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?

Doesn't Add Up

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

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.

Which Is Cheaper?

Age 14 to 16
Challenge Level

Why do this problem?

This problem introduces inequalities in the context of car parking charges which we hope will be familiar to students. After working on this problem, perhaps students could go on and explore Which is Bigger?, which explores the same mathematics in a more abstract and algebraic way.

The tasks can be tackled using informal numerical approaches but offer a great opportunity to show students the power of algebraic and graphical representations for making sense of real world situations involving comparisons. By switching between different representations, students can persevere and solve problems which would otherwise be out of reach to them.

Possible approach

Introduce the first part of the problem:

In car park A, it costs 80p to park for the first hour, and an extra 50p for each hour after that.

In car park B, it costs £1.50 to park for the first hour, and an extra 30p for each hour after that.

Give students some time to work in pairs to answer the question "Which car park should I use?". As they are working, circulate and listen in to different students' approaches.

Bring the class together to discuss their thinking. If any students chose to use diagrams, highlight their approach to the rest of the class; otherwise take some time to discuss how the car park charges could be represented algebraically and graphically, and how these methods lead very efficiently to a solution.

Next, set the second part of the problem:

There is a Park and Ride service where it costs 40p per hour to park, but you also have to pay 60p for the bus fare into town. Alternatively, I could park for free at the railway station and get the train to Mathstown - a return ticket costs £3.50.

Should I use the Park and Ride, the train, or one of the car parks?

Challenge students to use some of the algebraic and graphical techniques discussed to solve this.

Again, bring the class together and discuss their approaches.

The challenges at the bottom of the problem can be used in a number of ways:

Pairs could work on all four challenges and then contribute to a whole-class discussion
Pairs could work on one of the challenges and then present their work to other pairs
Graph plotting software (such as GeoGebra or Desmos) could be used to experiment or to check students' solutions to the challenges

The ideas introduced in this problem can be developed further in Which Is Bigger? where the algebraic structure is made explicit from the start.

Key questions

Is there a length of time for which the charges at both car parks would be the same?

How do you know that one of the car parks will always be cheaper after that point?

How could you represent the car park charges graphically?

Can you express the charges as a function of the number of hours parked?

Possible support

Encourage students to draw tables to represent costs for 1 hour, 2 hours, 3 hours...

Possible extension

Which Is Bigger? offers some challenging follow-up questions.

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