), with about 10-15% of students using it. It was typically used by students with a little above average mathematical competency, and had a success rate of about 90%.
This mixed success indicates that an important part of understanding fractions involves being able to explain or illustrate the working. Furthermore it shows that answers are not sufficient in themselves and should be backed up by explanation, justification, or at least labelling. This can be scaffolded in whole class discussion where students share and critique their own and others' strategies to develop a fuller understanding. To promote understanding students should be able to use a range of strategies to show addition and subtraction of fractions, and work towards non illustrative strategies to consolidate this understanding.