Through the Window
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
Problem
Through the Window printable sheet
The store in my town which sells windows calculates the price of windows according to the area of glass used and the length of frame needed.
Can you work out how they arrived at the prices of the windows below?
Getting Started
Are there any windows that use the same amount of glass? How do their frame lengths differ?
Are there any windows that use the same amount of frame? How do their glass areas differ?
Student Solutions
We had these submissions from Ben and Will at Colwich Primary School:
There are six windows shown and you could put them in pairs of windows which have the same perimeter but different areas, or the same areas but different perimeters.
We chose to work with the £150 and £140 windows which have the same perimeter, but different areas.
For the £150 window, A = 9 and P = 12. For the £140 window, A = 8 and P = 12.
This showed us that 1 square of A = £10.
For the £140 window, this means that A = £80 and P = £60, which means that 1P = £5.
So the answer is that a piece of glass costs £10 for every square of area, plus £5 for every square edge.
We checked our answers on all the other pieces of glass, and it worked every time.
Thank you very much for sending in these ideas.
We were also sent the following solutions from Emily, Ebony, Rose, Alex and Paul from Wold Newton Foundation School, which go into more detail about a similar method to the one Ben and Will used. These pictures can be clicked on to make them bigger:
(These children are using the notation 9² to mean 9m².)
These are really good ideas - I like the way everybody has decided to find pairs of windows that either have the same area or the same perimeter to help them. Having pairs of similar windows makes it really clear what the difference is between them, and you've used that to find the price of a single square metre of glass or a single metre of frame. Thank you for sharing these ideas with us!
Teachers' Resources
Through the Window
The store in my town which sells windows calculates the price of windows according to the area of glass used and the length of frame needed.
Can you work out how they arrived at the prices of the windows below?
Why do this problem?
This activity will help children become more familiar with the concepts of area and perimeter as well as giving them lots of practice at calculating both. Rather than being presented with all the necessary information in an immediately accessible way, learners will have to make sense of the activity and decide how to go about tackling it.
Possible approach
Project the image of the windows and ask the class to suggest what might be important for a store to consider in deciding on the price of their windows. Lead into the task by explaining that the store in your town sets the price of their windows according to the area of glass used and the length of the frame needed.
Give out copies of this sheet, which contains the image, and invite learners to work out (in pairs) how the prices have been calculated. Allow them time to investigate without saying anything more at this stage. As you move round the room, listen out for those children who have useful insights. You may wish to draw everyone back together for a short time to share some of these insights (a mini plenary). This will help those who need encouragement and also gives others chance to articulate their emerging ideas.
Leave time at the end of the lesson to gather everyone together once again to discuss the different ways they approached the problem.
Key questions
Are there any windows that use the same amount of glass? How do their frame lengths differ?
Are there any windows that use the same amount of frame? How do their glass areas differ?
Possible extension
You could encourage learners to create a formula for the total cost of a window, based on the area of glass and its perimeter. The problem Warmsnug Double Glazing is a more challenging version of this problem, which includes a wrongly-priced window.
The task Price Match follows on from this one and focuses on creating and using formulae.
Possible support
It may help to suggest that learners write the area and perimeter of each window on the sheet so they don't need to keep re-calculating.