Pizza Split

Many students attempt to solve problems like 2 ÷ 8 by changing the order of the numbers, as they would in multiplication or addition, and calculating 8 ÷ 2 instead. This activity prompts the students to discover that division is not commutative, that is, they explore examples using a fraction kit and diagrams to confirm that 8 ÷ 2 and 2 ÷ 8 in fact give the inverse of each other (8 ÷ 2 = 4/1 and 2 ÷ 8 = 1/4).
To do this activity independently, the students need to be able to:
• write symbols for fractional amounts, for example, three-quarters as 3/4 and four-thirds as 4/3.
• make simple equivalent fractions, such as 1/4 and  2/8
• divide whole numbers into fractional parts, using materials such as a fraction kit
• change mixed fractions (such as 1 1/2) into top-heavy fractions (such as 3/2) and vice versa.
With a guided teaching group, give each pair in the group a fraction kit (foam or plastic circles cut into various-sized fractions) so that they can explore the problems using materials that they can manipulate. If you don’t have commercial ones available, the students can make their own using the Fraction Pieces. The students can choose the circles they need to solve each problem and then either cut them up to share out or shade each person’s share.
Look out for students who interpret Mena’s diagram in question 1 as 2 ÷ 8 = 2/16 because they see 16 pieces of pizza in the diagram with each person getting 2 of them. A fraction is a part of a whole amount, and you need to define what the whole is to make the size of the fraction meaningful. It would be correct to say “2 sixteenths of 2 pizzas”, but also ask: How much of 1 pizza is that? It’s important that the students understand that, in the context, 1 whole refers to the number 1, not to the whole group of pizzas. It is critical that the students understand this
re-unitising if they are to progress in fractions. Emphasise this by asking questions such as:
If you ate 1 piece of pizza in Mena’s diagram, what fraction of 1 whole pizza would you have eaten? (1/8)
If you ate 2 pieces of pizza, what fraction of 1 whole pizza would you have eaten? (2/8)
Would you still have eaten 2/8 of a pizza if you ate 1 piece from 1 pizza and 1 from the other? Show me. (If I take 1/8 from each pizza, it’s the same size as if I take 2/8 from 1 pizza.)
Can you show me what 1/16 of 1 whole pizza would look like? (Half the size of one of the eighth pieces)
What about 2/16 of 1 whole pizza? (The same size as one of the eighth pieces)
You may need to help some students decide which fractions in their kit they should use to solve the problems. Ask questions such as:
When Mena shows 3 pizzas divided among 4 people, her diagram shows the pizzas cut into quarters. How did Mena know to cut her pizzas into quarters? (If you’re sharing between 4 people, each person will get a quarter.)
How did Zac know to cut his pizzas into thirds when he showed 4 pizzas divided among 3 people?
(If you’re sharing between 3 people, each person will get a third.)
What fraction pieces would be useful to choose to show 5 ÷ 2? 2 ÷ 5? (5 ÷ 2: halves because 5 is shared between 2 people; 2 ÷ 5: fifths because 2 is shared between 5 people)
The students will find it difficult to see the inverse pattern unless they write their fractions as common fractions rather than mixed ones. If they have written 4 ÷ 3 = 1 1/3 , ask them to write it as a common fraction as well, that is, 4/3.
In question 6, if the students can’t make the pattern work, encourage them to write their fractions in their simplest form. For example, in question 1, Mena works out 2 ÷ 8 as 2/8, but in order to see it as an inverse of 8 ÷ 2 = 4/1, she would need to write 2/8 as the equivalent fraction of 1/4, which she realises in question 5.
To encourage reflective discussion and generalisation, ask:
Were there any questions in this activity whose answer surprised you? How?
What patterns did you notice? How did you use them to predict the answers? What would you predict as the answer to 5 ÷ 15? How do you know?
A generalised solution to this is to create fifteenths, that is, cut each pizza into fifteenths. Each share will be 5/15 because 1/5 will come from each pizza. This generalises the link that 5 ÷ 15 = 5/15.  Note that 5 ÷ 15 is an operation or process, while 5/15 is a number.

Extension
 

The students could investigate whether the order of the numbers matters in addition, subtraction, and multiplication. They could explain and summarise their findings for all four operations, using a chart or a computer slide-show presentation.
Some students could explore further the ideas that a fraction is a part of a whole amount and that the size of the fractional piece depends on the size of the whole. Ask:
Which is bigger, 2/8 of 1 pizza or 2/16 of 2 pizzas?
Is 1/8 of a small pizza the same as 1/8 of a large pizza?
Can a quarter ever be bigger than a half?
 

Answers to Activity

1. Yes, they are both right. can be written as . They are equivalent fractions.
2. a. Yes. 8 ÷ 2 does not give the same answer as 2 ÷ 8.
b. 8 ÷ 2 = 4; 2 ÷ 8 = 1/4 (or 2/8)
3. a. i. 12
ii. 3. (12 ÷ 4)
iii. Practical activity.
A possible answer is:

can be rearranged as

iv. 3 ÷ 4 = 3/4
b. i. Practical activity.
A possible answer is:

can be rearranged as

ii. 4 ÷ 3 = . (This can also be written as 1 1/3.)
4. Yes, the pattern does work. 5/2 is 5 ÷ 2 and 2/5 is 2 ÷ 5.

can be rearranged as:

5 pizzas ÷ 2 = 5 halves of a pizza each
and

can be rearranged as:

2 pizzas ÷ 5 = 2 fifths of a pizza each
5 ÷ 2 = 5/2 (which can be written as 2 1/2)
2 ÷ 5 = 2/5
The number on the bottom of the fraction (the denominator) tells you what sort of fraction it is. For example, 5 on the bottom means you have fifths, 4 means you have quarters, 3 stands for thirds, 2 stands for halves, and 1 stands for wholes.
The number on the top of a fraction (the numerator) tells you how many you have:
in 3/4, you have 3 quarters; in 1/4 , you only have 1 quarter.
5. Yes, the answers do follow the pattern. 2/8 is 1/4 (one-quarter) and is 4 wholes. All whole numbers could be written as something over 1. For example, 15 could be written as 15/1 or 345 as 345/1 , but we don’t do this unless we need to.
So for 8 pizzas divided between 2 people, each person gets 4 whole pizzas each (4/1 ). With 2 pizzas among 8 people, each person gets 2/8 or 1/4 each.
6. Yes. Problems will vary. The pattern will work for all of them, although you may have to change some of your fractions to their simplest equivalent form. For example, you may have worked out that 4 ÷ 12 = 4/12 , which does not seem to match
12 ÷ 4 = 3. However, 4/12 is equivalent to 1/3, and 3 is 3/1.