Sharing shapes

This task is about drawing lines to share some shapes equally.

For each shape below, draw lines to show how to share the shape equally.

i)	Draw lines to show how to share this square equally between 3 people.
ii)	What fraction would each person get? _____

i)	Draw lines to show how to share the rectangle below equally between 9 people.

ii)	What fraction would each person get? _____

i)	Draw lines to show how to share this hexagon equally between 3 people
ii)	What fraction would each person get? _____

i)	Draw lines to show how to share these two squares equally between 5 people.

ii)	What fraction of a square would each person get? _____

Teaching and learning:

Knowing that shapes and sets can be partitioned into equal parts is important for understanding the part-whole relationship between the numerator and denominator in fractions.

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.

Diagnostic and formative information:

Partitioning two squares
Most of the students who correctly partitioned the two squares in question d) either partitioned each square into 5 equal parts, or partitioned each square into 2 and a half (the halves join to form the fifth equal part). The latter method makes it harder to identify the fractional value of the part.

	Common error	Likely misconception
a) & c)		Students do not recognise the need for partitions to be equal, or they cannot construct the correct angles to make them equal. One-fifth of students tried to partition the square as if it were a circle and half of these students correctly identified the fraction for a) to be ¹/₃ . Over half the students tried to partition the hexagon using vertical or horizontal lines rather than recognising the rotational nature of the shape and partitioning accordingly.
b)		Students create one more partition than required. Students are including the partition for the one (numerator) and are creating 1 + 9 partitions. This could either be a misconception based around prior learning to construct a ratio 1:9 or may be based around a miscount while trying to construct the partitions, i.e., students count the interior lines instead of the parts.
d)	¹/₅ or ²/₁₀	Students give the fraction of both squares not one square. They are treating both squares as the whole and finding the fraction of both squares – not the fraction of one of the squares. The language here is important as it distinguishes what the whole is that the fractions relate to. Over a third of students gave this response.

Next steps:

Some students have had limited experiences with fractions and partitioning and rely on the methods of cutting up they are familiar with. If they have only ever divided up "round pizza shapes" they may think this is the only way to divide shapes up. This becomes more difficult when the partitioning cannot be derived by a halving strategy, or the partitions are different shapes and cannot easily be overlaid to check for equal-size. 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).

By partitioning shapes into an odd number of parts and by using a number of shapes, students can develop a more robust understanding of partitioning. This variety ensures that they are not just memorising how to partition certain shapes, but they have the ability to partition any simple shape and understand the importance of these parts being equal-sized.

For students who answered ¹/₅ or ²/₂₀ for question d), the issue here is "What is the referent whole (whole being referred to)?" Ask "How much of the square would each person get?" and then "How much of a (single) square would each person get?"
Discuss how the fractions are different, and why they may be different. Ask what is the whole each and get students to compare them. Students should recognise that when the whole is two squares they get ¹/₅ and when it is one square they get ²/₅ (remembering that there is still two squares to share, but the fraction relates to fraction of a square). Students may also notice that the fraction is twice as big when the referent whole is half.

Other resources
For similar ARB resources, click on the link or use the keywords, fractions AND partitioning. Click on the link for further information about partitioning and how it supports fractional understanding.

		Y6 (06/06)
a)	[Accept other partitions that create 3 equal parts] ¹/₃	very easy moderate (moderate – both correct)
b)	[Accept other partitions that create 9 equal parts] ¹/₉	easy moderate (moderate – both correct)
c)	[Accept other partitions that create 3 equal parts] ¹/₃	difficult moderate (very difficult – both correct)
d)	[Accept other partitions that create 5 equal parts] ²/₅	difficult very difficult (very difficult – both correct)

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Sharing shapes

Sharing shapes