Shaded fraction shapes
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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about recognising the fraction that has been coloured of different shapes.
Task administration:
This task can be completed with pen and paper or online (with auto marking).
Level:
3
Curriculum info:
Key Competencies:
Keywords:
Description of task:
Students identify the shaded fraction of a number of different shapes.
Curriculum Links:
Key competencies
This resource involves recording the strategies students use to find fractions of sets, shapes, and quantities. This relates to the Key Competency: Using language, symbols and text.
For more information see http://nzcurriculum.tki.org.nz/Key-competencies
Answers/responses:
Question a)
7/10 (very easy) and
explanation (e.g.,):
- by counting the boxes that are blue and there were 7 and the total of boxs are 10 so it 7/10 .
Question b)
1/5 (easy) and
explanation (e.g.,):
- I made it like there were inisable lines and there were 5 and 1 was shaded so it was 1/5 .
- because it looks like 5 rectangles wiuld fit in.
- because 1/2 is shaded half of it so if it was 1/8 it would be smaller (halving to get to 1/8)
- we masured the width and 5 were able to fit
- because one out of five has been shaded, and we used our fingers to estimate how many pieces there were in total
- i took the shape and duplecated it
- because you had to multiply 1/5 to equal one whole
- I made it like there were inisable lines and there were 5 and 1 was shaded so it was 1/5 .
Question c)
5/8 (easy) and
explanation (e.g.,):
- because five squares are filled in and alltogether there are 8
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Mentaly moved the shapes together, and count how many squares remaining.
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I knew it was 5/8 because the other squre is suppusided to be half
Question d)
3/16 (easy) and
explanation (e.g.,):
- because you divie it into quarters and so their is 16 boxes and only 3
- 4x4 for numerator and the 3 squares for the denominator
- we took away a qurter from a quarter
- you could put 16 little squares in and three were shaded
- There are 4 quaters and in each quater there is another 4 quaters and 3 are shaded in.
- because in each quarter you could fit in 4 squares so 2 times 8 equals 16 so if there was 3 green squares than that would be 3/16
NOTE:
Students strategies are written (and spelt) verbatim.
Accept any equivalent fraction for the answers above. The explanation is provded to indicate the completeness of an explanation (which may need modelling/supporting more explicitly) and the understanding of the part-whole nature of fractions.
Teaching and learning:
This task is about identifying the fraction represented by shaded shapes/regions. Finding the shaded fraction involves students knowing that fractional parts must be equally sized in order to combine them.
- Question a) is evenly partitioned and students can solve by counting the number of shaded pieces and then placing them "over" the total number of pieces.
- Question b) involves working out a fractional measure of a shaded region.
- Questions c) and d) require students to re-construct the partitions to work out the shaded fraction.
See the Fractional thinking concept map for more information about the different representations of fractions.
Diagnostic and formative information:
| Common error | Likely misconception | |
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b) c) d) |
1/2 4/7 2/5 |
Naming fraction using only the number of pieces
Students count the shaded pieces of each shape, and put that number over the total number of pieces, irrespective of the size of the pieces. This indicates that they do not understand that the same fractional parts must be equal-sized.
Students who gave indictaed this kind of misconception described some of their strategies as:
|
|
a)
b)
c) d) |
7/3 or 3/7*
n/a
4/3 or 3/4* 2/3 |
Identifying a fraction as a ratio (shaded to unshaded pieces)
Students count the number of shaded pieces and put over the number of unshaded pieces irrespective of the size of the pieces. Similar to the above misconception, but also does not understand that a fraction is a part of the whole. It is a good example of the potential danger of relying on remembering a procedure to construct a fraction rather than understanding what the fraction represents.
Students who gave indicated this kind of misconception described some of their strategies as:
*Also another related error involves thinking that fractions are always less than zero (reversing the numerator and denominator).
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| b) | 1/4 or 1/8 |
Measurement error
Students have used the shaded part and tried to unitise to work out the fraction shaded, but through overlapping parts or loose estimation they have answered this incorrectly - note 1/4 is notably more accurate than 1/8 , and with a re-check it is likely that students could correct for this.
Students who gave indicated this kind of misconception described some of their strategies as:
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| c) | 5/4 |
Not keeping the same the (referent) whole
Students who answered 5/4 for c) described some of their strategies as:
Both strategies involve treating the shape as two shapes and then adding the fraction of them together)
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| d) | 3/4 |
Not unitising correctly
Some students thought that it looked like 3/4 ignoring the wider shape (e.g., theres a half and 2 qualtes on has 3 bits shaded), and others had a more (sophisticated strategy that did not work), e.g., "we figured out 3/4 of 3/4 and then simplified it (calculation error)"
NOTE: 3/4 of 3/4 is more appropriate at Level 4/5 of the curriculum - as application of a muliplicative strategy with fractions, however by visually re-unitising the shape to solve this problem, it is accessible at Level 3).
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Next steps:
Students who cannot identify the shaded fraction of the shapes may need to develop their understanding about partitioning and fractions as part-whole relationships.
See the Fractional thinking concept map for more information about the different representations of fractions.
Naming fraction using only the number of pieces
Students who constructed the name of the fraction by counting up the shaded pieces and putting that number over the total number of pieces (irrespective ot the size of the pieces) need to be aware when naming a fraction that each piece needs to be equal-sized. Get students to partition a number of shapes into a range of parts. Encourage students to explain how they know the partitions are even, and how they could justify this to somebody else (they may need to fold, or cut and overlay the pieces). Then build up the partitioned pieces to make non-unit fractions and have students name them. Encourage students to recognise the relationship between the bottom number (denominator) and the number of equal-sized pieces the shape is partitioned into. Exploring a range of partitioning tasks should help with this. Students could explore the importance of the pieces being equal-sized by partititioning to construct fractions (Fraction shapes or Sharing shapes).
Students who constructed the name of the fraction by counting up the shaded pieces and putting that number over the total number of pieces (irrespective ot the size of the pieces) need to be aware when naming a fraction that each piece needs to be equal-sized. Get students to partition a number of shapes into a range of parts. Encourage students to explain how they know the partitions are even, and how they could justify this to somebody else (they may need to fold, or cut and overlay the pieces). Then build up the partitioned pieces to make non-unit fractions and have students name them. Encourage students to recognise the relationship between the bottom number (denominator) and the number of equal-sized pieces the shape is partitioned into. Exploring a range of partitioning tasks should help with this. Students could explore the importance of the pieces being equal-sized by partititioning to construct fractions (Fraction shapes or Sharing shapes).
NOTE: Drawing fractions is important and can help eliminate misunderstandings around equal sized fractional pieces.
Identifying fractions as ratios
Students who constructed the name of the fraction by counting up the shaded parts and putting that number over the number of unshaded parts are setting up the fraction as a ratio. A fraction compares a part to the whole, whereas a ratio compares a part to another part. Ratio is an important understanding that students need to develop. The occurrence of this misconception provides a real opportunity to explore the difference between ratios and fractions explicitly.
Explore a range of partitioning and constructing fractions exercises and draw students attention to what "the whole" (shape) is that they are working with, reinforce that they are looking for a part of that whole (although, sometimes the part is larger than the whole) and that constructing fractions involves working with equal-sized pieces.
Measurement error and not unitising correctly
Students who indicated this error may simply need some more exposure to partitioning exercises and working with fraction examples that do not have the parts already partititioned (so they can make the decision about how to divide the whole. This is also true for the both question b) unitising based on length, and question d) - unitising based on area. Ensure that student re-check their work and that they are not overlapping the unit they are using to measure to the whole. If student are able to show their lines of partition this may help clarify this error.
Not keeping the same (referent) whole
This error involves students not keeping the same whole when working out the fractional part shaded. Resources that involve partitioning two (or more squares) amongst several people explore this idea - called the referent whole. See Sharing shapes and Sharing cake and pizza and read about the referent whole.
See the Fractional thinking concept map for more information about the different representations of fractions.