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