Looking for Concrete "Fraction Experiences"
Steve Thomas
mok...@earthtreasury.org
> Looking for ideas on how we can give kids (and adults) concrete
> experiences with the concept of fraction.
You do not have to *give* them such experiences. You need to draw
attention to the experiences they have had all their lives.
You do eat slices of pie, cake, and pizza, and chocolate bars marked for
breaking, I trust. You use coins, and can get dollar coins, half dollars,
quarters, tenths, twentieths, and hundredths. You can talk about the
divisions of hours, minutes, seconds, yards, feet, inches, meters,
decimeters, centimeters, gallons, quarts, pints, cups, fluid ounces,
tablespoons, teaspoons, liters, milliliters, pounds, ounces, kilograms,
grams...
> Special bonus points for anyone who can come up with an example of
> division with fractions (ex: 1/3 divided by 1/2)
1/2 goes into 1 twice. In fact it goes into any whole number N by dividing
N objects into 2 pieces each, giving 2N pieces. Similarly, it goes into
1/3 twice 1/3, or 2/3. If you divide a circle into sixths, you can easily
see that a third of the circle (two pieces) is two-thirds of half the
circle (three pieces), in just the same way that, for example, two beads
is 1/4 of eight beads.
It has been done in detail, and is available on various OER sites, some of
which are given at
http://wiki.sugarlabs.org/go/Open_Education_Resources
I have written about this on other mailing lists. I will do a Turtle Art
version of this some time soon, after I do a bit more on the concept I
have been working on most recently, figurate numbers. I have several such
lessons at
http://wiki.sugarlabs.org/go/Activities/TurtleArt/Tutorials#Mokurai.27s_Tutorials
Tony Forster did a TA visualization for fractions that I plan to carry
further.
http://tonyforster.blogspot.com/2009/12/turtle-fractions.html
Others are welcome to join with Tony and me.
So here is the outline. You will have to take this more slowly with
children, of course.
* Cut a pie in pieces, and color some of the pieces, as Tony did. That
gives the basic idea of a fraction. Point out that when you cut a pie in,
say, 8 pieces, you are doing 1 divided by 1/8.
* Cut more than one pie in the same number of pieces each. This lets us
talk about "improper" fractions and mixed fractions (integer plus
fraction), and converting between them. We can also introduce rational
numbers at some stage of child development.
* Cut a pie in pieces, and cut the pieces into smaller pieces
(multiplication of the simplest fractions, such as 1/2 times 1/3). Some
fractions can be described using the bigger pieces, and some require the
smaller pieces. Talk about reducing fractions to lowest terms. (You will
need other materials in order to talk about Greatest Common Divisors. I'll
do something on that.) Take some time on multiplying fractions. Then
notice that, for example, if you divide a pie into sixths, three of the
pieces make a half. 3 times 1/6 is 1/2, so 1/2 divided by 3 is 1/6, and
1/2 (= 3/6) divided by 1/6 is 3. (Assuming prior understanding that if the
product of, say, 2 and 3 is 6, then 6/3 = 2 and 6/2 = 3.)
* Cut several pies. For example, cut two pies into three pieces each, and
then color pairs of pieces. How many groups of two pieces make two pies?
Congratulations, you have just divided 2 by 2/3.
* Work other examples, dividing whole numbers by fractions, then fractions
by other fractions, choosing cases that come out even to start with.
* Now look at examples where one fraction does not go evenly into the
other. What do you have to do to make sense of the remainder? Say you have
a pizza cut into 8 pieces, and you have hungry pizza eaters who want three
slices each. How many can you accommodate? Well, two, with two slices left
over. Two slices is 2/3 of three slices, so that comes to 2 2/3 portions.
None of this requires Turtle Art. You can cut pies or cakes, or pieces of
construction paper to do all of this. Oh, yes. How many pieces do the
local pizza parlors cut pizzas into? What fractions can you make from
those pieces? Can you find pictures of pizzas from directly above, so that
they appear as circles? (Yes.) What else? Craters on the moon? The whole
moon? Circular swimming pools, fountains, ponds?
It remains an open question whether the children will discover the
invert-and-multiply rule for dividing fractions by themselves, whether
they will need broad hints, or whether they will have to be told. It would
be interesting to me to hear how they would explain these ideas to each
other. I will be interested to hear your results.
> Thanks,
> Stephen
> _______________________________________________
> IAEP -- It's An Education Project (not a laptop project!)
> IA...@lists.sugarlabs.org
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Edward Mokurai
(默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر
ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://wiki.sugarlabs.org/go/Replacing_Textbooks
Walter Bender
> IAEP -- It's An Education Project (not a laptop project!)
> IA...@lists.sugarlabs.org
> http://lists.sugarlabs.org/listinfo/iaep
>
There are few examples here:
http://wiki.sugarlabs.org/go/Math4Team/Florida#Instructional_Block_10
In Caacupe, they use the variant of Abacus that they invented, which
allows for adding and subtracting fractions.
-walter
--
Walter Bender
Sugar Labs
http://www.sugarlabs.org
David Corking
> Special bonus points for anyone who can come up with an example of division
> with fractions (ex: 1/3 divided by 1/2)
I found a couple of promising lesson plans with a web search:
http://mypages.iit.edu/~smile/ma9703.html
http://mypages.iit.edu/~smile/ma8805.html
If your students are familiar with cuisenaire rods, a little coaching
should help them to develop demonstrations or calculation methods with
the rods.
Also, how about pouring a liter a water into cups marked with a fill
line at two-tenths of a liter. (two-tenths is one fifth, the student
fills five cups, so the reciprocal of one fifth is five.) And so on
with two liters, then half a liter ...
Above present division as "quotition" (how many times x goes into z)
"Partition" (sharing z equally among y groups) is still up for Steve's
bonus points. I don't think elementary/middle school students should
be expected to think of sharing a third of a pizza among half a
person, but there are probably many natural examples that don't occur
to me at the moment. An unnatural one and perhaps intimidating one, is
figuring out how to pay a dividend to a holder of a fractional share
of stock!
David
Mike South
Looking for ideas on how we can give kids (and adults) concrete experiences with the concept of fraction.
Special bonus points for anyone who can come up with an example of division with fractions (ex: 1/3 divided by 1/2)Thanks,Stephen
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Maria Droujkova
1/2 goes into 1 twice.
> Special bonus points for anyone who can come up with an example of
> division with fractions (ex: 1/3 divided by 1/2)
When the divisor OR the quotient are whole, people do use fraction division in life. Many people will conceptualize "How many halves are there in three pizzas?" or even "How many quarters are there in three halves?" (the last one is a stretch) as division.
However, after messing with fraction division for about a year (see http://naturalmath.wikispaces.com/Divide+a+fraction+by+a+fraction ) I believe people who don't have PURE MATH purposes avoid conceptualizing division of a fraction by a fraction, when it's not immediately clear the result is a whole number. Instead, they conceptualize it as TWO operations (multiplication and division) where at least one number is whole.
The pure math purposes have to do with extensions of operations. In mathematics, figuring out how operations work for all types of numbers and even non-number entities is a very strong value. As such, we want to subtract greater numbers from smaller ones, take square roots of negatives, and multiply anything whatsoever (zeros, ordered arrays, transformations, etc.) This extension value definitely tramps any muggle values such as cognitive accessibility or ease of calculation. There are strong mathematical reasons for holding the extension value dear. We just have to realize these reasons don't necessarily apply to eating pizzas, or even to math-rich professional practices such as nursing (let me know if you want "Proportional Reasoning in Nursing Practice" study).
In fact it goes into any whole number N by dividing
N objects into 2 pieces each, giving 2N pieces. Similarly, it goes into
1/3 twice 1/3
There - you conceptualize it through whole-number steps. These steps are entirely sufficient for dividing pizzas.
You only need to re-conceptualize these steps (at a significant cognitive cost, as my teaching experiments indicate, if you go beyond the example of 1/2) as division by a fraction if you are going for the mathematical value of figuring how fraction division works.
There are no utilitarian or artistic purposes, that I could find in more than a year of looking for them, in conceptualizing the separate steps as division by a fraction. In practice, nurses, pizza cooks, carpenters and so on don't "really" divide by fractions - they work with numerators and denominators separately.
I would suggest exploring reasons behind the math value of stretching operations, for example, talking about how inefficient it would be to program operations separately for different types of variables.
Cheers,
Maria Droujkova
919-388-1721
Make math your own, to make your own math.
David Corking
experience as an engineer, I am nearly always happy to picture in my
head multiplying by the reciprocal, rather than create any mental
picture of dividing by a fraction. On reflection I probably think
about it as the inverse of a scaling operation: the quotient is what I
must have started with if I scaled the divisor by the dividend.
However, I wonder if you would allow me to clarify the following
point, as it seems to me that division involving non-natural numbers
is important outside pure math.
> In practice, nurses, pizza cooks, carpenters and so
> on don't "really" divide by fractions - they work with numerators and
> denominators separately.
This may not be the case when dealing with decimal fractions or
percentages, when they might wish to crank a formula, or hit the
division key on a calculator. Therefore I think it is worthwhile to
give students a chance to develop a mental picture of division by a
fraction, even if they choose to forget it later, and rely on the
algorithm.
I see from your wiki page that you might not agree with me that
percentages and decimals are special cases of fractions. However, at
least in some curricula, fraction arithmetic is taught first, and the
others follow.
A couple of examples:
(1) if a toy car completes a five foot track in three-quarters of a
second, what is its speed?
(2) if you want to take 20% sales tax, or value added tax, off an
invoice to find the pre-tax price, then you divide by 1.2
(3) devise a formula to convert lap times measured in problem (1) into
speeds in miles per hour, or kilometres if you prefer.
I think we will find numerous other examples in science, engineering
and finance. (I think we could describe some of these calculations as
non-integer ratios, or as the need to reverse or invert a
multiplication.)
I think authentic and concrete concepts of division of all real
numbers can be useful to help us to use calculators confidently, and
to use, develop, and adapt formulae that include division.
(Aspiring electrical technicians will probably also want to extend the
concept to complex numbers.)
Thanks for your patience. David
Maria Droujkova
This may not be the case when dealing with decimal fractions or
> In practice, nurses, pizza cooks, carpenters and so
> on don't "really" divide by fractions - they work with numerators and
> denominators separately.
percentages, when they might wish to crank a formula, or hit the
division key on a calculator.
I agree completely, and I should have specified this case.
People conceptualize decimals as "single numbers" for all operations, unlike fractions that are considered "pairs of numbers" for many purposes.
Therefore I think it is worthwhile to
give students a chance to develop a mental picture of division by a
fraction, even if they choose to forget it later, and rely on the
algorithm.
Do you teach division by decimals through division by fractions? Maybe some parts of it, e.g. why division by 0.01 is the same as multiplication by 100?
I see from your wiki page that you might not agree with me that
percentages and decimals are special cases of fractions. However, at
least in some curricula, fraction arithmetic is taught first, and the
others follow.
A couple of examples:
(1) if a toy car completes a five foot track in three-quarters of a
second, what is its speed?
(2) if you want to take 20% sales tax, or value added tax, off an
invoice to find the pre-tax price, then you divide by 1.2
(3) devise a formula to convert lap times measured in problem (1) into
speeds in miles per hour, or kilometres if you prefer.
I think we will find numerous other examples in science, engineering
and finance.
Yes, grown-ups divide by decimals (as you said, usually using computers) all the time.
David Corking
> the inverse of a scaling operation: the quotient is what I
> must have started with if I scaled the divisor by the dividend.
Sorry. The last phrase is gobbledygook.
I could have said: the quotient is the number that, when scaled by the
divisor, gives the dividend.
Rebecca Hanson
mok...@earthtreasury.org
> (Aspiring electrical technicians will probably also want to extend the
> concept to complex numbers.)
Addition and subtraction of complex numbers are easier in rectangular
coordinates than polar coordinates. Multiplication and division are the
opposite.
a + bi + c +di = (a + c) + (b + d)i
1/(r,theta) = (1/r, -theta)
But it can be done, of course.
1/(a + bi) = (a - bi)/(a^2 + b^2)
(a + bi)/(c + di)
= (a + bi)(c - di)/(c^2 + d^2)
= ((ac + bd) +(bc - ad)i/(c^2 + d^2)
I should do these up in Turtle Art, both as illustrations of coordinate
systems and as arithmetic, and add them to my Tutorials.
http://wiki.sugarlabs.org/go/Activities/TurtleArt/Tutorials#Mokurai.27s_Tutorials
> Thanks for your patience. David
--
mok...@earthtreasury.org
> On Wed, Jul 13, 2011 at 3:38 AM, <mok...@earthtreasury.org> wrote:
>>
>> > Special bonus points for anyone who can come up with an example of
>> > division with fractions (ex: 1/3 divided by 1/2)
>> 1/2 goes into 1 twice.
>
>
> When the divisor OR the quotient are whole, people do use fraction
> division
> in life. Many people will conceptualize "How many halves are there in
> three
> pizzas?" or even "How many quarters are there in three halves?" (the last
> one is a stretch) as division.
>
> However, after messing with fraction division for about a year (see
> http://naturalmath.wikispaces.com/Divide+a+fraction+by+a+fraction ) I
> believe people who don't have PURE MATH purposes avoid conceptualizing
> division of a fraction by a fraction, when it's not immediately clear the
> result is a whole number. Instead, they conceptualize it as TWO operations
> (multiplication and division) where at least one number is whole.
>
> The pure math purposes have to do with extensions of operations.
and providing consistent rules for them, maintaining algebraic identities
such as commutativity, associativity and distributivity.
ab = ba
(ab)c = a(bc)
a(b + c) = ab + ac
etc.
> In
> mathematics, figuring out how
to make (a matter of definition, not discovery)
> operations work for all types of numbers and
> even non-number entities is a very strong value.
Although non-commutative (matrices) and even non-associative (octonions)
structures are of great importance.
> As such, we want to
> subtract greater numbers from smaller ones, take square roots of
> negatives,
> and multiply anything whatsoever (zeros, ordered arrays, transformations,
> etc.) This extension value definitely trumps any muggle values such as
> cognitive accessibility or ease of calculation. There are strong
> mathematical reasons for holding the extension value dear. We just have to
> realize these reasons don't necessarily apply to eating pizzas, or even to
> math-rich professional practices such as nursing (let me know if you want
> "Proportional Reasoning in Nursing Practice" study).
>
[Mokurai again]
>> In fact it goes into any whole number N by dividing
>> N objects into 2 pieces each, giving 2N pieces. Similarly, it goes into
>> 1/3 twice 1/3
>
>
> There - you conceptualize it through whole-number steps.
in order to arrive at the general rule, which we can then extend to
fractions.
> These steps are
> entirely sufficient for dividing pizzas.
Fractions are completely equivalent to whole numbers of equal-sized
pieces. It makes no difference to the result whether we use 3/8 or talk
about 3 pieces of 1/8 each. But I would like to lead children through the
two-step process to the algebraic rules, particularly
(a/b)/(c/d) = ad/bc
without going through (a/b)/(c/d) = (ad/b)/c in between.
> You only need to re-conceptualize these steps (at a significant cognitive
> cost, as my teaching experiments indicate, if you go beyond the example of
> 1/2) as division by a fraction if you are going for the mathematical value
> of figuring how fraction division works.
Well, that's what I want to find out. Are we making the individual steps
sufficiently simple so that they can become obvious? What kind of practice
is required to make them obvious? The most complicated mathematics is made
up from steps so simple and obvious that even an utterly stupid computer
can cope with them. Children can actually understand them, and put them
together into more complicated ideas that become equally obvious over
time. It's just like learning language, which starts with memorization of
words and patterns, and soon becomes habit with sufficient practice. Only
the children can tell us how much practice is sufficient, and what kinds.
> There are no utilitarian or artistic purposes, that I could find in more
> than a year of looking for them, in conceptualizing the separate steps as
> division by a fraction.
Not in most of ordinary daily life, as opposed to work in engineering,
science, statistics, and such. But it does turn up occasionally in
cooking, sewing, carpentry and a few other areas.
* If my pancake recipe calls for 1 3/4 cups of flour, and I only have 1
1/4 cups on hand, how much egg, milk, blueberries, and so on should I add
to make a partial batch? Well, obviously I should multiply every
measurement in the recipe by (1 1/4)/(1 3/4). Multiplying top and bottom
by the denominator of the two fractions (4) gives 5/7.
* If it takes 2 3/4 yards of cloth to make this item, how many can I make
from a bolt of cloth 20 yards long?
> In practice, nurses, pizza cooks, carpenters and so
> on don't "really" divide by fractions - they work with numerators and
> denominators separately.
> I would suggest exploring reasons behind the math value of stretching
> operations, for example, talking about how inefficient it would be to
> program operations separately for different types of variables.
But we have to do that anyway. Computer processors are built with separate
integer and floating point hardware or microcode. If we want rational or
complex arithmetic in a programming language, or vector and matrix
operations, there must be a library or other segment of code for each.
You are correct about finding the maximal structure where integer
arithmetic holds, however.
> Cheers,
> Maria Droujkova
> 919-388-1721
>
> Make math your own, to make your own math.
>
>
[Mokurai again]
>> , or 2/3. If you divide a circle into sixths, you can easily
>> see that a third of the circle (two pieces) is two-thirds of half the
>> circle (three pieces), in just the same way that, for example, two beads
>> is 1/4 of eight beads.
>>
>> It has been done in detail, and is available on various OER sites, some
>> of which are given at
>>
>> http://wiki.sugarlabs.org/go/Open_Education_Resources
>>
>>
>
David Corking
> So David you're saying that if you consider:
> 3 divided by 2/3, the quotient will be 4.5 because 2/3 of that is 3?
> That's nice. Can you give us a concrete example for that?
Yes.
(1) A student made a scale model of the classroom, using a scale of
1/60. Using a ruler, she measured the diagonal of the model as 25 cm.
To calculate the diagonal of the classroom, I reverse the scaling
operation, so I divide by 1/60. When I divide 25 by a sixtieth, the
quotient will be 1500, because a sixtieth of 1500 is 25 (to me, "of"
means "scaled by" which means "multiplied by".) The diagonal of the
classroom is 1500 cm or 15 m.
(2) The toy car covers the five foot track in three-quarters of a
second. To divide five by three-quarters, I want a number that, when
scaled by three-quarters, gives five. So I would estimate that the
speed is bigger than five, probably seven or eight feet per second. I
can use this estimate to check a more accurate calculation, which I
can do with a calculator, or with the algorithm of divide by the
numerator ( 5 / 3 ) then multiply by the denominator ( 4 * five-thirds
which is twenty thirds or six and two-thirds feet per second.)
(3) Electrical calculations are practical, but not concrete for
everyone. 14 year olds might want to try this one:
Q:A circuit has a resistance, R, of 0.66 ohms, and is powered by a 3
volt battery. Make a mental estimate of the current, I, in amperes,
using the formula I = V / R.
A: 0.66 is about two-thirds, so I want 3 divided by 2/3. What number
would I scale by 2/3 to get 3? 4.5, so V is about 4.5 amperes.
The above text is a bit dense. I suspect it is easier to find the
inverse scaling model by guided discovery, than to absorb it from a
lecture or textbook.
Maria wrote:
> Do you teach division by decimals through division by fractions?
I don't teach, so I hope I didn't respond out of turn. I try to help
with Etoys, and I am a volunteer classroom assistant, but with an age
group that hasn't encountered fraction multiplication yet. My
suggestions are merely from my own knowledge of applied math, rather
than teaching experience.
Hope that helps, David
Laura Combs
Ok…my puny math brain needs to read that another 4 times. Because we are hot on fractions, I am going to think really hard about it!
Thanks, Maria
--
Tina Pasteris
conceptual approach to teaching division with fractions.
http://www.homeschoolmath.net/teaching/f/dividing_fractions_2.php
Tina Pasteris
Rebecca Hanson
Splitting/how many each (for one) (e.g. to do 486 divided by 2 - most people woudl split 468 into two parts)
Chunking/how many of the divisor in the divident (e.g. in calculation 39 divided by 13 most people would think - how many 13s in 39)
Scaling (insights into equivalent divisions - badly neglected in much western teaching).
Each primitive structure would require different concrete scenarios. I think we've covered them between us now. If we can find concrete situations which don't fit with any of these three they may indicate a further primitive structure.
If people find this difficult don't worry - I've been thinking about it a lot over the last 2 years and am not sure sure I properly understand it yet.
Rebecca
http://mathseducationandallthat.blogspot.com/2011/04/developing-scaling-primitive-in.html
Here is a link to free videos made by Maria Miller that provide a conceptual approach to teaching division with fractions. http://www.homeschoolmath.net/teaching/f/dividing_fractions_2.php
Tina Pasteris
On 7/12/11 7:23 PM, Steve Thomas wrote:
Looking for ideas on how we can give kids (and adults) concrete experiences with the concept of fraction.Thanks,
Special bonus points for anyone who can come up with an example of division with fractions (ex: 1/3 divided by 1/2)
Stephen
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David Corking
http://groups.google.com/group/naturalmath/msg/8fa9efef95a1dd68 ) :
> Essentially I think there are 3 common primative structures for division:
>
> Splitting/how many each (for one) (e.g. to do 486 divided by 2 - most
> people woudl split 468 into two parts)
>
> Chunking/how many of the divisor in the divident (e.g. in calculation 39
> divided by 13 most people would think - how many 13s in 39)
I agree. I read that the terms that some teachers now use are
"partition" (sharing equally among a given number) and "quotition"
(splitting into groups of equal sizes.)
I found this Australian explanation very helpful:
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N22502P.htm
> Scaling (insights into equivalent divisions - badly neglected in much
> western teaching).
I feel quite strongly about this, but I won't defend it further here:
scaling is Multiplication.
Division is the Inverse of Multiplication, so it is the inverse of scaling.
For me, I am afraid I am far too old to recall my childhood
perceptions of multiplication and division. That said, for me, scaling
is the most powerful concept to guide me in extending multiplication,
and division, from natural numbers (1,2,3....) beyond fractions to all
real numbers.
Regulars at naturalmath and iaep are probably already familiar with
Keith Devlin's 2008 column that inspired my strong feelings:
http://www.maa.org/devlin/devlin_09_08.html
wka...@aol.com
I feel quite strongly about this, but I won't defend it further here: scaling is Multiplication. Division is the Inverse of Multiplication, so it is the inverse of scaling.
http://www.maa.org/devlin/devlin_09_08.html
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Bradford Hansen-Smith
| The problem seems to be too many math specialist. The only way this discussion makes sense for me is to put it in the largest possible context. That must be what I would call Whole; it cannot be any less or any more that what it is, which is inclusive. For there to be original differentiation within the Whole there must first be division which yields a multiplication of parts that can then be added and subtracted between each other in various combinations of systems differentiation. Here is a fundamental "in reverse of the usual order taught" that needs some serious consideration. What is the conceptual bases for all this discussion about multiplication and division? If it is just talking about what parts do to other parts there can be no resolve. Brad --- On Fri, 7/15/11, wka...@aol.com <wka...@aol.com> wrote: |
Sue VanHattum
Whether or not math is repeated addition is getting to be a tired argument. Some of us who have thought deeply about it disagree with Devlin. We probably would all agree that math is not just repeated addition, but have differences about where to start.
I agree with Rebecca that it takes years to come to grips with the different underlying models of division. I think there were 3 in Liping Ma's book, but I sure don't remember them.
Warmly,
Sue
blogging at Math Mama Writes
Subject: Re: [NaturalMath] Looking for Concrete "Fraction Experiences"
From: wka...@aol.com
wka...@aol.com
Rebecca Hanson
Rebecca Hanson
Sue VanHattum
Warmly,
Sue
Bradford Hansen-Smith
| True, it is important to look at how parts interact. It is equally important to understand the relationship of each part to the Whole; for that has much to do with what sets the dynamics between parts. In the case of division of fractions, what is the more comprehensive understanding? Brad --- On Fri, 7/15/11, Rebecca Hanson <rebecca...@gmail.com> wrote: |
Rebecca Hanson
Maria Droujkova
Cheers,
Maria Droujkova
1-919-388-1721
Make math your own, to make your own math
Rebecca Hanson
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Maria Droujkova
I think scaling can correspond both to addition and to multiplication.
Cheers,
Maria Droujkova
919-388-1721
Make math your own, to make your own math.
Rebecca Hanson
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