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I am working with a college student who was placed into a pre-college level math class. She's in a unit that covers a lot of unit conversions (simple ones, like 5 km = ___ miles or 3 yards = ___ cm ). She says she's confused by what the teacher showed her in class so I spent time working with her to reinforce the method they learned in class (the "Train Track Method" is what she calls it). I like the idea because to me it makes a lot of sense. Organizing the conversion this way helps the students focus on cancelling the units first, before filling in the numbers. For example, for 3 yards, I want to cancel the yards, so I know I'd multiply by feet/yards to get yards on the bottom of the fraction to cancel. Then to get to inches I'd multiply by inches/feet, then finally cm/inches. Then once all the units are cancelling out properly, then I can worry about filling in the numbers and handling them.

Well, after two different 1-hour long meetings with her, she still isn't getting it. She says it's totally foreign to her. Every other student I have shown this idea to or practiced it with was able to get it and do very well on this class's unit conversion quiz. But she is still stumped. She emailed me asking me for other helpful resources for these kind of problems, but everything I Google seems to be a re-hashing of the same method, or the exact same explanations I was already saying to her.

What are some good resources (handouts, webpages, videos, etc.) to help a student that is really struggling with basic conversions?

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    $\begingroup$ $1$ yard is $3$ feet, but to convert from yards to feet, the relation $y = 3f$ is incorrect. For example, substituting $f = 3$ gives 9 yards = 3 feet. The correct relation is actually $y = f/3$. $\endgroup$
    – Toby Mak
    Oct 21, 2020 at 23:14
  • $\begingroup$ @TobyMak I see what you are saying: she shortens the textual expression "1 yard is equal to 3 feet" to "1y = 3f", where "y" and "f" are dimensions, and confuse it with algebraic expression "1y = 3f" where "y" and "f" are values respectively in yards and feet. $\endgroup$
    – Rusty Core
    Oct 22, 2020 at 18:05
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    $\begingroup$ @TobyMak ah ok, I didn't understand what you were referring to until RustyCore rephrased. I don't think that's the girl's mistake, since she isn't trying to manipulate things algebraically with variables. This is a very low level math class, they haven't started to use variables yet. $\endgroup$
    – ruferd
    Oct 22, 2020 at 19:01
  • $\begingroup$ The videos that seem like exactly the same explanation to you might seem different to her. It is worth pointing her to a few good ones to see if any of it helps. $\endgroup$
    – Sue VanHattum ♦
    Oct 22, 2020 at 19:16
  • $\begingroup$ The concept of units is powerful. Unfortunately in the U.S. this concept is not introduced early enough. Worse, it's actively abused at the lower levels. My youngest grandchild had a word/picture problem that showed "3 violins + 1 trombone = ?" My answer was A Bad Quartet . $\endgroup$
    – user10216038
    Dec 13, 2020 at 17:28

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Could try section 0.9 of my book: http://lightandmatter.com/lm/ . The discussion question at the end of the section is meant to immunize against common mistakes. I go through each of these with my students and do a reality check to get them to see how absurd the results are, before they're even allowed to shout out what the actual mistake is.

It sounds like she's American. You can also work with her to make sure she can visualize the various metric units look like. Many people in the US have no clue how big a centimeter is, how much weight a kilogram of beans is, and so on. If they don't know these things, then they can't check for reasonableness.

Most likely what's happening with your student is that she doesn't really understand conceptually what multiplication and division mean. This probably needs to be dealt with at a more basic level, like Alice, Bob, and Carol sharing applies from a bag.

Some people have an easier time with money, and you can then make the analogy between money and other units. E.g., if I change my dollars into pennies, do I end up with more pennies, or fewer? Then, if they get that: if I change my kilograms into grams, do I end up with more grams, or fewer? I tell a joke with this, which is that when we change to a bigger unit, the number gets smaller; it's compensation, like the guy who needs a really big pickup truck because he's compensating for ... something else that's really small.

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    $\begingroup$ I will take a look into the section you suggested, thank you. We are American, and yes, she is very unfamiliar with a lot of basic units (even pints, cups, & yards which are Customary Units in the US). I thought about trying to draw a scale model, like a line segment that is labelled as 1 ft, and asking her to mark an inch on it, to see if she'd understand why we multiply by 12 to go from feet to inches. I will try that approach next, but you're right, if she doesn't understand multiplying/dividing, she will struggle with this too. $\endgroup$
    – ruferd
    Oct 22, 2020 at 14:57
  • $\begingroup$ "I tell a joke with this ... it's compensation" — this joke would not help me if I had a problem converting grams into kilograms. IMO, it is much more productive to memorize the prefixes, like K is for "kilo" which means "1000" and then replace prefixes with their values: 1kg = 1 * (kilo * g) = (1 * 1000) * g. Sometimes case is important, MB is a "mega"byte, while "ms" is a "milli"second and kb is "kilo""bit". I see "m" vs "M" and "b" vs "B" misuse everywhere. No one who has learned math and physics outside of the U.S. would confuse "m" with "M". $\endgroup$
    – Rusty Core
    Oct 22, 2020 at 18:26

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