At a conference I recently attended, the keynote speaker Dr. Randy Charles spoke of “two inversions”. The first I was very familiar with, that of changing

Mathematical tool -> Problem solved by tool

to

Problem solved by tool -> Mathematical tool

That is, motivating mathematics by presenting problems which force the students to develop the tools they are supposed to be learning in the first place.

He spoke of another inversion which I was not familiar with:

Standards -> Understandings

to

Understandings -> Standards

A “standard” might be “the student knows how to add two fractions” while an understanding might be “why do the denominators need to be the same when adding two fractions?”

This portion of the talk was somewhat foggy, and honestly the analogy with reversing problem solving and mathematical tools isn’t the best since the problem-solving approach leads to an obvious change in sequence while the understandings approach leads to a change in … well, I’m not sure, really. Perhaps it is a change in assessment. I have in the past included problems like “Why does the graph of tan(x) have asymptotes at and ?” rather than just “Graph this function” on tests. Does every understanding needed be an explicit question? I’m not sure how this is an “inversion” rather than just “we should be thinking about this more.”

However, I did want to give my honors students a fraction test that involved very few numbers, and I think it gives an idea of what Dr. Charles was talking about.

1. Why is the top of a fraction called a “numerator” and the bottom called a “denominator”?

2. Why can each fraction be represented in an infinite number of ways?

3. Explain in your own words a method for reducing a fraction.

4. When is reducing fractions important? When is reducing fractions not important?

5. Why do the denominators need to be the same when adding two fractions?

6. Describe the error made below, and explain why it is an error.

7. Why do the denominators **not** need to be the same when multiplying two fractions, and why does multiplication involve multiplying both the numerators and denominators?

8. How can an integer (like 6 or -2) be written as a fraction? Why does this work with any integer?

9. Why does a fraction equal zero when a zero is in the numerator?

10. Why can’t you have a zero in the denominator (division by zero)?

11. Explain, without referring to the rule “dividing by a fraction is the same as multiplying by the reciprocal,” why dividing by ½ is the same as multiplying by 2.

[Link to the test in Word document format.]

Filed under: Education, Mathematics

Sue VanHattum, on March 10, 2010 at 2:08 pm said:Nice! What level course is this for? (You mentioned honors, but nothing else?)

Jason Dyer, on March 10, 2010 at 2:47 pm said:Honors Algebra II / Trigonometry.

Some are still fuzzy around the edges with their fractions, but they’re also at the mathematical maturity level they have a fair shot at the questions above. Hopefully it will help sharpen their thinking.

Sue VanHattum, on March 10, 2010 at 3:11 pm said:I think this is a great collection of questions for anyone who ‘should’ know fractions. I think I’ll give it out in my beginning algebra class (at community college), as a worksheet instead of a test.

I’m curious what you have in mind for the 2nd part of #4.

Jason Dyer, on March 10, 2010 at 4:06 pm said:I would probably put that computers and calculators don’t care, so for a calculation destined for a floating point approximation it doesn’t matter anyway.

I’m sure there are other good answers. Sometimes when trying to spot a pattern reducing fractions can make it harder, compare:

1/8 1/4 3/8 1/2 5/8 …

vs

1/8 2/8 3/8 4/8 5/8 …

CalcDave, on March 10, 2010 at 8:41 pm said:One not on here but has often baffled me is why we insist on “rationalizing the denominator.” I guess it’s easier to get a common denom. if you have an integer there, but otherwise it seems like a lot of unnecessary work (especially when our students like the decimal approximations anyways).

joshg, on March 11, 2010 at 7:29 pm said:I’d be interested to hear what sorts of answers you get, especially if any surprisingly good ones show up. (Especially for question 11!)

Great idea, I should give this a try. Fractions is a great topic to use this kind of questioning on, since it’s one of those big things people get hung up on and trip over for the rest of their lives.

Androski, on March 13, 2010 at 6:53 pm said:I have used many of these same questions in my high school algebra 2 classes.

Here are a couple of musings about fraction division that I have pondered:

A curiosity related to fraction division is the fact that you can divide straight across (horizontally) to divide fractions (similar to multiplying fractions). The problem is that this method often creates complex fractions unless the first numerator is divisible by the second numerator and the same divisibility would be desirable for denominators. Example: (1/4 divided by 1/2) = (numerators 1/1=1, denominators 4/2=2), placing these together the answer is 1/2.

Another interesting method for dividing fractions is to get common denominators. If you get common denominators, then you can forget about the denominators and you only need to divide the two numerators to get the answer. Example: (2/3 divided by 1/9) = (6/9 divided by 1/9) = (just divide the two numerators, 6 divided by 1) = 6. In my mind, this relates to making the parts of equal size where they can be divided into groups. I suppose this method is not popular because it takes extra steps to get a common denominator. Has anyone ever seen it taught this way or in any text?

jd2718, on March 14, 2010 at 2:23 pm said:You might like playing with this idea:

we can rewrite any rational number as an integer times a unit fraction.

It shifts the view, perhaps, on the expected answers to several of these questions.

Jonathan

Math Teachers at Play #24 « Let's Play Math!, on March 19, 2010 at 4:55 am said:[…] How well do you understand fractions? Challenge yourself with Jason’s Conceptual fraction test. […]

L, on December 20, 2011 at 7:02 pm said:The denominators don’t need to be the same when adding two fractions!

Jason Dyer, on December 20, 2011 at 10:08 pm said:I’m hoping it’s clear the intent was “so that you can do the adding by adding the numerators and keeping the denominator the same,” but point taken.