Entries from MathManipulatives

About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a model of a room that was part of my lab. The real space was furnished like a standard living room, albeit a rather shabby one, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible to their larger counterparts: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory of the small room to figure out where to find the toy in the large one.

The three-year-olds were, as we had expected, very successful. After they observed the small toy being placed behind the miniature couch, they ran into the room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the room to retrieve the large toy, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.

Their failure to use what they knew about the model to draw an inference about the room indicated that they did not appreciate the relation between the model and room. I soon realized that my memory study was instead a study of symbolic understanding and that the younger children's failure might be telling us something interesting about how and when youngsters acquire the ability to understand that one object can stand for another.

here's the anti-constructivist moment:

[The] capacity [to] create and manipulate a wide variety of symbolic representations .... enables us to transmit information from one generation to another, making culture possible, and to learn vast amounts without having direct experience--we all know about dinosaurs despite never having met one. Because of the fundamental role of symbolization in almost everything we do, perhaps no aspect of human development is more important than becoming symbol-minded.

symbols aren't 'natural'

The first type of symbolic object infants and young children master is pictures. No symbols seem simpler to adults, but my colleagues and I have discovered that infants initially find pictures perplexing. The problem stems from the duality inherent in all symbolic objects: they are real in and of themselves and, at the same time, representations of something else. To understand them, the viewer must achieve dual representation: he or she must mentally represent the object as well as the relation between it and what it stands for.

A few years ago I became intrigued by anecdotes suggesting that infants do not appreciate the dual nature of pictures.

[snip]

.... the Beng babies, who had almost certainly never seen a picture before, manually explored the depicted objects just as the American babies had.

The confusion seems to be conceptual, not perceptual. Infants can perfectly well perceive the difference between objects and pictures. Given a choice between the two, infants choose the real thing. But they do not yet fully understand what pictures are and how they differ from the things depicted (the "referents") and so they explore: some actually lean over and put their lips on the nipple in a photograph of a bottle, for instance. They only do so, however, when the depicted object is highly similar to the object it represents, as in color photographs....

[snip]

it takes several years for the nature of pictures to be completely understood. John H. Flavell of Stanford University and his colleagues have found, for example, that until the age of four, many children think that turning a picture of a bowl of popcorn upside down will result in the depicted popcorn falling out of the bowl.

Andrew makes Barney fly

Martine came in and said, 'He wants you to make Barney fly.' She'd been sitting in the family room when Andrew had put his Barney on top of the plane, and then flung plane & Barney up into the air, apparently thinking Barney would fly around the room.

Andrew had been very unhappy with the outcome, and was now appealing to me. Clearly he believed that making Barney fly was one of those things, like operating the TIVO, only adults know how to do.

I was flattered, but also dumbfounded. What goes on inside this child's head? was my exact thought.

I was thinking....does he not understand gravity?

Does he not understand toys?

What's with this kid???!!

The Scientific American article makes me think that Andrew, although he can read, hasn't completely figured out the dual nature of symbolic representation.

He probably couldn't understand the plastic airplane as being TWO THINGS:

• an airplane
  
  AND
  
  
• a symbolic representation of an airplane

What I'd like to know is: what does he think about Barney?

is this a shoe?

lost in translation

Assuming this work on manipulatives & symbolic representation is correct, the constructivist obsession with manipulatives looks to be another instance of a good idea lost in translation. Constructivism is majorly obsessed with manipulatives, that's for sure. I understand that the TERC curriculum is basically just a huge box of manipulatives, with no textbook or 'consumables'--workbooks--at all.

Following in Piaget's footsteps, constructivists believe children don't reach the stage of 'formal operations' until age 11; from 7 to 11 they're in the Period of Concrete Operations. (Often you'll see the word 'developmental' used to designate constructivist curricula. Apparently that's a reference to Piaget.)

Wayne Wickelgren says this is nonsense; children can handle abstract concepts long before age 11. But constructivists are the people time forgot, and they're still basing their pedagogy on work done in the 1950s.

That's bad enough in itself, seeing as how the field of cognitive science was just getting started around that time, and Piaget's work hasn't fared so well over the past 60 years.

But the more glaring misstep, it appears, is that they failed to grasp the nature of the concrete.

The reason constructivists think children should spend their grade school years working with manipulatives is that manipulatives are concrete. But they're not. Manipulatives are symbolic objects that require the child to have mastered the concept of dual representation.

Skinnies and bits are not concrete. They are symbolic representations of the Hindu-Arabic numeral system. Worse yet, they are more intellectually demanding, and hence more confusing, symbolic representations than pencil marks on paper.

They're harder to understand, not easier.

Lost in translation.

question

My question is: why should pencil and paper be less challenging than manipulatives?

I can see why pencil and paper wouldn't be any more challenging than manipulatives, but why should pencil and paper be easier? Do pencil marks somehow not involve dual representation? That's what the authors seem to imply, but they don't say so directly.


CA state study on manipulatives
Fraction Manipulatives
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
NewStudyOnManipulatives
New Study on Manipulatives Part 2

NewDimensionalDominoes 04 Aug 2005 - 16:22 CarolynJohnston


DanK has added some more dimensional dominoes worksheets at his dimensional dominoes page. If you're interested in learning a little bit about dimensional analysis and perhaps teaching it to your kids -- it's great for checking work -- have a look (and please don't be shy about giving feedback and asking questions!).

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DougSundsethNumberLines 30 Sep 2005 - 21:37 CatherineJohnson





blank number lines (pdf file)

symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)

number lines: all positive numbers (pdf file)

number lines: all negative numbers (pdf file)

update

If anyone is interested in, or has time to, critique these study sheets, that would great. (There's no pressing need for this; I'm reasonably certain these are accurate, especially since the second document came straight from the pages of Mathematics 6.

addition & subtractions of integers review sheet

integers problems from RUSSIAN MATH

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RonAharoniOnTheFifthOperationOfArithmetic 14 Sep 2006 - 14:53 CatherineJohnson


Carolyn has kindly left my two favorite passages in Ron Aharoni's What I Learned in Elementary School for me to blooki.

Here's the first:

What Arithmetic Should Be Covered in Elementary School?

The embarrassingly simple answer is: the four basic operations—addition, subtraction, multiplication, and division.

Yet, this seemingly simple answer is deceptive in two ways. One is that there are actually five operations. In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.

I've thought about this observation every day since reading Aharoni's article. I probably can't explain why. At least, I can't at the moment. (Good thing I'm not taking the Regents, I guess.)

But it reminded me of a post Carolyn wrote early on:

Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.

These are the fraction tiles I like:




You can order extra tiles, too, which I have done. I've used these over and over again, with Christopher, and with at least two of his friends.

Worth their weight in gold.


Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

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FractionManipulativeLessonOnReciprocals 14 Sep 2006 - 14:26 CarolynJohnston


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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FractionTilesByDoug 14 Dec 2005 - 21:39 CatherineJohnson

• downloadable fraction manipulatives (color - pdf file)

• downloadable fraction manipulatives (color - pdf file)

• downloadable fraction manipulatives (black & white - pdf file)
  

ideas for using these

from Carolyn:
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.

from Catherine:
9" x 12" self-adhesive foam boards at Office Depot
I also happen to own a laminator, which I guess places me in a Special Autism Category. But if you have a laminator, it's incredibly quick. They have those 'fake laminator' pages too....the 'cold' ones, at Staples. You don't need a laminating machine to use them
self-adhesive lamination, letter size sheets at Staples

from KDeRosa:
Rubber cement will do the trick without wetting the paper and ruining it [if you're using non-adhesive foam board].

from Doug:
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.

from Susan
...for the lazy parent you can just print several times on heavy card stock in different colors. They'd be stiff enough to last for a bit. I did that with the Saxon pie manipulatives from Saxon 6/5. I just scanned the pages and bought heavy duty colored card stock and printed. Laminating sounds good, too. I totally forgot about that cold stuff at hobby stores. Just printing out works best for me, though. If they wreck it I just print again.


This is cool: brightly colored card stock at Staples



(hmm. I wonder if Doug feels like making one more fraction sheet with no color or graytone at all??)

fraction tile pdf files attached to Comments page

I've posted a link on Math lessons, on Favorite Math Supplements for Kids, and on the book-style index page, too. (Just noticed: I need links to Doug's number lines & Dan's dimensional dominoes in the index, too.]]

extended response

from Doug:

I chose the tile colors using a particular scheme.

What is that scheme?

(Catherine isn't allowed to answer, since I told her what it was in the email.)

I'm going to try this on Christopher!

(This strikes me as a Math Challenge that's actually FUN. Or could be, anyway.)

I say 'or could be' because my philosophy on What Is Fun For A Child is to find out from an actual child whether a thing is fun or not.

Do you ever wonder whether the people at Everyday Math have ever actually met a child?

fraction tiles in black & white

keywords: Doug's fraction manipulatives

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DougsDistributivePropertyGraphic 26 Oct 2005 - 23:04 CatherineJohnson




this is beautiful!





In my next life, I want to come back as a graphic designer.

distributive property manipulatives?

Any thoughts on whether distributive property manipulatives might be a good idea?

I'm thinking something extremely simple, maybe a large grid and those round plastic number 'counters' (they look a little like thin Poker chips). Or you could just use paper, or pennies or buttons. Anything.

You could ask the child to show you:

3 ( 2 + 4)

as an array like Doug's, or possibly as the two different arrays, which might be best. The child would see the equivalence because he or she had just made the equivalence with his own hands.

Temple on using your hands

I talked to Temple (Grandin) again last night. She had lots of horror stories of professional architects who can't make scale drawings. These are all young people who learned to do scale drawings on the computer.

She is adamant that there is a motor component to seeing. She thinks you can't get your visual processing right if you haven't....done things with your hands (can't be more specific than that).

The mistakes these people made were all perceptual mistakes, not cattle handling mistakes. (We're talking about the meatpacking industry.)

I really think there's something to this. (It's making me curious about that strange-looking Borenson fellow with his hands-on algebra, I must say....)

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JDFishersDistributivePropertyManips 27 Oct 2005 - 13:35 CatherineJohnson


Incredible!





I can't wait to try these with Christopher!


hmm

I wonder if this image is slightly to big for the front page....

I have to go get Christopher & his friend Joe; back later. (If I need to cut this image down slightly, I'll do it when I get back.)

Extremely cool!


J.D.'s full-size graphic is here.

J.D.'s model for distributive property with subtraction

I just noticed J.D. also included a model for the distributive property used with subtraction, also in the Comments section. (Will jpeg it & post up front ASAP.)

NAEP analysis

And...I just discovered J.D. has posted an analysis of the new NAEP scores.

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MathemagicalManipulatives 27 Oct 2005 - 13:33 CatherineJohnson



J.D. just coined a term for really bad manipulatives.

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DougsWorksheetGenerator 28 Oct 2005 - 19:16 CatherineJohnson






Random Worksheet Generator

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DanVisualFractionMultiplication 30 Oct 2005 - 22:11 CatherineJohnson



from Dan K

[This is] my attempt to show why you need to multiply the denominators in fraction multiplication.

3/4 x 5/6

In the attached powerpoint file, we have one big figure that represents 3/4. That is shaded in light blue. Then, we have a smaller figure that represents 5/6. That is shaded in orange.





In the second slide, we combine these figures by multiplying them together. Every place that was shaded light blue to begin with is "covered" with a 5/6 section. The amount of the resulting figure that is shaded orange is the answer to 3/4 * 5/6. You can see that the original large figure has now been subdivided into 24 sub- sections, 15 of which are shaded orange.




hmm

I'm having trouble 'reading' this. (I say visuals are abstract, and they are!)

The first thing I stumble over is the 5/6 being smaller....then I'm wondering what would happen if we made the 3/4 small and the 5/6 large, and we covered up the 5/6 orange with 3 of the THREEFOURTHS chunks....and I'm thinking, OK, it would be the same thing.....

But I'm still not getting why we're multiplying the denominator, apart from needing the same unit in which to express the answer.

Here's the thing that constantly stumps me, and I don't know why it should.

With addition & subtraction of fractions, it makes sense to me that we need the same unit in which to express the answer, and that the unit should be the same denominator, AND that we find the same denominator by using multiplication (or division, as the case may be) to find an equivalent fraction.

For some reason, I'm just not getting why we use multiplication to create the same denominator in the multiplication of fractions.

I don't know what my problem is, or why I'm not 'seeing' it here....

(I need to look at the other two answers.)

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WhatToDoWithLinnkingCubes 17 Nov 2005 - 01:19 CatherineJohnson



I met a retired teacher at a bowling party the other day, who told me a story about math manipulatives.

He'd been a middle school teacher, and for some reason he was put in a 3rd grade class and given a crate full of math manipulatives. He had no idea what to do with them. He was especially stumped by the 'linking cubes,' which looked like Legos, but clearly were not Legos.

I had the same experience two summers ago when I orderd the Saxon Math manipulatives that go with the Saxon Kindergarten book. The Geoboard especially stumped me, but the linking cubes were a mystery, too.

Now I get it.






source:
Harcourt School Publishers Multimedia Math Glossary

I like Harcourt's glossary. The images are clean and sharp, and you can ask the website to speak the name of each concept or rule, which is fantastic for a child like Andrew who (probably) never hears the same word the same way twice.

A recorded word is much more likely to be a 'stable stimulus.'


On the other hand, their illustration of multiplication (in Grade 2) is visually wrong, I think. (Anyone else?)

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DougsExcelProblemWorksheet 18 Nov 2005 - 18:20 CatherineJohnson


Doug to the rescue again.....

We were talking about the problems a lot of our kids have re-copying problems from worksheets, and....you guessed it.

Doug has made new worksheets!

They're here (scroll to the bottom of the Comments section):

Doug also left this comment:

I've promised him cash for performance ($1 if he can get under 4 minutes with 100% correct — so largely a token), but he was pretty happy with his feeling of accomplishment and our praise last night. Now that I think about it, maybe I should cobble up a certificate too.

Needless to say, I vote yes on the certificate.

The very thought of a beautiful, professionally designed Homework Award Certificate, instead of the garish, Page Splattered reward stickers you buy at Staples—I'm feeling faint.

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GeometryFractionKit 19 Dec 2005 - 20:38 CatherineJohnson







I wonder if I can still get one of these in time for Christmas?


Or this:

Your calculator can't help you

(We don't need no stinkin' calculators!)

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LegosForVolumeAndArea 03 Jul 2006 - 19:32 CatherineJohnson







After Ed did his final teaching-to-crammery unit on volume formulas, "teach conceptual understanding of volume formula" popped onto my to-do list.

I don't know whether I acquired much comprehension of volume formulas when I was in school. I don't think I did. Since I've been reteaching myself math I've spent quite a bit of time staring at array models trying to grok the fact that 3 x 4 really is 3 "of" 4 and vice versa.

I love array models.






source:
joannegoodwin 2nd grade class



Saxon 8/7 has a number of fantastically effective lessons in which students have to figure out how many 1-inch square sugar cubes will fit into a square or rectangular box. I haven't checked Prentice - Hall (I may never be able to look at that book again) but I'm guessing Christopher didn't spend a lot of time thinking about how many small cubes can fit into a larger cube or prism.

So I was gearing up to search through Saxon 8/7, find all the lessons with sugar cubes, figure out a schedule, harrass Christopher to sit down with me and go over them, etc. when it struck me that a cottage-cheese size six-dollar plastic container of tiny Legos would do the job.

I bought two little white bases plates, each measuring 8 x 2cm and a bunch of white & see-thru red squares. Christopher likes to build Legos &mdsah; at least he's back to liking them at the moment — so he can spend some time building Lego rectangular prisms & cubes and I won't have to spend hours tracking Lessons in Saxon 8-7 and then tracking Christopher to get him to come do them.

I also found the coolest little Lego thingie for fooling around with area & circumference of circles! It's a Lego square with a Lego circle on top, that turns. don't know what it's supposed to be. I'll see if I can find a picture.

[pause]

I have no idea what I bought. It's minute — diameter is 3 cm — and we won't spend much time with it.

But it's all I need for some hands-on distributed practice.


I'll also have him take a look at this webpage for Houghton Mifflin Math:


The volume of a rectangular prism can be found by counting the number of cubic units or by using a formula. The formula for finding the volume of a rectangular prism is V = l w h.




The volume of this rectangular prism is 30 cubic meters.

Another way to measure a solid figure is to find the surface area. Surface area is the sum of the areas of all the faces of the solid figure. Since opposite faces of rectangles or cubes have the same area, you can also multiply each area by 2 and then find the sum of the areas of the faces.





The surface area of this rectangular prism is 52 yd2.

Perimeter, area, volume, and surface area are measurements of geometric figures. Perimeter and area are measurements of plane figures and surface area and volume are measurements of solid figures.


This one's nice, too:

To find the area of a complex figure, separate it into simpler figures, find the areas, and then add the areas together.





And these two:


Figures with the same perimeters can have different areas.





Figures with the same area can have different perimeters.





teachtocrammery



-- CatherineJohnson - 19 Jun 2006

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DougFractionTilesRevisited 05 Jul 2006 - 23:07 CatherineJohnson




Here's a cool lesson to do with Doug's fraction manipulatives (pdf file):





source:
Primary Mathematics 2B US Edition





Doug's fraction tiles & number lines (pdf files)

blank number lines

symmetric number lines

positive number lines

negative number lines

fraction manipulatives

Doug's fraction tiles w/equiv decimal & percent

fraction tiles with equiv decimal & percent b&w



-- CatherineJohnson - 05 Jul 2006

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PicturesOfNumbers 20 Jul 2006 - 17:23 CatherineJohnson




This blog looks fantastic.


good grief

More synchronicity. This is becoming a daily occurrence.

I just pulled up my two-year old Excel chart for Jimmy's weight, and today I find a blog with a post called Fixing Excel's Charts.

I'm going to pretend this didn't happen.

-- CatherineJohnson - 18 Jul 2006

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FractionManipulativesAndNumberLinesByDougSundseth 14 Sep 2006 - 14:44 CatherineJohnson




Doug Sundseth created these fraction manipulatives a year ago. They're incredibly useful, so download when you need them and let your friends know, too. (pdf files)

• fraction manipulatives (color)

• fraction manipulatives (color)

• fraction manipulatives (black & white)
  

Doug's manipulatives and number lines are easy to find on Our Favorite Math Supplements for Kids (link in sidebar)

Alternatively, if you search Blog Posts this post should be the first to come up.

Last but not least, I've finally added links to Doug's fraction manipulatives and number lines on the "Book-Style Index" page.

So these things are about as findable as I can make them at the moment. (Apparently I've been inspired by my purchase of a used copy of Steve Krug's Don't Make Me Think at the Mercy College bookstore.)



KTM Guest: fraction manipulative website

I'm guessing this is from Suzanne:

Math Playground has an online version of the fractions manipulative. You can change the number of parts, shade in what is needed, and toggle the fraction labels. The link is: Math Playground fraction bars.

Math Playground looks wonderful. Thank you!


Doug's number lines

• blank number lines (pdf file)
  
  
• symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)
  
  
• number lines: all positive numbers (pdf file)
  
  
• number lines: all negative numbers (pdf file)
  
  

These are good, too:

• number lines from Schoolhouse Teach (pdf file)

Links to Doug's number lines are also on these pages:

• number lines on Favorite Supplements page

• fraction manipulatives on Favorite Supplements page
  

Carolyn's lesson on reciprocals

Here is Carolyn's lesson on fraction manipulatives and reciprocals.



Now I need to pull together one post with all the stuff on unit multipliers we've put together over the last year and a half...








archive of posts on math manipulatives

original post about Doug's fraction tiles
original post about Doug's number lines



-- CatherineJohnson - 14 Sep 2006

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LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

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