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Students start to develop fluency with addition and subtraction of fractions,
develop understanding of the multiplication of fractions and of
division of fractions in limited cases (unit fractions divided by whole
numbers and whole numbers divided by unit fractions)

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Students start to understand fraction equivalence, and understand addition and subtraction with like denominators.

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Students should be fluent when working with fractions and be able to recognize fractions fluently.

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Students develop understanding of division of fractions and extending the
notion of number to the system of rational numbers, which includes
negative numbers.

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Develop understanding of fractions as numbers.

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Understand a fraction as a number on the number line; represent
fractions on a number line diagram.

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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.

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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.

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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.

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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?

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Express a fraction with denominator 10 as an equivalent fraction
with denominator 100, and use this technique to add two fractions
with respective denominators 10 and 100. (Note: Students 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.) For example, express 3/10 as 30/100, and add 3/10 +
4/100 = 34/100.

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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.

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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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.
(In general, a/b + c/d = (ad + bc)/bd.)

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Solve word problems involving addition and subtraction of fractions
referring to the same whole, including cases of unlike denominators, e.g.,
by using visual fraction models or equations to represent the problem.
Use benchmark fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers. For example,
recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

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Interpret a fraction as division of the numerator by the denominator
(a/b = a ÷ b). Solve word problems involving division of whole
numbers leading to answers in the form of fractions or mixed
numbers, e.g., by using visual fraction models or equations to
represent the problem. For example, interpret 3/4 as the result of
dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when
Fifth Grade – Standards
1. D eveloping fluency with addition and subtraction of fractions,
developing understanding of the multiplication of fractions and of
division of fractions in limited cases (unit fractions divided by whole
numbers and whole numbers divided by unit fractions) – Students apply
their understanding of fractions and fraction models to represent the
addition and subtraction of fractions with unlike denominators as equivalent
calculations with like denominators. They develop fluency in calculating
sums and differences of fractions, and make reasonable estimates of them.
Students also use the meaning of fractions, of multiplication and division,
and the relationship between multiplication and division to understand and
explain why the procedures for multiplying and dividing fractions make
sense. (Note: this is limited to the case of dividing unit fractions by whole
numbers and whole numbers by unit fractions.)
2. E xtending division to 2-digit divisors, integrating decimal fractions into
the place value system and developing understanding of operations
with decimals to hundredths, and developing fluency with whole
number and decimal operation – Students develop understanding of why
division procedures work based on the meaning of base-ten numerals and
properties of operations. They finalize fluency with multi-digit addition,
subtraction, multiplication, and division. They apply their understandings
of models for decimals, decimal notation, and properties of operations
to add and subtract decimals to hundredths. They develop fluency in
these computations, and make reasonable estimates of their results.
Students use the relationship between decimals and fractions, as well as
the relationship between finite decimals and whole numbers (i.e., a finite
decimal multiplied by an appropriate power of 10 is a whole number), to
understand and explain why the procedures for multiplying and dividing
finite decimals make sense. They compute products and quotients of
decimals to hundredths efficiently and accurately.
3. D eveloping understanding of volume – Students recognize volume as an
attribute of three-dimensional space. They understand that volume can be
quantified by finding the total number of same-size units of volume required
to fill the space without gaps or overlaps. They understand that a 1-unit by
1-unit by 1-unit cube is the standard unit for measuring volume. They select
appropriate units, strategies, and tools for solving problems that involve
estimating and measuring volume. They decompose three-dimensional
shapes and find volumes of right rectangular prisms by viewing them as
decomposed into layers of arrays of cubes. They measure necessary
attributes of shapes in order to solve real world and mathematical problems.
Mathematical Practices
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 wholes are shared equally among 4 people each person has a
share of size 3/4. If 9 people want to share a 50-pound sack of rice
equally by weight, how many pounds of rice should each person get?
Between what two whole numbers does your answer lie?

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Apply and extend previous understandings of multiplication to
multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b
equal parts; equivalently, as the result of a sequence of operations
a × q ÷ b. For example, use a visual fraction model to show (2/3) ×
4 = 8/3, and create a story context for this equation. Do the same
with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it
with unit squares of the appropriate unit fraction side lengths, and
show that the area is the same as would be found by multiplying
the side lengths. Multiply fractional side lengths to find areas of
rectangles, and represent fraction products as rectangular areas

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Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on
the basis of the size of the other factor, without performing the
indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater
than 1 results in a product greater than the given number
(recognizing multiplication by whole numbers greater than 1 as
a familiar case); explaining why multiplying a given number by
a fraction less than 1 results in a product smaller than the given
number; and relating the principle of fraction equivalence a/b =
(n×a)/(n×b) to the effect of multiplying a/b by 1.

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Solve real world problems involving multiplication of fractions and
mixed numbers, e.g., by using visual fraction models or equations to
represent the problem.

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Apply and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.
(Note: Students able to multiply fractions in general can develop
strategies to divide fractions in general, by reasoning about the
relationship between multiplication and division. But division of a
fraction by a fraction is not a requirement at this grade.)
a. Interpret division of a unit fraction by a non-zero whole number,
and compute such quotients. For example, create a story context
for (1/3) ÷ 4, and use a visual fraction model to show the quotient.
Use the relationship between multiplication and division to explain
that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and
compute such quotients. For example, create a story context for 4
÷ (1/5), and use a visual fraction model to show the quotient. Use
the relationship between multiplication and division to explain that
4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by
non-zero whole numbers and division of whole numbers by unit
fractions, e.g., by using visual fraction models and equations to
represent the problem. For example, how much chocolate will each
person get if 3 people share 1/2 lb of chocolate equally? How many
1/3-cup servings are in 2 cups of raisins?

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Make a line plot to display a data set of measurements in fractions
of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade
to solve problems involving information presented in line plots. For
example, given different measurements of liquid in identical beakers,
find the amount of liquid each beaker would contain if the total
amount in all the beakers were redistributed equally.

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Understand the concept of a unit rate a/b associated with a ratio a:b
with b ≠ 0, and use rate language in the context of a ratio relationship.
For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar,
so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15
hamburgers, which is a rate of $5 per hamburger.” (Note: Expectations
for unit rates in this grade are limited to non-complex fractions.)

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Interpret and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual fraction
models and equations to represent the problem. For example, create
a story context for (2/3) ÷ (3/4) and use a visual fraction model to show
the quotient; use the relationship between multiplication and division
to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general,
(a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3
people share 1/2 lb of chocolate equally? How many 3/4-cup servings
are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land
with length 3/4 mi and area 1/2 square mi?

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Compute unit rates associated with ratios of fractions, including
ratios of lengths, areas and other quantities measured in like or
different units. For example, if a person walks 1/2 mile in each 1/4
hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles
per hour, equivalently 2 miles per hour.

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Apply and extend previous understandings of multiplication and
division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to
rational numbers by requiring that operations continue to
satisfy the properties of operations, particularly the distributive
property, leading to products such as (–1)(–1) = 1 and the rules for
multiplying signed numbers. Interpret products of rational numbers
by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers (with non-zero divisor) is a rational
number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret
quotients of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide
rational numbers.
d. Convert a rational number to a decimal using long division; know
that the decimal form of a rational number terminates in 0s or
eventually repeats.

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Solve real-world and mathematical problems involving the four operations
with rational numbers. (NOTE: Computations with rational numbers extend
the rules for manipulating fractions to complex fractions.)

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Solve multi-step real-life and mathematical problems posed with positive
and negative rational numbers in any form (whole numbers, fractions,
and decimals), using tools strategically. Apply properties of operations
to calculate with numbers in any form; convert between forms as
appropriate; and assess the reasonableness of answers using mental
computation and estimation strategies. For example: If a woman making
$25 an hour gets a 10% raise, she will make an additional 1/10 of her salary
an hour, or $2.50, for a new salary of $27.50. If you want to place a towel
bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you
will need to place the bar about 9 inches from each edge; this estimate
can be used as a check on the exact computation.

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Students will start to understand what fractions are. Students will begin by focusing on unit fractions and fractions as numbers.