### Surds

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

### The Root of the Problem

Find the sum of the series.

### Fit for Photocopying

Photocopiers can reduce from A3 to A4 without distorting the image. Explore the relationships between different paper sizes that make this possible.

# Origami Shapes

### Why do this problem

This offers an opportunity to explore a situation which requires multiple applications of Pythagoras' Theorem, working with surds and finding areas of less common shapes as well as extend manipulation of surds.

### Possible Approach

Give time to explore what happens when you fold A4 paper. Discuss how the lengths of sides stay in the same ratio if the paper is folded in half perpendicular to its long side and that is ratio is maintained no matter how many folds are made. Can students see a pattern in the lengths of the sides of the resulting rectangles if the paper is considered to have a short side of $1$ unit to start with?

Now move into the problem, encouraging learners to share ideas and verify each other's working and results at stages throughout the process. Using larger sheets of A3 (why does this work) to share ideas and to note lengths as part of a classroom display that tells the story of the solution will motivate learners to ensure their so-workers also understand what is happening.

### Key questions

• Why would a starting point of any paper"A" paper work?
• What information do you need to find the answer?
• What do you know?

Napkin

### Possible extension

You might utilise the second part of this problem for the extension activity and focus on the first part as the main part of the lesson.