In some situation, they may see the big board as a big square----and actually that is----1 square contribution.
Thus it is reasonable to consider those squares "in the middle", or those squares whose sizes vary from 2*2 to 7*7.
Starting from finding a 7*7 square, students can easily see that there are 4 of them. Using similar schemes, students can see that there are 9 squares for 6*6 squares, and so on.
As a result, the total number of squares is calculated by 1+4+9+16+25+36+49+64=204.
Additional:
Look at the image:
There are 8 squares of different sizes. If students thinking through the processes of:
1) What is a square:
2) How many different styles/kinds/sizes of squares are there in a chess board?
3) How many of each kind of squares?
The following table is an informative one to show:
Additional:
Look at the image:
1) What is a square:
2) How many different styles/kinds/sizes of squares are there in a chess board?
3) How many of each kind of squares?
The following table is an informative one to show:
Good start, Shan! But I would also like you to add to this blog post: 1) What tools might help you and your students visualize the other sizes of squares and how they could be counted? 2) How could you extend this puzzle?
ReplyDeleteThank you for the comments. I'll post another blog on this topic later today.Your comments remind me that this is how students stop thinking----when the main question is answered, some "branch" questions or in-depth unposed questions are ignored. I guess this is because students normally do not understand, especially in math, why they need to learn this subject.
DeleteWell
I don't know what had happened but my machine was shut down by itself......
DeleteSorry for the unstructured comment above.
Thanks for the great diagrams and chart you've added! These will really help students if they need help on a 'way in' to solving the puzzle -- and they're beautiful too.
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