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09 Oct 2005 - 16:33

a question on teaching fractions from Lone Ranger

Lone Ranger put this question on the Request Page:

Nuts and bolt help for my 8 year old daughter for all you math experts....So, we are working in book 4A in Singapore Math and are learning about fractions of a set. We move into problems such as 2/3 of 27. She can easily build this problem with beads or draw it so we move on to the the algorithm. My daughters asks me why 27 can be written as 27 over 1 and I am stumped. I cannot figure out how to "show " her like I can with improper fractions or mixed numbers. I told her it is a division problem but she wasn't ready to understand that. Any ideas?

This is one of those situations where I sputter a bit, trying to think up a good way to teach this to an 8 year-old. Here's my best shot at it (of course a good teacher has at least 3 ways to explain anything, so hopefully others out there will have other approaches):

You could say that the denominator represents the number of pieces that a unit is broken up into, and the numerator represents the number of such pieces that you have: she understands this intuitively when the denominator is a number greater than 1, I expect. Do comparative examples, perhaps, where you go from saying that 4/2 means you've split a unit into two pieces and you have four of them, to saying that 2/1 means you've split a unit into 1 piece and you have two of them.

So I guess that's one way to try to teach it; now we need two more.

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that's also the first thing that occurs to me.

let's see: how about this. "a/b = c" is
(almost [-- b \not= 0]) equivalent to "a = bc",
we already have "a" and "b" in our example: 27/1.
so the equivalence just mentioned tells us that
the "answer" ("c") in 27/1 = c is the same as in
21 = 1c -- "what do i have to multiply 1 by to get 27?".

now we need one more.
hint: the reciprocal of the reciprocal of 27.

-- VlorbikDotCom - 09 Oct 2005

Try a number line. To represent 1/3 we split the unit space into three equal parts and then we count one of them. For 2/3 we split the unit space into three equal parts and count of two.

Now we need to extend this representation to improper fractions. For 7/4 we cut each unit into four equal pieces and count out seven pieces. Finally we get to 3/1. Here we leave each unit space alone (cut it into one piece that equals itself) and then count unit spaces.

So a whole number can be represented as a fraction. Why bother? Because sometimes it helps us solve a problem.

-- CharlesWilliams - 09 Oct 2005

Wow, that's neat (Lesley's explanation--!)

I'll go look up the way RUSSIAN MATH teaches it, but I can warn you that by the time you get to the end of RUSSIAN MATH, they are demonstrating math concepts 'mathematically,' by which I mean to say semi-proofs. (Is that the right terminology?)

They use a sequence of mathematical steps & reasoning.

-- CatherineJohnson - 09 Oct 2005

You can't be serious about teaching algebraic-type stuff to a third grader.

I would suggest also teaching the idea of division. Point out that the number at the bottom divides the number at the top. Write 12/3. Then take 12 pieces of something and split them into four groups. Then write 12/2 and split into two groups. Then write 12/1. No more splitting. See where I am going with this?

-- CharlesH - 09 Oct 2005

Doug's number lines are here.

-- CatherineJohnson - 09 Oct 2005

I just checked through Mathematics 6 & I don't think they ever formally teach, or even show, that a whole number can also be thought of as that number divided by 1.

-- CatherineJohnson - 09 Oct 2005


let's see: how about this. "a/b = c" is (almost [-- b \not= 0]) equivalent to "a = bc"

I think this is too big a leap for an 8 year old -- you'll get all mired up trying to explain why this is true, and lose sight of the original question.


Write 12/3. Then take 12 pieces of something and split them into four groups. Then write 12/2 and split into two groups. Then write 12/1. No more splitting. See where I am going with this?

Not really, although I just had another thought as a result of reading what you wrote..

the 'quotitive' explanation of division says that x/y can be interpreted as the number of times y will fit into x.

So, LoneRanger, you can just explain to her that just as 4/2 can be thought of as the number of times 2 goes into 4, 27/1 can be thought of as the number of times 1 goes into 27.

-- CarolynJohnston - 09 Oct 2005

Saxon Math comes at this in a terrific way, I think.


Saxon spends quite a bit of time underlining the identity property of multiplication.

1 x a = a

I find this to be an easy idea for a child to grasp and believe; it's probably one of those inborn math facts animals know, too. (That's my guess.)

So start there, with 1 x a = a. Or, for an 8 year old, a bunch of examples.

1 x 2 = 2
1 x 3 = 3
1 x 100 = 100
1 x 1001 = 1001

Note: I've found kids love this kind of sequence, where THE RULE IS ALWAYS THE SAME AND IS ALWAYS OBEYED!




Justice is served.


The books also spend a huge amount of time teaching the kids 'fact families,' in order to establish the idea that multiplication and division are inverse operations. This is a fantastic aspect of Saxon.

A typical fact family:

3 x 2 = 6
2 x 3 = 6
6 / 2 = 3
6 / 3 = 2

Practically every lesson in Saxon 6/5 has the kids do fact families during either the mental math portion of the lesson, the written mixed review, or both. The problems are written something like this:

'Create a multiplication & division fact family using the numbers 2, 3, & 6.'


A Saxon kid gets it completely drilled into his head that division & multiplication are inverse operations, SO since a Saxon kid knows absolutely that 1 X a = a, you can have him do a four-fact family to show that:

1 x a = a
a x 1 = a
a/a = 1
a/1 = a

At the bottom of that list you see that--voila!--any number can be divided by 1.

If this is too abstract for an 8-year old (it probably is), you simply illustrate this idea using numbers:

1 x 3 = 3
3 x 1 = 3
3/3 = 1
3/1 = 3

I think the four-fact families (which Saxon does with addition & subtraction, too) are fantastic. Incredibly effective.

SINGAPORE MATH doesn't use them, as far as I know. Singapore seems to use 'number bonds' instead, which are the triangular flash cards with one of the 3 numbers of the fact family written in each corner. Parker & Baldridge actually recommend the number bonds over fact families, I believe.

I completely disagree, based on Christopher and on me.

Christopher & I spent so much time doing 4-fact families, that we both have DEEP BELIEF & INSTANTANEOUS RECOGNITION that multiplication & division are inverse operations.

-- CatherineJohnson - 09 Oct 2005

right; i'm not advocating that one spring
variables on an eight-year-old kid out of the blue.
rather, demonstrate with examples:
12/3 = 4 "because" 12 = 4*3 etcetera.
evidently this is the "quotitive" property.
live and learn.

-- VlorbikDotCom - 10 Oct 2005

Well.....Singapore Math does have kids doing what I would call algebra (i.e. there is an 'unknown' that must be solved for) in 3rd grade math.

I think V is right that you can show this to a 3rd grader, but I suspect you need the long-ish build-up Saxon gives it. (That is, a lot of practice of the idea of inverse operations.)

-- CatherineJohnson - 10 Oct 2005

I'd also like to point out that Singapore gives the kids a fair amount of practice in this area. We're only in Level 1 and we've had to "write four sentences with these numbers" several times already. I expect to see the concept in the later books.

I have triangle flash cards, but they're a lot less visually cluttered than the ones you showed from Paula's Archives. Mine just have the numbers; the parts are in blue and the whole is in pink. I think the colors change in the mult/div set, but I haven't looked at them at the store.

I think both approaches are valid (fact families vs. number bonds), and that they're just two different ways of teaching the same thing. As long the student reaches automaticity, and understands the inverse operations idea, then it's fine.

Just because Everyday Math takes a good thing that Singapore or some other program uses and "includes" it in their program, doesn't make the good programs fuzzy. It's a whole shift in philosophy that makes Everyday Math fuzzy. Singapore, Saxon, Rod & Staff, and all the other good ones -- they all teach math explicitly. That's the difference.

-- BrendaM - 10 Oct 2005

On the fraction pedagogy question, I thought of something we covered in 1B last week when they introduced multiplication and division (yes, they do it that early).

The book was explaining multiplication in terms of numbers of sets of a given quantity (like I'm assuming many books do). So, you've got 3 sets, or groups, of 4, and that makes 12 all together.

Can you explain that another way to display the whole number 27 is to show it as 1 set of 27, or 27 groups of 1?

It's the same number, wearing different clothes. If you're working with fractions, then a whole number needs to put on its "fraction work clothes" and it will look different than it usually does. (I'm going off the top of my head here, but maybe an 8yo girl will understand the clothes analogy.)

-- BrendaM - 10 Oct 2005

Thank you everyone. I'm going to implement these ideas today. We too talk about fractions being in "Halloween's still you but disguised" I think what threw her was the idea of 1 as a denominator as it had never come up before, and she couldn't visualize what that meant. I appreciate your help!

-- LoneRanger - 10 Oct 2005

"Note: I've found kids love this kind of sequence, where THE RULE IS ALWAYS THE SAME AND IS ALWAYS OBEYED!




"Justice is served."

I've had exactly this experience with my son (who's quite a bit younger -- in kindergarten). He's fascinated by n-n=0, and loves to be asked random versions of this.

Recently, he's figured out that x*10^n + y*10^n = z*10^n. I think the next concept I try will be (n+1)-n=1.

Obviously, I'm not using variables with him, but the concept should extend eventually to explicit variables when he gets a bit older.

-- DougSundseth - 10 Oct 2005

Title: a question on teaching fractions from Lone Ranger
TopicType: WebLog
SubjectArea: ElementaryMath, FractionsDecimalsAndPercents, SingaporeMath
LogDate: 200510091231