ANN Numeracy Discussion List:
“How to See Fractions”

Any information, especially on using manipulatives, to help students see or visualize fractions before they start working with them.
Jane Ellis
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Date: Tue, 15 Aug 1995 14:57:49 -0400
From: fsoucie@k12.oit.umass.edu (Frances Soucie (Berkshire T & E/ALP))
To: maltt@world.std.com
Subject: Re: how to "see" fractions
>
>Any information, especially on using manipulatives, to help students see or
>visualize fractions before they start working with them.
>Jane Ellis
>
In a fourth grade I used an overhead projector. I had a clear plastic box and placed it on the projector with a layer of quackers. The number of quackers was the denominator and as the students took turns eating a few quackers at a time the recount became the new numerators.

I also like saving the plastic bread ties that are found on baked goods. Not the twisties but the various colored flat ones with the slit which show the price. These are inexpensive manipulatives and can be used to demonstrate many operations in math.

quackers??? yes! the ducky ones!
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From: "Calvin E. Piston"
Organization: John Brown University
To: numeracy@world.std.com
Date: Tue, 15 Aug 1995 14:12:45 GMT-5
Subject: Re: how to "see" fractions
> Any information, especially on using manipulatives, to help students
> see or
> visualize fractions before they start working with them.
> Jane Ellis
>
A great resource is Math and the Mind's Eye: Unit IV Modeling Rational Numbers.

This is published by The Math Learning Center, PO BOx 3226 Salem Oregon 97302
**********************************
* Calvin Piston *
* Department of Mathematics *
* John Brown University *
* Siloam Springs, AR 72761 *
* 501 - 524 - 7272 *
* FAX: 524 - 9548 *
**********************************
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Date: 16 Aug 1995 09:51:18 -0700
From: "Virginia Thompson"
Subject: Re: how to "see" fractions
To: numeracy@world.std.com

We used a Fraction Kit idea in the FAMILY MATH book that provides a very elementary beginning. It has been extremely successful in family settings with both children and adults. (pp. 120-124, FAMILY MATH) The book is available through the Lawrence Hall of Science EQUALS/FAMILY MATH programs and in many catalogs and book stores.

Many others use manipulatives such as Pattern Blocks, Cuisenare Rods, and cubes to develop concepts in fractions.

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Date: Wed, 16 Aug 1995 15:38:38 -0400
From: gyarnell@k12.oit.umass.edu (Glenn Yarnell Jr. (HCC/SABES/ALP))
To: maltt@world.std.com
Cc: numeracy@world.std.com
Subject: Re: how to "see" fractions

>
>Any information, especially on using manipulatives, to help students see or >visualize fractions before they start working with them.
>Jane Ellis
>
>
Can't claim credit for these two ideas...that goes to Leslee, who's sitting here reading as I peck away....

Cooking and recipes: the portions of the cups, doubling or halving, etc...plus goodies afterward!

Hershey bars: the way they are segemented. S'pose any candybar that's 'blocked' like that would work.

glenn, via leslee's ideas...both obviously very hungry.

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Date: Thu, 17 Aug 95 08:45:45 EST
From: "Mark Schwartz"
To: numeracy@world.std.com
Subject: Re[2]: how to "see" fractions

It has been a while, but I recall in may teaching days using Dienes blocks. Dienes developed a wide array of materials, which can introduce children to complex ideas using simple manipulative games ... this includes an introduction to probability concepts (without the statistical language). I don't know how popular they are now, but I strongly urge you to do some research and find the literature. I've seen them work ... it's powerful ... mark

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From:
To: numeracy@world.std.com
Date: Thu, 17 Aug 1995 11:08:36 GMT+400
Subject: Re: how to "see" fractions
Jane:

Believe it or not, I have found that using pies is a great way to introduce the concept of fractions. We have done it over coffee break in the class. The adults had fun with it. We used pies cut into a variety of sizes to demonstrate the parts of a whole. Most had a great laugh but found that it now made sense. You couldn't eat it until you knew what fractional part of the pie you were eating.

Mary Thompson

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Date: Mon, 21 Aug 1995 23:09:56 -0400 (EDT)
From: Lynne Mikuliak
To: SusanG2@aol.com
Subject: Re: how to "see" fractions

Jane,

Before I got the time to answer your fraction question Mary Thompson brought up pies, and cooking, measurment, and Hershey bars were mentioned. Great ideas all. I use cakes. I bake two sheet cakes and use them to demonstrate one whole cake cut into halves, fourths, etc. As the students are pretty interested in what is going to happen to the cakes AFTER this lesson I usually have their total attention. I use the second cake to demonstrate thirds, sixths, etc. I frequently repeat as I cut, "Is this still one whole cake?" I ask them to please remember the whole cake when we go farther into fractions. When we eventually get to that nasty fraction borrowing part I remind them of the whole cake cut into eight pieces was still a whole cake that was expressed as 8 over 8. I'm not totally sure that the cakes help that much but they are so totally charmed that a teacher baked something for them that they talk about it for the rest of the year.

Lynda Ginsburg from the National Center for Adult Literacy did a good fraction workshop last year using cut up pieces of construction paper. Cheap resources for over strained program budgets. I'm going to incorporate some of that this year but I'm going to have the students cut up their own paper. I think this will engage the tactile learners as well as give everyone their own set of fraction manipulatives, until they loose them, give them to their kids, or decide that it is "childish". (That can be a problem with manipulatives.) Do let us know what you find that works well, and share your own ideas too. Best wishes,

Lynne

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Date: 22 Aug 1995 16:36:48 -0700
From: "Virginia Thompson"
Subject: Re: how to "see" fractions
To: numeracy@world.std.com, SusanG2@aol.com

The construction paper strips are the basis of the fraction kits in FAMILY MATH. A convenient size is strips of 3" by 18". You can easily cut 4 from a 12" by 18" strip. If you want to use smaller paper, you can try 2" by 12" strips out of 8" by 12" sheets of paper. However the sixteens will be quite narrow, using this size paper.

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Date: Wed, 23 Aug 95 15:45:49 EST
From: "Mark Schwartz"
Subject: Re[2]: how to "see" fractions

I'm going to be a little presumptuous in this note, but I have to talk a little about seeing fractions, particularly with adults.

Years ago, when I taught adults in a remedial environment, they forced me to teach them to UNDERSTAND fractions. If you're working with adults, you will know exactly what I mean by "they forced me".

When I asked them what they wanted to know, they said to explain to them what it means - what is a fraction?

We worked through several scenarios until they were satisfied and until their work demonstrated they they really understood what was happening. In essence, there wasn't any operation with fractions that they couldn't do.

Now the hard part. Somewhere in my house are my "worksheets", which are somewhat descriptive of the methodology we finally developed ... and it was we. If I can locate them, I will a post a note and give you an address to write to for snail mail and I'll get it to you.

As a tease (this actually works), I started by writing the symbol "1/2" on the board and asked "what does this MEAN?" The answers "1/2", "half of something", "a part of something", "break it in two", "you mean find the lowest common denominator", ... and a host of other answers are unacceptable. The answer that finally emerged (with some prodding and probing from me and what I was looking for is "this means take some thing, break it into two parts and take, pay attention to, or do something with one of the two parts."

This, by the way is not the complete answer, but since this is done as a verbal,exploratory exercise, I sort of paced the information. The next level of exploration, after playing with this as a base definition (and it usually comes out in the conversation) is to realize that the thing must be broken into two EQUAL parts (or that whatever is in the denominator must be broken into "x" equal parts). This becomes quite critical since it is the essence of being able to have students clearly understand WHY one needs a common denominator in order to add fractions with different denominators.

Enough for now. I'll be off-net until Sep. 11th but hope to generate some interest in exploring this a little bit when I'm back ... thanks ... mark

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Date: Tue, 05 Sep 1995 08:40:51 -0500 (CDT)
From: Mary Ann Shope
Subject: RE: Re[2]: how to "see" fractions

I agree with Mark. Our experience has been that adults don't know where numerators or denominators come from or what they mean. We have almost stopped using the term fraction and instead talk about part/whole relationships. MA Shope

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Date: Mon, 20 Nov 1995 08:21:47 -0700
From: Murray Meszaros
To: numeracy@facteur.std.com
Subject: ideas for teaching fraction -Reply
Dear Eileen, Rebecca, and Sarah

You've probably already been told that a fair bit of discussion has already gone "under the bridge" on topics related to this. No discussion has hit directly on this that I can remember.

As things develop, are you willing to share with the rest of your Internet friends?

Murray Meszaros
Utah Literacy and Adult Education Resource Center
mmeszaro@usoe.k12.ut.us

>>> 11/17 1937 >>>
We ( a group of teachers at Dorcas Place) would like ideas on new ways to teach the connections between fractions, percents, and decimals. We're new to the numeracy list and are looking for math buddies.

Eileen, Rebecca, Sarah
Dorcas Place
270 Elmwood Ave
Providence, RI 02907
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Date: Mon, 20 Nov 1995 15:34:01 -0700 (MST)
From: "M. Adele Megann"
To: JATDP@aol.com
CC: numeracy@facteur.std.com
Subject: Re: ideas for teaching fraction

One way to show the connection between fractions and decimals is to use graph paper. I make a master with several blocks of ten by ten outlined in black. One "bar" is one/tenth, or 0.1. One block is one/hundredth, or 0.01. This is especialy useful with addition. Use different colours of highlighter for each addend. Highlighters are good, because you can fill in a "bar" quickly.

You can use place value blocks for tenths, hundredths, and thousandths. You might be able to find a picture of a thousand-cube to make up worksheets.

I assume you'll teach fractions first. To me, they are the most basic. Adults students will, of course, have practical experience with decimals and percents, but I think fractions is the key to understanding them.

Anyway, when I get to decimals, I go back the the place value chart I gave them at the beginning of the year, and extend it to the right. I begin every year with place value.

Get a money tray, and remove everything except the hundreds, tens, ones, dimes, and pennies. You can do addition, subtraction, multiplication and division of decimals using those denominations. I think you Americans still have one-dollar bills! We have dollar coins only, but it works the same, as long as you establish the the one-dollar is the basic unit. You can relate it back the the place value chart.

Because percentages are always out of a hundred, the graph paper is ideal. Discuss the origin of the word and the symbol to give them context. After fractions and decimals, they'll probably find percents easy!

Good luck. Please ewrite me if you'd like some clarification.

Adele Megann
Transitional Vocational Program
Mount Royal College
--------------------------/\__/\--------------------------------
Adele Megann & Linus_____; o o ; mamegann@freenet.calgary.ab.ca
Alberta, Canada _/`_____ =^= / Newfoundlander Abroad
---------------<_______>__m_m_>---------------------------------
On Fri, 17 Nov 1995 JATDP@aol.com wrote:

> We ( a group of teachers at Dorcas Place) would like ideas on new ways to
> teach the connections between fractions, percents, and decimals. We're new to
> the numeracy list and are looking for math buddies.
> > Eileen, Rebecca, Sarah
> Dorcas Place
> 270 Elmwood Ave
> Providence, RI 02907
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Date: Tue, 21 Nov 1995 07:01:55 +0001 (EST)
From: Esther D Leonelli
Subject: Re: ideas for teaching fraction -Reply
To: numeracy@facteur.std.com

On Mon, 20 Nov 1995, Murray Meszaros wrote:

> Dear Eileen, Rebecca, and Sarah

>
> You've probably already been told that a fair bit of
> discussion has already gone "under the bridge" on
> topics related to this. No discussion has hit directly on
> this that I can remember.

>

Murray is right, there was a fair amount of discussion on the topic of teaching fractions this summer. I have created a digest of the Numeracy "fraction" messages which should appear in the next issue of _The Math Practitioner_ (coming out soon). If you want to receive the digest of "fraction" messages, please write to me OFF the list:

edl@world.std.com

Esther
________________________________________________
Esther D. Leonelli
Community Learning Center, 19 Brookline Street, Cambridge, MA 02139
617-349-6363 Fax: 617-349-6339

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Date: Tue, 21 Nov 1995 07:28:22 -0500 (EST)
From: Lynne Mikuliak
To: JATDP@aol.com
Cc: numeracy@facteur.std.com
Subject: Re: ideas for teaching fraction

Hi, and welcome to Numeracy. I would take Esther up on her offer to write for a digest of discussion on teaching fractions. People had many good ideas. I'd also suggest starting to work with money, actual coins and a dollar bill. We call a fifty cent piece "half a dollar" and a quarter, a quarter. Using the coins as actual manipulatives the relationship among fractions and decimal parts of one whole dollar can easily be seen. When I start decimals I always say, "You already know decimals because you already know MONEY." Most students also understand that 50% off means "half off" which can also be shown with money. As Philadelphia's sales tax is 7%, putting seven pennies next to the dollar shows this relationship in a concrete way. As you move on and teach more complex decimals and percent you can mention the relationships you already "saw" with the money. Good luck, and please do share your best ideas.
Lynne Mikuliak - Community Womens Education Project - Phila., Pa.

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Date: Tue, 21 Nov 95 13:32:58 EST
From: "Mark Schwartz"
To: numeracy@world.std.com
Subject: fractions
Sender: numeracy-approval@facteur.std.com
Time to poke my nose into the numeracy world again.

As a federal employee, somewhat distanced in time but not in thought, from the adult classroom, I want to share my experiences as well, because these are ideas that -- at the time of my teaching -- adults taught me.

One basic concept that I learned from students revolves around the idea of finding a common denominator in adding/subtracting. I realize that there are many facets to fractions, but this one is surprisingly fundamental and robust and hardly ever addressed.

Consider why a common denominator is needed at all. Really; think about it.

For example, I like the idea of using objects, manipulatives, and other tactile approaches. I like the idea of using money, and showing 7 pennies and a dollar bill as a percent/fraction relationship. However, embedded in that example is the "hidden" construct of the pennies and the dollar ALREADY using a common and "equal" basis for the fractional relationship.

In fact, most manipulatives do this. I posted a note earlier in which I commented on re-phrasing things like 1/2 as a symbolic statement of "take some thing, break (divide, etc.) it into 2 parts and take (or do something with) 1 of those 2 parts". Notice that I did not include the concept of 2 "equal" parts. But, the reality is that that's what is meant. When you write 3/4, the understood event is that the denominator represents 4 equal pieces.

If you let students play with this idea, or if you lead them into a discussion of it (and let them lead themselves out), the discussion usually trails through all kinds of concepts -- not just fractions -- and it becomes an aha! experience to realize the basis for needing a lowest common denominator.

As I said above, think about it. This can really be demonstrated with objects. For example, if you start with various lengths of strips of paper and you don't use the same length strips or do use the same length strips but tear (or cut) them into non-equal parts, you can get into some amazing little paradoxical situations about fractions, which are remarkable learning experiences in addition to learning fractions.

The other point to note about this is that you can readily shift into grouping of objects, factors, sets of factors, prime factoring, etc. This sets up the framework not only for a methodology for finding the LCD for any set of numbers, but also sets up ideas (without identifying them as such) about algebraic factoring.

Enough for now ... let's talk.

Mark_Schwartz@ED.Gov

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From: SouthWoods@aol.com
Date: Tue, 21 Nov 1995 17:51:53 -0500
To: numeracy@facteur.std.com
Subject: Re: ideas for teaching fraction

Have you ever seen a percent circle? It is a circle with 100 small lines on its circumference, and then 10 long lines, The short lines are hundredths and the long ones tenths (didivding the circle into 10 equal parts.) THe advantage of the circle is that it clearly ties fractions to decimals to percents. It you cut the circcle into quarters of fourths, it is the quarter circle that the have used in fractions and it has 2 long lines 2/10 and 5 more shrt lines for 5/100 wonderful for showing that .25 - .2 + .05. I also tie the percent sing to the cent sign telling my students that 2% = 2c (my cents sign won't print) = .02. THey seem to be able to see that better and have less trouble with the turning percents into decimals. Good luck Eileen Simons