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Ladder and Cube

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?

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Doesn't Add Up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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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.

At Right Angles

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem challenges students to use what they know about gradients to make general statements about the gradients of perpendicular lines.
This is a suitable preparation for students who are about to move on to:

Possible approach

This problem follows on from How Steep is the Slope?

Introduce students to the existence of tilted squares, by playing the game Square it or using the activities Eight Hidden Squares and Ten Hidden Squares.

Hand out this 8 by 8 square dotty grid and ask students, perhaps working in pairs, to find all the different tilted squares which can be drawn with vertices at the dots. Clarify that two squares are the same if one could be cut out and placed exactly on top of the other. There are twelve different tilted squares. There is no need to share this information with the students; ideally challenge the students to justify that they have found all the possible tilted squares.

Once students have had a chance to find most of the tilted squares, ask them to report back. Agree a way of describing the squares they have found, perhaps by reference to the way you move to get from one vertex to the next. The interactivity can be used to check that the shapes identified are indeed squares. Keep a record of the squares found, perhaps in a table that can be added to later.

Once all twelve have been identified, and there is agreement that there are no more, ask students to find the gradients of adjacent sides in each square. Add this information to the table with the square descriptors.

Bring the class together to discuss what they have noticed about the gradients. Can they predict the gradient of a side of a square if they are given the gradient of the adjacent side?

Ask students to describe a method for checking whether two lines are perpendicular to each other.

Now set the task at the end of the problem where students must decide whether two lines given by their coordinates are perpendicular. Students could follow this up by making up their own sets of lines (which must contain some which are perpendicular and some which are not) and challenging their partners to identify the perpendicular ones.

Key questions

How do you think the computer decides whether a shape is a square?
How do you decide whether two lines are perpendicular or not?

Possible extension

This problem could be followed up by Square Coordinates and then Tilted Squares for a geometrical introduction to Pythagoras' Theorem.

Alternatively, students can move into algebra by investigating the relationship between the equations of parallel and perpendicular lines on a co-ordinate grid in Parallel Lines and Perpendicular Lines

Possible support

If students are struggling to draw tilted squares, they could work at computers using the interactivity provided in the problem. They could complete squares of different sizes and list how to get from one vertex to the next. These results could be used in the next stage of the lesson.

If students are struggling to describe gradients, they could take a look at How Steep Is the Slope?