FractionManipulativeLessonOnReciprocals < Kitchen

21 Oct 2005 - 03:35

a fraction manipulative lesson on reciprocals

The other night, Ben was working on his math, and I was doing something else. He paused in his work and asked me: "Mom, what's a reciprocal?"

I guess it's no surprise that he doesn't know what a reciprocal is, since it wasn't taught in his Everyday Math classes in elementary school, and since (apparently) it's not introduced in the early part of Saxon 6/5, the curriculum I was supplementing from last year. But in Saxon 8/7, which he's using this year (he tested into it, I swear), knowledge about reciprocals, and the role they play in division of fractions, is assumed.

So I'm doing reactive teaching again, but at least this time I'm reacting to the curriculum of my own choosing.

Saxon has had problems in the mixed practice the last few nights that go straight to the heart of why the reciprocal gets involved in fraction division. The questions are like this: how many 3/8s are there in 1? How many 4/5ths are there in 1?

Here's a demonstration I devised for him on the 3/8ths problem, using the tile fraction manipulatives that Catherine and I have recommended here (warning: pies won't work for this very well).

This sort of question seems to throw him off, so I start by asking other questions that sound more familiar, like: How many 2s in 8? and How many 3s in 9? Then I point out that he is getting the answer by dividing, so by analogy, we'd want to divide 1 by 3/8.

Everyone knows the rule for fraction division: Ours is not to reason why, just invert and multiply. But of course, we are modern traditionalists here, and procedural knowledge is only the beginning of our demands. We want our kids to have an understanding, too, of why they are inverting and multiplying. I taught Ben the invert-and-multiply rule, but then I wanted to convince him that the answer that the invert-and-multiply rule gives you, 2 and 2/3rds, is the right answer.

We attacked the question of how many 3/8ths go into 1 directly, using the manipulatives. The manipulatives were all placed on a sheet of paper, so I could write curly braces and labels next to the tiles. I drew a diagram below of what we do with the tiles (note that the 3/8th tiles are not really single blocks, they are 3 1/8 blocks in a row; I have to tell him to think of them as a single unit.. The labels and curly braces help with this).

It's easy for him to see that two 3/8ths will fit into the 1; I stick them below the 1 tile, and label them as "2 3/8ths". A third 3/8th will overhang the end, though. So I take the extra 3/8ths unit and break it apart into thirds (pointing out that that's what I'm doing). Two of those thirds will fit into the rest of the space in the 1. So this gives us a total of 2 and 2/3rds 3/8-units that will fit into the 1.

I don't expect that this is the end of this; we'll do this a bunch more times and hopefully it will sink in. The trick is to get the kid thinking of the divisor (in this case 3/8ths), however weird a fraction it is, as being a unit. I hope Saxon keeps this sort of problem coming for a while.

Doug Sundseth's downloadable fraction manipulatives & number lines

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THOSE ARE BEAUTIFUL!!!!

-- CatherineJohnson - 21 Oct 2005

how did you do them?

is the line made with the computer line draw thingie???

kind of looks like it

-- CatherineJohnson - 21 Oct 2005

Have you gotten to Lesson 25 yet?

I just did it myself last night, and it was great.

It was the first time Saxon actually gave a bunch of practice of this concept.

Saxon 6/5 teaches this lesson, but just once, and you forget it the instant you've done it. I actually didn't understand it the first time I did it, and taught it to Christopher. (Saxon obviously 'spirals' fraction division in the exasperating preview-instead-of-teaching-to-mastery mode of constructivist curricula.)

I think you've seen the 7-fact family sheet I made up from SINGAPORE MATH.

They teach this idea in connection with ratios & proportions, which was pretty interesting:

Ratio of 2:3 ‘There are 2 girls for every 3 boys’

(part to part)

2 girls for 3 boys
3 boys for 2 girls
2 + 3 = 5 girls & boys altogether
3/5 are boys (part to whole)
2/5 are girls (part to whole)
girls are 2/3 of boys
boys are 3/2 of girls

The curious thing is that, IIRC, they don't point out that reciprocals are involved in the last two facts: girls are 2/3 of boys, and boys are 3/2 of girls. They show it in a drawing, but they don't explicitly make the connection to reciprocals.

I noticed it while we were doing the lesson with my neighbor last year, and made up the sheet. (The lesson was way over Christopher's head at the time.)

-- CatherineJohnson - 21 Oct 2005

btw, the secret here is practice, practice, practice.

Although I understood the Saxon approach, I really couldn't retain it, and would lapse back into reciprocal confusion 5 seconds after re-reading the sequence.

I'd just keep having him do this night after night.

Drawing it is really good, too, but I think you should use quadrille paper. If the drawn units aren't perfectly proportional, you don't see it.

Actually, I'd done so many drawings that the step you just showed had become naturalized for me.

I was having a huge amount of trouble remembering the second Saxon step:

How many 3/4 are in 3?

Step 1: find how many 3/4 are in 1
Step 2: multiply that by 3

-- CatherineJohnson - 21 Oct 2005

This is also a great topic for practicing the idea of forming a unit.

-- CatherineJohnson - 21 Oct 2005

I've been wanting to teach my Singapore Math kids the concept of forming a unit, but I don't think my knowledge of it is flexible enough at this point to do it on the fly (i.e. to jump into the middle of a two-step word problem and instantly identify, in a clear way, the changing units).

-- CatherineJohnson - 21 Oct 2005

I'm sending Ron Aharoni the link to this post.

-- CatherineJohnson - 21 Oct 2005

source:
Primary Mathematics 6A Textbook, U.S. Edition, p. 24

I'll pull this up front later.

-- CatherineJohnson - 21 Oct 2005

If I hadn't done this lesson, I wouldn't have thought of ratios in this 'unit' sort of way....obviously I've always 'understood' that a ratio of 2 boys to 3 girls is also a ratio of 3 girls to 2 boys...but I never connected this back to fractions, and certainly not to reciprocals.

-- CatherineJohnson - 21 Oct 2005

from Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa

The Quotient of Two Numbers

1. A quotient greater than 1. ... In this case the quotient shows how many times larger the dividend is than the divisor. The number 8 is 1.6 times bigger than the number 5.
2. A quotient equal to 1. ... In this case, the divisor and the dividend are equal numbers.
3. A quotient less than 1. In this case, the quotient shows what part of the divisor the dividend is. And so, the number 6 is 3/4 of the number 8.

math and language, part 2

-- CatherineJohnson - 21 Oct 2005

The funny thing is, I have now spent so much time obsessing over reciprocals that they make sense.

What I don't get is: multiplication.

Which is a real kick in the teeth, since the textbooks all get around the explaining-division problem by saying division is the operation by which we find a missing factor.

Meanwhile, I'm not grokking factors.

-- CatherineJohnson - 21 Oct 2005

What I don't grok about multiplying fractions is why it makes sense to multiply the denominators.

Multiplying the numerators now makes sense to me, ever since some textbook (I've forgotten which one) explained the numerator multiplication as a simple matter of whole number multiplication, in just the way everyone's been talking about it here.

That is, if you take 2/3 x 3/4, you've got two one-thirds and three one-fourths, the same as you have 2 ones and 3 ones when you multiply 2 x 3.

You end up with 6.

What I'm not getting is why the denominator changes in the precise way it does; why it works to change the denominator by multiplying.

-- CatherineJohnson - 21 Oct 2005

I should probably experiment with doing zillions of hand-drawings of multiplication of fractions.

-- CatherineJohnson - 21 Oct 2005

To make 3 1/8 pieces into one unit, you can also scotch tape them together.

That makes them seem like one thing because they are one thing once they're taped together!

-- CatherineJohnson - 21 Oct 2005

  2     3
= -  x  -
  3     4

= (2÷3)x(3÷4)

rearrange

= 2x3÷3÷4

regroup

= (2x3)÷(3x4)

   2x3
=  ---
   3x4

   1   
= ---
   2

-- KDeRosa - 21 Oct 2005

I love these little lessons. Especially since I'm exactly at this place with one kid. These are very helpful in teaching confused kids (and moms)

If you don't have the tiles you can just print out Carolyn's lovely bars.

-- SusanS - 21 Oct 2005

Catherine, I made the picture with the Open Office drawing tool, then exported to a jpeg and cropped and rescaled with the GIMP tool. We're strictly Linux around here. :)

I really like that idea of taping the 3 1/8ths together to make a 3/8ths unit!

I also wanted to point out that in order to do this demo you need 2 sets of fraction manipulatives, but they're only about 7 bucks apiece.

-- CarolynJohnston - 21 Oct 2005

sheesh

-- CatherineJohnson - 21 Oct 2005

yup, taping them together INSTANTLY trips the THIS IS ONE THING brain-wire

-- CatherineJohnson - 21 Oct 2005

KDeRosa:

I will mull.

-- CatherineJohnson - 21 Oct 2005

I'll try it with a bit different orthography:

2/3 x 3/4 = (2 x 1/3) x (3 x 1/4)

[rearrange]

= (2 x 3) x (1/3 x 1/4)

[remember that "times" can be replaced with "of" when convenient]

= 6 x (1/3 of 1/4)

[1/3 of 1/4 = 1/12; that is, 3 x 1/12 = 3/12 = 1/4]

= 6 x 1/12

= 6/12

= 1/2

Conceptually I think it helps to deal with the numerators and denominators separately.

-- DougSundseth - 21 Oct 2005

Doug!

You're just the person I was thinking of!

I'm thinking of......FRACTION MANIPULATIVES!!!

YUMMY, PROFSESIONALLY-DESIGN FRACTION MANIPULATIVES!!!

-- CatherineJohnson - 21 Oct 2005

Here are the ones I just made up.

I think we can all see the difference between Amateur Fraction Manipulatives and Professionally Designed Fraction Manipulatives.

-- CatherineJohnson - 21 Oct 2005

OK, that didn't work.

Let me try again.

-- CatherineJohnson - 21 Oct 2005

This is bizarre

-- CatherineJohnson - 21 Oct 2005

alright

the Word document downloading should be amateur fraction manipulatives

-- CatherineJohnson - 21 Oct 2005

OK, I'm printing out the two multiplication explanations.

BUT BEFORE I READ THEM I'M WRITING ONE PAGE OF MY LETTER-TO-AGENT.

-- CatherineJohnson - 21 Oct 2005

I MEAN IT

-- CatherineJohnson - 21 Oct 2005

Hey, those are pretty good! Print em out, paste em on foamcore, label em, cut em out with an exacto knife, and you've got some nice cheapo fraction tiles.

How did you get them to be evenly spaced?

-- CarolynJohnston - 21 Oct 2005

The word doc opens just fine for me.

I think color is useful, but I might have a clever idea about how to use it. (Run for the hills!) Give me a bit and I'll see if I can work something up.

-- DougSundseth - 21 Oct 2005

.pdf file sent to Catherine's e-mail address. Let me know if you would like any changes.

-- DougSundseth - 21 Oct 2005

How did you get them to be evenly spaced?

Carolyn, check on Open Office; you might have the same function.

It's the 'table drawing' tool, and it allows you to automatically space out the grid bars in a row (or a whole table) evenly.

So you can just drop in as many grid lines as you want, then have the program space them right.

It's great; I use it all the time, for other things, too.

-- CatherineJohnson - 21 Oct 2005

When I get to it, I'll print fraction titles on them, too.

I think it's probably a great idea (wish I'd thought of this sooner) to have some with fraction titles & some without).

btw, this is another Saxon tip: his manipulatives are ALWAYS labeled with the equivalents.

The 1/4 piece will be labeled:

1/4
.25
25%

I did that with my fraction tiles, using the Brother P-Touch labeler. On the backs of each tile I have the equivalents.

-- CatherineJohnson - 21 Oct 2005

Doug--MUST GO FOR A WALK!

CAN'T WAIT TO SEE YOUR TILES!

WE'VE BEEN USING YOUR NUMBER LINES ON EVERY EXTENDED RESPONSE PROBLEM

AND I'VE PRINTED OUT ZILLIONS OF THEM FOR THE SINGAPORE MATH CLASS!!

-- CatherineJohnson - 21 Oct 2005

btw, I found the article that cites research saying we have number lines inside our heads, naturally

-- CatherineJohnson - 21 Oct 2005

Carolyn

Is foamcore that same stuff you can buy with adhesive on one side?

You can get 8 x 10 boards of that stuff.

I keep threatening to write the distributive property on it in big letters and post it on the wall.

-- CatherineJohnson - 21 Oct 2005

Foamcore is the stuff that architects use to make models. It's got no adhesive on it that I know of (unless they make it now with adhesive on one side -- it's been a few years). It's like drywall, except it's got about 1/8" of styrofoam in the middle. You can get it at any hobby shop.

-- CarolynJohnston - 21 Oct 2005

hmm

there's something like that sold at Staples...I'll find it

-- CatherineJohnson - 21 Oct 2005

New version dropped. It includes fractions and decimals (with macrons to indicate repeating decimals).

Foamcore is also used for posters, and is usually easy to find in craft stores.

-- DougSundseth - 21 Oct 2005

The fractions are incredible!!!!!

BEAUTIFUL!!!!

-- CatherineJohnson - 21 Oct 2005

Rubber cement will do the trick without wetting the paper and ruining it.

-- KDeRosa - 21 Oct 2005

RUBBER CEMENT!

I FORGOT!

-- CatherineJohnson - 21 Oct 2005

heck

Staples doesn't have, online, the cool 8 x 11 1/2" adhesive foam boards.

-- CatherineJohnson - 21 Oct 2005

9" x 12" self-adhesive foam boards at Office Depot

-- CatherineJohnson - 21 Oct 2005

Doug's fraction tiles fraction labels (pdf file)

Doug's fraction tiles w/percent, decimal, & fraction (pdf file)

I asked Doug if he could make a black and white version, and he did!

Doug's fraction tiles w/percent, decimal, & fraction, BW (pdf file)

-- CatherineJohnson - 21 Oct 2005

Another option for sticking these down would be to print them on label paper. You can (or at least could) get 8-1/2 x 11" labels at office supply stores, then stick them down on foamcore, cardstock, or whatever.

-- DougSundseth - 21 Oct 2005

Did you get the greyscale version?

-- DougSundseth - 21 Oct 2005

oh golly, I didn't!

I'll look!

Check out the homepage!

-- CatherineJohnson - 21 Oct 2005

Label paper is a great idea.

-- CatherineJohnson - 21 Oct 2005

Everyone—

check out the black & white fractions!

-- CatherineJohnson - 21 Oct 2005

Doug, those are sweet!

You're pretty good with the old graphics program, whatever it is.

I think I'm going to make some of those even though I already have the regular ones.

-- CarolynJohnston - 21 Oct 2005

Thank, I used Adobe Illustrator, which is probably unreasonable to buy for casual use. OTOH, I suspect there are some inexpensive draw programs available. (I wouldn't choose a paint program for something like this; wrong paradigm.)

-- DougSundseth - 21 Oct 2005

I like Open Office's Drawing tool (which I used to create the jpeg I included in this post) better than Microsoft's equivalent, but I wouldn't expect either to measure up to the Adobe package.

-- CarolynJohnston - 21 Oct 2005

WebLogForm
Title:	a fraction manipulative lesson on reciprocals
TopicType:	WebLog
SubjectArea:	FractionsDecimalsAndPercents, MathManipulatives, SaxonMath, TipsAndTricks
LogDate:	200510202334

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a fraction manipulative lesson on reciprocals

Comments

Ratio of 2:3 ‘There are 2 girls for every 3 boys’

The Quotient of Two Numbers

WE'VE BEEN USING YOUR NUMBER LINES ON EVERY EXTENDED RESPONSE PROBLEM

Everyone—