Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Homes

## Homes

**Why do this problem?**

### Possible approach

### Key questions

### Possible extension

### Possible support

## You may also like

### Train Carriages

### Let's Investigate Triangles

### Teddy Town

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 5 to 7

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

A brand new road has been built on the edge of a village, and six new houses are being designed. The houses can be any of these three types:

- Detached houses
- Semi-detached houses
- Terraced houses

Have a think about what each of these might mean. Do you live in one of these types of house?

Detached houses are built separately and aren't joined to another house. Semi-detached houses are joined to one other house, so these types of house come in pairs. When there are three or more houses joined together they're called terraced houses.

The developers are trying to decide how many of each type of house to include. They could build houses that are all the same type, or they could build a mixture of types of houses. Don't worry about the order of the houses for the moment - we're just interested in how many of each type we can build.

Can you think of a combination of houses that could be built?

Are there any combinations that aren't possible? Why?

Here is one possible combination of houses:

This picture shows three terraced houses, two semi-detached houses and one detached house.

How could we record this combination? Rather than drawing the houses or writing the types of house, is there an easier way of keeping track of the combinations we've tried?

Once you've decided how to record your ideas, see if you can find all the possible combinations for the types of house that can be built. How do you know that you've found them all?

Once you've had a go at that, have a think about how we could change this question. You might think, "I wonder what would happen if...?"

For example, what would happen if more than six houses were being built? Or less than six? Does anything interesting happen if the total number of houses is odd instead of even?

What if we did care about the order of the houses? Try this with a smaller number of houses at first - if we wanted to build four houses, what are the different possibilities?

See if you can think of any more ideas for ways to change this question. Good luck!

This activity is an engaging way for younger children to practise their number bonds to six and their understanding of odd and even numbers. It also provides an opportunity for children to begin thinking about working systematically, as well as giving pupils the freedom to record their ideas in different ways.

As a class, talk through the different types of house and ask children what they notice. If they don't suggest it, point out that semi-detached houses have to come in pairs, and terraced houses have to come in groups of three or more. Ask the children for suggestions of what types of house we could build if there are going to be three houses on the road. Write or draw these options on the whiteboard, explaining that for this task we aren't worried about the order of the houses, only the number of different types of house. Do pupils have any ideas for how we could record these ideas? How can we tell that we've found all of the possibilities?

In pairs, provide pupils with pictures of the different types of house (at least six pictures of each type) and some paper on which they can record their ideas. Give children time to work together to find different combinations of six houses. As you walk around the room, discuss with pupils how they are
keeping track of their ideas. How are they making sure they don't get the same combination twice? At the end of the lesson, ask some pairs to discuss the way they approached the problem. How can we work systematically to be sure we've found all of the options?

How are you keeping track of your ideas?

Tell me about this arrangement.

How do you know you've found all the different possibilities?

Is it possible to have three semi-detached houses? Or two terraced houses? How do you know?

Depending on how the pupils have approached this activity, you may be able to ask them for ideas as to what other things they could now explore. If not, you can suggest looking at the possible combinations for five or seven houses.

Using multilink cubes as well as pictures of the different types of house might be help some children imagine the houses being 'attached'. If there are children who find recording their ideas difficult, taking photos of the different combinations might allow these children to focus on the activity rather than the recording.

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?

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?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?