It’s unclear to students when we can leave off the multiplication sign, and in the heat of the question (fractions = brain freeze), they thought 2 and 1/5 was 2 times 1/5, because why not. If the question was (2x)^2, the answer of 2^2 * x^2 would be OK. Explicitly addressing the question of when we can leave off the multiplication sign helps a bit (there’s a good question near the beginning of the Exeter Math 1 problem set on this.

This is why I hate writing mixed numbers. Everywhere else in math when two things are beside each other with no symbol, it means multiply them. But then all of a sudden in this context we mean addition? I teach first year university and refuse to let my students use mixed numbers.

Sue is absolutely right. This is a classic example of the Student’s Universal Law of Distribution here. Sue – what do you mean when you say that you show people with a triangle? I just always (kind of boring, I know) resort to reminding them that squaring something means multiplying it by itself.

11/5 squared, 121/25 =4 21/25, combine the fraction back together; better, have student illustrate the 2 1/5 as three boxes, div in to 5 parts each. Scratch out 4 of the last box. Let them think about the image and discuss possibilities.

Triangle demo, why a^2 + b^2 does not = (a+b)^2 (Actually I think I use it more often for when they say sqrt(a^2+b^2)=a+b, which is almost the same but not quite.):

I draw a 3-4-5 triangle, with legs of 3 inches and 4 inches labeled. I ask them what the third side is. They all quote the Pythagorean theorem at me. We come up with 5, which is sqrt(3^2+4^2), and which is not 3+4. I show them that 3+4 would be taking the long way around the corner, instead of going straight along the hypotenuse.

If I saw the above mistake, I’d also address the issue of the “invisible plus sign”, and show them that the distributive property would give them the same answer that they get by the usual method of converting to improper fraction first.

Thoughts: Whenever you use mixed fractions, DON’T LEAVE OUT THE PLUS SIGN. So much easier.

Vladislav Sekulic

Hm, this is interesting, because I didn’t think the answer was wrong at all. I think the problem here is that there is a disjunction in math notation, between that of pre-university and post-university (roughly speaking). In university/college-level math, I’ve *never* seen fractions expressed this way. In fact, fractions are rarely, if ever, used. “2 and a fifth” would always be expressed as 2.5 or 5/2. In that case, the notation in the question is clearly interpreted as 2 * 1/5.

In other words, many university-level students would get this one “wrong”, too. Is the problem the student, or the lack of consistency in notation?

Vladislav Sekulic

^ hehe, oops. It would, of course, be expressed as 2.2 or 11/5. *That* was a brain-freeze. ðŸ™‚