## Pizza pans

The purpose of this activity is to engage students in using ratios to find a common ratio (scale factor).

## About this resource

This activity assumes the students have experience in the following areas:

- Finding the areas of polygons and circles.
- Naming parts of circles, such as circumference, diameter, and radius, arc length.
- Manipulating rational algebraic expressions.
- Applying ratios.

The problem is sufficiently open-ended to allow the students freedom of choice in their approach. It may be scaffolded with guidance that leads to a solution, and/or the students might be given the opportunity to solve the problem independently.

The example responses at the end of the resource give an indication of the kind of response to expect from students who approach the problem in particular ways.

# Pizza pans

## Achievement objectives

GM5-9: Define and use transformations and describe the invariant properties of figures and objects under these transformations.

## Supplementary achievement objectives

NA5-4: Use rates and ratios.

## Required materials

See **Materials that come with this resource** to download:

*Pizza pans activity*(.pdf)

## Activity

Task: A family-sized pizza is baked in a pan that has twice the base area of a standard pizza pan.

Find out how much bigger the radius of the family pizza pan is than the standard; that is the scale factor for this enlargement.

The following prompts illustrate how this activity can be structured around the phases of the mathematics investigation cycle.

### Make sense

Introduce the problem. Allow students time to read it and discuss it in pairs or small groups.

- What are the important words and symbols? (Students will need to interpret the meaning of base area and radius.)
- Can I imagine (visualise) what the pizzas look like?
- Where else in my life/the world can I see this happen? (Students may have encountered enlargement in other situations.)
- What will an answer to this problem look like? (It will provide a scale factor as a number.)

### Plan approach

Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.

- What are the maths skills I need to work this out?
- What else do I need to know to get started?
- How could I show this problem using a diagram? (Can students sketch a hypothetical standard-sized pizza and a family-sized pizza with twice the area?)
- What rules will I need to solve this problem? (Students need to know or access the formula for the area of a circle.)
- How many examples of circles should I try?
- How might algebra help me with this problem?
- What tools (digital or physical) could help my investigation?

### Take action

Allow students time to work through their strategy and find a solution to the problem.

- Does my scale factor seem correct? Is it close to my estimation?
- Is there another possible way to solve the problem? Could I use algebra?
- Does the scale factor work for many examples (cases)?
- How do my results look different from others? Why could this be?
- How can I prove that my scale factor is correct?

### Convince yourself and others

Allow students time to check their answers, and then either have them pair share with other groups or ask for volunteers to share their solution with the class.

- Show and explain how you worked out your solution.
- Can others see how I worked it out?
- How does my solution answer the question?
- Would my answer work in a different situation? (Ideally, students might generalise that a scale factor of √2 doubles the area of the original figure.)
- What scale factor would I use to triple and quadruple the area?

## Examples of work

The student finds a decimal scale factor using concrete examples in their calculations.

The student finds an exact (surd) way to record the scale factor, using concrete examples in their calculations.

The student finds an exact (surd) way to record the scale factor using algebraic generalisations.

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