Fractions of cake
| a) |
Ariana ate one-third of an apricot cake and Peter ate one-third. Draw or show how to work out how much cake they ate altogether.
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b) |
Maraea ate one-quarter of a rēwena bread and John ate a half. Draw or show how to work out how much they ate altogether.
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c) |
Joshua ate one-sixth of a carrot cake. Draw or show how to work out how much cake was left.
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d) |
Some friends ordered 3 pai fala (pineapple pie). Josef ate half of one pai fala. Draw or show how to work out how much pai fala is left for his friends.
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| Y6 (11/2007) | ||
| a) |
2/3 or two-thirds Working involving adding a third (1/3) and a third (1/3):
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difficult difficult |
| b) |
3/4 or three-quarters Working involving adding one-quarter and one-half:
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difficult moderate |
| c) |
5/6 or five-sixths Working involving subtracting one-sixth from a whole:
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difficult difficult |
| d) |
21/2 or two and a half [also accept other equivalent answers, e.g., 5 halves] Working involving subtracting a half from 3 pai fala.
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difficult moderate |
This resource encourages the use of diagrams or representations to promote understanding about fractional relationships. Additionally, this representation of fractions being added or subtracted should support students to check the reasonableness of their answer – providing students maintain what the referent whole is in their working. These questions asked students to explain the strategies they used to add and subtract fractions with simple and similar denominators. It is also important to encourage students to explore other fractions (including improper fractions) using the strategies they have come up with. An important point for representing fractions is that students should use a range of representations using shapes other than a circle – fractions can be represented in many ways – this in itself could be an exploration with students (e.g., How many different ways can you show 3/4 ?).
Strategies
Diagrams were used by about three-quarters of students for both the addition and subtraction problems. Of these diagrams about half were correct or accurate representations of the problem. However, even correct diagrams had a mixed success rate with only about half of them leading to correct answers for b), c) and d), and about three-quarters for a). Almost 80% of students who labelled their diagrams also wrote the correct answers. This mixed success indicates that diagrams are not sufficient in themselves and should be backed up by explanation, justification, or at least labelling with fractional notation. This can be scaffolded in whole class discussion where students share and critique their own and others' strategies to develop a fuller understanding.
| Common error | Likely misconception | |
| b) |
1/2 and 1/4 or half and quarter |
Cannot combine (add) fractions with different denominators Students do not know how to combine the fractions with different denominators, so write them as two fractions. |
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a) b) |
1/2 1/3 |
Fractions size misconception: fractions of the same name must be equal Many students who answered a) with half or 1/2 also drew an inaccurate diagram. Many students who answered b) with a third or 1/3 drew an accurate diagram, but did not maintain the rule that fractions of the same name must be the same size, e.g.,  represents a half and 2 quarters, but if fraction size is not consistent each could be a third (especially if the diagram is less accurate). |
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a) b) |
2/6 2/6 |
Whole number misconception about fractions: adding numerators and denominators 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 misconception is more likely when fractional notation is used rather than words. |
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a) b) c) d) |
1/3 1/4 1/6 1/2 or half |
Identifies the incorrect part involved in the solution Students identify the part that remains rather than the part being "eaten" (addition) or the amount being removed rather than the amount remaining (subtraction). For example question a) when asked to add 1/3 and 1/3 an accurate diagram is drawn:  but the students name the remaining (unshaded 1/3) part rather than the part that was eaten (shaded 2/3). |
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a) b) c) d) |
2 or 6 2 or 3 5 or 6 3 or 5 |
Writing the number of pieces rather than the fraction Students refer only to the number of pieces shaded not their size. Some students simply count up all the pieces ignoring what they represent or how large they are (and therefore what fraction they represent). |
| c) | 1/5 |
Referent whole misconception: Not keeping the whole (referent whole) Students believe that when you take one piece away from  what is left is  Each part of this new whole (with a piece missing) is a one fifth. Students with this misconception may have drawn accurate or inaccurate diagrams, indicating the importance of an explanation as well as a "picture". |
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a) b) d) |
2/6 2/6 5/6 |
Referent whole misconception: finding the fraction of a different whole Students have worked out the fraction of all the cake that has been eaten (i.e., 2/6 of all the cake that was there has been eaten) rather than the fraction of a cake eaten (2/3). For question b) this may also be combined with the misconception about fraction size. |
Cannot combine (add) fractions with different denominators
For students who cannot add fractions of different denominators encourage them to draw the
fractions concerned on one diagram, e.g., identify and name the fractional parts and then discuss how much of the whole shape is now shaded. They can be encouraged to draw any lines on the shape that may help them name the fraction of the shape shaded.
Fractions size misconception: fractions of the same name must be equal
Students with this misconception may need to revisit how fractions or parts of shapes (and sets) can be made by sharing the region (or set) into equal parts. Use partitioning to help them construct and name fractions.
Whole number misconceptions about fractions: adding numerators and denominators
Students who have a whole number misconception and are adding both the numerator and denominator need to develop a part-whole understanding of fractions before trying to devise a system to add or subtract fractions. 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).
Writing the number of pieces rather than the fraction
For students who gave the number of pieces as their answer, encourage them to find out what fraction the pieces are and incorporate that into their answer, e.g., for question b), 
could also be 3 pieces (each of 1/4). Asking students what the whole shape is, and then what part they are finding, can support them to develop a more part-whole understanding about fractions.
Identifies the incorrect part involved in the solution
Students who identify the opposite part of the problem they were trying to solve may need to articulate how they see the problem they are solving. It is important that students can make sense of the language of mathematics in order for them to understand mathematical problems. This can be consolidated by students writing their own maths problems.
Referent whole misconception : varying the referent whole
When adding, subtracting or comparing fractions it is important to keep the same whole. This is referred to as the referent whole. For example, if we are adding 1/2 and 1/4 the referent whole should be 1 – so we would be adding 1/2 of one and 1/4 of the same sized one. If we were adding fractions of a different whole then the fractions would not be of the same size and it would be very difficult to add and know what fractions they represent. For example:
1/2 + 1/2 = ?
How would we add these?
Students using inaccurate diagrams
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 (e.g., we can add/subtract fractions because they are the same size). Additionally, students can be encouraged to use easier shapes to represent the fractions. For example, rectangles can be easier to show (and compare) a range of fractions:
1/6 is considerably easier to draw as a rectangle and the representation supports understanding rather than distracting from the fraction addition problem by adding another geometric issue of equally distributing a circle into 6 equal parts
Students using accurate diagrams
For students who showed their working using an accurate diagram, encourage them to write the fractional notation to show what they have worked out with the diagrams, look at other students' strategies, and 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 1/6 , I know that there are 6 sixths in a whole which means there is five left" and for question d)
"I know that there are 3 cakes, so I take the half away from one of the cakes, and there is a half left. So there are the two cakes I didn't take half from and the half left of the third cake, which is two and a half."
Encourage students to reflect on the rules they may see for adding fractions (i.e., that to add or subtract fractional parts the parts must be of equal size).
Students who drew diagrams to show their working predominantly used circles to show the fractions. Encouraging the use of a range of representations should support students to solve different problems more efficiently, e.g., as well as a rectangle for almost all fractions, a hexagon could be used for thirds or sixths, a pentagon for fifths, etc. This should also help focus the activity on the learning issue of adding fractions rather than the geometric issue of shape and space.
Numeracy resources
Book 7: Teaching Fractions, Decimals and Percentages, 2006: Hungry birds, (p.11), Advanced counting/ Early additive part-whole.