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.
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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.htmlCan
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.
AveragesSports
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 line5:(-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, combinedUse 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 multiplicationUse
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).
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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