### Homes

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

### Train Carriages

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

### Let's Investigate Triangles

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

# 3 Blocks Towers

## 3 Blocks Towers

Take three different colour blocks, maybe red, yellow and blue.
Make a tower using one of each colour.
Here's one with red on top, blue in the middle and yellow on the bottom.

Now make another tower with a different colour on top.

How many different towers can you make?
How will you know that you have found them all?

You can click below to see how two learners started this task.

Jemima says:

I used multilink cubes to make the towers. First of all, I looked for towers that were red at the top.

Elias says:

I drew coloured squares for the blocks. I put them in any old order to start with:

Then I moved red to the bottom and moved yellow and blue up to make a new tower:

I did the same again, moving yellow to the bottom:

I did this again, but realised that would give the first tower.  So I went back to my first tower and kept the bottom cube the same, but swapped the top and middle.

Did you start the problem in the same way as either of these children?
What do you think about each method?

You may like to print off one of these sheets for recording three-block towers.

### Why do this problem?

There are two parts to this problem. Firstly, making as many different towers as you can think of, and secondly, making sure that all possible ones have been made. In order to find all possible solutions, systematic working is essential and the two different starting approches given as examples (Jemima's and Elias' methods) can support children in understanding what working systematically might 'look like' and make them curious about other ordered way of working.

By providing opportunities for your class to reflect on different ways of solving a problem and talking about them, learners can begin to appreciate the problem-solving journey and not just the answer.

### Possible approach

It would be helpful to have a lot of interlocking cubes (or stackable blocks) in three different colours for this task.

Start with a whole class activity in which you invite a child to make a three-block tower for all to see.  Then ask someone else to make a different three-block tower. Invite other leaners to explain why the two towers are different. Repeat the process so that another different tower is made, again asking someone to explain why this third one is different from the first two. At this point, introduce the task and give learners time to work in pairs.  You may like to give out this sheet to aid recording.

Once everyone has had chance to create a number of solutions, but not to have completed the task, share some different ways of approaching it.  You can do this by drawing on methods that your learners have come up with, or by using the examples of Jemima and Elias in the problem. (If the latter, this sheet of the problem and two methods my be useful.) Invite everyone to look at each method and understand it, before facilitating a general discussion about the two different approaches.  Did any pair use one of these methods?  What do they like about each?  What do they not like? Give time for pairs to continue to work on the problem, but invite them to choose one of the approaches they have heard about, if they so wish.

In the plenary, you could ask a couple of pairs to explain why they changed their approach, or not.  Allow time to then ask one pair to try to convince everyone that they have found all the possible towers. (You may want to have warned a few pairs beforehand so they have chance to refine their arguments.)

### Key questions

How will we make sure we remember all the ways we have got so far?
How can we be sure we have the different solutions?
If we cut out each individual solution, how could we order them to help?

### Possible extension

Working with four blocks is a natural extension to this problem and could provide more of a focus on working systematically. This sheet could be used for recording solutions.

### Possible support

Children who find this way of working difficult could be encouraged to colour their solutions on the three-block sheet provided, cut each out and order them into a pattern.