Sharing cake and pizza
a)

At a party 7 square cakes are shared equally amongst 4 friends.
i) Show how to work out how much cake each person gets. 



b)

At a party 8 pizzas are shared equally amongst 3 friends.
i) Show how to work out how much pizza each person gets. 


Y8 (11/07)  
a) 
i)
ii)

Working, e.g.,
or other working that demonstrates an understanding of partitioning between 4. 1^{3}/_{4} or ^{7}/_{4} [also accept 1.75 or equivalent] NOTE: Also accept ^{1}/_{4} as it is not strictly incorrect, see note below 
easy
moderate

b) 
i)
ii)

Working, e.g.,
or other working that demonstrates an understanding of partitioning between 3. 2^{2}/_{3} or ^{8}/_{3} [also accept 2.66666 and 2.7] NOTE: Also accept ^{1}/_{3} as it is not strictly incorrect, see note below 
easy
difficult 
NOTE: Although ^{1}/_{4} and ^{1}/_{3} are correct for a) ii) and b) ii) respectively, this resource is about encouraging students to explore a single cake/pizza as the referent whole resulting in a more useful and rich investigation into fractions as a division relationship.
Classroom discussion
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. See Mathematical classroom discourse. for more information on this assessment strategy.
Common error  Likely misconception  
a) ii) b) ii) 
7 pieces 8 pieces 
Students do not write the answer as a fraction. They have partitioned the cake/pizza into ^{1}/_{4}s or ^{1}/_{3}s and then counted the number of parts and written as a whole number rather than a fraction. 
a) ii) b) ii) 
1 and a bit 2 and a bit 
Calculation error. Students have represented the partition, and identified that each person gets 1 (or 2) and a bit (this could be ^{1}/_{3} , ^{1}/_{2} , ^{1}/_{4} , etc). However they have not worked out the fractional name for each "bit". 
Students who do not write the answer as a fraction, need to be asked what fraction of a cake/pizza each piece is and how they know this. Then ask them if one piece is ^{1}/_{4} , then what fraction is 2 pieces? (2 quarters), 3 pieces? (3 quarters) … 7 pieces (7 quarters). Ask them to draw what ^{7}/_{4} could look like. This could be on a number line or as a shaded shape.
For students who had trouble naming the fractional part, more exploration with simple fractional partitions should help identify how to name or identify the part. Partitioning resources at Level 2 or 3 may be useful for this.
Using all the pizza/cake as a referent whole
Over 10% of students answered ^{1}/_{4} for question a) and ^{1}/_{3} for question b). This answer is actually correct if the referent whole is taken to be all the cake or all the pizza. However it is important to move students on to recognising there are 7 (or 8) cakes (or pizzas) and that shared amongst 4 (or 3) people is ^{7}/_{4} (or ^{8}/_{3}). This helps students develop an understanding that a fraction is also the result of a division (partition). For students who answer this ask them what the whole was they have used and then ask them "How much of a cake (pizza) does each person get?". This is also true for students who answered ^{7}/_{28} or ^{8}/_{24} .
Discuss how the answers, ^{1}/_{4} and ^{7}/_{4} (or ^{1}/_{3} and ^{8}/_{3}), are different, and what makes them different. Students should recognise that when the whole is 7 pizzas the solution is ^{1}/_{4} and when the whole is 1 pizza the solution is ^{7}/_{4} . Students may also notice that the fraction is 7 times as big when the referent whole is one seventh the size.
Exploring more efficient partitioning
For students who could partition each cake and pizza into 3rds or 4ths and correctly identify what fraction of a cake/pizza each person got, ask them if there is a way to do fewer cuts when sharing the cake/pizza up. They could start by exploring the least number of cuts and identifying the parts they have created, e.g., sharing 7 cakes between 4 people only involves 4 straight cuts – indicating it is more efficient than cutting up each cake.
Then get students to work what parts they have: 1 whole, 1 half, and 1 quarter (and then how to add them). This uses efficiency to drive different partitioning and requires students to add different fractions to work out the total share. For information about adding or subtracting fractions click Fractional thinking concept map.
For further information about partitioning and how it supports fractional understanding, click on the link Fractional thinking concept map, and go to the section called Partitioning and Divided quantities.