## A parabolic investigation

The purpose of this activity is to engage students in an investigation of parabolas in a practical context.

## About this resource

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

- Factorising and expanding quadratic expressions.
- Graphing relations and functions on a Cartesian plane.

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.

# A parabolic investigation

NA5-7: Form and solve linear and simple quadratic equations.

NA5-9: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.

## Required materials

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

*A parabolic investigation activity*(.pdf)

## Activity

Task: A rectangular sheet of card with a perimeter of 80 cm is made into an open-topped box by folding in 2 cm x 2 cm squares from each corner. Investigate the relationship between x, the length of one side of the card, and C, the capacity of the box.

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.

- Do I understand the situation and the words? (Students may need to build a single open box to make sense of the requirements.)
- What varies in the problem, and what stays the same? (The size of the cut-out squares and the perimeter of the rectangular paper stay constant. The dimensions of the rectangular paper vary.)
- Does this look or sound like a problem I have worked on before? (Students may have encountered optimisation problems before.)
- What will my solution look like? (The solution gives the dimensions of the rectangular paper that yield the open box with the greatest volume.)

### 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? (Students will need to recognise that rectangles with equal perimeters can have different areas. That means the boxes are likely to have different volumes.)
- What could the solution definitely not be? (Set sensible bounds for the sides of the rectangular paper.)
- How could I show this problem using numbers, calculations, pictures, graphs, tables, or materials?
- Do I know how to calculate the areas of rectangles and the volumes of cuboids?
- 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.

- Have I shown my workings in a step-by-step way? Have I organised my results systematically to look for patterns?
- Have I got enough examples? Have I got all the examples that are possible?
- How could I make sure that I haven’t missed anything?
- Are there relationships in my results? (What happens to volume as the rectangular paper gets "skinnier"?)
- Can I use algebra to represent area and volume with x as a variable?
- Can I solve equations to find the maximum volume possible?
- Could I graph the relationship between x and volume? What will the graph show?

### 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.

- What is the solution? Can I justify that I have found the maximum volume?
- Is my working clear for someone else to follow?
- How would I convince someone else that I am correct?
- Could I have solved the problem in a more efficient way, such as by using algebraic formulae?
- How might an efficient algebraic or graphic strategy be used for similar problems?

## Examples of work

The student makes and measures concrete examples to lead them to solve the problem. They link the practical task to a generalised model.

The student carries out directed calculations that will lead them to find a quadratic relationship.

The student carries out an algebraic investigation that allows them to describe a quadratic relationship.

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