Mathematical Explorations Six

Mathematical Explorations Six

Ages 11 to 12

Exploring Geometry & Area Pebbles
Stage: 2 and 3 
Use this activity to introduce the youngsters to an investigation that mixes both shape and space work with number work. You could also introduce learners to this extended piece of work to help you look at perseverance and persistence. Great Squares
Stage: 2 and 3
The activity is good for pupils working with squares and triangles. It will increase their familiarity with the properties of these shapes and is a good introduction to 'thinking outside the box'. Isometric Areas & More Isometric Areas
Stage: 3
This problem invites students to look at area in a slightly different way from usual, using a triangle as the basic unit of area rather than a square, to reinforce the concept that area is about the space enclosed within a shape. Along the way students have the opportunity to derive and justify a formula that they will not have met before.


Investigating patterns and permutations

Triangle numbers

This problem allows students to investigate the relationship between triangle numbers expressed as a sum and triangle numbers expressed visually. Students may focus on either representation to arrive at a conclusion. The problem can be concrete with students focusing on small numbers or can be extended to give a general result.

Skills and concepts experienced: Generalising. Visualising. Creating expressions/formulae. Mathematical reasoning & proof. Triangle numbers. Interactivities. Patterned numbers. Curious.

Tower of Hanoi

The Tower of Hanoi is a well-known mathematical problem which yields some very interesting number patterns. This version of the problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights.

Initially working on the building blocks gives students the opportunity to then work on harder mathematical challenges than they might otherwise attempt.

The problem is structured in a way that makes it ideal for students to work on in small groups.

Skills and concepts experienced: Practical Activity. Creating expressions/formulae. Generalising. Resilient. Visualising. Resourceful. Possible Roadshow. Patterned numbers. Mathematical reasoning & proof.

1 Step or 2 steps

This problem provides an opportunity to draw out from students techniques for problem solving.

A key to solving this problem is to simplify to smaller staircases, and use these earlier results explain what is happening and hence calculate later results.

Skills and concepts experienced: Sequences. Curious. Generalising. Integers. Mathematical reasoning & proof. Combinations. Processing and representing data. Visualising. Fibonacci sequence.