The purpose of this activity is to engage students in applying their knowledge of fractions in a geometric context.

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

- Describing and classifying polygons by their properties.
- Identifying congruent polygons.
- Finding areas of rectangles.
- Expressing part-whole relationships using fractions.

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.

Here are two equilateral triangles. One has sides that are 3cm long and the other has sides that are 6cm long.

What fraction of the large triangle’s area is the small triangle?

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 in pairs or small groups.

- Do I understand what is being asked? (Key words like equilateral triangle, sides, fraction and area need to be discussed.)
- What are the important words and symbols? (The meaning of measures is required, such as 6 cm means “six centimetres.)
- What will my solution look like? (The solution will say the fraction that the small triangle is of the large triangle by area. The answer needs to be justified using a physical or diagrammatic model.)

### 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 maths ideas are involved in the problem? (Length and area, and fractions are key ideas.)
- What do I already know about these ideas? (Measurement and fraction knowledge and skills are needed.)
- What strategies might I use? Make a model? Draw a diagram? Act it out? Look for a pattern?
- What tools might be useful? (A physical representation such as grid paper and rulers will be helpful.)
- What order should I do things to solve the problem?
- How will I know when I solve it?

### Take action

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

- Is my strategy working or should I try something else?
- Have I tried drawing or making triangles the right size?
- Am I showing my workings in a way that helps me see patterns?
- What patterns can I see?
- Have I got all the maths I need, or do I need to ask for help?
- Do my answers seem correct? How can I check my answers?

### 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 answer?
- Is my working clear for someone else to follow?
- How might I convince someone else I am correct?
- Could I have solved the problem in a more efficient way? How?
- What could I find out next? What maths do I need to work on?
- Would my finding work on other shapes?

### Examples of work

#### Work sample 1

The student fills the larger equilateral triangle with a tessellation of smaller equilateral triangles and uses this to pattern to express the area relationship as a fraction.

Click on the image to enlarge it. Click again to close.

#### Work sample 2

The student creates a physical model of the large triangle and folds it into equal parts. They use a fraction to express the part-whole relationship.