Develop Understanding of Fractions

Transcription

11/1/13 DEVELOPING UNDERSTANDING
OF FRACTIONS WITH THE
COMMON CORE
GRADES 3 - 5
November 2, 2013
Presented by
Julie Joseph
Tulare County Of fice of Education
Visalia, California
1
GOALS/AGENDA
¡ Develop an understanding of the standards for
fractions outlined in the Common Core State
Standards.
¡ Develop strategies for supporting students in
modeling fractions and understanding fractions
conceptually.
2
8 MATHEMATICAL PRACTICE STANDARDS
1.  Make sense of problems and persevere in solving them.
2.  Reason abstractly and quantitatively.
3.  Construct viable arguments and critique the reasoning
of others.
4.  Model with mathematics.
5.  Use appropriate tools strategically.
6.  Attend to precision
7.  Look for and make use of structure.
8.  Look for and express regularity in repeated reasoning.
3
1 11/1/13 CCSS - FRACTIONS
!"#$%&'()*+*,"#-.)*/01*
Grades 1 & 2
Grade 3
Grade 4
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
Geometry 1.G
Number and Operations—Fractions 3.NF
Grade 1
Reason with shapes
and their attributes
!"# Partition circles and
rectangles into two
and four equal
shares, describe the
shares using words
halves, fourths, and
quarters, and use
the phrases half of,
fourth of, and
quarter of. Describe
the whole as two of,
or four of the
shares. Understand
for these examples
that decomposing
into more equal
shares creates
smaller shares.#
Develop understanding of fractions as numbers.
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.
2. Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
a. 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.
b. 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.
3. Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same
size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4,
4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a
visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are
equivalent to whole numbers. Examples: Express 3 in the form 3 =
3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a
number line diagram.
d. Compare two fractions with the same numerator or the same
denominator by reasoning about their size. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with the symbols >, =, or <,
and justify the conclusions, e.g., by using a visual fraction model.
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n ! a)/(n ! b) by using visual fraction models, with attention to how the
number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or
numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two
fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g.,
by using a visual fraction model.
!
Geometry 2.G
Grade 2
Reason with shapes
and their
attributes.
3. Partition circles and
rectangles into two,
three, or four equal
shares using the
words halves, thirds,
half of, a third of,
etc., and describe
the whole as two
halves, three thirds,
four fourths.
Recognize that
equal shares of
identical wholes
need not have the
same shape.
!
Measurement and Data 3.MD
Represent and interpret data.
4. Generate measurement data by measuring lengths using rulers marked
with halves and fourths of an inch. Show the data by making a line plot,
where the horizontal scale is marked off in appropriate units— whole
numbers, halves, or quarters.
!
Geometry 3.G
Reason with shapes and their attributes.
2. Partition shapes into parts with equal areas. Express the area of each
part as a unit fraction of the whole. For example, partition a shape into 4
parts with equal area, and describe the area of each part as 1/4 of the
area of the shape.
4
!
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each
decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 +
1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction,
and/or by using properties of operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like
denominators, e.g., by using visual fraction models and equations to represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 !
(1/4), recording the conclusion by the equation 5/4 = 5 ! (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For
example, use a visual fraction model to express 3 ! (2/5) as 6 ! (1/5), recognizing this product as 6/5. (In general, n ! (a/b) =
(n ! a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and
equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there
will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your
answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two
4
fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62
meters; locate 0.62 on a number line diagram. (4Students who can generate equivalent fractions can develop strategies for adding fractions with
unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.)
7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two
decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions,
e.g., by using the number line or another visual model.
4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition
and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the
difference in length between the longest and shortest specimens in an insect collection.
Formatted by Educational Resource Services, Tulare County Office of Education, Visalia, California (559) 651-3031 www.commoncore.tcoe.org
A CLOSER LOOK AT THE
FRACTION STANDARDS
Read 3.NF.1-3
Develop Understanding of Fractions as Numbers
!"#$%&'()*+*,"#-.)*/01*
Grades 1 & 2
Grade 3
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Grade 4
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
Geometry 1.G
Grade 1
Reason with shapes
and their attributes
!"# Partition circles and
rectangles into two
and four equal
shares using words
halves, fourths, and
quarters, and use
the phrases half of,
fourth of, and
quarter of. Describe
the whole as two of,
or four of the
shares. Understand
for these examples
that decomposing
into more equal
shares creates
smaller shares.#
Develop understanding of fractions as numbers.
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.
2. Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
a. 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.
b. 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.
3. Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same
size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4,
4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a
visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are
equivalent to whole numbers. Examples: Express 3 in the form 3 =
3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a
number line diagram.
d. Compare two fractions with the same numerator or the same
denominator by reasoning about their size. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with the symbols >, =, or <,
and justify the conclusions, e.g., by using a visual fraction model.
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n ! a)/(n ! b) by using visual fraction models, with attention to how the
number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or
numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two
fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g.,
by using a visual fraction model.
!
Geometry 2.G
Grade 2
Reason with shapes
and their
attributes.
3. Partition circles and
rectangles into two,
three, or four equal
shares using the
words halves, thirds,
half of, a third of,
etc., and describe
the whole as two
halves, three thirds,
four fourths.
Recognize that
equal shares of
identical wholes
need not have the
same shape.
!
4. Generate measurement data by measuring lengths using rulers marked
with halves and fourths of an inch. Show the data by making a line plot,
where the horizontal scale is marked off in appropriate units— whole
numbers, halves, or quarters.
!
Geometry 3.G
Reason with shapes and their attributes.
2. Partition shapes into parts with equal areas. Express the area of each
part as a unit fraction of the whole. For example, partition a shape into 4
parts with equal area, and describe the area of each part as 1/4 of the
area of the shape.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each
decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 +
1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction,
and/or by using properties of operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like
denominators, e.g., by using visual fraction models and equations to represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 !
(1/4), recording the conclusion by the equation 5/4 = 5 ! (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For
example, use a visual fraction model to express 3 ! (2/5) as 6 ! (1/5), recognizing this product as 6/5. (In general, n ! (a/b) =
(n ! a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and
equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there
will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your
answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two
4
fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62
meters; locate 0.62 on a number line diagram. (4Students who can generate equivalent fractions can develop strategies for adding fractions with
unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.)
7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two
decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions,
e.g., by using the number line or another visual model.
4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition
and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the
difference in length between the longest and shortest specimens in an insect collection.
!
Formatted by Educational Resource Services, Tulare County Office of Education, Visalia, California (559) 651-3031 www.commoncore.tcoe.org
5
PROGRESSION:
T H E MEA NING OF FRAC T IONS
¡  In Grades 1 and 2, students use fraction
language to describe partitions of shapes
in to equal shares. In Grade 3 they start to
develop the idea of a fraction more
formally, building on the idea of
partitioning a whole into equal parts. The
whole can be a shape such as a circle or
rectangle, a line segment, or any one finite
entity susceptible to subdivision and
measurement. In Grade 4, this is extended
to include wholes that are collections of
objects.
Fraction Progression
page 2
6
2 11/1/13 FRACTIONS PROGRESSION
¡ U nderstanding the arithmetic of fractions
draws upon four prior progressions that
informed the CCSS:
§ equal partitioning,
§ unitizing,
§ number line,
§ and operations.
Phil Daro
7
3.NF.1
3.NF.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.
¡  “Students start with unit fractions (fractions with numerator
1)….”
¡  “Next, students build fractions from unit fractions, seeing the
numerator 3 or ¾ as saying that ¾ is the quantity you get by
putting 3 of the ¼’s together.”
¡  “They read any fraction this way, and in particular there is no
need to introduce “proper fractions” and “improper fractions”
initially; 5/3 is the quantity you get by combining 5 parts
together when the whole is divided into 3 equal parts.”
8
“3-5 Number and Operations – Fractions” Progression, page 2
UNITIZING
One is One…or is it?
http://ed.ted.com/lessons/one-is-one-or-is-it
Discuss:
Why is the idea of the unit important in fractions?
9
3 11/1/13 “UNIT (ONE)”, A SIMPLE BUT POWERFUL
CONCEPT
The following quotations are from Sheldon’s Complete Arithmetic (1 886)
Quotation 1
¡  A unit is a single thing or one; as one apple, one dollar, one hour, one.
Quotation 2
¡  Like numbers are numbers whose units are the same; as $7 and $9.
¡  Unlike numbers are numbers whose units are dif ferent; as 8 lb. and 12
cents.
Can you add 8 cents and 7 cents? What kind of numbers are they?
Can you add $5 and 5lb.? What kind of numbers are they?
Principle:
Only like numbers can be added and subtracted.
Why do we need to line numbers up when we do addition ?
10
Liping Ma
WITH MULTIPLICATION AND DIVISION, THE
CONCEPT OF “UNIT” IS EXPANDED:
Quotation 1
¡  A unit is a single thing or one.
Quotation 2
¡  A group of things if considered as a single thing or one is also
a unit; as one class, one dozen, one group of 5 students.
There are 3 plates each with 5 apples in it. How many apples
are there in all?
What is the unit (the “one”)?
Liping Ma
11
WITH FRACTIONS, THE CONCEPT OF “UNIT” IS
EXPANDED ONE MORE TIME:
Quotation 1
¡  A unit is a single thing or one.
Quotation 2
¡  A unit, however, may be divided into equal parts, and each of
these parts becomes a single thing or a unit.
Quotation 3
¡  In order to distinguish between these two kinds of units, the
first is called an integral unit, and the second a fractional
unit.
What is the fractional unit of 3/4 ? of 2/3?
Principle:
Only like numbers can be added and subtracted.
Computing 3/4 + 2/3, Why do we need to turn the
fractions into fractions with common denominator?
12
Liping Ma
4 11/1/13 UNITS ARE THINGS YOU COUNT
¡ O bjects
¡ G roups of objects
¡ 1
¡ 1 0
¡ 1 00
¡ ¼ unit fractions
¡ N umbers represented as expressions
Phil Daro
13
PRINCIPLE:
A D D I N G A N D S U B T R AC T I N G “ U N I T S ”
¡ 3 pennies + 5 pennies = 8 pennies
¡ 3 ones + 5 ones = 8 ones
¡ 3 tens + 5 tens = 8 tens
¡ 3 inches + 5 inches = 8 inches
¡ 3 “¼ inches” + 5 “¼ inches” = 8 “¼ inches”
¡ 3/4 + 5/4 = 8/4
¡ 3(x + 1) + 5(x + 1) = 8(x + 1)
You can compute the sums and differences of like
terms
Phil Daro
14
PRINCIPLE:
ADDING AND SUBTRACTING “ UNIT S”
¡  2 dimes + 3 quarters
¡  27 inches + 2 feet
¡  8 ones + 9 tens
¡  12 seconds + 1 minute
¡  1/4 + 1/3
¡  What must happen before we can compute the sum or
difference?
¡  Unlike Terms may be added and subtracted, but
computing the sum or difference is only possible when
we have Like Terms (Common Units)
15
Phil Daro
5 11/1/13 FRACTION STRIPS
¡  Materials
§  Six different colors of paper cut into 1” x 8” strips. Each child will
need 6 strips, one of each color.
¡  Task Description
§  Give each student six strips of paper, one of each color.
§  Specify one color. Tell students that this strip will represent the
whole. Have students write “one whole” on the fraction strip.
§  Specify a different color and have students fold it into two equal
pieces. Have students draw a line on the fold. Ask students what
they think each of these strips should be called. Discuss how we
write fractions. Have students label their strips using both the word
and the fractional representation – 1/2 and one-half.
§  Repeat this process for thirds, fourths, sixths, and eighths.
§  Students should have 6 fraction strips.
16
Adapted from Georgia Department of Education, Grade 3 Unit 6: Reasoning
and Comparing Fractions
EXPLORING FRACTIONS WITH FRACTION
STRIPS
Small Groups – Discuss and Record
¡  What observations do you have about the fractions strips?
Group Discussion
§  How many halves does it take to make a whole strip?
§  How many thirds does it take to equal one whole?
§  How many fourths, sixths, eighths?
§  What patterns do you notice?
§  What does the numerator represent?
§  What does the denominator represent?
§  If you made a 1/9 fraction strip, how many ninths would it take
to make a whole?
17
UNIT FRACTIONS
¡ “ The goal is for students to see unit fractions
as the basic building blocks of fractions, in
the same sense that the number 1 is the basic
building block of the whole numbers; just as
every whole number is obtained by combining
a sufficient number of 1s, every fraction is
obtained by combining a sufficient number of
unit fractions.”
18
“3-5 Number and Operations – Fractions” Progression, page 3
6 11/1/13 COUNTING BY FRACTIONS
¡  “Students should come to think of counting fractional parts in
much the same way as they might count apples or any other
objects.”
¡  Example: “ …tell students what type of piece is being shown
and simply count them together: “one-fourth, two-fourths,
three-fourths, four-fourths, five-fourths.” Ask, “If we have fivefourths, is that more than one whole, less than one whole, or
the same as one whole?”
19
Teaching Student-Centered Mathematics: Grades 5-8. Van de Walle and Lovin, page 67.
3.NF.3
3.NF.3: Explain equivalence of fractions in special cases, and compare fractions by
reasoning about their size.
a.  Understand two fractions as equivalent (equal) if they are the same size, or the
same point on a number line.
b.  Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3.
Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c.  Express whole numbers as fractions, and recognize fractions that are equivalent
to whole numbers.
d.  Compare two fractions with the same numerator or the same denominator by
reasoning about their size. Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with
the symbols <, =, or <, and justify the conclusions, e.g., by using a visual
fractions model.
20
COMPARING FRACTIONS
Materials:
¡  Fraction Strips: Cut along the folds
¡  Sandwich bags or envelopes to store the strips.
I. Small Groups – Discuss and Record
¡  What obser vations do you have about the separated Fraction
Strips?
¡  Do you see any special relationships among the dif ferent colored
strips?
Additional Questions
¡  Place a 1/2 strip on your desk. How many strips or combinations
of strips are the same size as 1/2?
¡  When fractions are the same size, they are called equivalent.
What other equivalent fractions can you create?
21
7 11/1/13 COMPARING FRACTIONS
(CONTINUED)
II. Whole Group Discussion:
¡  What relationships did you discover about fractions?
¡  What equivalent groups of fractions did you discover?
Comparing Fractions
22 and
COMPARING FRACTIONS
(CONTINUED)
III. The friends below are playing red light-green light. Who is
winning? Explain your reasoning.
Use your fraction strips to determine how far each friend has
moved.
Mary – 3/4
Harry – 1/2
Larry – 5/6
Sam – 5/8
Angie – 2/3
23
3.NF.2
Develop understanding of fractions as numbers
3.NF.2: Understand a fraction as a number on the number line; represent fractions
on a number line diagram.
a.  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.
b.  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.
24
8 11/1/13 LOCATING ½ ON A NUMBER LINE
3
Why do
students
respond
this
way?
Initially, students can use an intuitive notion of congruence (“same
Area representations of 14
size and same shape”) to explain why the parts are equal, e.g., when
they divide a square into four equal squares or four equal rectangles.
Students come to understand a more precise meaning for “equal
parts” as “parts with equal measurements.” For example, when a
In each representation the square is the whole. The two
ruler is partitioned into halves or quarters of an inch, they see that
squares on the left are divided into four parts that have the
each subdivision has the same length. In area models they reason
same size and shape, and so the same area. In the three
about the area of a shaded region to decide what fraction of the
squares on the right, the shaded area is 14 of the whole area,
3
even though it is not easily seen as one part in a division of
whole it represents (MP3).
the square into four parts of the same shape and size.
The goal is for students to see unit fractions as the basic building
3
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Students
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25
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Students
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squares on the left are divided into four parts that have the
parts”
as “parts
equal
measurements.”
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the subdivision
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unitthe
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models
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isthe
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ainto
shaded
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squares on the left
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thatwhole
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each
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In
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they
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numbers,
so
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the
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onearea.
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in the
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same
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the square
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0goal
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to
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etc.,
all
ofdecide
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about
of students
region
tofractions
what
fraction
of the
squares
on into
the right,
the shaded
4 of the whole area,
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seeare
unit
as
the
basicas
building
even 0though 1it is not2easily seen
3 as one
4 part 5in a division
6 etc.of
Students
might think
of same
the number
line as
infinite1 ruler.
whole
itof represents
blocks
fractions,
in(MP3).
the
sense that
thean
number
is the basic1
the square into four parts of the same shape and size.
To goal
construct
a unit
fraction
on ajust
number
linethe
diagram,
e.g. is3 ,
The
is of
forthe
students
to
see unit
fractions
as
basicnumber
building
building
block
whole
numbers;
as every
whole
students
partition
the
unit
interval
into
3
intervals
of
equal
length
blocks
of
fractions,
in
the
same
sense
that
the
number
1
is
the
basic
obtained by combining a sufficient number
of 1s, every fraction1 is
1
and recognize
has
lengthnumber
. They
locate
the number
building
block
ofthat
the each
whole
numbers;
asofevery
whole
number
is
3just
3 on
obtained
by combining
a sufficient
unit fractions.
the number
line by marking
off thisnumber
length from
and locate
other
obtained
by combining
a sufficient
of 1s,0,every
fraction
is
The number line marked off in thirds
fractionsbywith
denominator
3 by marking
the number
of lengths
obtained
combining
a sufficient
number off
of unit
fractions.
0
1
2
3
4
The
number
line
and
number
line
diagrams
On
the
number
line,
3.NF.2
indicated by the numerator.
0
1
2
3
4
5
6
7
8
9 10 11 12
the whole
is the
unit interval,
is, the
interval the
fromunit
0 toon1,amea3
3
3
3
3
3
3
3
3
3
3
3
3
Students
sometimes
have that
difficulty
perceiving
numThe
number
line
and number
line diagrams
On
the off
number
line,
sured
by
length.
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this
whole
to
the
right
marks
the
whole
ber
line diagram.
When
locating
a fraction
on afrom
number
line
diathe
whole
is that
the unit
interval,
that
is,
the
interval
0 tonumbers,
1, meanumbers,
so
theuse
intervals
between
consecutive
whole
gram,bythey
might
asthis
thewhole
unit the
entire
portion
ofoffthe
number
3.NF.2 Understand a fraction
sured
length.
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to
the
right
marks
the
whole
The number
line on the number line;
as a number
from
to 1,is1shown
to 2, 2ontothe
3, etc.,
are all
the same
length, the
as shown.
line0that
diagram,
for of
example
indicating
number
represent
fractions
on2a number
numbers,
so thatthink
the intervals
between
consecutive
whole
numbers,
0
1
3 line diagram.
4
5
6 etc.
Students
might
of the
line
as an infinite
ruler.
3
3 when
show
onnumber
a are
number
line
marked
0 1to
The number line
4 etc.,
from
0 construct
to asked
1, 1 toto2,
to 3,
thediagram
same
length,
asfrom
shown.
a Represent a fraction 1 � on a number line diagram
To
a 2unit
fraction
onall
aareof
number
linerepresentations
diagram,
e.g.
Materials:
3,
4.
Although
number
line
diagrams
important
for
0 by defining
1
6 etc.
the2 interval3 from 04to 1 as5the whole
and
Students
might think
ofunit
the interval
number into
line 3asintervals
an infinite
ruler. length
students
partition
the
of
equal
students
as theya develop
an understanding
of aline
fraction
as a number,
1
partitioning it into � equal parts. Recognize that each
Fraction
Strips
To construct
fraction
on ¡ 
a1 . number
diagram,
e.g.1 on
3,
and
that unit
each
hasNF
length
locate
the
number
part has size 1 � and that the endpoint of the part
in recognize
the early
stages
ofunit
the
Progression
use
represen3 They they
3
students
partition
the
interval
into
3 intervals
ofother
equal
length
basedThe
at 0 number
locates the
number
1
�
on
the
number
line.
the
number
line
by
marking
off
this
length
from
0,
and
locate
other
¡ 
Sentence
Strip
marked
line marked off in thirds
tations
such that
as area
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andthestrips
of paper.
1
and
recognize
eachmodels,
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locate
number
3 They
3 on
fractions
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by marking
the number
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b 0Represent a1fraction � �2 on a number
3 line diagram
4 by
These, like
line diagrams,
can
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representing
into
three
8”locate
units.
the
number
by
marking3.NF.2
off this length
from
0, and
other
Theoffnumber
line1marked
off
thirds that the
marking
� lengths
� from
0. in
Recognize
indicated
byline
theaspect
numerator.
an important
of fractions.
0
1
2
3
4
5
6
7
8
9 10 11 12
fractions
with
denominator
3
by
marking
off
the
number
of
lengths
resulting
interval
has
3
3
3
3
3
3 size
Task:
0
1
23 �3 � 3and 33that3 its3 endpoint
43
Students
sometimes
have3.NF.2
difficulty
perceiving
the unit
on a numThe number
line reinforces
the analogy
between
fractions
and
locates
the
number
�
�
on
the
number
line.
indicated
by the numerator.
0
1
2
3
4
5
6
7
8
9 10 11 12
ber
line numbers.
diagram.
When
on number
a number
diawhole
Just
as locating
5 is the apoint
on the
lineline
reached
¡ fraction
Use
your
fractions
3
3
3
3
3
3
3
3
3
3
3
3
3
Students
sometimes
difficulty
perceiving
the
unit
on number
a num5
gram,
they might
ashave
thelength
unit the
entire
offrom
the
3.NF.2 Understand a fraction as a number on the
by marking
off 5 use
times
the
of the
unitportion
interval
0, so your
3 is
Number line representation of 53 number line;
strips
toa number
mark
off
ber
line diagram.
When
locating for
a fraction
on
dialine
is shown
on
thethe
diagram,
indicating
theline
number
thethat
point
obtained
in
same way example
using a different
interval
as the
represent fractions on a number line diagram.
3 the unit the number
gram,
they
might
use as
entire
portion
number
One part of a diviline
in
thirds.
3.NF.2
1 of the
3 when
asked
to
show
on
a
number
line
diagram
marked
from
0
to
Understand
a fraction
as a number on the number line;
basic unit of length, namely
the interval from 0 to 3 .
4
sion of athe
unit ina Represent
fraction
1 � on a number line diagram
line
that is shown
online
the
diagram, for
indicating
the number
represent fractions
on3 aparts
number
line diagram.
4. Although
number
diagrams
areexample
important
representations
for
intothe
of from
by terval
defining
interval
0 to 1 as the whole and
3
3students
when asked
to show
number line diagram
marked
0 to
equal length
4 on
as they
develop
an aunderstanding
of asome
fraction
as afrom
number,
partitioninga itfraction
into � equal
parts.
Recognize
each
a Represent
1 � on
a number
linethat
diagram
Equivalent
fractions
Grade
3 students
do
preliminary
rea4.
Although
number
line
diagrams
are
important
representations
for
¡ 
What
observations
can
0
1
2
3 1 as theofwhole
4the part
partdefining
has size
� and that
by
the1 interval
fromthe
0 toendpoint
and
insoning
the early
stages
of the NF
Progression
they usefor
other
represenabout
equivalent
fractions,
in
preparation
work
in
Grade
based at 0 locates
number
1 �Recognize
on the number
line.
students
as they
develop
an understanding
ofmake?
a fraction
as aofnumber,
partitioning
it into the
� equal
parts.
that each
you
tations
such
as
area
models,
tape
diagrams,
and
strips
paper.
parts
4.theAsearly
students
experiment
number line
diagrams
they
discover
point
on the number
part 5has
size 1 � and the
that
the 53endpoint
of thelinepart
in
stages
of thediagrams,
NFon
Progression
they
use other
represenb Represent a fraction � � on a number
line diagram by
These,
like number
beon
subdivided,
that many
fractionsline
label the samecan
point
the numberrepresenting
line, and are
based at 0 locates the number 1 � on the number line.
tations
such as
areaofmodels,
tape diagrams, and strips of paper.
marking off � lengths 1 � from 0. Recognize that the
an important
aspect
fractions.
resulting interval
has� size
and that
endpoint
b Represent
a fraction
� on�a�number
lineits
diagram
by
These,
number
diagrams,
be subdivided,
representing
Thelike
number
lineline
reinforces
the can
analogy
between fractions
and
locates the
� 1
��
onfrom
the number
line. that the
marking
off number
� lengths
0. Recognize
an
important
aspect
ofasfractions.
Draft,
8/12/2011,
comment
commoncoretools.
wordpress.
com .
whole
numbers.
Just
5 isatthe
point on the number
line reached
resulting
interval has size � � and that its endpoint
26
Diagrams
fromfractions
Fraction Progression,
page 3
5
The number
theof analogy
between
locates
the
number
�
�
on
the
number
line.
by marking
off 5 line
timesreinforces
the length
the unit
interval
from
0, so and
is
3
Number line representation of 53
whole
numbers.
Just
as 5same
is the
point
on athe
numberinterval
line reached
3.NF.3abc
the point
obtained
in the
way
using
different
the
Explainasequivalence
of fractions
special cases,
and
5
One part of aindiviby
marking
5 times
the length
of the unit
basic
unit ofoff
length,
namely
the interval
frominterval
0 to 13 . from 0, so 3 is
Number
line inrepresentation of 53
sion of
the unit
compare
fractions
their size.
the point obtained in the same way using
a different
interval as by
the reasoning about
terval into 3 parts of
One part of a divi1
equal length
basic unit of length, namely the interval from 0 to 3 .
sion of the unit inEquivalent fractions Grade 3 students do some preliminary reaterval
into
a Understand two fractions
as equivalent
if they
0
1 3 parts of
2 (equal)
3
4
soning about equivalent fractions, in preparation for work in Grade
equal length
Equivalent
Grade
students
do diagrams
some
rea-size, or the same
arepreliminary
the
point on
a number line.
5 parts
6 As students
8 fractions
4.
experiment
on3number
line
they same
discover
0
1
2 the point 53 3on the number
4 line
soning
about
equivalent
fractions,
in
preparation
for
work
in
Grade
that
3 many
4 fractions label the same point on the number line, and are
5 parts
4. As students experiment on number line diagrams they discover
the point 53 on the number line
b number
Recognize
that many fractions label the same point on the
line, and and
are generate simple equivalent fractions,
1 2 com2. 4, 4 6 2 3. Explain why the fractions
Draft, 8/12/2011, comment at commoncoretools.e.g.,
wordpress.
FRACTION STRIPS ON A NUMBER LINE
4
ions. For example,
dents can also use
numbers as fracn the number line
4
, , etc. so
2,
equivalent,
Draft, 8/12/2011, comment at commoncoretools.are
wordpress.
com .
students compared
3 In Grade 3 they
same denominator.
denominator, the
the fraction with
made of more unit
s shorter than the
s of 14 as opposed
one with the larger
e, that in order for
the pieces must be
hat have the same
ator is greater. For
of 17 is less than 2
tant in comparing
o the same whole.
ns as points on the
order in terms of
the number line—
e.g., by using a visual fraction model.
EQUIVALENT FRACTIONS ON A NUMBER
c Express whole numbers as fractions, and recognize
LINE to whole numbers.
fractions that are equivalent
1 as a fraction:
Using the number line and fraction strips to see fraction
equivalence
1
2
0
1
4
2
4
3
4
2
2
1
4
4
1
1
6
1
6
1
6
1
2
¡  Use your fractions strips to mark off your number line in
2.MD.3
Estimate lengths using units of inches, feet, centimefourths.
ters,
and
meters.
¡  Now
mark
off your number line in halves.
27
¡  What observations can you make?
3.NF.3d
Explain equivalence
of fractions in special cases, and
¡  What fractions
are equivalent?
compare fractionsDiagrams
by reasoning
about their size.
from Fraction Progression, page 4
d Compare two fractions with the same numerator or the
same denominator by reasoning about their size. Recognize that comparisons are valid only when the two
fractions refer to the same whole. Record the results of
comparisons with the symbols , =, or , and justify
the conclusions, e.g., by using a visual fraction model.
The importance of referring to the same whole when
comparing fractions
9 11/1/13 4.NF.3
Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a.  Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
b.  Decompose a fraction into a sum of fractions with the same denominator in more
than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8
+ 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c.  Add and subtract mixed numbers with like denominators, e.g., by replacing each
mixed number with an equivalent fraction, and/or by using properties of operations
and the relationship between addition and subtraction.
d.  Solve word problems involving addition and subtraction of fractions referring to the
same whole and having like denominators, e.g., by using visual fraction models
and equations to represent the problem.
28
UNITIZING LINKS FRACTIONS TO WHOLE
NUMBER ARITHMETIC
¡ S tudents’ expertise in whole number
arithmetic is the most reliable expertise they
have in mathematics
¡ I t makes sense to students
¡ I f we can connect difficult topics like fractions
and algebraic expressions to whole number
arithmetic, these difficult topics can have a
solid foundation for students
Phil Daro
29
ADDING AND SUBTRACTING FRACTIONS:
VIA C OMP OS IT ION A ND DEC OMP OS IT ION
¡  Let’s start with the number 5
¡  What do we have 5 of (What are we counting)?
¡  How else can we write 5?
Why do we use 5, rather than 3+2, or 1+1+1+1+1?
30
10 11/1/13 ADDING AND SUBTRACTING FRACTIONS:
¡  So when working with the unit “ones”, we can express that like
this:
1 + 1 + 1 + 1 + 1 = 5
What unit would we be working with here?
Show what this expression looks like using your fraction strips.
Adapted from Fraction Progression, page 6
31
¡  How else can we express this?
32
5/6 + 3/6
How many ways can we do this?
33
11 11/1/13 CONSTRUCTING TASK: FRACTION
ADDITION
Part 1 - In pairs:
§  Using the paper fraction strips, demonstrate a model of the following
expressions.
§  1/4 + 1/4
§  1/3 + 1/3
§  2/3 + 2/3
§  Even if you know the “answer,” prove it with the model. Can the
answer be written in other ways?
§  Show the same expressions on your number line.
34
Adapted from Georgia Department of Education, Grade 5 Unit 4
CONSTRUCTING TASK: FRACTION
ADDITION
¡  In groups:
§  How do we know that our answers are correct?
§  How do we know that our points on the number line are labeled
correctly?
§  Are there other names for some of the points marked on our number
line?
§  Are there any patterns or rules for adding these types of fractions?
Explain what you have discovered.
35
CONSTRUCTING TASK: FRACTION
ADDITION AND SUBTRACTION
Part 2 - In pairs:
§  Using the paper fraction strips, demonstrate a model of the following
expressions.
§  1/2 + 1/3
§  2/3 – 1/2
§  2/3 + 1/2
§  Even if you know the “answer,” prove it with the model. Can the
answer be written in other ways?
§  Show the same expressions on your number line.
36
12 11/1/13 CONSTRUCTING TASK: FRACTION
ADDITION AND SUBTRACTION
¡  In groups:
§  How do we know that our answers are correct?
§  How do we know that our points on the number line are labeled
correctly?
§  How can we change these problems to make them easier, like in part
I?
§  Is there one fraction name for your equation points on the number
line?
§  Did you identify any patterns or rules for adding and subtracting
these kinds of fractions? Explain what you have found.
37
DISCUSS
How is this similar and how is
this different from how your
curriculum currently addresses
fractions?
38
RESOURCES
¡  Common Core Connect
http://commoncore.tcoe.org
¡  E-mail
§  Julie Joseph– jjoseph@ers.tcoe.org
39
13

Develop Understanding of Fractions

Transcription

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