Making sticks

This problem encourages children to use counting-on techniques, but offers you an opportunity to introduce them to the idea of multiples.

Having cubes available for the children to use is a necessary part of this problem, as it makes it accessible to all. One way to introduce it would be for children to work in pairs, one of them making blue sticks (each of two cubes) and one making red sticks (each using three cubes), although of course the colour isn't important. Then, pose the questions in the problem for them to investigate together. (Depending on the kind of cubes you have, you may want the children to actually attach the sticks to each other, as just lining them up may mean that you cannot get them close enough together.) You could ask children to record their working, perhaps on squared paper by colouring squares.

Talking to the group about total lengths of blue sticks which match lengths of red sticks allows you to model the appropriate language, for example "$6$ is a multiple of $2$ and $6$ is also a multiple of $3$". However, it has a lot of scope to be taken further - the open-ended nature of the activity also allows children to make a generalisation about all the lengths of sticks that can be made from both blue and red. Although many may not be able to verbalise this formally, they will certainly be able to look for patterns in the numbers that are possible and this can lead to a fruitful discussion.

This work would make a lovely display, for example using sticky red and blue squares on a large grid.

How many cubes have you used in this line? And this line?

Can you find any other lines that are the same length as each other?

What is the next line that can be made from both red and blue sticks? How do you know?

Some pupils could investigate sticks of two different lengths, for example 2 and 5; or even three different lengths.

Some children may have difficulty keeping track of the number of sticks they have joined together. It would be worth you talking about strategies to help with this, such as counting in threes once a long line has been constructed.