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?

Finding 3D Stacks

3D Stacks

This activity may be used as a follow-on for those who have successfully worked at Doplication.

The picture below shows a 2 by 3 by 4 arrangement of red spheres, which you might see as two layers of 3 by 4:

These two layers together could also be viewed as representing six cubes:

In the top picture the centre of each cube is shown with a light or dark brown sphere.

How many spheres are there altogether in this 2 by 3 by 4 arrangement?
How did you count them?

Use your system of counting to find the total number of spheres for other sizes, for example 3 by 3 by 3;  3 by 4 by 4; 4 by 4 by 5. etc.

How would you find the number of spheres for any sized arrangement?

Why do this problem?

This task acts as an extension to Doplication. It gives pupils chance to explore spatial awareness, number relationships and patterns deeply, and has the potential to be a meaningful context in which learners can create and use a formula.

Possible approach

You could show the group the images in the problem and invite them to 'say what they see'.  Facilitate discussion amongst learners so that the structure of the arrangement is clear.

Ask learners to work on the questions in pairs and then bring them together at a suitable point to check that everyone agrees with the total number of spheres.  Encourage them to articulate how they are going about counting and record the different ways on the board for all to see.  (Pairs may then stick with their own approach or adopt someone else's.)

Once they have had the opportunity to count several examples (a spreadsheet might be useful for recording these), challenge pupils to move from the particular to the general.  Give them time to compose instructions for counting the spheres for any size grid.  Once language-based instructions have been created, help them move to a formula.  Depending on the children's experience, you could do this as a whole group by choosing one method and working on it together.

Key questions

Tell me about what you have found out.
Can you describe how you arrived at these numbers?
How would you give someone else instructions for counting the total number of spheres?

Possible extension

Learners could compare the 3 by 3 by 3 arrangement of this challenge with the 3 by 3 of Doplication; then the 4 by 4 and 5 by 5 and so on.

Possible support

The visualisation may be difficult for some so using construction materials may be helpful.