Gym Bag
Can Jo make a gym bag for her trainers from the piece of fabric she has?
Age
11 to 16
Challenge level
Problem
Image
The easiest bag for her to make is a simple draw-string bag, consisting of a cylinder of fabric on a circular base.
Measuring her trainers, she found that they were nearly 30cm long and that a piece of string wrapped round both of them measured almost 60cm.
She decided that she should allow 1cm for all seams and an extra 2cm for the slot to contain the draw string at the top (so the seam allowance is 3cm in total for the top).
Student Solutions
The drawstring bag is made from 2 shapes: a circle and a rectangle, which together make a cylinder.
The circle must be big enough to contain the trainers, which have a circumference of nearly 60cm.
Circumference of a circle= x diameter
So the diameter = 60 /Ï€
We now have to add an extra 1cm either side for the seams. So the new diameter = (60 / π) + 2
As for the rectangle, it must fit around the circle as a cylinder, and be as long as the trainers, so it must measure 60cmx30cm. We then need to leave 1cm each side for the seam, plus an extra 2cm on the opposite side to the one that will be attached to the circle, for the drawstring (a total of 3 extra cm).
For simplicity's sake, we can say that this takes up the same room as a rectangle measuring 62cm x 34cm.
Jo has a piece of fabric measuring 100cm x 50cm. We need to see if the rectangle and the circle will fit on this piece of fabric. First the rectangle:
We can see from the space left that the circle will fit on the piece of fabric if the diameter of the circle is less than 38cm.
The diameter of the circle = (60/Ï€) + 2 = 21.1
So both shapes will fit on the fabric, and her gym bag will fit her trainers!
The circle must be big enough to contain the trainers, which have a circumference of nearly 60cm.
Circumference of a circle= x diameter
So the diameter = 60 /Ï€
We now have to add an extra 1cm either side for the seams. So the new diameter = (60 / π) + 2
Image
As for the rectangle, it must fit around the circle as a cylinder, and be as long as the trainers, so it must measure 60cmx30cm. We then need to leave 1cm each side for the seam, plus an extra 2cm on the opposite side to the one that will be attached to the circle, for the drawstring (a total of 3 extra cm).
Image
For simplicity's sake, we can say that this takes up the same room as a rectangle measuring 62cm x 34cm.
Image
Jo has a piece of fabric measuring 100cm x 50cm. We need to see if the rectangle and the circle will fit on this piece of fabric. First the rectangle:
Image
We can see from the space left that the circle will fit on the piece of fabric if the diameter of the circle is less than 38cm.
The diameter of the circle = (60/Ï€) + 2 = 21.1
So both shapes will fit on the fabric, and her gym bag will fit her trainers!
Teachers' Resources
Why do this problem?
This problem provides students with an opportunity to engage in mathematical modelling, using practical activity as a way of investigating a problem which focuses on nets. Many students find it difficult to relate the net of a solid to its 3-D appearance or to mentally unpack a solid to visualise its net, and the modelling approach will help them with this, without getting bogged down in calculation.This problem could be linked with the Design Technology curriculum, and used to support textiles work, particularly isses to do with seam allowance.
Possible approach
Equipment required:- It would be really helpful to have a cylindrical drawstring bag available so students can see what the finished gym bag ought to look like.
- Lots of scrap paper, material and card
- Squared paper
- Rulers, compasses, scissors
- Sellotape and glue
The cylindrical part is a rectangle with length equal to the circumference of the base.
Students should work out what the net is first, and see how it makes up into a cylindrical bag before considering the seam allowances required. Scrap paper and sellotape would be ideal at this stage.
They will then need to think about the relative sizes of the rectangular section and the circular section. How will they ensure that the length of the rectangle is the same as the circumference of the circular base? How will they ensure that the finished article actually fits the trainers, if this is possible? Squared paper would be useful at this stage.
Once they are ready to think about seam allowance, it might be better if they used fabric of some kind. Working in groups would help at this stage, with each group using their own piece of fabric 100cm x 50cm. This is an ideal opportunity to bring the Textiles teacher into the Maths classroom to provide expert advice.
Key questions
What shapes do you need for the gym bag?How do you make sure the gym bag will be big enough for the trainers?
Is there a 'best' size for the gym bag? What criteria might you use to judge?
Possible extension
If the piece of fabric 100cm by 50cm is big enough, what is the minimum sized piece of fabric which would do?If it isn't big enough, how big a piece of fabric would be needed?
Possible support
For students who struggle to understand how the 3-D gym bag is related to the 2-D net, it helps to have a collection of cylindrical objects which they can take apart and put back together again.For students who struggle with working out if the fabric is big enough, practical activity will help them to decide.
Students who struggle with the seam allowance will need fabric to work with.