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Why do this problem?
This problem offers a good opportunity for students to discuss patterns and find convincing arguments for their solutions.
Reuben Hersh has written that:
"In the classroom, convincing is no problem. Students are too easily convinced. Two special cases will do it."
This problem offers an opportunity to ensure that students are justified in generalising from the particular cases that they have selected.
"Have a look at this image. Can you work out the coordinates of the centre of square number 3?"
"I wonder if you can now work out the coordinates of the centre of square number 20 from the image, without working out the centres of the squares in between."
"Spend a short while thinking about it on your own, then discuss it with your partner, and together develop a convincing explanation for your answer to share with the class."
As students are working, if they get stuck you could offer the following hints:
"How do you move from one square to the next?
What do you notice about the x coordinates of the centres?
What do you notice about the y coordinates of the centres?"
While pairs are talking, circulate and eavesdrop on discussions, drawing attention to mistakes and making a mental note of any students with clear explanations.
Bring the class together and invite those students with interesting or elegant strategies to present their ideas to the rest of the class.
"In a while I'm going to choose a square and ask you to work out the coordinates of one of the vertices. Can you find a quick and elegant strategy?"
"Again, you may want to start by working on your own before discussing it with your partner."
Finally bring the class together and challenge them with a few examples. Students could be asked to display their solutions on their mini-whiteboards. Allow some time for discussion of their strategies.
Students can work on Alison's Triangles and More Squares from Charlie in a similar way.
Find a general symbolic expression for the coordinates of the vertices of the $n$th square or triangle.
Before working on this problem students could develop fluency in using coordinates by working on Cops and Robbers and fluency with linear sequences by taking a look at Shifting Times Tables.