Counting Counters

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

Cuisenaire Squares

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?

Window Frames

Window Frames

Here is a window that has six panes of glass. It is three panes wide and two panes high. Can you see how many small pieces of edging I have used?

The pieces of frame go around the edge of the panes, so we call them 'edging pieces'. They are all the same length and there are 17 of them.
The picture on the right shows the edging pieces - there are 9 green pieces going across and 8 blue pieces going up.

Challenge 1

Explore other rectangular windows which are one window pane wider than they are high.
How many edging pieces do you need? (We need 17 in the example above.)
How many panes of glass are there altogether? (There are 6 in the example above.)
Explore the numbers you get as you make the windows bigger.

Challenge 2

In this picture, we can see the inside edging pieces coloured blue and the ones surrounding the shape coloured green:

This gives us 7 inside blue pieces and 10 surrounding green pieces, making 17 altogether.
Have another look at the rectangular windows from Challenge 1, which are one window pane wider than they are high.
What numbers can you get for the inside and surrounding pieces?

Challenge 3

Using the numbers you've got from exploring rectangular windows (that are one wider than they are high), what things can you find out about those numbers? Below you can see how others started exploring.

This idea from Louis uses sloping lines:

Louis thought of using sloping lines to cut across the windows and help him count the edging pieces. Here's a bit of his picture which so far shows that he has 2 green and 4 blue edging pieces.

Freya used her tables knowledge:

Freya thought of dividing each of the numbers she had got by 6 and seeing what she had left over. She worked out that 17 divided by 6 is 3, and there is 1 left over.

Petros used this idea for the areas of the windows:

Petros compared the number of panes in the whole window with the number of edging pieces needed.

Kia found a different way of looking at the little squares:

Kia looked for small squares where each one was made of 4 edging pieces. She coloured these small squares blue, and she coloured the leftover ones red.

Why do this problem?

This task provides an introduction to the ideas of area and perimeter, while allowing children to practise their counting skills and spot patterns in numbers. It encourages a systematic approach and provides an opportunity for pupils to explain how they know they have all the solutions. This task gives children the opportunity to pursue their curiosity, asking questions like "I wonder what would happen if I...?"

If curiosity is something you're developing in your classroom then the videos mentioned at the bottom of this page may be useful.

Possible approach

Begin by showing pupils a model similar to the blue and green version in the second picture. Discuss what children notice. Can they talk about the 'panes' and the 'edges'? How many of each can they see? Explain how the blue and green lines are different, and make sure that all pupils understand what the words 'height' and 'width' mean.

When pupils have understood these ideas, give them time to make the window frames and count the 'edging pieces' and 'panes'.

After each challenge, bring the whole class back together to discuss what number patterns they can see. The pictures in Challenge 3 can be used as prompts to encourage children to think of different ways of exploring these patterns.

Key questions

Tell me about how you are doing this challenge.

Possible extension

Use other windows when the longer side may be more than 1 unit longer than the narrower side. Encourage children to explore patterns, keeping the length and width of the windows no bigger than 6.

Ask children to investigate the following challenge: If you have 90 edging pieces to use, what window sizes can you make if you are allowed to use the pieces to make more than one window? Can you make windows which are all different where the number of edging pieces adds up to exactly 90?
e.g.
1 by 3 = 10, 2 by 4 = 22, 4 by 6 = 58 makes 90 altogether
1 by 6 = 19, 2 by 4 = 22, 4 by 5 = 49 makes 90 altogether

Wire Cages is a 3D version of this task.

Possible support

Some pupils may need support with working out which edging pieces to count. Providing simple models for children to move around would be helpful for all pupils.

Teacher support for curiosity

This task was created to help younger pupils pursue their curiosity within maths lessons. If you'd like any support with sparking curiosity in your pupils, these videos might be helpful.

Firstly "The Rise & Fall of Curiosity", particularly the extract [23.50 - 37.15] on "adult encouragement answering and teacher behaviour."
Secondly, "The Hungry Mind: The Origins of Curiosity", particularly the extract [8.22 - 12.29]  on "Children asking questions"

First can also be found at - https://www.youtube.com/watch?v=X-0NOrIU67w