Parent activity: division models
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Maria Droujkova
Oct 18, 2008, 11:31:44 PM10/18/08
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This is a collection created over several weeks by homeschooling parents participating in the Living Math discussion group - you will see their names next to their models. The goal was to put together many different ways division happens. Different division models answer questions such as, "Why do we flip fractions when we divide?" or "Why is it that a negative number divided by a negative number results in a positive number?" I edited the models and added a few references.
I love running "collection activities" like that with groups of people, either online or in physical space. You can create something wonderful together, more often than not.
~*~*~*~*~*
Division Models!
Sharing 1
Find what each person gets. Natural for dividing by whole numbers ("sharing among people"). Fun examples: pirate booty; pizzas. A hundred pirates looted two thousand gold pieces; what is the share of each?
Extension of Sharing 1 into fractions: a third of a worm gained 2g weight. How much is the gain of each (whole) worm? Can probably be found in sampling. Don Cohen from "Calculus for seven year olds" has an activity expanding a sharing model into (infinite) strings of fractions, say, for sharing 6 cookies among 7 people: http://www.mathman.biz/html/prob2.html
Can use for dividing negative numbers by positive, but not for dividing by negatives, then? Some researchers, or kids, also call this and the next model "splitting".
Sharing 2
Find how many people can get the given share. Natural for dividing by whole numbers and fractions, when the answer is whole. E.g. if you have three pizzas and want to give 1/6 per person, how many people can you feed. You have 21 cookies, and want to give each child 2, how many kids can you share them between? - Sue. How about negative numbers?
Partitive model
Breaking up into equal sized groups - Vicki. Natural for dividing by whole numbers of fractions when the answer is whole. How many thirds are in five apples? How many pairs can twenty kids form? 5 foot board cut into 2/3 foot sections will give you how many sections? (5 ÷ 2/3 = ?) - LauraLyn 4 ÷ ½ means "How many halves in 4? A concrete model can be used if you make an area model worth 4 and want to know how many halves are in 4 (partition 4 into half sized pieces) A string can represent a linear measurement (4 inch piece has how many half inch pieces in it?) and a favorite model of money (how many half dollars are in $4?). The length of a race will be ¾ of a mile. Each runner will run ¼ mile. How
many runners are needed to complete the race? The easiest model for kids to immediately connect with (IMO) is money. This gives you these fractions to work with ½, ¼, 1/10, 1/20, 1/100 and you can solve many problems and investigate patterns.- Vicki. Steve Demme's explanation in Math U See, but he called this "gozinta" (goes into) - how many halves "gozinta" 4. - Julie.
Velocity, unit price and other intrinsic rates or "intensive quantities" (per)
Find the velocity given distance and time. Works with all real numbers. Humans don't have grounding time metaphors, and kids may not measure distance much, so it's less intuitive for some, depending on driving/measuring experience. Liping Ma uses in her book—we're building a road of a certain length (say 5 miles) and we know we can only build so much a day (say 2/3 of a mile). How many days will it take to build the road? (again 5 ÷ 2/3 = ?) - LauraLyn. Can be extended into negative numbers with past time and reverse direction.
Averages
Sports applications such as figuring out a player's batting average, or a horse's odds of winning a race. Or odds for cards (like poker hands) and rolls of a pair of dice. - Kari.
Number line
5:(-1/3). Place a marker on 5 units positive (to the right). Place a marker on -1/3 (to the left). If we continue to add markers of amount -1/3 to the left, we are multiplying -1/3 a positive number of times, and it is obvious that the result is going in
the wrong direction and will never reach the 5 marker on the right. So we must reverse direction by multiplying the -1/3 a negative number of times, and it will reach the 5 marker after -15 increments. - John. I've done something like this with kids physically taking steps across blocks in a small group, with negative and positive indicating opposite directions. A very good explanation of this is in a book called Playing with Infinity. - Julie.
Multiplicative invariance and other number rules, combined
Use several rules of operations together. Let's say the students first learn how to divide 5 / 3. They learn how to do 6 / (-2) and do all their integer operations. (Models could be used with those). They learn decimal operations. Here we'd study 5 / 0.2 and 5 / 0.3 and find that they are the same as dividing 50/2 and 50/3. One would need to justify this idea, of course. Then it's just a natural extension of the previously studied ideas that 5 / (-0.3) is equal to 50 / (-3) and then use what you know about integers. - Maria M. For example, in problems such as "divide 163 by 0.3," children are taught to change the decimal divisor 0.3 to the whole-number divisor 3 by multiplying by 10, divide 163 by 3, and, multiply the result by 10. What constitutes the understanding of this procedure is the awareness that the equality relation between the dividend 163, the divisor 0.3, and their quotient is not invariant under the change of the divisor 0.3 to 3, and that the "multiply by 10" transformation — applied to the quotient of 163÷3 — is an appropriate compensation for this change. - The Rational Number Project (a research group; this is from their review of the Multiplicative Conceptual Field here http://cehd.umn.edu/rationalnumberproject/90_1.html)
Iterations: Repeated subtraction ("the measurement model")
Many elementary level textbooks introduce division as repeated subtraction - how many 2s are in 6? - Research ideas for the classroom: Middle grades mathematics (a book). This model uses either the "number as a collection of objects" metaphor, or the "number as an object of a particular size" metaphor. C:B=A can mean either of the two possibilities. One: the repeated subtraction of collections of size B from an initial collection of size C until the initial collection is exhausted. Or two: The repeated subtraction of parts of the size B from an initial object of size C until the initial object is exhausted. The result, A, is the number of times the subtraction occurs. - Where Mathematics Comes From (a book).
Fraction bars or measuring sticks model
For ½ ÷ 1/3, it is easy to use the [fraction] bars to illustrate as follows: Lay out ½ bar. Underneath, lay out the 1/3 bar. Take another 1/3 bar and fold it in half to fill the remaining space of the ½ bar. Easy to see that you have 1 ½ of the 1/3 bars under (or covering) the half bar. - Vicki. C:B=A Dividing up: the splitting up of a physical segment C into A parts of length B. Iteration: the repeated subtraction of physical segments of length B from an initial physical segment of length C until nothing is left of the initial segment. The result, A, is the number of times the subtraction occurs. - Where Mathematics Comes From (a book).
Inverse multiplication
Use any multiplication model with a known multiplier and a known product. For example, given the area of a rectangle is 6 square feet and one side 1.5 feet, find the other side. Kids would use multiplication and approximation, trying different numbers and seeing if they get the result close to the desired product.
Difficulties with division
...My own mind seems to have trouble with it - we're dividing another number that is being divided, so it's a nested operation. - Julie.
An friend of mine had switched from engineering to teaching. He was making little progess teaching division in a multi-ethnic school, using the time honoured illustration of cutting a cake into different size slices. One day he switched from cake to pizza, and hey presto! the situation was transformed, and he never looked back. - John.
...Middle grades students often believe that expressions such as "4km per hour" and "4 hours per km" mean the same thing - Research ideas for the classroom: Middle grades mathematics (a book).
--
Cheers,
MariaD
I write, 'In the beginning was the Deed!' - Goethe, Faust
naturalmath.com: a sketch of a social math site
groups.google.com/group/naturalmath: a mailing list about math maker activities
I love running "collection activities" like that with groups of people, either online or in physical space. You can create something wonderful together, more often than not.
~*~*~*~*~*
Division Models!
Sharing 1
Find what each person gets. Natural for dividing by whole numbers ("sharing among people"). Fun examples: pirate booty; pizzas. A hundred pirates looted two thousand gold pieces; what is the share of each?
Extension of Sharing 1 into fractions: a third of a worm gained 2g weight. How much is the gain of each (whole) worm? Can probably be found in sampling. Don Cohen from "Calculus for seven year olds" has an activity expanding a sharing model into (infinite) strings of fractions, say, for sharing 6 cookies among 7 people: http://www.mathman.biz/html/prob2.html
Can use for dividing negative numbers by positive, but not for dividing by negatives, then? Some researchers, or kids, also call this and the next model "splitting".
Sharing 2
Find how many people can get the given share. Natural for dividing by whole numbers and fractions, when the answer is whole. E.g. if you have three pizzas and want to give 1/6 per person, how many people can you feed. You have 21 cookies, and want to give each child 2, how many kids can you share them between? - Sue. How about negative numbers?
Partitive model
Breaking up into equal sized groups - Vicki. Natural for dividing by whole numbers of fractions when the answer is whole. How many thirds are in five apples? How many pairs can twenty kids form? 5 foot board cut into 2/3 foot sections will give you how many sections? (5 ÷ 2/3 = ?) - LauraLyn 4 ÷ ½ means "How many halves in 4? A concrete model can be used if you make an area model worth 4 and want to know how many halves are in 4 (partition 4 into half sized pieces) A string can represent a linear measurement (4 inch piece has how many half inch pieces in it?) and a favorite model of money (how many half dollars are in $4?). The length of a race will be ¾ of a mile. Each runner will run ¼ mile. How
many runners are needed to complete the race? The easiest model for kids to immediately connect with (IMO) is money. This gives you these fractions to work with ½, ¼, 1/10, 1/20, 1/100 and you can solve many problems and investigate patterns.- Vicki. Steve Demme's explanation in Math U See, but he called this "gozinta" (goes into) - how many halves "gozinta" 4. - Julie.
Velocity, unit price and other intrinsic rates or "intensive quantities" (per)
Find the velocity given distance and time. Works with all real numbers. Humans don't have grounding time metaphors, and kids may not measure distance much, so it's less intuitive for some, depending on driving/measuring experience. Liping Ma uses in her book—we're building a road of a certain length (say 5 miles) and we know we can only build so much a day (say 2/3 of a mile). How many days will it take to build the road? (again 5 ÷ 2/3 = ?) - LauraLyn. Can be extended into negative numbers with past time and reverse direction.
Averages
Sports applications such as figuring out a player's batting average, or a horse's odds of winning a race. Or odds for cards (like poker hands) and rolls of a pair of dice. - Kari.
Number line
5:(-1/3). Place a marker on 5 units positive (to the right). Place a marker on -1/3 (to the left). If we continue to add markers of amount -1/3 to the left, we are multiplying -1/3 a positive number of times, and it is obvious that the result is going in
the wrong direction and will never reach the 5 marker on the right. So we must reverse direction by multiplying the -1/3 a negative number of times, and it will reach the 5 marker after -15 increments. - John. I've done something like this with kids physically taking steps across blocks in a small group, with negative and positive indicating opposite directions. A very good explanation of this is in a book called Playing with Infinity. - Julie.
Multiplicative invariance and other number rules, combined
Use several rules of operations together. Let's say the students first learn how to divide 5 / 3. They learn how to do 6 / (-2) and do all their integer operations. (Models could be used with those). They learn decimal operations. Here we'd study 5 / 0.2 and 5 / 0.3 and find that they are the same as dividing 50/2 and 50/3. One would need to justify this idea, of course. Then it's just a natural extension of the previously studied ideas that 5 / (-0.3) is equal to 50 / (-3) and then use what you know about integers. - Maria M. For example, in problems such as "divide 163 by 0.3," children are taught to change the decimal divisor 0.3 to the whole-number divisor 3 by multiplying by 10, divide 163 by 3, and, multiply the result by 10. What constitutes the understanding of this procedure is the awareness that the equality relation between the dividend 163, the divisor 0.3, and their quotient is not invariant under the change of the divisor 0.3 to 3, and that the "multiply by 10" transformation — applied to the quotient of 163÷3 — is an appropriate compensation for this change. - The Rational Number Project (a research group; this is from their review of the Multiplicative Conceptual Field here http://cehd.umn.edu/rationalnumberproject/90_1.html)
Iterations: Repeated subtraction ("the measurement model")
Many elementary level textbooks introduce division as repeated subtraction - how many 2s are in 6? - Research ideas for the classroom: Middle grades mathematics (a book). This model uses either the "number as a collection of objects" metaphor, or the "number as an object of a particular size" metaphor. C:B=A can mean either of the two possibilities. One: the repeated subtraction of collections of size B from an initial collection of size C until the initial collection is exhausted. Or two: The repeated subtraction of parts of the size B from an initial object of size C until the initial object is exhausted. The result, A, is the number of times the subtraction occurs. - Where Mathematics Comes From (a book).
Fraction bars or measuring sticks model
For ½ ÷ 1/3, it is easy to use the [fraction] bars to illustrate as follows: Lay out ½ bar. Underneath, lay out the 1/3 bar. Take another 1/3 bar and fold it in half to fill the remaining space of the ½ bar. Easy to see that you have 1 ½ of the 1/3 bars under (or covering) the half bar. - Vicki. C:B=A Dividing up: the splitting up of a physical segment C into A parts of length B. Iteration: the repeated subtraction of physical segments of length B from an initial physical segment of length C until nothing is left of the initial segment. The result, A, is the number of times the subtraction occurs. - Where Mathematics Comes From (a book).
Inverse multiplication
Use any multiplication model with a known multiplier and a known product. For example, given the area of a rectangle is 6 square feet and one side 1.5 feet, find the other side. Kids would use multiplication and approximation, trying different numbers and seeing if they get the result close to the desired product.
Difficulties with division
...My own mind seems to have trouble with it - we're dividing another number that is being divided, so it's a nested operation. - Julie.
An friend of mine had switched from engineering to teaching. He was making little progess teaching division in a multi-ethnic school, using the time honoured illustration of cutting a cake into different size slices. One day he switched from cake to pizza, and hey presto! the situation was transformed, and he never looked back. - John.
...Middle grades students often believe that expressions such as "4km per hour" and "4 hours per km" mean the same thing - Research ideas for the classroom: Middle grades mathematics (a book).
--
Cheers,
MariaD
I write, 'In the beginning was the Deed!' - Goethe, Faust
naturalmath.com: a sketch of a social math site
groups.google.com/group/naturalmath: a mailing list about math maker activities
Maria Droujkova
Mar 22, 2010, 12:02:06 PM3/22/10
to natur...@googlegroups.com
I am replying to a thread from a year and a half ago, where I summarized division models for the purpose of explaining things like, "Why do you invert and multiply when dividing fractions?" The models are brief, and probably take some knowledge of teacher slang to understand.
To this day, there is no fraction division model I like!
Today, Chris Harberk sent me this Voice Thread in Twitter, using a device called "Ratio Tables." It is a powerful tool. Here's the video: http://voicethread.com/#u6703.b5120.i39561
It's not a hands-on or manipulative model, but a pure math (formal) model. These are important to master, and fun for some.
I would still very much like to find a good hands-on model, grounded in everyday experiences, for dividing something like 2/3 by 4/5.
Cheers,
Maria Droujkova
http://www.naturalmath.com
Make math your own, to make your own math.
To this day, there is no fraction division model I like!
Today, Chris Harberk sent me this Voice Thread in Twitter, using a device called "Ratio Tables." It is a powerful tool. Here's the video: http://voicethread.com/#u6703.b5120.i39561
It's not a hands-on or manipulative model, but a pure math (formal) model. These are important to master, and fun for some.
I would still very much like to find a good hands-on model, grounded in everyday experiences, for dividing something like 2/3 by 4/5.
Cheers,
Maria Droujkova
http://www.naturalmath.com
Make math your own, to make your own math.
wka...@aol.com
Mar 27, 2010, 9:43:18 PM3/27/10
to natur...@googlegroups.com
See below
-----Original Message-----
From: Maria Droujkova <drou...@gmail.com>
To: natur...@googlegroups.com
Sent: Mon, Mar 22, 2010 12:02 pm
Subject: [NaturalMath] Re: Parent activity: division models
From: Maria Droujkova <drou...@gmail.com>
To: natur...@googlegroups.com
Sent: Mon, Mar 22, 2010 12:02 pm
Subject: [NaturalMath] Re: Parent activity: division models
I am replying to a thread from a year and a half ago, where I summarized division models for the purpose of explaining things like, "Why do you invert and multiply when dividing fractions?" The models are brief, and probably take some knowledge of teacher slang to understand.
To this day, there is no fraction division model I like!
Today, Chris Harberk sent me this Voice Thread in Twitter, using a device called "Ratio Tables." It is a powerful tool. Here's the video: http://voicethread.com/#u6703.b5120.i39561
It's not a hands-on or manipulative model, but a pure math (formal) model. These are important to master, and fun for some.
To this day, there is no fraction division model I like!
Today, Chris Harberk sent me this Voice Thread in Twitter, using a device called "Ratio Tables." It is a powerful tool. Here's the video: http://voicethread.com/#u6703.b5120.i39561
It's not a hands-on or manipulative model, but a pure math (formal) model. These are important to master, and fun for some.
I would still very much like to find a good hands-on model, grounded in everyday experiences, for dividing something like 2/3 by 4/5. I doubt that one exists because when in everyday experience will one ever have 2/3 of something and then want to divide that by 4/5? This appears to me what Paul Lockhart refers to as numbers being a creation of our mind and we can thus manipulate them symbolically in many ways.
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Maria Droujkova
Mar 27, 2010, 9:58:15 PM3/27/10
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On Sat, Mar 27, 2010 at 9:43 PM, <wka...@aol.com> wrote:
I would still very much like to find a good hands-on model, grounded in everyday experiences, for dividing something like 2/3 by 4/5. I doubt that one exists because when in everyday experience will one ever have 2/3 of something and then want to divide that by 4/5? This appears to me what Paul Lockhart refers to as numbers being a creation of our mind and we can thus manipulate them symbolically in many ways.
Let me qualify what I want here. In everyday life, people do NOT conceptualize their actions as division by fractions - not in the context of models. Someone doing carpentry might need to divide by a fraction, and they will do it formally and symbolically, not through models. Someone working with models intuitively, without formal math, will always go for two operations instead of dividing by a fraction.
However, I want to find a hands-on division model that extends into fraction division somewhat "naturally." For example, "time and money" model extends into negative number multiplication, as losing $5/day means you had $35 MORE a week ago: -5*(-7)=35
Maybe it's a useless desire to have, or even harmful in some ways.
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