Copyright © University of Cambridge. All rights reserved.

This was an engaging problem. It encouraged you to think about factors and multiples and showed how these principles may be applied to real-life situations! It also prompted you to think about different kinds of shapes.

Mrs. Heffernan's students from Paton School, Shrewsbury, Massachusetts, USA targeted this problem using the "chunking" method of division. There are twenty four students, and one group of three students makes a triangle. So, to find out the number of triangles they can make, we need to know how many groups of three there are.

Caitlin, from Deer Hill School worked out the correct answer, also noting that no-one should be left out. Nadia from Wimbledon High School and Ryan from Rhu Primary both sent excellent solutions to this problem. Nadia says:

Well for the triangle one the answer is eight because there
are $24$ children and you need three children for each triangle so
you divide $24$ by $3$ and you get $8$.

Then for the question that asks you what different shapes could be made the answer is a square, a rectangle, a diamond and many more.

It might be better not to use the word "diamond" in maths - can you think of the mathematical names of shapes that we might think look like a diamond? Ryan suggests that parallelograms are possible too. What other four-sided shapes could we make?

Carolyn, from Pigeon Mountain Primary School suggested some four-sided shapes, including parallelograms, and rhombuses. Priya and Ashleigh, from Penrhos College also mentioned a trapezium for another four-sided shape. Mrs. Heffernan's students pointed out that a rectangle is actually a type of parallelogram; it is a special version, because all of the angles are $90^{\circ}$. Similarly a square is a special type of rhombus, with right angles.

The problem then asks about the numbers of other shapes that the class can make. Again, Mrs. Heffernan's class used the same method ("chunking") to find the answers. Nadia, and Ryan also submitted correct solutions. Ryan explains:

They can make six four-sided shapes ($24\div 4=6$) , four
hexagons ($24\div 6=4$) and three octagons ($24\div 8=3$). To make
five pentagons Mrs. Trimmer can help.

Like Ryan, Nadia also thought that Mrs Trimmer could join in - that's a good idea. Sarah from Greenlands Secondary School also suggested this, as did Matthew from Stambridge. Caitlin, and Margaret from Deer Hill School, and Ryan and Trystan also sent in correct solutions, with great reasoning.

Kieran from Newman Primary School, and Ebony from Gordon Primary suggested that one child could hold two corners of a pentagon, which would be another good way around the difficulty. Priya and Ashleigh suggested that four children could be left out so that four pentagons could be made.

Can you think of any other ways that the class could make shapes with the string? What if some people (or even all of them!) held two pieces of string, one in each hand? What shapes could they make? How many?

Well done, and thank you for your solutions.