Every year in 5th grade, when we begin classifying quadrilaterals, students will continually call a rhombus a diamond. It never fails. While doing a Which One Doesn’t Belong in 3rd grade yesterday, the same thing happened, so Christopher’s tweet came at the most perfect time! (On Desmos here: https://t.co/rZQhu2SGnR)

Of course I had to pop into the same classroom today and try it out! The lower right was so obviously a diamond to me that I was curious to see if students saw the same thing and if it changed their reasoning about the rhombus as a diamond.

Here are pictures of the SMARTboard after our talk:

After great discussions around number of sides, rotations, decomposition and orientation, they finally got to the naming piece. Honestly, I was surprised names didn’t come up as one of the first things. It started with a student saying the square didn’t belong because it is the only one that doesn’t look like a diamond. The next student said the lower left was the only one “that didn’t have a name.” When I asked him to explain further, he named the square, rhombus, and diamond. Because I knew at the end of our talk I wanted to ask about the diamond vs rhombus, I wrote the names on the shapes. Another classmate added on and said the lower left “may not have a name but it is kite-shaped and looks like it got stuck in a tree sideways.” I asked the class what they thought about the names we had on the board and it was a unanimous agreement on all of them. Funny how quickly they abandoned their idea from yesterday, so I reminded them….they were not getting off the hook that easy;)

“Yesterday you were calling this rhombus a diamond, what changed your mind?”

Students explained that the lower right actually *looks *like a real diamond and the rhombus doesn’t now that they see them together.

“Can we call both of them a diamond?” I asked. I saw a few thinking that may be a great idea. I had them turn and talk to a neighbor while I listened to them.

We came back and they seemed to agree we couldn’t call them both a diamond because of the number of sides. They were really confident in making the rule that the quadrilateral one had to be a rhombus and the pentagon was the diamond. I pointed to the kite and asked about that one, since it has four sides. “Could we call this a rhombus?” They said no because the sides weren’t equal, so not a rhombus. And because it didn’t have five sides, not a diamond either.

Thank you Christopher! All of these years of trying to settle that rhombus vs diamond debate settled right here with great conversation all around!

Next up, this one from Christopher…

howardat58I seems clear that the orientation of the figure has a lot to do with the name applied to it. The top right one, the original “diamond” is actually a square.

Makes me think of “The sum of 3 and 5 is 8″……… “Now let’s do some division sums”.

Also, when is a trapezoid a parallelogram? or are they mutually exclusive categories?

And when they get a pack of playing cards, what then?

And finally, do we still have “scalene” triangles?

Mathemaitcally, what is most important, at some point or other, is to establish the inclusive nature of mathematical definitions.

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Simon GreggHow great, to switch a conversation which is usually a matter of the teacher telling the (unconvinced) students what the naming convention is, into one where the students make their way to this themselves.

I tend to avoid quadrilateral classification because it gets a little namey and boggling (rhombus is special case of kite, rhombus is special case of parallelogram, square is special case of rhombus that is also a rectangle), but this conversation takes it away from that kind of thing.

I wonder why Christopher chose a square as his rhombus, rather than one without equal angles? If it was “thinner” it would look diamondy.

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howardat58Kids do like arguing about tomatoes, as “vegetable” or “fruit”.

On the other hand they don’t have a problem with “golden delicious” as a special case of “apple”.

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mathmindsblogPost authorYes! Have you read this Howard? https://linesoftangency.wordpress.com/2013/07/30/consider-the-strawberry/ Reminded me of you comment!

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howardat58I just read the two posts and the comments. Delightful, and very thorough, especially the topological bit. There are two things in mathematics concerned with the classification and description problem (well, probably more than two, if we get into mathematical logic!!). One is the idea of “special case”, where one says “square” is a special case of “rectangle”, and so squares fit the definition of “rectangle”. Likewise “circle” as a definition is a special case of “ellipse”. So, is a circle an ellipse?

The other thing is the idea of degeneracy, where a point is a degenerate circle, that is, a circle with zero radius.A triangle in which the sum of the lengths of two of the sides is equal to the length of the third side is degenerate, you get a line segment ! This idea of degeneracy is as hard for older,serious math students to get hold of as is the square/rectangle.

it is all in aid of making theories tidy, so there are no annoying exceptions, but it fits poorly with everyday life.

Many thanks for the link!

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Simon GreggThanks Kristin – that Chris Lusto post was very entertaining. And thinking about it, it’s not just the structure of the shape naming logic that’t the problem, teachers that make it hard work for students too. Think about the posters that get put up:

https://goo.gl/Ky3gjn

Then we say, after years of this kind of thing, well, squares are rectangles too. And by the way “diamonds” are rhombuses actually. And a square is a rhombus that’s a rectangle.

It could all be a nice logical exercise, if we hadn’t muddied the waters to start with, and hadn’t got quite so many names we wanted to teach. It will be appealing for some students, but for others it’s going to be a real turn-off.

Which is why this WODB lesson is nice: in this noisy shape naming logic space, you’ve created a little quiet tent where just one distinction is considered at the students pace – and it makes sense to them.

To Chris’s assertion that “a truth’s usefulness is a function of the cognitive level at which it becomes both comprehensible and important — but not before” we can add that some ways of encountering the truth make it a lot more comprehensible!

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howardat58Just looking at the Danielson diamond edition, clearly the top left is the odd one out, as it is the only one all of whose whose sides are parallel to the edges of the paper !!!!!

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Dana SpielbergerLooking at the image, it was clear to me that the lower right shape was a diamond. That said, it was more interesting to read what the students thought it was and the discussion that ensued. The best part was all the thinking and reasoning behind why the shapes are different/similar. This one image inspired loads of thinking, learning and reasoning; amazing!

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