In case you missed it I gave my Geometry class this at the end of the period on Monday.
It didn’t go over very well and that piece has been bothering me for the last couple days.
I am not expecting to come up with the magical fix to making this sorta question great. However, I want to use this post to look at what I was trying to get at in this problem and where it went wrong.
Let’s start by looking at the objectives I had in mind with this thing
- Position a figure in a coordinate plane.
- Prove a geometric concept by using coordinate proof.
To proves this students need to know one or more of the following
- Definition of a rectangle
- Distance Formula
- Slope
A majority of the struggle probably came from the definition of a rectangle. When thinking about this I immediately jumped to proving lines perpendicular and congruent distances. A lot of my students were still stuck in the two column proof mindset.
There isn’t really anything groundbreaking in proving a shape is a rectangle. For some reason I was expecting this problem to challenge student’s thinking and encourage them never to just jump to conclusions about a shape.
What I am really interested in though is how to re-vamp this problem to meet the same objectives but approach them from a different angle.
The first type of problem that comes to my mind appears in my Fence Problem Part II post.
I am still working on how exactly this problem could be modified into a rectangle… But hey, it doesn’t necessarily have to be a rectangle.
Feel free to share your suggestions/comments, I would love some input!
Quadrant Dan 
For your fence post problem, I hope there is some discussion among the kids about whether there is only 1 triangle with the required area and/or what is shared by all such triangles. Then, I would like to hear thoughts on what the “best” such triangle is and what makes it “best.”
For another great coordinate geometry investigation, you could spring off this numberphile video: All Triangles are Equilateral, which I saw in a post from Mike Lawler. Can the kids use coordinate geometry to investigate where the proof goes wrong? I’d suggest they follow Mike’s son and use a right triangle (even 3-4-5 as in the blog video).
Choosing a right triangle and setting it in the coordinate plane carefully (right angle at the origin, legs along the axes) makes the equations really nice and, I think, forms a useful demonstration of the power of coordinate geometry. If they play with the picture, there are further rewards as they see interesting relationships between triangle side lengths and the lengths of other key line segments.
A further tidbit out of this whole sequence is that you can also have an interesting discussion about what it means for something to be a proof. The numberphile argument looks, smells, and feels like a solid proof, but we know it is wrong. Something subtle was left out, but there are always subtle things left out of real proofs. Furthermore, we start with the feeling that, where two lines intersect is minor, technical and obvious. Then, it is suddenly a lot more difficult when we try to prove it. Then it turns out the original idea was wrong.
A classic progression: “obvious, hard, false.”
BTW, happy holidays; hope you and your students are getting plenty of math amidst the holiday cheer!
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Dan,
You should definitely look at the pooltastic problem that was developed from Howard County public schools. The problem is right up your alley for the objectives you are looking to cover. I used it for slopes, perpendicularity, and parallel lines. I made it a group worked problem and we spent time talking about the mathematical practices. We had one day to share how each group attacked the problem and then they needed to individually reflect on other groups’ methods. Check it out:
https://commoncoregeometry.wikispaces.hcpss.org/Unit+1
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Here is a set of problems that bring together the ideas of coordinate geometry, Pythagorean triples, area and perimeter of parallelograms.
1. The points (1,2) and (4,6) are two vertices of a square that has area 25 sq. units. What are possible points for the other two vertices? Prove that this is a square by showing that the points selected form a shape that has all the characteristics of a square. What is the perimeter of the square?
2. The points (1,2) and (4,6) are two vertices of a rectangle that has 100 sq. units. What are possible points for the other two vertices? Prove that this is a rectangle by showing that the points selected form a shape that has all the characteristics of a rectangle. What is the perimeter of the rectangle?
3. The points (-3,2) and (1,5) are two vertices of a parallelogram with area 12 sq. units. What are possible points for the other two vertices? Prove that this is a parallelogram by showing that the points selected form a shape that has all the characteristics of a parallelogram. What is the perimeter of the parallelogram?
4. The points (-2, -1) ad (10,4) are two vertices of a square that has 169 sq. units. What are possible points for the other two vertices? Prove that this is a square by showing that the points selected form a shape that has all the characteristics of a square. What is the perimeter of the square?
5. The points (-2, -1) ad (10,4) are two vertices of a rectangle that has 338 sq. units. What are possible points for the other two vertices? Prove that this is a square by showing that the points selected form a shape that has all the characteristics of a square. What is the perimeter of the square?The points
6. The points (-2, -1) ad (10,4) are two vertices of a parallelogram that has 60 sq. units. What are possible points for the other two vertices? Prove that this is a parallelogram by showing that the points selected form a shape that has all the characteristics of a parallelogram. What is the perimeter of the parallelogram?
Compare and contrast the above problems. What do 1 and 2 have in common? What do 1 and 4 have in common? What do 1, 2 and 3 have in common? What do 1 and 4 have in common? What do 4 and 5 have in common? What do 3 and 6 have in common? What do all 6 problems have in common?
Create your own problem similar to the ones above.
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This is awesome. I am excited to look deeper into these questions. Thank you!!
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