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My nephew is preparing for a $4$-th grade state test. They need to learn topics like reflection about $x$ or $y$-axis of a point( say $(3,5)$ reflected about the $y$-axis).

I tried to explain but he's not getting it. He's smart, but very distracted. How can I explain in an intuitive way?

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Cartesian coordinates in 4-th grade? Isn't that too much for a 4-th grade kid? Is the state test some kind of olympiad? – Beni Bogosel Apr 9 '12 at 16:32
@BeniBogosel - Admittedly, it's towards the end of the sample test. And it isn't tested as much as basic arithmetic........ but - 4th graders are 8 and 9 years of age. They have decent minds, so Why not? – Adel Apr 9 '12 at 16:34
up vote 6 down vote accepted

Use the 'Battleship game' (or other) and put (virtual) boats in the four quarters to get his attention. Because the pleasure of sinking is great! :-)
Introduce horizontal and vertical mirrors if needed (and/or negative coordinates) explaining that the boats are seen or not in mirrors.

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The chess board also woks nicely for this. – N. S. Apr 9 '12 at 15:58
+1 because I love that game... :) and yes, a very nice example. – J. M. Apr 18 '12 at 16:17
@J.M.: Thanks J.M.! Battleship and Chess are great games and only few kids could resist to the pleasure of sinking your boats! :-) – Raymond Manzoni Apr 18 '12 at 20:02

I would say something like "Starting at the origin, $(3,5)$ means go $3$ units to the right (the $x$ direction), and then $5$ units up (in the $y$ direction). If you reflect about the $y$-axis, you have to go $3$ units to the left and then $5$ up, i.e. you're at the point $(-3,5)$.

Hopefully that helps. When I was learning it helped me to think of the Cartesian plane as a map I had to naviagate

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Thank You Very Much - I like this a lot, it's like a "treasure map" analogy – Adel Apr 9 '12 at 15:46
@Adel, you have to make sure that he understands relative numbers well. You might start with the number line taking his street as model. With his house as the origin, +3 is three houses down and -4 is four houses up. – lhf Apr 9 '12 at 15:47

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