Two 4th Grade girls were playing “Guess My Rule.” All the green shapes fit Molly’s rule, all the red ones were excluded by her rule. Christi was trying to figure out the rule.

**Christi**: Oh, I’ve got it! They’re all triangles.

**Molly**: Nope!

**Christi**: But, look, it’s true! They’ve all got three sides.

**Molly**: They all have three sides, but they’re not all triangles.

**Christi**: WTF?

**Molly**: Yeah, L has three sides, but it’s not a triangle.

Why not? What makes this seem untriangley to Molly?

(Relevant: Young Children’s Ideas About Shapes)

## 10 replies on “Triangles and 3-gons”

Because it’s half a rectangle?

The orientation and fact it’s a (seemingly) right triangle? Could be something that’s not familiar as a “triangle.”

I may never understand why pre-school resources on shapes so often limit themselves to regular polygons, plus circles and non-square rectangles. When the only things you’ve seen labelled as “triangles” are equilateral triangles, it’s so easy to make Molly’s mistake.

It’s not isosceles?

I had this exact issue come up with my 6-year-old son just a few minutes before reading. He sees a right triangle (Like triangle L above) as “half a triangle.” In his mind, the whole triangle is what happens when you reflect it on the y axis.

Orientation matters. When I show him the same triangle with the hypotenuse side down, he sees it as a triangle.

I held a rectangle with the the short side down. he called it “a rectangle.” Turned it 45 degrees and he called it “a leaning rectangle.” Another 45 degrees and he called it a “fallen-down rectangle.”

great article in TCM about this a decade ago. http://www.nctm.org/publications/article.aspx?id=21431

It’s probably about the point not being in the middle, like it’s supposed to be. “Real” triangles have reflectional symmetry, and point up.

It’d be very interesting to throw some non-polygons in there; rounded edges, open instead of vertices.

…Or even throw in some polygons that are not convex for more fun

Because it’s a right angle and not looking like J and N 🙂

For what it’s worth, when I asked the kid, she told me that L wasn’t a triangle because it didn’t go down on both sides. So her vision of a triangle involves some sort of symmetry.

The combination of the right angle and the lack of symmetry can be very confusing to a child who’s never had the rule explicitly explained. If the child is told, “All shapes with three sides are triangles,” then L might not pose a problem. If the child is just shown pictures of (almost always isosceles) triangles and told “These shapes are triangles” with no kind of explanation, then a lot of triangles won’t be recognized as such. This is one thing I think Sesame Street definitely did right: they would have segments of “what shapes can you find?” in which every square, rectangle, triangle, and circle was highlighted on the screen. Since there were different kinds of triangles, it was easier not to fall into the trap of “triangles are symmetrical.”

Interesting anecdote: My mother, a high-school math teacher, taught me the kinds of triangles when I was 4 or so. (I may have asked her what she taught, leading to types of triangles as a kind of “learning about shapes” that is done in geometry classes.) I didn’t pick up on scalene, but I remember saying that isosceles triangles “look like a pizza,” because if you straighten out the crust part of a pizza slice, you generally get an isosceles triangle. (That the non-congruent side of an isosceles triangle can sometimes be longer than the others was quite the revelation to me in high school!) I also tended to think of equilateral triangles as “perfect” triangles, and the other kinds of triangles as “sloppy.” I drove myself nuts trying to draw equilateral triangles freehanded.