Six to Four
2 out of 3
Move four sticks so there are exactly four triangles.
Problem
Move four sticks so there are exactly four triangles.
Getting Started
How many matchsticks do we need to make one triangle?
How many to make two?
Is there more than one answer to each of these questions?
Student Solutions
Thank you to everyone who sent in solutions for this one.
Charlotte , Natasha , Abbie , Adiva and Kristina (Moorgate CP School)
Harriette, Holly and Ellie (The Mount School, York)
Luke (Tattingstone School)
Christina (Marlborough Primary School)
Only the two very best solutions are shown below. They are best because they use all the match- sticks in the triangles and there are exactly four triangles (and only four triangles) in each.
Helen, Lydia and Antonia (The Mount School York)
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
The problem as written could be offered as a quick starter activity, perhaps at the beginning of the day as the children come into class. However by introducing associated activities it can provide a rich context for mathematical talk and mathematical thinking.
Possible approach
If you have an IWB you might choose to draw up 12 moveable thick lines to represent matches. If not an OHP with spent matches would work well.
Make the hexagonal pattern as in the problem and ask the children how may matches there are, and what shapes they can see. How many of each shape can they see? Ask them to visualise what happens if they take just one match away. Ask how many matches there are now, and what shapes they can see. Did they visualise correctly? Did it matter which match they chose?
Give each pair of children twelve spent matches or equivalent to make their own model. Ask them to visualise what happens if they take two matches away, and then to do it. Again ask how many matches there are now, and what shapes they can see. Did it matter which matches they chose?
Do the same again, this time taking away three matches. Again ask what shapes, how many, and whether it matters which matches they take. Draw attention to the possible different sizes of triangles and how many matches are needed for each.
Then pose the problem and allow some time for the children to have a go. Point out that although they are taking away matches as before, this time they are replacing them.
After some time bring the class together to share their solutions and talk about how they did it. Listen for explanations that use ideas from the previous discussion.
Key questions
How many triangles are we trying to make?
What is the most number of matches we would need?
Possible extension
Possible support
Isometric paper can be a useful tool in recording the positions of the matches for children with poor fine motor skills. They too can make up their own questions and will delight in posing them to you. They will need to be very clear in the instructions they are giving, so this is a good opportunity to emphasise the accuracy of mathematical language.