This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
This fun problem encouraged you to work
systematically, and develop a way to record and organise your
findings. For some numbers of sticks, there are several different
possibilities, so it is important to approach the problem
carefully, to ensure all of the options are explored.
Cong from St Peter's Roman Catholic School,
Aberdeen was very thorough indeed, and sent in a fantastic solution
to this problem. You must have worked very hard, Cong!
Cong sent us a table of all the different
triangles you can make with 4 sticks up to 20 sticks. Making a
table was a very good idea. However, it's a bit long to put on a
webpage, so I've made a list of all the triangles in Cong's table
so you won't have to scroll too far to read them all!
Cong has made each triangle in turn in a very
systematic way. Can you see how Cong has started with the longest
side possible each time and worked down? He always lists the side
lengths with the longest first as well. This makes it very easy to
check. Well done!
Cong summarised the number of possible
triangles with each number of sticks in another table:
He also showed the same information on a bar
chart. Notice that there are no possible triangles with four
sticks. Holly from Anston Brook Primary School said:
Children from Riversdale Primary School
examined the "rules" for the longest side of any triangle:
Well done for spotting this. Can you see why
this is the case? This is actually an example of the "triangle
inequality". Do not worry, this sounds more difficult than it is!
The triangle inequality applies to any triangle, and states that
the length of a given side must be less than or equal to the sum of
the two other sides. So, imagine that you have a triangle with
sides of three, four and five. The longest side, five, is less than
the sum of the two other sides ($3+4=7$). So, this is an example of
the triangle inequality in action!
What does this mean for the problem with
sticks? Well, firstly it provides a quick way to check if the
triangles you have thought about are possible. Secondly, it can
also save you some time when working out all of the possibilities.
For example, you may be able to see that, for this task with
sticks, the triangle inequality means that the length of any side
cannot be greater than half of the total number of sticks. If the
length of a side used is more than half of the total number of
sticks, this would not agree with the triangle inequality, and a
triangle could not be made. Why not have a go at this to see for
yourself to check you understand why the triangle inequality is
Cong also spotted a general "rule":
Yes, I see what you mean, Cong. Children
from Riversdale Primary School also submitted a similar table. They
also made a similar observation:
Odd numbers of seven and above always have more triangles than
the even numbers adjacent to them. Odd numbers in the three times
tables have even more as
they have an equilateral triangle as well.
In addition, Ellie, Emma, Olivia and Ibrahim
from Lakeside looked at "rules" for being able to make equilateral
Well done for these fantastic solutions