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What is a fraction?

130 miles 2 hours = 65 miles per hour

135 miles 2 hours = ? miles per hour

6 pies 3 people = 2 pies per person

2 pies 3 people = ? pies per person

What is a fraction?2 pies 3 people = ? pies per person

Guess and check:

pies per person x people = pies

0 x 3 = 01 x 3 = 3

Answer is between 0 and 1. We need some new

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Cut each pie into 3 pieces

Each person now gets 2 of the 6 pieces

But what is each of the pieces?

Unit Fractions

Is this 1 unit shown 3 times?

Or 1 unit broken into 3 pieces?

1/3 is the number that multiplied by 3 gives 1

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1/n is the number

that multiplied by n gives 1.

11

=n

n

It takes 3 one thirds to make 1.

2/3 means 2 of what it takes 3 of to make 1.

A fraction is a point on a number line.

0 1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3

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Expanding the number system

A fraction is a number; not two numbers.

They were developed to answer questions

that cant be answered with whole

numbers.

We want fractions to behave just like wholenumbers.

2/3The denominator tells us our noun - what we

have.

The numerator tells us our adjective - how

many we have.

2 dogs + 3 hats = ???

2 fifth + 3 fourths = ????

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Equivalent Fractions

3 pies and 2 people.

What if we divide each pie into 6 pieces?

How many pieces does each person get

altogether?

nnumberwhole

nonzeroanyfor

bn

an

b

a=

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Addition and Subtraction

4 sevenths + 2 sevenths = 6 sevenths

7 ninths - 5 ninths = 2 ninths

If we have the same fractional unit, we count,

add and subtract fractions like whole

numbers

Different fractional unitsConvert to equivalent fractions with like

units.

Teachers and students must understand

why this must be done, before they learn

the algorithm.

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More Fraction Basics 6.2

A mixed numberis a whole number plus a

fraction.

The mixed number21/8 is an abbreviation

for 2 + 1/8.

An improper fraction is a fraction a/b where

ab

Conversions3 _ is how many halves?

Area Model:

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Another Area Model

Measurement ModelConvert 14/5 to a mixed number.

Draw a number line. Mark off fifths.

10/5 gets us to 2

Then 4/5 more gets us to 14/5

5

42

5

42

5

4

5

10

5

14=+=+=

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?12

391numbermixedWhat=

Note: 39112 = 32 R7

12

73212

7)3212(

12

391=

+

=

Fraction Division Equivalence2 pies 3 people = ???

= 2/3 pies per person

baba =

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Adding with unlike denominators

See example 2.4 page 140

What would you say to a student who does

the following:

1/3 + 2/5 = 3/8

Problems to Try 6.1, 6.2Read 4A pp 42-69

Page 137 HW set 24 2c, 3, 4b

Read 6.2

Read 5A pages 33-43

HW Set 25 - 3ab, 4

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Multiplication of Fractions

What does it mean to multiplyfractions?

How can we definemultiplication of fractions in a

way that is consistent with ourdefinition of multiplication ofwhole numbers?

6 x _ or _ x 6We want them to have the same value.

We can think of this as 6 groups of _.

We often read _ x 6 as 1/2 of 6.

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How much is 3/7 of 4/5?

Imagine a rectangular cornbread. Imaginethat it has been sliced into 35 equal pieces

3/7 of 4/5 of the cornbread =

3/7 of 4/5 of 35 pieces =

3/7 of 28 pieces =

12 pieces =

12/35 of the original cornbread

Fraction as AreaConsider 1/2 x 1/3

Start with a 1 inch square:

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Divide it into 3 equal horizontal pieces

and shade one them. This is 1/3 of the

square.

Divide the square vertically into 2

equal pieces. Shade _ of the 1/3.

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The blue shaded area represent _ of 1/3 of the square. Itsarea is 1/6 of the square.

The blue shaded region is a rectangle of width _ and height1/3.

Our area formula is length times width or

_ x 1/3 = 1/6

Your Turn: Draw a rectangle array

model to represent

2/3 x 4/5

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Division of Fractions

Yours is not to reason why, just invert and

multiply. (O.P.)

If you believe in things that you dont

understand, then you suffer (S.W.)

Division of FractionsHow did we define division of whole

numbers?

6 2 = ?

2 x ? = 6

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2

5

3

7 = ?

? =3

7

2

5

=3

7

2

5

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=7

3

3

7

2

5

2

5

7

3

3

7

2

5 =

2

5

3

7 = ?

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2

5

3

7

2

5

7

3 =

So

WHY?

Model Drawing PracticeRead 5A pages 56-59

Page 60 Problems 1-8

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Problems to Try

as time permits

Read 5A pages 33-60

HW Set 26 (1,2), 5, 9,11

HW Set 27 3, 4, 5b, 6