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A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### Lap Times

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the track.

### Why do this problem:

This problem forces attention on what happens when we calculate area by dissection into known shapes and the care we should take in our justification. In problem solving diagrams can often be misleading, and not just diagrams we draw for oursleves. There is a need to question and check that the representation is meaningful and conveys the message accurately. This problem brings that idea to the fore.

### Possible approach :

You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

Many students will need to dissect the 8 by 8 square into the four pieces to begin to get a feel for the possible source of the discrepancy. Although students want to know the answer (that's what motivates the problem-solving effort) it is the process of finding a route to an answer which provides the real long-term benefit.

Once someone says out loud where to look for the discrepancy the problem is over.

To avoid this, encourage students who are ready to develop a convincing argument to share and then to move on to the challenge of creating similar problem. When ready move to a whole group discussion emphsising the need for convincing arguments. The extesnion activity also gives scope to those who see what is happening quickly.

### Key questions :

• What assumptions have been made?
• What is the area of each piece?
• How would you convince someone else?

### Possible extension :

Do you recognise the numbers involved in this problem? Can you account for why this problem works with those numbers?

Try the problems Muggles Magic and Lying and Cheating.

### Possible support :

Explore the area calculation for a Parallelogram and then for a Trapezium by dissection.