Shared by Tracy on twitter, and a great conversation ensued.

@TracyZager @mpershan My first thought on this to see where students are making the mistake. pic.twitter.com/7X7spt0BMI

— Bryan Anderson (@And02B) April 6, 2014

@mpershan @Anderson02B I think I would go concrete right away, and link it to sharing. If this were a brownie, which piece would you want?

— Tracy Johnston Zager (@TracyZager) April 6, 2014

@mpershan @Anderson02B @TracyZager This one is a bit different, tho. Kids seem to cue in on the equal widths here.

— Christopher Danielson (@Trianglemancsd) April 6, 2014

https://twitter.com/BHS_Doyle/status/452804242672476160

@mpershan @TracyZager I'd bet that most 7th grade students have never seen the problem with a shape other than a circle or rectangle

— Bryan Anderson (@And02B) April 6, 2014

@BHS_Doyle @mpershan @Anderson02B @TracyZager So this task needs revision and should only ask about 1/5 or 4/5.

— Christopher Danielson (@Trianglemancsd) April 6, 2014

## 9 replies on “Three Fifths of a Triangle”

Thanks, Michael! The question came from Pam Buffington, who shared the new EDC fractions formative assessment at ATOMIM, Maine’s NCTM affiliate.

It seemed obvious to me that it wasn’t 3/5, but then, as I tried to frame a reply to Michael Doyle’s comment, I realized that I wasn’t 100% sure. Yes, the end pieces and middle piece are not fifths. But is it

possiblethat the shaded portion is 3/5?Ifthe slices are equal width,andthe triangle is equilateral, then 68% of the triangle is shaded. (That was fun.) I wonder if removing the equilateral assumption could get us closer to 60%…Ok, I’ve got it. I’ll leave my question as an exercise for interested readers.

But I agree that revision is called for. Using 1/5 or 4/5 makes for a problem that assesses the basic equal sharing issue, and doesn’t get tangled up in whether this really could be 3/5.

My 9 year old son asked me a similar question when he was going to bed the other night. He asked, “mom, can an isosceles triangle be divided int three equal parts? 4? 5?” So we started talking about the difference between congruent triangles and equal areas.

One of the solutions to divide a triangle into equal areas is to divide one side into equal lengths and take slices going from those portions to the opposite vertex. That way, each “slice” has equal base and equal height. (or imagine the triangle as a circular sector but flatten out the arc.)

It might be useful to show that picture in contrast to this one.

Does it not depend on the purpose of the question and the context in which it is studied? Distinguishing eminently plausible false statements from true statements is an important skill, and could be the learning outcome of this question. Skipping this sort of question removes the need for deductive reasoning at all. “The two triangles are congruent because they look so much alike that they probably are” would become acceptable reasoning and would remove the interest in a large amount of Euclidean geometry.

I was having fun playing around with math. I don’t know the grade levels, but I think this question was designed for younger children to test their understanding of fraction, and not to test their understanding of triangles. If it were to have the purposes you mention, it would need to be one where they show work, not just yes or no.

“Is the shaded part 3/5?” is a terrible question. The wording of the question does not ask “Is the shaded part 3/5 of the area?” or “Is the shaded part 3/5 of the width?”.

If the test writer put this choice in a legal document, the test taker would have the choice of how to interpret the choice. A court (in my jurisdiction) would probably accept both “Yes” and “No” as being correct — and the test writer would not get to choose. (By the way, I am not a lawyer, but I have written easements.)