Ccs 5th math

COMMON CORE STATE STANDARDS for MATHEMATICS

Mathematics | Grade 5
In Grade 5, instructional time should focus on three critical areas: (1)
developing fluency with addition and subtraction of fractions, and
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); (2) extending 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 operations; and (3)
developing understanding of volume.

(1) 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) 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) Students recognize volume as an attribute of three-dimensional
space. They understand that volume can be measured 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 determine volumes to solve real world and mathematical problems.
GRADE 5 |
33


Grade 5 Overview
Operations and Algebraic Thinking Mathematical Practices

• Write and interpret numerical expressions. 1. Make sense of problems and persevere in
solving them.
• Analyze patterns and relationships.
2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique
the reasoning of others.
Number and Operations in Base Ten
4. Model with mathematics.
• Understand the place value system.
5. Use appropriate tools strategically.
• Perform operations with multi-digit whole 6. Attend to precision.
numbers and with decimals to hundredths.
7. Look for and make use of structure.

8. Look for and express regularity in repeated
Number and Operations—Fractions reasoning.

• Use equivalent fractions as a strategy to add
and subtract fractions.

• Apply and extend previous understandings
of multiplication and division to multiply and
divide fractions.

Measurement and Data

• Convert like measurement units within a given
measurement system.

• Represent and interpret data.

• Geometric measurement: understand concepts
of volume and relate volume to multiplication
and to addition.

Geometry

• Graph points on the coordinate plane to solve
real-world and mathematical problems.

• Classify two-dimensional ﬁgures into categories
based on their properties.
GRADE 5 |
34


Operations and Algebraic Thinking 5.OA

Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and
evaluate expressions with these symbols.
2. Write simple expressions that record calculations with numbers, and
interpret numerical expressions without evaluating them. For example,
express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7).
Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921,
without having to calculate the indicated sum or product.

Analyze patterns and relationships.
3. Generate two numerical patterns using two given rules. Identify
apparent relationships between corresponding terms. Form ordered
pairs consisting of corresponding terms from the two patterns, and
graph the ordered pairs on a coordinate plane. For example, given the
rule “Add 3” and the starting number 0, and given the rule “Add 6” and the
starting number 0, generate terms in the resulting sequences, and observe
that the terms in one sequence are twice the corresponding terms in the
other sequence. Explain informally why this is so.

Number and Operations in Base Ten 5.NBT

Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents
10 times as much as it represents in the place to its right and 1/10 of
what it represents in the place to its left.
2. Explain patterns in the number of zeros of the product when
multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplied or
divided by a power of 10. Use whole-number exponents to denote
powers of 10.
3. Read, write, and compare decimals to thousandths.

Read and write decimals to thousandths using base-ten numerals,
number names, and expanded form, e.g., 347.392 = 3 Ð 100 + 4 Ð
10 + 7 Ð 1 + 3 Ð (1/10) + 9 Ð (1/100) + 2 Ð (1/1000).

Compare two decimals to thousandths based on meanings of the
digits in each place, using >, =, and < symbols to record the results
of comparisons.
4. Use place value understanding to round decimals to any place.

Perform operations with multi-digit whole numbers and with
decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard
algorithm.
6. Find whole-number quotients of whole numbers with up to four-digit
dividends and two-digit divisors, using strategies based on place
value, the properties of operations, and/or the relationship between
multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.
7. Add, subtract, multiply, and divide decimals to hundredths, using
concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method and explain the
reasoning used.
GRADE 5 |
35


Number and Operations—Fractions 5.NF

Use equivalent fractions as a strategy to add and subtract fractions.
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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In
general, a/b + c/d = (ad + bc)/bd.)
2. 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.

Apply and extend previous understandings of multiplication and
division to multiply and divide fractions.
3. Interpret a fraction as division of the numerator by the denominator
(a/b = a Ð 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 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?
4. Apply and extend previous understandings of multiplication to
multiply a fraction or whole number by a fraction.

Interpret the product (a/b) Ð q as a parts of a partition of q
into b equal parts; equivalently, as the result of a sequence of
operations a Ð q Ð 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.)

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.
5. Interpret multiplication as scaling (resizing), by:

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.

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 =
(nÐa)/(nÐb) to the effect of multiplying a/b by 1.
6. Solve real world problems involving multiplication of fractions and
mixed numbers, e.g., by using visual fraction models or equations to
represent the problem.
7. Apply and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.1

Interpret division of a unit fraction by a non-zero whole number,
GRADE 5 |

1
Students able to multiply fractions in general can develop strategies to divide frac-
tions 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.
36


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.

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.

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?

Measurement and Data 5.MD

Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a
given measurement system (e.g., convert 5 cm to 0.05 m), and use
these conversions in solving multi-step, real world problems.

Represent and interpret data.
2. 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.

Geometric measurement: understand concepts of volume and relate
volume to multiplication and to addition.
3. Recognize volume as an attribute of solid figures and understand
concepts of volume measurement.

A cube with side length 1 unit, called a “unit cube,” is said to have
“one cubic unit” of volume, and can be used to measure volume.

A solid figure which can be packed without gaps or overlaps
using n unit cubes is said to have a volume of n cubic units.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in,
cubic ft, and improvised units.
5. Relate volume to the operations of multiplication and addition and
solve real world and mathematical problems involving volume.

Find the volume of a right rectangular prism with whole-number
side lengths by packing it with unit cubes, and show that the
volume is the same as would be found by multiplying the edge
lengths, equivalently by multiplying the height by the area of the
base. Represent threefold whole-number products as volumes,
e.g., to represent the associative property of multiplication.

Apply the formulas V = l Ð w Ð h and V = b Ð h for rectangular
prisms to find volumes of right rectangular prisms with whole-
number edge lengths in the context of solving real world and
mathematical problems.

Recognize volume as additive. Find volumes of solid figures
composed of two non-overlapping right rectangular prisms by
adding the volumes of the non-overlapping parts, applying this
GRADE 5 |

technique to solve real world problems.
37


Geometry 5.G

Graph points on the coordinate plane to solve real-world and
mathematical problems.
1. Use a pair of perpendicular number lines, called axes, to define a
coordinate system, with the intersection of the lines (the origin)
arranged to coincide with the 0 on each line and a given point in
the plane located by using an ordered pair of numbers, called its
coordinates. Understand that the first number indicates how far to
travel from the origin in the direction of one axis, and the second
number indicates how far to travel in the direction of the second
axis, with the convention that the names of the two axes and the
coordinates correspond (e.g., x-axis and x-coordinate, y-axis and
y-coordinate).
2. Represent real world and mathematical problems by graphing points
in the first quadrant of the coordinate plane, and interpret coordinate
values of points in the context of the situation.

Classify two-dimensional figures into categories based on their
properties.
3. Understand that attributes belonging to a category of two-
dimensional figures also belong to all subcategories of that category.
For example, all rectangles have four right angles and squares are
rectangles, so all squares have four right angles.
4. Classify two-dimensional figures in a hierarchy based on properties.

GRADE 5 |
38

Ccs 5th math

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