### Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

### Cuisenaire Rods

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

# Cut it Out

## Cut it Out

An equilateral triangle can be dissected into four (equal) smaller equilateral triangles.

Can you dissect an equilateral triangle into six smaller ones, not necessarily all the same size?

Is it possible to dissect a larger equilateral triangle into any number of smaller equilateral triangles?

Are there any numbers that are impossible?

### Why do this problem?

At the most basic level, this problem provides an opportunity to reinforce what an equilateral triangle is, and whilst that will be sufficient for some, this question requires much more sophisticated thinking than that. For the high flyers in your class, the fact that it is a 'tough nut' (ie no children's solution has so far been submitted) might provide the extra incentive to persevere when the going gets tough.

### Possible approach

This problem requires some time to do properly. If you can introduce it whilst doing 2D shape as a main topic, you might want to offer it as a 'simmering' activity for a small able group who can return to it over several days, perhaps when they have completed their allotted tasks.

The question is in three parts and these are best introduced one at a time. Give out sheets of equilateral triangles and allow some time for the children to explore the first question either individually or in pairs. Some children might prefer to draw their triangles on isometric paper. Listen for statements which are accompanied by some sort of justification, for example "Four is the smallest because after one big triangle the next size up has two triangles along each side." Encourage the children to work systematically.

If you think it appropriate, make available a section of display space where they can place their diagrams, cut out and sorted into 'families'. This way the individual findings can be a contribution to a greater whole.

### Key questions

What is the smallest number of triangles we can break one large one into?
How do you know?
Is there a largest number?

### Possible extension

Some children will want to find a general rule. This is a sophisticated piece of generalising.

### Possible support

You may want to give out triangular grid or dotty paper and ask the children to draw equilateral triangles of different sizes and count how many smaller triangles are in each. This will help them to see that 1, 4, 9, 16 are useful numbers to use.