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?
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
Find a great variety of ways of asking questions which make 8.
If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...
Arrange the numbers $1$ to $6$ in each set of circles below.
The sum of each side of the triangle should equal the number in the centre of the triangular shape.
Once you've had a chance to think about it, click below to see how three different pupils began working on the task.
"I used counters which had $1$ to $6$ on them.
I put the counters in a triangle in any old way, then I added up the sides.
Then I moved the counters around to try and get the right total on each side."
"I noticed that three of the numbers are odd ($1, 3$ and $5$) and three of the numbers are even ($2, 4$ and $6$). I thought this might help.
I know that $9$ is an odd number so it can be made using odd + odd + odd or using even + even + odd."
"If I want a small total on each side, I'll need small numbers in the corners of the triangle."
Can you take each of these starting ideas and develop it into a solution?
A practical version of this activity is included in the Year 3/4 Brain Buster Maths Box which contains hands-on challenges developed by members of NRICH and produced by BEAM. For more details and ordering information, please scroll down this
Try not to direct learners too much at this stage but make sure they understand that they can use any resources or equipment that they might find helpful. (You might want to have these sheets of blank triangles available should anyone request them: Word document or pdf).
Once you feel that most children have made progress and understand the problem well (this does not necessarily mean that they have found all the solutions), give out this (.doc or pdf). Suggest to the class that
when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". It might be appropriate to read through each method as a whole class before giving pairs time to work on each
one. Alternatively, you may prefer to allocate a particular starting point to each pair.
Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages
of each. Which way would they choose to use if they were presented with a similar task in the future? Why?
(You may find that conversations arise about the number of different solutions for each total. Encourage children to articulate what they think is the same and what is different in this context. They might not all agree!)
Tell me about this approach. What do you think s/he was doing?
How do you think this will help to solve the problem?
What do you think s/he would have done next?