Tumbling Down

Why do this problem?

The idea of a fraction wall may be familiar to some but the animation in this task will fuel their curiosity and encourage exploration of sequences of fractions, and equivalent fractions, in a meaningful way.

Possible approach

Watch the video all the way through with the class, asking them to resist from talking as it plays and to think about what they see.  Give chance for pairs to chat and as they do so, move around the room and listen to their conversations.  Write up snippets of what you hear on the board and draw learners' attention to this, merely explaining that this is what you have overheard.  (You could read out what is written if children will struggle to read it independently.)

Play the clip again all the way through and ask the group to watch again.  This time suggest they can talk while they watch if they wish.  Give them chance to discuss further in their pairs and challenge them to describe what they notice.  You could share some points and then play the video again, stopping and starting as necessary as you facilitiate the whole class discussion. Try to acknowledge all contributions, even if learners offer something that you had not thought of, or that seems irrelevant.

Through the discussion, draw out the key features of the video, such as:

• The bottom row is a whole block, the second row up is split into two halves, the third row is split into three thirds, the fourth is split into four quarters etc.
• As the video plays, the rows 'fall into' each other
• Some of the lines overlap as the rows fall.

(Your class is likely to develop their own vocabulary and way of describing what they see - it does not matter if they don't refer to 'rows', for example.) Learners might also remark on the symmetry in the final image.

After this general discussion, pose the questions about the numbers of vertical lines and give learners time to consider the reasons that some lines 'fall into' each other.  You could organise the class in groups of three or four at this stage, or stick with pairs.  Bring the whole class together to share thoughts.  The key idea here is equivalence and learners might justify this using the image, or in a more abstract way without reference to the picture necessarily.  You might wish to share different ways that pairs/groups have recorded their thinking.

Give time for the final challenge and in the plenary begin by giving opportunities for a few pairs/groups to share their thoughts.  If appropriate, encourage a move from the particular to the general - can anyone describe what will happen if you keep on introducing more and more rows to the starting image?

Key questions

What do you see?
What do you notice?
Why do some lines 'fall into' others?
How would you add to the picture to continue the pattern?
What would happen if your new picture was animated?

Possible extension

Learners who are confident at comparing, ordering and recording fractions might like to explore the Stage 3 task Farey Sequences which focuses on the same mathematical structure.

Possible support

Using strips of paper, for example 30cm long, would enable learners to create their own model.