Shaded fractions
Y6 (06/06) | ||
a) | ^{1}/_{2} or ^{2}/_{4} or ^{4}/_{8} or ^{6}/_{12} or ^{8}/_{16} | difficult |
b) | ^{3}/_{4} or ^{6}/_{8} or ^{12}/_{16} | difficult |
c) | ^{1}/_{4} or ^{2}/_{8} or ^{4}/_{16} | difficult |
d) | ^{1}/_{3} or ^{2}/_{6} or ^{6}/_{18} | difficult |
NOTE: Accept as correct if students write the name instead of the notation for the fraction.
About a quarter of students simply counted the number of shaded pieces and recorded it in one of the following three formats.
Common error | Likely misconception | |
c) d) |
4 6 |
Cannot write fraction notation Students count the number of shaded pieces and write this as a whole number rather than a fraction. |
a) b) c) d) |
^{6}/_{11} ^{7}/_{10} ^{4}/_{13} ^{6}/_{14} |
Uneven size 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. |
a) b) c) d) |
^{6}/_{5} ^{7}/_{3} ^{4}/_{9} ^{6}/_{8} |
Writing a fraction as a ratio of 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. |
Students who count the number of pieces and write this as their answer may not recognise or know how to construct fractions, or they may have no understanding of the fraction as being a part of a whole and may need to further explore partitioning and fractions as part-whole relationships.
Partitioning
Get students to partition a variety of different 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).
It is important that students build up many experiences of partitioning starting with:
- halving of basic shapes, then halving multiple times to derive other parts;
- partitioning a variety of shapes: squares, two squares, rectangles, circles, hexagons (which may be easier to partition for younger students), etc;
- partitioning shapes into a different number of pieces (e.g., 3, 5, 6, 7, 9, etc).
Encourage students to recognise the relationship between the bottom number (denominator) and the number of equal-sized pieces the shape is partitioned into (numerator).
Fractions as part-whole relationships
Get students to identify what fraction of a very simple shape is shaded,
Ask "What fraction of the whole shape is shaded?
[Now draw a horizontal line across the shape.]
[Now show the same shape shaded in a different way.]
Ask "What fraction of the whole shape is shaded?"
Encourage students to recognise that the fraction of the whole shape that has been shaded is unchanged.
Experiment by showing the same shape, but drawing lines to make smaller and larger pieces, each time asking "What if I draw a line here…what fraction of the whole shape is shaded?" e.g.,
Remind students that they are finding a part of a whole by asking "What is this part?" and "What is the whole?"
Students who construct the fraction by merely counting up the number of shaded pieces and putting that number over the total number of pieces need to be aware when naming a fraction that each piece needs to be equal-sized. For example, when something is divided into quarters each of the 4 pieces has to be equal-sized. They should be encouraged to draw lines to create equal-sized partitions to help them work out or explain their strategy. See Fractions as part-whole relationships above.
Although some students may feel they are not supposed to draw lines on to the shapes, encouraging their diagrammatic representation of fraction is important and can help eliminate misunderstandings.
Students who construct the name of the fraction by merely counting up the number of shaded pieces and putting that number over the number of unshaded pieces are setting up the fraction as a ratio. Here the fraction is constructed by putting the number of shaded pieces on the top and the number of unshaded pieces on the bottom. A fraction compares a part to the whole, whereas a ratio compares a part to another part. Exploring Fractions as part-whole relationships should help students recognise how fractions are constructed.
NOTE: Ratios are a concept that students will need to develop, so it is important to affirm students' ideas in this area. The difference between a ratio and a fraction could be discussed and made explicit. This may be a good opportunity to explore this difference.
Other resources
For similar ARB resources about fractions of regions click on the link or use the keywords, fractions AND 2-dimensional shapes. Click on the link for further information about fractions as part-whole relationships.