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Challenge 1:
31 edge pieces
12 glass panes
I got an odd number of edge pieces
Challenge 2:
There are 14 outside edge pieces and 17 inside edge pieces. Same number of edge pieces as above.
Challenge 3:
All the number of edge pieces are odd.
Formula to work out edge pieces without counting:
(Number of panes ×2) + (the height + the width)
Good ideas! I wonder how Dhruv has worked out that formula?
We also received lots of solutions from Ananya, Kanaa, Twisha, Niya, Valerie, Ved, Anika, Kimaya, Swara, Mrunmayee, Ayansh, Aarav and Shivashree from Ganit Kreeda, Vicharvatika. They used matchsticks to help them with this task.
Challenge 1:
All of the children worked out that the number of panes is equal to the product of the height and the width of the window.
Kanaa noticed that the total number of matchsticks goes up by 4 more each time the window size increases. She has used colours in the second picture to show why this happens. In the first picture, the column with the header 'number of frames' actually refers to the number of panes of glass. You can click on Kanaa's pictures to make them bigger.
Mrunmayee used these calculations to work out the number of edging pieces in the 4 by 5 window:
4 × 5 = 20
20 + 20 = 40
40 + 4 + 5 = 49
Aarav, Niya and Ayansh generalised this method, using h and w to represent the height and width of the window:
Total number of vertical sticks = (h + 1) × w
Total number of horizontal sticks = (w + 1) × h
Total number of sticks = (h + 1) × w + (w + 1) × h
They also found out that the total number of vertical sticks is always bigger than the total number of horizontal sticks by 1. These are some good ideas - I wonder if we can use the fact that the width of the window is always 1 more than the height to help us write a simpler formula for the total number of sticks?
Ayansh came up with another formula for the total number of sticks, which is the same as Dhruv's from earlier:
Number of sticks = 2 × panes + height + width
Well done for working this out! I wonder if this always gives the same answer as (h + 1) × w + (w + 1) × h?
Challenge 2:
Kanaa found the number of outside edging pieces by adding double the width to double the height. You can click on this picture to make it bigger.
Aarav explained that this is because for every window boundary, there are 2 horizontal lines each made up of sticks equal to its width and 2 vertical lines each made up of sticks equal to its height.
To find the inside edging pieces for the 5 by 6 rectangle, Kanaa added the width 4 times and the height 5 times. I wonder if this method can be developed into a general formula to find the number of inside edging pieces for a rectangle of any size?
Thank you all for sending in your ideas. You've worked very hard on this task. To see some more of these children's ideas, take a look at Ganit Kreeda's full solutions.
If anybody would like to add to any of these solutions or send in any new suggestions, please email us.