kitchen table math, the sequel: multiplying fractions

Wednesday, January 17, 2007

multiplying fractions

UPDATE: POST MODIFIED TO KEEP PREVENT ME LOOKING LIKE AN IDIOT (I work better on paper)

So last night my 6th grader was doing multiplication of mixed numbers. For example:

1 5/10 x 2 3/6 = ?

Based on what she was shown we tackled it the long way.

1 5/10 x 2 3/6 = ? (First change to mix fractions) (Note: I know you can reduce, but ignore that for now)

15/10 x 15/6 = ? (multiply out) >>> Insert multiplication problem here

225/60 = ? Reduce >>> Insert long division problem here.

Of course, after she finished I noted a few wrong answers, mostly attributed to multiplication errors. She also was very resistant to showing her work in an organized manner on a separate sheet, but that's a different matter. I tried to show her how to factor first and then cancel out, but by this time she had shut down out of frustration.

1 5/10 x 2 3/6 = ? (First change to mix fractions)

15/10 x 15/6 = ? (factor all components)

(5 x 3) x (5 x 3)
----------------- = ? (cancel out)
(2 x 5) x (2 x 3)

(3 x 5)
------- = ?
(2 x 2)

15 / 4 = Final Answer

What I really want to know, is which is the appropriate order in which to teach fractions and factoring. Factoring and cancelling seems to me to be a lot easier and allows me to do the arithmetic in my head, but perhaps students need to learn the long way first, to make sense of the short way.

What do you guys think? How does Singapore math do it? Help me, because I am sure there are going to be more problems tonight.

Catherine Johnson said...

hmmm....

Well, I'm no expert, BUT here's what I would recommend based on "what comes next."

What comes next is "arithmetic with letters," i.e. problems like (I have to spell this out in letters - can't do a superscript):

x-squared divided by x

Christopher had a terrible time dealing with variables-with-exponents divided by other variables-with-exponents until Ed wrote these problems out as factors:

(x)(x) divided by (x)

Ed also stressed that "canceling" is actually dividing a number by itself and getting 1.

(I'll check tonight to see if Christopher still knows that.)

So....I tend to think that teaching this now gives you a better lead-in to algebra and exponents than simply having her multiply the two numerators, multiply the two denominators, and then reduce.

I assume she's learned factoring by now - right?

I wonder whether her curriculum has a reason for teaching the calculation this way first and then showing the kids it can often be done more simply by canceling?

Catherine Johnson said...

THE single best piece of advice I ever got from anyone at ktm-1, I think, was from Steve who talked about a teacher he had in high school who had students justify every step in every calculation.

After a year and a half in "Phase 4" math here in Irvington, I COMPLETELY get why the NCTM decided procedural math is the enemy.

Our kids are being taught almost pure procedure, and they're being taught pure procedure with far too little distributed practice & curricular coherence for meaning to emerge from procedure.

But I don't think the answer is discovering your own knowledge or writing explanations of how you solved a problem.

Catherine Johnson said...

I'm glad you wrote this post, because it reminds me I need to do this with Christopher.

Last night he had to learn, in one fell swoop, how to solve equations with variables on both sides of the equation.

(aside: The math teacher-parent told me the kids were given homework problems with variables on both sides of the equation without ever having been taught how to do them in class. Christopher was out sick that week.)

He was slogging through the equations, but he'd obviously forgotten why it's possible to "do the same thing to both sides," which meant that he was also forgetting to do the same thing to both sides.

Tonight I'm going to have him write the identity property, etc., next to each step.

Of course I'll have to look them up first, because I've forgotten the names of the various properties.

sigh

Catherine Johnson said...

Ed has now interviewed either one or two talented high school kids (applying to Princeton), both of whom said their parents thought the math teaching in their schools was purely procedural.

In both cases the parent had remediated by asking his child to derive every formula he used.

Catherine Johnson said...

Rory -- This is the kind of problem Christopher had trouble doing:

http://www.mathdork.com/images/exponents.JPG

All textbooks explain this problem by writing:

X x X (in the numbeator)
X x X x X x X x X (denominator)

Then you cancel.

And the reason why you can cancel flies out of your head!

Anonymous said...

Singapore teaches canceling first. But let's back up a minute.

In the beginning was the rule that says any time you multiply a number by one you get the same number. This holds true with fractions as well. Does the child recognize fractions that are equivalent to one? 2/2 etc?

I had my child do some intermediate steps before "canceling out"

(5 x 3) x (5 x 3)
-----------------
(2 x 5) x (2 x 3)

5 x 3 x 5 x 3
--------------
2 x 5 x 2 x 3

Now rearrange the numbers(commute!) so that you can see how many times you've multiplied by "one"

5 3 5 3
-- x -- x -- x --
2 2 5 3

(I used the preview window and the above does NOT look how I want it)

When you choose to "cancel out" is a matter of computational convenience rather than "mathematical truth." While Singapore stresses the ability to quickly do computations, in general this doesn't happen until the child is proficient at the justification which he must do the long way.

The heart of the matter is the multiplicative identity and it's far more important that the kid understand the reasoning behind canceling out AND be able to apply it rather than whether he does it sooner or later in a particular computation.

The thing that always drives home the point of doing it the easy way is to allow the kid to do it the hard way...many times on his own. After scrawling out hairy multiplication calculations in the margin they are much more receptive to short cuts!

Instructivist said...

"Last night he had to learn, in one fell swoop, how to solve equations with variables on both sides of the equation."

For which grade level is that?

LynnG said...

I'm confused.

Maybe it's the spacing, but it looks like you converted 5 3/10 to an improper fraction of 15/10

But isn't it 53/10? and the mixed number of 3 5/6 would convert to 23/6

which leaves no canceling as 53 is prime and so 23. I must be reading this wrong.

Barry Garelick said...

I got the same thing as Lynn G but the spacing may be off.

Instructivist said...

"5 3/10 x 3 5/6 = ?

Based on what she was shown we tackled it the long way.

5 3/10 x 3 5/6 = ? (First change to mix fractions)

15/10 x 15/6 = ? (multiply out) >>> Insert multiplication problem here

225/60 = ? Reduce >>> Insert long division problem here.

15 / 4 = Final Answer"

Am I missing something here?

You must convert to improper fractions. That'll give you:

53/10 times 23/6

Independent George said...

Wait - my browser might not be displaying it correctly - is it 5 and 3/10 times 3 and 5/6, or 5 times 3/10 times 3 times 5/6?

5 and 3/10 = 53/10; 5 * 3/10 = 15/10
3 and 5/6 = 23/6; 3 * 5/6 = 23/6

Independent George said...

jinx!

LynnG said...

Anyway, when I started fraction multiplication with Singapore 5A, we did a lot of prep work with factoring and the commutative property of multiplication. 5x3 = 3x5

After a review of that, it was a logical step to set up this type of problem

2 x 5
------
15 x 6

and have her see that 2 x 5 is the same as 5 x 2. We can then switch the order and simply reduce like we would an "regular" fraction

5 x 2
----- or
15 x 6

5 x 2
-- -- which is reduces to:
15 x 6

1 x 1
-- -- which is easy to multiply
3 x 3

1
--
9

Does this help?

They need to have a good grasp of the commutative property, know how to find equivalent fractions, and be able to reduce fractions. This fraction mulitiplication then seems to make more sense.

Tex said...

Catherine,

Would constructivist educators say that writing out your explanation is the same as justifying each step? Are you saying they focus too much on the writing part?

TurbineGuy said...

I f**ked up. Post corrected. Principle is the same.

Rory <-- red faced

SteveH said...

"I think, was from Steve who talked about a teacher he had in high school who had students justify every step in every calculation."

Yep. He didn't even like the idea of canceling, because students used it in the wrong places. If they saw the same thing in the numerator and denominator, the kids would cancel them, whether or not they were factors of the fraction. Kids would cancel the 5's in the following expression

(x+5)/5

He drilled into our heads that a/a = 1 is not canceling. Actually, there is no such thing as canceling. I think he was the one who said that there is no such thing as cross-multiplication.

The answer to the question is that students have to be able to show that multiplying mixed fractions works out any which way you do it.

5 3/10 x 3 5/6 = ?

I would recommend that they first change it to improper fractions:

53/10 * 23/6 = ?

My son likes the shortcut of multiplying the denominator times the whole number and adding in the numerator. All of this goes over the denominator. He doesn't like it when I force him to explain exactly why he can do this.

You could have kids show that they get the same answer if they multiply the following:

(5 + 3/10)(3 + 5/6)

Then, I would look to see if these fractions can be reduced. If you want to factor the numbers in the numerators and denominators, that's fine, but students should practice this enough to reduce or simplify any fraction in their heads.

You can't do that in this case, so you could multiply them next, like this:

(53*23)/(10*6)

Then see if there are any common terms to cancel - oops, I used that word. Intermediate reducing or canceling is always helpful. What works to simplify the problem one time may not work the next time.

This is not like long division, where there is one method that is powerful and efficient. If you want kids to be ready for algebra, you have to focus on the basic rules, like a*1 = a, and a/1 = a. I remember thinking that these were stupid and trivial rules until I saw how they worked on complex expressions. If you really understand the rules and how to apply them, then combining and simplifying expressions like this:

a*x^3/(x-2) + 55 - 5*(x-2)^3

is easy.

LynnG said...

Why was the problem written

1 5/10 x 2 3/6

why not reduce first?

1 1/2 x 2 1/2 or as an improper fraction:

3/2 x 5/2 or 15/4

strange that the problem was written that way.

Catherine Johnson said...

sheesh, just lost my comment

sigh

Catherine Johnson said...

For which grade level is that?

The course is so hideously jumbled these kids are going to end up being behind

A mom who dropped her older child out of this course, which has apparently always been conceived as a "killer" course, told me the kids who dropped down sometimes ended up with higher scores than the kids who stayed & struggled on

The Phase 4 math course in Irvington is probably the worst course I've seen in my life no matter who's teaching it

Catherine Johnson said...

While Singapore stresses the ability to quickly do computations, in general this doesn't happen until the child is proficient at the justification which he must do the long way.

The heart of the matter is the multiplicative identity and it's far more important that the kid understand the reasoning behind canceling out AND be able to apply it rather than whether he does it sooner or later in a particular computation.

Perfect.

This is the heart of the matter for Christopher's class.

I shouldn't even call it "procedural" teaching.

It's really shortcuts.

3 years of Accelerated Shortcuts

Catherine Johnson said...

Would constructivist educators say that writing out your explanation is the same as justifying each step? Are you saying they focus too much on the writing part?

My sense is that constructivist educators - I could be wrong here - want a fair amount of what they think of as "originality."

They really do seem to seek "invented" or "discovered" strategies that, of course, won't have names like "multiplicative identity" unless the child just so happens also to invent the phrase "multiplicative identity."

When I look at examples of children's explanations of how they solved a problem on the web, I don't see simple lists of properties used to justify each step.

Does EM teach the commutative property as the commutative property?

I should look to see if TRAILBLAZERS does this.

Saxon Math teaches the properties and their names very early on.

Catherine Johnson said...

RUSSIAN MATH also asks students, fairly often, to write out each step in a computation and write the property that justifies the step on the same line.

Catherine Johnson said...

"Shortcut Math" is, if you don't mind my saying so, probably the right term for the kind of math education to which both the NCTM and mathematicians object.

TurbineGuy said...

Actually I just made up the problem because I was at work. My goal was to just come up with some numbers that could be factored and cancelled out.

I should of posted last night when I had the work in front of me, but I was too busy performing a smackdown of one of my commentors over at parentalcation.

Catherine Johnson said...

smackdown!

I have to see this!

but first I have to take a walk in freezing-cold 20-degree weather!

SteveH said...

lynng is right. The rule could be: "The sooner you can simplify or reduce a fraction, the better.", although I could probably come up with an example where that wasn't true.

3 4/7 * 2 7/10 = ?

25/7 * 27/10 = ?

each can't be simplified, but

25*27/7*10 = ?

can now be factored to get

(5*5*3*9)/(7*5*2) = ?

or

My high school teacher made a point that we really should write it as

5/5 * (5*3*9)/(7*2) = ?

then

1 * (5*3*9)/(7*2) =

to get

(5*3*9)/(7*2) = ?

135/14

In any case, kids should know that simplifying isn't necessary. Do it at the end or don't do it at all, unless the problem states that it has to be in its simplest form.

That reminds me of all of the algebra problems that said: "Simplify this".

Sometimes it wasn't clear what that meant. Do we expand or do we factor or do we ask another student what they are doing?

Instructivist said...

"...unless the child just so happens also to invent the phrase "multiplicative identity."

Double LOL!

Barry Garelick said...

I don't remember being given an explanation of why one could cancel when multiplying fractions (in 6th grade) but I liked doing it, and that was all that mattered then.

When I had algebra, teachers got down on us for calling it cancelling, telling us it was "dividing out", which I suppose is more descriptive. In any event, algebra made a lot of things about arithmetic a lot more obvious. I'm an example of "learned the principle/concept later".

Perhaps this is frowned upon, but if learning the concept interferes with learning the procedure, then something's gotta give. That's sort of Saxon's philosophy. I used to be quite haughty about the sanctity of mathematical concepts until having to deal with a daughter who is ADD.

SteveH said...

"Would constructivist educators say that writing out your explanation is the same as justifying each step?"

This is not the same as writing down the mathematical rule, like a/a = 1. They want students to write it down with words that somehow convey a common-sense, or gut-level understanding of what is going on. Of course, there is nothing more common-sensical and accurate than a/a = 1, but kids don't discover that. They would prefer something like:

5*6/5 = 6

"Well, like you can separate the 5's so that, like, you get 5 divided by 5. If 5 is the whole and you have five of the thing, then that is like having the whole pie and that is equal to one. Then, like, if you have one of the 6, then that is just 6."

Catherine Johnson said...

Steve -

wow!

I'm so word-oriented myself I hadn't even thought of having Christopher write a/a = 1

I was going to have him write "multiplicative identity."

In fact, at this point, it would be MUCH more useful for him, in terms of procedural and conceptual learning, to write a/a=1.

One possible problem is the fact that I don't think he's been shown many a/a = 1 type principles. I'm afraid he's mainly learned these things, when he's learned them, in words.

More learning sacrificed to SPEED SPIRALING.

You're absolutely right that constructivists don't want kids writing a/a = 1.

They don't even want kids writing multiplicative identity, I don't think.

Catherine Johnson said...

Saxon often does a simple word-description of what's been done.

Here's an example from Lesson 113 in Algebra 1.

The distance traversed by a car traveling at a constant speed is directly proportional to the time spent traveling. If the car goes 75 kilometers in 5 hours, how far will it go in 7 hours?.

solution We will use four steps.

Step 1: D = kT write the equation

Step 2: 75 = k(5) k = 5 solve for k

Step 3: D = 15T put k in the equation

Step 4: D D = (15)(7) T = 200 K solve for D

I bet it would help Christopher just to write these word labels.

I think it might help him keep track of where he is in the computation.

I'm just this year starting to be able to get him to write each step directly below the preceding step.

Catherine Johnson said...

I used to be quite haughty about the sanctity of mathematical concepts until having to deal with a daughter who is ADD.

Well this is the CRUX OF THE PROBLEM having only teachers in their 20s.

None of them has been mugged by reality yet.

Karen A said...

"I'm an example of "learned the principle/concept later"."

My 12th grader made a similar comment the other day to the effect that she thought that procedural fluency with math helped her to eventually grasp the concepts.

My 8th grader catches on to the concepts much quicker; she always has. But (am I beating a dead horse here?), we took her to KUMON for 2 1/2 years (3rd through 5th grade) so that she could become procedurally fluent.

Barry Garelick said...

Karen,

Beating a dead horse is sometimes necessary to get one's point across, I'm very sorry to say.

Karen A said...

Barry--

I absolutely agree with that. My daughter's third grade teacher fit Catherine's description of being young and fresh out of ed school. She "didn't know what she didn't know." After several requests for more practice, more repetitions, etc., we finally gave up and took our kid to KUMON. We felt that we didn't have the mental discipline to consistently "after school" her; we needed the external motivation of KUMON.

However, I did continue to "preach my message" to that teacher; I think at some later point, some of what I was trying to convey to her finally sunk in. Of course, it was too late for my daughter's class, but I am optimistic that it helped future classes.

Catherine Johnson said...

I may have to suck it up and haul Christopher back to KUMON.

He doesn't want to do it, and I don't really blame him; it's overkill for him.

KUMON is an example of educational rigidity.

The program is great, but they don't allow individuals to pay for it and then use it as they see fit.

I have a friend who's been taking her child to KUMON for over a year. Her child never passes the end-of-level test because she does the problems too slowly.

She's probably just being methodical; who knows.

The point is, she needs practice in skills beyond whatever level she's on (subtraction? - I think she's having trouble moving from vertical problems to horizontal problems or vice versa).

But she's (probably) stuck.

Nevertheless, I'm probably going to have to tell Christopher we're doing KUMON over the summer.

He's just getting nothing at school.

Catherine Johnson said...

I may have to suck it up and haul Christopher back to KUMON.

He doesn't want to do it, and I don't really blame him; it's overkill for him.

KUMON is an example of educational rigidity.

The program is great, but they don't allow individuals to pay for it and then use it as they see fit.

I have a friend who's been taking her child to KUMON for over a year. Her child never passes the end-of-level test because she does the problems too slowly.

She's probably just being methodical; who knows.

The point is, she needs practice in skills beyond whatever level she's on (subtraction? - I think she's having trouble moving from vertical problems to horizontal problems or vice versa).

But she's (probably) stuck.

Nevertheless, I'm probably going to have to tell Christopher we're doing KUMON over the summer.

He's just getting nothing at school.

KathyIggy said...

"The point is, she needs practice in skills beyond whatever level she's on..."

Exactly the problem we ran into after about 9 months in KUMON Math. Meg made it from 3A to halfway through C, but we really wanted it for help in fractions. That's still a level and a half away, and it seems they've really slowed her down though she knows mult facts very well. The cost was also increasing. So we'll try to go back to Saxon 54 on our own, though KUMON did provide the motivation. KUMON has helped Meg in math this year, but she's in a special ed class now and they're covering stuff she knows easily. Next year in Jr High, sped uses SRA's CMC and can make the instruction much more individualized as to where Meg's gaps are.