Pizza Place has three tables of the same size.

The Chicken N Chips bar has four of the same tables and can seat 24 people altogether.

How many people can Pizza Place seat?

One third of the seats at Chicken N Chips are empty and a half of the places at Pizza Place are empty.

If 18 more people want to eat out, is there room for them at the two restaurants?

In this problem students find fractions of sets. As they find equal parts using sharing, division or basic facts, and combine parts to find other fractions, the inverse relationship between multiplication and division is explored. Having students see and use the fact that division undoes multiplication and that multiplication undoes division, is fundamental to working with fractions with understanding.

To solve this problem, students must see how information about one situation can be used to solve the problem in another.

### The Problem

Pizza Place has three tables of the same size. The Chicken N Chips bar has four of the same tables and can seat 24 people altogether. How many people can Pizza Place seat?

One third of the seats at Chicken N Chips are empty and a half of the places at Pizza Place are empty. If 18 more people want to eat out, is there room for them at the two restaurants?

### Teaching Sequence

- Introduce the problem by playing a guessing game:
*I am a number which is half of 6 and 6, what am I?*

Have students explain whether they doubled 6 and then found half of 12, or if they could immediately see 'the trick'.

With discussion, help students to see that doubling and halving are inverse operations. One ondoes the other. - Pose another 'What am I?' involving quarters, highlighting how halves of halves can help find quarters.
- Read the problem with the class.
- Brainstorm for ways to approach the problem, including asking if they've seen a problem like this before.
- As the students work on the problem, ask questions that focus on their use of mental strategies and their ability to justify the reasonableness of their answers.
*How are you solving the problem?*

Is your answer reasonable? Why do you think that?

Can you think of a way to check your answer? - Ask the students to record their solutions so that they can be displayed and shared with others.
- Have students share solutions.

Together talk about the number operations that your students choose to use, highlighting connections between multiplication and division, and fractions.

#### Other Contexts

People in cars, children working in groups

#### Extension to the problem

The next day, one third of the seats at Chicken N Chips are empty and a half of the places at Pizza Place are empty. If 18 more people wanted to eat out but they did not want to share a table with the people who were already in the restaurants, how many would have to be turned away from the two restaurants?

#### Solution

Students will use a range of strategies to show:

24 people sitting at four tables in Chicken N Chips means there are 6 per table.

Pizza Place can seat 3 x 6 = 18 people.

One third of 24 is 8. So there are 8 seats vacant in Chicken N Chips.

One half of 18 is 9, so there are 9 seats vacant in the Pizza Place.

8 + 9 = 17 spare seats.

18 people are looking for a meal and 18 – 17 = 1, so one person will have to go elsewhere.

#### Solution to the Extension:

Think how the tables could be filled. At Chicken N Chips there are 8 spare seats, so this means 24 – 8 = 16 seats filled. As each table seats 6 and there are four tables, the 16 could be seated at three tables or four.

At Pizza Place, there are 9 people. They could be seated at two or three tables but not one.

There are three possible answers to this problem. First, all of the tables in both restaurants could be being used, so no further people could be seated. On the other hand, one table in one of the restaurants might be free. In this case 6 people could be seated. And finally, there might be a spare table in both restaurants. That would mean that 12 people could be seated.

The number turned away would then be 18, 12 or 6.