Download FI17.HANDOUT.Bars Area Model Number Line

Fractions with
Bars, Area Model and
Number Line
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The Structure of Fractions
3.NF 1. Understand a fraction 1/B as the quantity formed by 1
part when a whole is partitioned into B equal parts;
B=
6
1/B
1/6
1/6
0/6
1/6
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The Structure of Fractions
3NF 1. Understand a fraction A/B as the quantity formed by A parts
of size 1/B.”
1/b
1/B = 1/6
1/6
1/6
A=0 and B=6
0/6 = 0
A= 4
4 parts of
1/6 = 4/6
A=4
B=6
4/6
A=6 and B=6
6/6 = 1
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The Structure of Fractions
3NF 1. Understand a fraction 1/B as the quantity formed by 1
part when a whole is partitioned into B equal parts; understand
a fraction A/B as the quantity formed by A parts of size 1/B.”
1/B = 1/12. The length of the blue line is A (in this case 4) parts
of 1/B or 4 equal parts of 1/12; 1/12 + 1/12 + 1/12 + 1/12 =
4/12
• 1
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The Structure of Fractions
3NF 2a. Represent a fraction 1/b on a number line
diagram by defining the interval from 0 to 1 as the whole
and partitioning it into b equal parts. Recognize that
each part has size 1/b and that the endpoint of the part
based at 0 locates the number 1/b on the number line.
3NF 2b. Represent a fraction a/b on a number line
diagram by marking off a lengths 1/b from 0. Recognize
that the resulting interval has size a/b and that its
endpoint locates the number a/b on the number line.
0/6 1/6
1=6/6
Starting at 0, 1/6 it is reproduced 6 times to get a long
segment equal to 6/6 or 1.
The Main Topics of 4th Grade
4.NF
A big idea in 4th grade is the fundamental facts about
equivalent fractions (a fraction is not changed when its
numerator and denominator are multiplied by the same
nonzero whole number.)
Spend time comparing, adding, and subtracting
fractions with common denominators and the meaning
of multiplying a fraction by a whole number.
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The fraction value is not changed when its numerator and denominator
are multiplied by the same nonzero whole number. Or why 2/3 = 8/12
The square is vertically
divided into three rectangles
of equal
area, and 2/3 of the area
is represented by the
thickened rectangle.
n X a is equivalent to a
nXb
b
4X2= 8
4 X 3 = 12
2/3 = 8/12
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(n × a)/(n × b) example: (4X2)/(4X3) = 8/12
2 = 4X2
3
4X3
2 = 8
3
12
2/3
2/3
4/4
If the square is vertically
divided into three
rectangles of equal
area, then 2/3 is
represented by the
thickened rectangle.
8/12
The 2/3 is now divided
horizontally into 4 parts of
equal area, making a total
of 12 congruent parts. The
thickened rectangle now
has a value of 8 parts out
of 12 total parts or 8/12.
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Grade Four Fractions
Examples from CCSS Content Standards
for Mathematics
1) Break apart a fraction into smaller fractions with the
same denominator, or bottom number, in more than
one way. For example:
3⁄8 = 1 ⁄ 8 + 1 ⁄ 8 + 1 ⁄ 8 = 2 ⁄ 8+ 1 ⁄ 8
2) Explain why a fraction is equal to another fraction
3) Add and subtract mixed numbers (whole numbers
mixed with fractions, such as 1 1/5) with the same
denominators
4) Multiply a fraction by a whole number
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Fraction Bars
Individual Work
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n
x a) / (n x b) by using visual fraction models, with attention to
how the number and size of the parts differ . . .
•Lay the 1/4 bar on your desk and then find bars with
more parts but with the same amount of shading. Write
down a number sentence to describe your actions.
•Repeat this with the 1/2, 3/4, and 2/3 bars. When done,
choose bars of your own choice, continue writing down
number sentences that describe your actions.
•Be ready to share your thinking and noticings.
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Fraction Bars: Work and discuss in pairs,
and then in your small group.
4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n
x a) / (n x b) by using visual fraction models, with attention to
how the number and size of the parts differ . . .
• Find the 1/2 bar. With your bars, show that
1/2 = 2/4 = 3/6 = 6/12 What do you notice?
• Sketch a bar for 2/5. Divide the parts of your bar to
show that 2/5 = 5/10.
• Sketch a bar for 3/8. Divide the parts of your bar to
show that 3/8 = 6/16.
• What are some general methods you found for forming
equivalent fractions? Be ready to report out to the large
group.
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Fraction Bars
4.NF.2 Compare 2 fractions with different
numerators and different denominators . . . . . . .
Locate the 1/3 fraction bar and the 1/4 fraction.
Privately decide which fraction is greater and write
down a number sentence that matches your thinking,
using greater than or less then notation.
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Fraction Bars
4.NF.2 Compare 2 fractions with different
numerators and different denominators . . . . . . .
Locate and lay out the following pairs of fraction bars.




1/2 and 4/6
5/12 and 2/3
5/6 and 7/12
2/3 and 3/4
Write down a number sentence for each pair using
greater than or less than signs.
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Fraction Bars
Devise ways to use fraction bars to find these sums.
Be ready to share your thinking.
4.NF.B Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions
1/b.
a) 1/5 + 2/5 =
b) 3/4 + 2/4 =
c) 3/6 + 4/6 =
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Grade Five Fractions
Examples from CCSS Content Standards
for Mathematics
1) Interpret a fraction as division of the numerator (the
top number) by the denominator (the bottom
number)
2) Add and subtract fractions with different
denominators
3) Multiply a fraction by a whole number or another
fraction
4) Divide fractions by whole numbers and whole
numbers by fractions
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Fraction Bars
Devise ways to use fraction bars to find these sums.
Be ready to share your thinking.
5.NF.A Use equivalent fractions as a strategy to add and
subtract fractions.
5.NF.A.1 Add and subtract fractions with unlike denominators
(including mixed numbers) by replacing given fractions with
equivalent fractions in such a way as to produce an equivalent
sum or difference of fractions with like denominators.
a) 2/3 + 3/4 =
b) 2/3 + 5/6 =
c) 3/4+ 3/12 =
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Fraction Bars
Devise ways to use fraction bars to find these
differences.
5.NF.A.1 Add and subtract fractions with unlike denominators
(including mixed numbers) by replacing given fractions with
equivalent fractions in such a way as to produce an equivalent sum
or difference of fractions with like denominators.
a) 3/4 - 2/3 =
b) 5/12 - 1/6 =
c) 3/4 - 3/12 =
d) 1 ½ - ⅚ =
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Stop
Go to “What is the product of 4 x 2/3?
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Grade Five Fractions
Examples from CCSS Content Standards
for Mathematics
1) Interpret a fraction as division of the numerator (the
top number) by the denominator (the bottom
number)
2) Add and subtract fractions with different
denominators
3) Multiply a fraction by a whole number or another
fraction
4) Divide fractions by whole numbers and whole
numbers by fractions
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Fraction Multiplication
Method Two: Area Model 2/3 X 3/4
The square is vertically
divided into three
rectangles of equal
area, and 2/3 is
represented by the grey
portion of the rectangle.
The square is horizontally
divided into four
rectangles of equal area,
and 3/4 is represented by
the grey portion of the
rectangle.
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Fraction Multiplication
Method Two: Area Model 2/3 X 3/4
Now lay the 3/4 over the 2/3
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Fraction Multiplication
Method Two: Area Model
2/3
3/4
The shaded part of the rectangle is the part
3 X 2 = 6 (part)
over the whole 4 X 3 = 12 (whole)
3 parts
of equal area
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Fraction Multiplication
Method Two: Area Model
3/5 X 2/3
The whole is 15 and the
part is 6 so the answer is 6/15
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Method Two Area Model
Fraction Multiplication 3 1/2 X 4 = 14
AND 4 X 3 1/2 = 14
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1+ ½ + ½ + ½ +
½ = 12 + 4 (1/2) = 12 + 2 = 14
1
1
1
1/2
1
1
1
1/2
1
1
1
1/2
1
1
1
1/2
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CCSS Math Task, Fraction Multiplication
Rob is calculating the area of this rectangle. His strategy is
to multiply the whole numbers first and then multiply the
fractions. Since 6 X 7 = 42 and 1/2 × 6 = 3, he concludes
that the area of the rectangle is 42 + 3 = 45 sq. units.
Is he correct? If your answer is yes, explain why he correct.
If you answer is no, write down the correct answer and tell
why it is correct.
6
7 1/2
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Fraction Multiplication, Method Three Egg Carton
Fractions. Use your egg cartons and yarn to solve
a selection (your choice) of the following.
a) 1/4 of 2/3
e) 1/3 X 6/12
b) 1/3 of 2
f) 3/4 X 1
c) 3/4 of 1/3
g) 9/12 X 1/9
d) 1/4 of 1/3
h) 8/12 X 1/4
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CCSS, 5th Grade
Stuffed with Pizza
Tito and Luis are stuffed with pizza! Tito ate one-fourth
of a cheese pizza. Tito ate three-eights of a pepperoni
pizza. Tito ate one-half of a mushroom pizza. Louis ate
five-eights of the chees pizza. Louis ate the other half of
the mushroom pizza. All the pizzas were the same size.
Tito says he ate more pizza than Luis, because Luis did
not eat any pepperoni pizza. Luis says they ate the
same amount of pizza.
• Who is correct?
• Show all your mathematical thinking.
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