Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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?

Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

CD Heaven

Why do this problem :

This problem creates a context where students are challenged to pick out detail from a large block of text, represent that data and the relationships between the data as a set of equations, which need solving and interpreting back into the practical context.

Possible approach :

This kind of work can be hard and not always rewarding for individuals not motivated just because it's a puzzle. It may be that the problem achieves most to develop a problem solving culture in the classroom if it is approached with plenty of discussion between students and in the spirit of a group effort. Less motivated students can still verify and critique the proposals of others.

Pair students to extract the relevant data and relationships from the text. Check orally that all the group has all the needed data and debate items that are not universally agreed upon.

Ask pairs to express these relationships as equations, again pooling all and discussing elements of contention.

Set students to work looking for a solution, which is then written up poster-size for display around the room.

Students peruse each others results and presentation of method, before a whole group discussion of the routes followed and the comparative efficiency of the methods deployed. Finally, ask different students to explain how the algebraic results work as a meaningful solution to the practical problem context.

Key questions :

• Pick out all the data that you are going to need from this text.
• Try to get clear about exactly how these values relate to each other and express those relationships as algebra (equation)
• What does it mean to solve those equations and can you see how to manage that ?

Possible extension :

Matchless is an interesting challenge which pushes students into a deeper understanding of simultaneous equations.

Possible support :

The style of approach suggested above should allow a wide range of ability within a group to function effectively as a problem-solving culture - where abler students explain more, and less able students providing verification of results and feedback on effectiveness of communication.