Session 130: Unpacking Multiplication and Division With Fractions

SAS Math Summit
Kim Zeugner
Kingswood Elementary, Cary NC
August 6, 2014
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Van de Walle – “Invert the divisor and multiply is
probably one of the most mysterious rules in
elementary mathematics.” (2007)
Division is usually considered the hardest
operation in mathematics
Fractions are usually considered the most
difficult representation of number
So……..Division with fractions produces a
great deal of angst in our students and our
teachers
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5.NF.3 Interpret a fraction as division of the
numerator by the denominator (a/b = a ÷ b).
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5.NF.7 Apply and extend previous
understandings of division to divide unit
fractions by whole numbers and whole
numbers by unit fractions.
Interpret division of a unit fraction by a nonzero whole number, and a whole number by a
unit fraction and compute such quotients.
Use a visual fraction model and the
relationship between multiplication and
division to explain that:
(1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
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Part-whole - a comparison of a number of parts
(represented by the numerator) to a number of
parts that make up the whole (represented by the
denominator).
Measurement – an amount of something as
compared to a designated whole measure.
Quotient - ¾ = 3 ÷ 4
Ratio – expressing the relative measure of two
quantities. 3/4 could mean 3 for every 4.
Operator – the fraction changes another quantity
through the effect of the numerator and denominator
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 Focus
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fraction as an operator
The numerator as multiplier
The denominator as divisor
2/3 of 1 means you take one and divide it
into 3 parts, each part 1/3. Then you take 2
of those parts (2 x 1/3) and you get 2/3.
One whole
1/3
1/3
1/3
Two-thirds
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The meaning of an operation does not change when you use a different
type of number.
DIVISION
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the process of ascertaining how many times one number
or quantity is contained in another.
the action of separating something into equal parts,
the finding of a quantity, the quotient, that when
multiplied by a given quantity, the divisor, gives another
given quantity, the dividend
12 ÷ 4 = _____ so _____ x 4 = 12
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Measurement model – you know the size of the
group, so how many groups can you make?
the process of ascertaining how many times one number or quantity is contained in another.
(How many inches in a foot)
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Partitive model – you know how many groups,
so what size is each group. (fair shares)
the action of separating something into equal parts
(You have a certain number of parts, so how big are the parts)
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Product Factor model – The dividend is a product
and the quotient is a missing factor.
the finding of a quantity, the quotient, that when multiplied by a given quantity, the divisor, gives
another given quantity, the dividend
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Dividend
A known
quantity to
separate into
equal groups
A known
quantity to
separate into
equal groups
A known
quantity =
Divisor
You know the
size of the
group.
Quotient
How many
groups can you
make?
You know the
number of
groups.
What size is the
group?
A known factor
x
An unknown
factor
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Measurement model in action.
This introduces students to what happens when
dividing a whole by a unit fraction and a unit
fraction by a whole.
Let’s experiment. Look at your handout, “How do
you divide a whole by a part?”
Take the rice bucket, tray, cups, and scissors.
In small groups, follow the directions. Stop at
“Extend your thinking.”
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Modification of Val Faulkner’s “Bean Party”
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Page 2 of the handout shows what happens
when you divide a fraction by a whole, again
using the measurement model.
Try it with your small group.
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The purpose is to have students see where
the unexpected quotients came from. Some
may start to see that dividing is multiplying
by the reciprocal because of patterns that
develop.
But do we understand it yet? This was the
measurement model only. What about
different situations?
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With a partner, take a can of play dough and
a division box. Also take some color discs
which will represent a group.
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You have 6 cookies.
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Each child will get 2 cookies.
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How many children can eat cookies?
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6÷2=3
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You have 6 cookies.
There are 2 children who will share the
cookies equally.
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How many cookies will each child get?
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6÷2=3
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The equation was the same,
but the situation was
different.
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You have 6 cookies.
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Everyone gets ½ cookie.
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How many children can eat?
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6 ÷ ½ = 12
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You have 6 cookies.
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This is ½ of a serving.
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How big is a serving?
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6 ÷ ½ = 12
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The partitive model is more
challenging when dividing by a
fraction.
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Now let’s think about that algorithm.
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“When you divide, you multiply the dividend
by the reciprocal of the divisor.”
Beyond just saying multiplication and division
are inverse operations, why does this work?
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6÷¾
(hint: 6 is ¾ of a group. What is the group size?)
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½÷¾
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1/3 ÷ ¾
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6/7 ÷ ¾
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Go back to thinking about a fraction as an
operator where the numerator is the
multiplier and the denominator is the
divisor.
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In each of these examples the numerator
became the divisor and the denominator
became the multiplier.
We “flipped” the operator.
6 ÷ 3/4 =
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(6 ÷3) x 4 =
6 x 4/3
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What if the parts you divide are the same
size? Wouldn’t that be like whole number
division? (6/1 ÷ 2/1 = 3)
So, if you find common denominators, the
quotient is the numerator in the dividend
divided by the numerator in the divisor.
½ ÷ ¾ = 2/4 ÷ 3/4 = 2 ÷ 3 = 2/3
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We tell students, “you don’t just divide
numerator by numerator and denominator by
denominator.” But is that true?
Can you do it?
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Yes you can!
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[email protected]
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