kitchen table math, the sequel: Fractions as Division Problems

Friday, October 12, 2007

Fractions as Division Problems

My daughter had trouble remember how to find a decimal when given a fraction. She knew it was a division problem, but often transposed the numbers. For example, if she needed to find the decimal equivalent of 5/8 she might divide the 5 into the 8 because it just seems more logical when you are 10 to do it that way. She needed help remembering which way to divide.

Her older (15) brother gave her a mnemonic devise he had learned.

Top dog goes in the house.

There you have it, problem solved. This was new to me too, but it really works, she remembers this easily, even though we only talked about it that one night. Now she never forgets. It's great. Try it with your kids.

The numerator (number on top, or the "top dog") goes inside the "house" the half box you draw for division.

Am I making sense here?


KarenA said...

"Top dog goes in the house."

I love it! You are making perfect sense--in my mind, it's a great way of remembering the procedure. The only downside is that I will now have "Who let the dogs out" playing over and over in my mind for the rest of the morning.

I think mnemonic devices can be very useful. Here's a quiz for the lawyers among us: who among you still remembers "MYLEGS" and what each letter stands for???

LynnG said...

I never learned that one.

But then I went to law school in Wisconsin, and WE DON'T TAKE A BAR EXAM!!

So I never had to memorize all that detail stuff.

My torts professor believed he could teach us everything we ever needed to know but just doing assault and negligence. If we ever did a real tort case we could look up the black letter law. He was a a conceptual understanding kind of guy.

He also admitted that if he'd been good at math, he'd have been a doctor.

SusanS said...

"Top dog goes in the house."

That is really cute. I've got to remember that one.

The way Saxon teaches it really helped it stick with my sped kid.

They write the fraction (5/8) and you point to the top number and say "five", then point to the "division bar" and say "divided by", then point to the bottom number and say, "eight."
It's just top to bottom.

I had to remind him to start at the top and do it, but it did stick and he was able to hang on to it.

It does help that Saxon calls the fraction bar a "division bar."

Catherine Johnson said...


and--miracle of miracles--I don't think C. ever gets confused about which number is the divisor and which the dividend!



I'm going to tell him this anyway

Catherine Johnson said...

I have had the theme from Last of the Mohicans running continuously inside my brain for 3 days now.

Catherine Johnson said...

Of course, that's a good thing.

KarenA said...

lynng--Okay, here goes. M is for Marriage, Y is for the one Year Rule, L is for Land, E is for Executors, G is for Goods, and S is for Surety (the six most typical types of contracts covered by the Statute of Frauds).

My Property I prof was so abstract in his teaching style that nobody in the class knew what he was saying. A third year student decided to take pity on us and set up what amounted to a weekly tutoring session. The prof would write the terminology all over the board (fee simple, fee simple absolute, fee tail, and so on), and draw all of these complex diagrams, which nobody understood.
Then, he would attempt to erase the board, but never completely, and then start writing again. It was just a mess!

Making matters worse, he was rather intimidating in his approach to the Socratic method, and as a result, a number of students ended up in tears during his class. As I think about it, he may have been one of the least effective teachers I have ever had.

LynnG said...

Sounds like fun. I had the Gang of 8 for contracts. They were a riot.

There is just nothing like WI for law school. I really enjoyed myself there. With no bar exam hanging over your head, it really freed up the teachers to teach what they wanted. Fortunately, I only had one really bad teacher. Criminal Procedure.

The Prof had just finished as the Head of the Department of Corrections. He got into huge shouting matches with students that tried to defend prisoner's rights and such.

I've heard that he took to heart much of the criticism he got that year (it was his first ever teaching). I've talked to people who graduated after me -- he was their favorite.

I was scared to death in that class. I wouldn't have been surprised if a fist fight had broken out between teacher and students (teacher would have won, I'd bet). We covered so little in lecture that sometimes I'd just skip and read the book instead.

Matthew K. Tabor said...

I've got to be a stick in the mud again.

The problem with the mnemonic is that it reminds one of the procedure but almost wholly ignores the concept. The more conceptually important something is, the less appropriate a mnemonic tends to be.

There are loads of Great Lakes mnemonics, ie. She Made Harry Eat Onions - they're fine because there's no conceptual information beyond the order of the Lakes west to east. Mnemonics are fine for the factual or procedural.

If the question is, "How do I set up a fraction when I'm given the numerator and denominator," a child doesn't understand conceptually what a fraction is about. This mnemonic really doesn't add much to the conceptual understand of fractions, though I'll admit it gets one through the motions properly.

And sometimes that's enough - getting a kid through the procedure can lay the foundation for conceptual understanding *after*, so it definitely can have value.

But, as we all know, fractions are hugely important and mnemonics run a risk of becoming a crutch.

Saxon's method is decent - it gives a procedure that's easy and also demonstrates that one number [numerator] is to be evaluated by another number [denominator].

Anonymous said...

My son never had a problem with a heuristic understanding of fractions. He had problems rewriting a long division problem using a fraction bar or rewriting an expression that used the division symbol as long division in much the same way he confused d/b for years.

Over the past few weeks we've been working a lot with proving that the a/b = a x 1/b, etc., so I'm fairly certain that at this point he understands what a fraction is "conceptually" (at least one of the things that a fraction represents) but he had his arithmetic down cold before we got here and while I didn't use the "top dog" reminder, which I like very much and may use it with another kid, I did use something called "the swing" reminder: The divisor starts underneath the fraction and can arc to the left (long division) or arc to the right (division symbol). See, that wasn't nearly as cool as "top dog."

LynnG said...

Matthew, you are absolutely right that the mnemonic sacrifices conceptual understanding to procedural proficiency.

Sometimes (many times?) conceptual understanding is gained later. I've used many procedures (as do most students)long before I fully appreciated the conceptual underpinnings.

I never understood the relationship of long division to algebra until a couple years ago (that David Klein article was fantastic). That was okay, because I could do long division. And I could do algebra.

This is one of the flaws of EM -- insisting on conceptual understanding first, always, and instead of procedural knowledge. I've got nothing against conceptual understanding, it's great really, but don't hold kids back to the artificial timetables.

Oh - the concept has been explained as well. At some point you don't want kids to have to continually think through the concept every time they are faced with a problem. Mnemonics can be useful like that.

Matthew K. Tabor said...


Absolutely - ramming home concepts first at the expense of the long-term isn't always a productive way to go about things. With a concept as seminal as the fraction, I think you've got to give a good effort at conceptual understanding first [preferably mixed with procedural], but if it doesn't take you get them through the procedure and try again later.

I'd be lying if I said I didn't recite "righty-tighty, lefty-loosey" in my head occasionally when I turn a knob.