Problems to develop mathematical processes and applications

Mathematics problem: Tilted squares

Problem outline

This problem hinges around finding areas of squares drawn on a square-dotted grid. It involves identifying patterns in the areas of squares with different tilts in order to be able to predict the area of a square with any tilt.

Why do this problem?

This is a rich task for sharing ideas and ways of working. It combines the need to adopt systematic approaches to collecting data with extension of knowledge of areas and a potential link to Pythagoras' Theorem. It also encourages identification of and managing variables and considering the constraints imposed by the environment. There is plenty of opportunity for learners to take different routes, depending on the variables they introduce, and to structure their findings in different ways.

For the problem itself and some associated teachers' notes

Tilted Squares (link opens in new window) (link opens in new window) from NRICH (link opens in new window) (link opens in new window).

Curriculum references: process

The guidance sections 'What teachers might do' offer suggested actions that can help to draw out pupils' skills in how to use mathematical reasoning. There is, however, a breadth of opportunities to develop a range of process skills including:

• Simplify the situation or problem in order to represent it mathematically, using appropriate variables, symbols, diagrams and models.
• Explore the effects of varying values, working logically towards results and solutions and recognising the impact of constraints.
• Calculate areas accurately and find effective ways of recording methods, solutions and conclusions to share.
• Make connections within mathematics, using knowledge of related problems and leading into new mathematics.
• Form convincing arguments, communicate findings effectively and engage with someone else’s mathematics.

Curriculum references: content

Algebra: Equations, formulae, expressions and identities.

Geometry: Geometrical reasoning; Measures and mensuration.

Working from a geometric context, reasoning about the properties of squares and calculating areas means that the problem can lead to the use of algebraic representation of generalisations.

Useful links

Other useful links including problems and articles from the NRICH website can be found in the 'Related links'.