A Number Divided By Another Number

One more instance of Math not being English

Lately I had a few students who had some trouble with fractions. Part of the problem was a very specific type of confusion at the time of setting up a division calculation. When asked “How much is 5 divided by 12?” for example, they would sometimes correctly calculate the result but most other times they would set up the division as 12 divided by 5. One of them asked me a few times if the result would be the same. When I tried to explain that division is not commutative, using some visual representations of fractions, he was not totally convinced. So I just pulled the calculator, I asked him to give me a couple numbers, and I did both divisions (let’s say, 17/4 and 4/17) with the calculator, showing him the results. We repeated the “experiment” with two other examples, and then he was convinced that division is not commutative. However, such discovery created some anxiety in him because he doubted he would choose the right order of calculation in any given problem. Actually, it took multiple repetitions on my part for him to finally learn how to set up the right calculation when a word problem involves the phrase “divided by.” At first he wanted to transliterate the written phrase (“105 divided by 15” for example) word by word, and number by number, in the exact same order into the division calculation by writing, from left to right, the 105 first, outside the division symbol, then the division symbol (the one that looks like a rotated “L”), and finally the 15, inside the division symbol. I would tell him that the order of the numbers in the phrase “105 divided by 15” is backwards to the order of the same numbers in the actual calculation but this only seemed to surprise him, and confuse him. Once more, I resorted to the calculator. I said: “O.K. just do the division.” When he asked: “In what order?” I said: “Do them both.” Once he had calculated both results by long-hand, I gave him the calculator, and I said, “Now do them both with the calculator.” As he punched the calculator keys, I directed his attention to what sequence in the calculator was giving him the same result as he had calculated long-hand. I said “Do you see when you enter into the calculator: ‘105, division symbol, 15, enter’ that gives you the same result as when you did ’15, division symbol, 105’ by long-hand?” When he saw this evidence he still said “It is confusing.” Then I said: “Yes, I know, but that is just the way it is, so you are going to have to remember it that way. In the calculator division the numbers go in the same order as they are in the phrase ‘105 divided by 15,’ whereas in the long-hand division, to get the same result, the numbers have to go backwards.” It took some repetition over a few tutoring sessions but he finally got it consistently right.

Look for Solutions with Less Math and More Logic.

One instance where often “less is more.”

The following question is an excellent guideline for solving math word problems:
“How can I solve this problem by doing the least possible amount of math?”

Oftentimes there are several pathways from the setting of a problem to its final solution. Some routes are safer, while some are riskier, more error-prone. Some routes are faster, while some are time-consuming. Some routes are clearer, while some may be confusing.
Usually, the routes with more elementary operations (especially long division), and bigger numbers, tend to be lengthier, longer, and riskier, because adding, multiplying, and dividing big numbers or expressions requires a laser-focus attention. There are plenty of opportunities for doing silly mistakes during these calculations. Besides, it is easy to lose sight of the big picture when worrying about the accuracy of the calculations.
Factoring whole numbers and algebraic expressions is a good habit because it allows you to simplify some expressions before diving into the calculations, so you can operate with smaller numbers, gaining time, and accuracy.
Using logic is a very good habit, too. Many problems lend themselves to solutions that involve more reasoning, and less calculation. This is usually a good thing because these solutions tend to be clearer, and shorter.
Organizing all the information about a problem in a way that makes sense to you, is an excellent habit because this way you keep track of where you are and what you are doing all the time through the problem, and having all these references available makes it easy to retrace your steps, and identify any possible mistakes.
Go visual at any opportunity. Pictures, drawings, charts, graphs, and tables often are a huge help in writing down the right equations, or even in avoiding equations altogether sometimes.
There are many problems you can solve with a drawing and a little logic. Just because the problem is a math problem, that does not mean you need to write down an equation to solve it.
Focus on your possibilities, on what you can do. Organize the information in a logical way, using a drawing, or a table. Above all, try to spend the least possible amount of time and energy doing long, detailed, time-consuming calculations. Instead, simplify the expressions, and ask yourself logical questions about the problem.