Converting Units Ratios & Proportional Relationships

Converting Units

What’s the misconception? But what does the student know? How would you bridge the gap?

4 replies on “Converting Units”

The student made is interpreting 5.375ish feet as 5 feet 3 inches, instead of recognizing that .3ish feet is NOT 3 inches (it’s about 4 inches).

I would maybe give them some context to why this problem is difficult.

This is why the decimal system was invented! because you can just use your calculator to divide, and the answer is right there. if you were converting from feet to centimeters, your answer would be 5 centimeters and 3 millimeters.

The imperial system does not work out so neatly. The Imperial system was invented by people who didn’t have calculators, and they had to divide, but they wanted to avoid messy fractions. For people who’ve never heard of a calculator, 12 is an awesome number because 12/2 is a whole number, and so is 12/3, and so is 12/4, and so is 12/6. (12/8 is almost as good because the answer is 1 and a half, which is also an easy quantity to understand.) So, with 12 inches in a foot–it’s easy to figure out how many inches there are in half a foot or a third of a foot, and so on.

This makes it easier for me to separate centimeters into the calculator-method and inches into the old-fashioned method. It also helps those students who ask “why does this have to be so complicated?” because you can blame it on history.

This would be such a great question and story to bring to the class to see how many approaches they could come up with for solving this problem. I might ask that question: Think or one minute, then show me on your fingers how many different ways you can think of to solve this problem. Then I’d start with someone who only had one way, and do my best to solicit different answers. Or, I might just show them the three methods outline below and have them vote on which way they like best.

So, you have a couple of options:
1) Use your calculator, but do extra work to convert a calculator answer into a non-calculator-friendly-unit-of-measurement. Once you get 5.3ish, ask them to take away the .3 and turn it into a fraction (because decimals are for calculators, fractions are for old-fashioned division) Now you multiply 3/10 of a foot by 12 inches/foot to get the number of inches.

2) You can also do it the non calculator way, since inches are not calculator-friendly.
a) Do division house-style division, with a remainder. The remainder is the number of inches.
b) Do some mental math: Once you have 64ish inches, subtract a number of inches to get a number that you can neatly divide by 12 (it helps a lot to have the 12s table memorized, but you can also guess and check). Once you’ve figured out how many inches to subtract, write it down–that’s the “inches” part of your answer.* Divide the remaining quantity by 12, and you have the feet part of your answer.

On a related note, the way this student is approaching the other problems bothers me. It seems as if he has memorized the conversion factors properly, but won’t be able to use them well in different contexts – i.e. how does knowing 2.54 help you convert from inches back to centimeters? How do you consistently keep track of whether to multiply or divide?
So, teaching that the units that go with 2.54 are really (cm / in) will go a long way towards future success.

The student understands that there are 2.54 cm in an inch, and that there are 12 inches in a foot, and that the imperial system expresses heights as a mix of feet and inches. The student can express their math clearly, so that their steps are obvious.

The student seems to understand the use of significant figures. They are correctly using absurdly precise intermediate values, and rounding off at the end. Their round off at the end is to units (inches) that are comparable to the original units (centimeters).

There a few mistakes.

The student has not been taught that most unit conversions are the same as multiplying by one. 163 cm / 2.54 cm is 64.173…, not 64.173 inches. 64.173… inches / 12 inches is 5.347…, not 5.347… feet. These unit conversions should be 163 cm * (1 inch / 2.54 cm) = 64.173… inches, and then 64.173… inches * (1 foot / 12 inches) = 5.3477… feet.

The student has not been taught that converting from a large number of inches to feet and inches requires using division and remainders. This is because there are 12 inches in a foot, instead of 10 inches in a foot, so things get awkward. 64.173… inches = 5.3477… feet. In other words, 5 feet plus some number of inches. 5 feet = 5 feet * (12 inches / foot) = 60 inches, so 64.173… inches = 5 feet plus 4.173 inches, which rounds to 5′ 4″.

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