### Rectangles with Dominoes

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

### Chain of Changes

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

### Making Shapes

Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?

# Poly Plug Rectangles

## Poly Plug Rectangles

This activity has been inspired by Doug Williams' Poly Plug resource.  You can find out more details, including how to order sets of Poly Plug, on the Mathematics Centre website.  However, you do not need sets of Poly Plug to have a go at this activity - please see below and take a look at the Teachers' Notes.

In this activity, the computer secretly makes a rectangle using equal rows of spots on the $5$ by $5$ grid.

The aim is for you to find the rectangle by testing spots on the interactivity below.

Click on 'Start' and the computer will tell you the total number of spots its rectangle uses.

In the left-hand 'test' grid, click on a spot to see whether it is part of the computer's rectangle.  If it is part of the rectangle, the spot will turn yellow.  If it isn't part of the rectangle, the spot will turn blue.

Once you think you know where the computer's rectangle is, you can create it on the 'decide here' grid.  Clicking 'Reveal' will show whether you are correct.

Which spot is a good one to test first?  Why?

If you had to use as few test spots as possible, how would that change the way you play?

Are there some total numbers of spots that are easier than others?

We would love to hear about the strategies you use for finding the computer's rectangle.

This text is usually replaced by the Flash movie.

### Why do this problem?

This problem reinforces children's knowledge of the properties of a rectangle.  It also provides opportunities for them to make hypotheses, explain their reasoning and test out their ideas.
As with most NRICH problems, we 'roadtested' this with a class of real children.  You can see annotated videos of our attempts below.   We learnt a lot ...

### Possible approach

If you have sets of Poly Plugs (see the Mathematics Centre website), please follow version 1 below.  If not, version 2 will be more appropriate.

Version 1
If this is the first time the children have handled Poly Plug, give them time to play (we found $15$ minutes was about right).  Observe what learners do and comment on anything mathematically interesting, drawing out ideas such as repeating patterns, symmetrical patterns, pictures etc.
Then, in pairs, ask children to take out all the red plugs and count out six blue/yellow plugs.  What can they make using their six plugs?  Share interesting ideas, and finish by drawing out the idea of a rectangle of six plugs, perhaps by showing a rectangle someone has made.

Show the interactivity and click 'Start' until the computer has chosen a rectangle made of six plugs.  Explain that the computer has chosen a rectangle using six plugs (or spots).  Can we find out where it is?   Tell the group that when they test a plug, it will turn yellow if, yes, it is part of the rectangle but blue if, no, it isn't part of the rectangle.

Invite children to identify a plug to test.  Click on the chosen plug and reinforce whether it is or isn't in the rectangle.  At this stage don't worry about the reasoning.  Every so often ask, "Do we know where the rectangle is?".  When this is certain, check on the right-hand grid.  (This itself can be a challenge as the children transfer the rectangle from one grid to the other.)

Repeat with a new rectangle of six plugs, this time asking for a justification for each plug to test.  Listen for explanations that indicate the children are visualising the possible locations of the rectangle.  You may have some sophisticated thinking where rather than confirming what they know, children test one of two or more possibilities.  (If appropriate emphasise that we're looking at filled rectangles, not just outlines.)

With a child, model the same activity using the Poly Plug so that one of you secretly visualises a rectangle of six plugs on the blank red grid and the other points to the holes to test.   The blue/yellow plugs are used to confirm.  Give the children some time to play this in pairs.  You may want to challenge them to discover the rectangle in as few plugs as possible.

Having had a go with six plugs, children could choose the number of plugs their rectangle contains.

Version 2
Either individually or in pairs, give children a copy of this large $5$ by $5$ grid and a selection of counters.  Invite them to make a pattern on the grid using any number of counters.  Observe what learners do and comment on anything mathematically interesting, drawing out ideas such as repeating patterns, symmetrical patterns, pictures etc.

Then ask learners to make something using exactly six counters of any colours.  Share interesting ideas and finish by drawing out the idea of a rectangle of six plugs, perhaps by showing a rectangle someone has made.

Show the interactivity and click 'Start' until the computer has chosen a rectangle made of six  spots.  Explain that the computer has chosen a rectangle using six spots.  Can we find out where it is?   Tell the group that when they test a spot, it will turn yellow if, yes, it is part of the rectangle but blue if, no, it isn't part of the rectangle.

Invite children to identify a spot to test.  Click on the chosen spot and reinforce whether it is or isn't in the rectangle.  At this stage don't worry about the reasoning.  Every so often ask, "Do we know where the rectangle is?".  When this is certain, check on the right-hand grid.  (This itself can be a challenge as the children transfer the rectangle from one grid to the other.)

Repeat with a new rectangle of six spots, this time asking for a justification for each plug to test.  Listen for explanations that indicate the children are visualising the possible locations of the rectangle.  You may have some sophisticated thinking where rather than confirming what they know, children test one of two or more possibilities.  (If appropriate emphasise that we're looking at filled rectangles, not just outlines.)

With a child, model the same activity using the large paper grid so that one of you secretly visualises a rectangle of six counters on the blank grid and the other points to the holes to test.   Blue/yellow counters are used to confirm. Some children might find this sheet of six blank grids useful. They could use it to record their own rectangle so that they don't need to rely on visualisation. Give the children some time to play this in pairs.  You may want to challenge them to discover the rectangle in as few counters as possible.

Having had a go with six counters, children could choose the number of counters their rectangle contains.

### Key questions

What shape is this?  How do you know?
What can we say about a rectangle?
Where could the rectangle go?  How do you know?
Where can't the rectangle go?  How do you know?
Which plug/spot shall we test now?  Why?

### Possible extension

This version of the interactivity includes diagonal rectangles.

### Possible support

This sheet contains six different rectangles each using six plugs which could be used if children find it difficult to imagine their own.  The sheet could be cut up into six cards and the child creating the rectangle could choose one card without showing it to his/her partner.  Alternatively, all six cards could be visible to both players all the time for comparing and checking.

### Annotated videos

With thanks to Class Two children and teachers at Bourn Primary School.

Clip 1: We gave the children some time to play with the Poly Plugs as these were new to them.  Not all used them to make patterns within the grid but almost everything was mathematical!  There was a lot of noise, most of it productive, and our feeling was that if we hadn't given them this time, they would not have been so focused later on.

Picture 2: Having given them time to play freely, we then suggested they make any pattern or picture they like using just six plugs.  We used this child's pattern to draw out the properties of a rectangle.

Clip 3: We then introduced the class to the interactivity.  We made the mistake of using a rectangle of four plugs, rather than building on the work they'd just done using six plugs to make a rectangle.  You can see the difference in their reactions in this clip and clip 4 which does use six plugs.  We re-wrote these teachers' notes after this experience.

Clip 4: In this clip, we encourage the children to find the rectangle more quickly and this time it contains six plugs.

Clip 5: We were keen that the children should share their thinking.  It was interesting, but unsurprising, that they were disappointed when a plug turned blue.

Clip 6: In this clip, we confronted their keenness to choose only positive examples, trying to help them to see that a blue plug could give them far more information than a yellow one.  We're not sure we won though!