# Grade 5 Math Content

** Number and Operations :
Rational Numbers **

The major focus of the work on rational numbers in grade 5 is on understanding

relationships among fractions, decimals, and percents. Students make comparisons
and

identify equivalent fractions, decimals, and percents, and they develop
strategies for

adding and subtracting fractions and decimals.

In a study of fractions and percents, students work with halves, thirds,
fourths, fifths,

sixths, eighths, tenths, and twelfths. They develop strategies for finding
percent

equivalents for these fractions so that they are able to move back and forth
easily between

fractions and percents and choose what is most helpful in solving a particular
problem,

such as finding percentages or fractions of a group.

Students use their knowledge of fraction equivalents, fraction-percent
equivalents, the

relationship of fractions to landmarks such as ½, 1, and 2, and other
relationships to

decide which of two fractions is greater. They carry out addition and
subtraction of

fractional amounts in ways that make sense to them by using representations such
as

rectangles, rotation on a clock, and the number line to visualize and reason
about fraction

equivalents and relationships.

Students continue to develop their understanding of how decimal fractions
represent

quantities less than 1 and extend their work with decimals to thousandths. By

representing tenths, hundredths, and thousandths on rectangular grids, students
learn

about the relationships among these numbers—for example, that one tenth is
equivalent

to ten hundredths and one hundredth is equivalent to ten thousandths—and how
these

numbers extend the place value structure of tens that they understand from their
work

with whole numbers.

Students extend their knowledge of fraction-decimal equivalents by studying how

fractions represent division and carrying out that division to find an
equivalent decimal.

They compare, order, and add decimal fractions (tenths, hundredths, and
thousandths) by

carefully identifying the place value of the digits in each number and using

representations to visualize the quantities represented by these numbers.

**Emphases**

Rational Numbers

• Understanding the meaning of fractions and percents

• Comparing fractions

• Understanding the meaning of decimal fractions

• Comparing decimal fractions

Computation with Rational Numbers

• Adding and subtracting fractions

• Adding decimals

**Benchmarks**

• Use fraction-percent equivalents to solve problems about the percentage of a

quantity

• Order fractions with like and unlike denominators

• Add fractions through reasoning about fraction equivalents and relationships

• Read, write, and interpret decimal fractions to thousandths

• Order decimals to the thousandths

• Add decimal fractions through reasoning about place value, equivalents, and

representations

**Patterns, Functions, and Change**

In Grade 5, students continue their work from Grades 3 and 4 by examining,

representing, and describing situations in which the rate of change is constant .
Students

create tables and graphs to represent the relationship between two variables in
a variety of

contexts. They also articulate general rules for each situation. For example,
consider the

perimeters of the following set of rectangles made from rows of tiles with three
tiles in

each row:

If the value of one variable (the number of rows of three tiles) is known, the

corresponding value of the other variable (the perimeter of the rectangle) can
be

calculated . Students express these rules in words and then in symbolic notation.
For

example:

For the first time in Grade 5, students create graphs for situations in which
the rate of

change is itself changing–for example, the change in the area of a square as a
side

increases by a constant increment–and consider why the shape of the graph is not
a

straight line as it is for situations with a constant rate of change.

Throughout their work, students move among tables, graphs, and equations and
between

those representations and the situation they represent. Their work with symbolic
notation

is closely related to the context in which they are working. By moving back and
forth

between the contexts, their own ways of describing general rules in words, and
symbolic

notation, students learn how this notation can carry mathematical meaning .

**Emphases**

Using Tables and Graphs

• Using graphs to represent change

• Using tables to represent change

Linear Change

• Describing and representing a constant rate of change

Number Sequences

• Describing and representing situations in which the rate of change is not
constant

**Benchmarks**

• Connect tables and graphs to represent the relationship between two variables

• Use tables and graphs to compare two situations with constant rates of change

• Use symbolic notation to represent the value of one variable in terms of
another

variable in situations with constant rates of change

**Data Analysis and Probability**

Students continue to develop their understanding of data analysis in Grade 5 by

collecting, representing, describing, and interpreting numerical data . Students’
work in

this unit focuses on comparing two sets of data collected from experiments.
Students

develop a question to compare two groups, objects, or conditions. (Sample
questions:

Which toy car goes farthest after rolling down the ramp? Which paper bridge
holds more

weight?). They consider how to ensure a consistent procedure for their
experiment and

discuss the importance of multiple trials . Using representations of data,
including line

plots and bar graphs, students describe the shape of the data—where the data are

concentrated, how they are spread across the range. They summarize the data for
each

group or object or condition and use these summaries, including medians, to back
up their

conclusions and arguments. By carrying out a complete data investigation, from

formulating a question through drawing conclusions from their data, students
gain an

understanding of data analysis as a tool for learning about the world.

In their work with probability, students describe and predict the likelihood of
events and

compare theoretical probabilities with actual outcomes of many trials. They use
fractions

to express the probabilities of the possible outcomes (e.g., landing on the
green part of the

spinner, landing on the white part of the spinner). Then they conduct
experiments to see

what actually occurs. The experiments lead to questions about theoretical and

experimental probability, for example, if half the area of a spinner is colored
green and

half is colored white, why doesn’t the spinner land on green exactly half the
time?

**Emphases
**Data Analysis

• Representing data

• Describing, summarizing, and comparing data

• Analyzing and interpreting data

• Designing and carrying out a data investigation

Probability

• Describing the probability of an event

• Describe major features of a set of data represented in a line plot or bar
graph, and

quantify the description by using medians or fractional parts of the data

**Benchmarks**

• Draw conclusions about how 2 groups compare based on summarizing the data

for each group

• Design and carry out an experiment in order to compare two groups

• Use a decimal, fraction, or percent to describe and compare the theoretical

probabilities of events with a certain number of equally likely outcomes

**Geometry and Measurement**

In their work with geometry and measurement in grade 5, students further develop
their

understanding of the attributes of two-dimensional (2-D) shapes, find the
measure of

angles of polygons, determine the volume of three-dimensional (3-D) shapes, and
work

with area and perimeter. Students examine the characteristics of polygons,
including a

variety of triangles, quadrilaterals, and regular polygons. They consider
questions about

the classification of geometric figures, for example:

Are all squares rectangles?

Are all rectangles parallelograms?

If all squares are rhombuses, then are all rhombuses squares?

They investigate angle sizes in a set of polygons and measure angles of 30, 45,
60, 90,

120, and 150 degrees by comparing the angles of these shapes. Students also
investigate

perimeter and area. They consider how changes to the shape of a rectangle can
affect one

of the measures and not the other (e.g., two shapes that have the same area
don’t

necessarily have the same perimeter), and examine the relationship between area
and

perimeter in similar figures.

Students continue to develop their visualization skills and their understanding
of the

relationship between 2-D pictures and the 3-D objects they represent. Students
determine

the volume of boxes (rectangular prisms) made from 2-D patterns and create
patterns for

boxes to hold a certain number of cubes. They develop strategies for determining
the

number of cubes in 3-D arrays by mentally organizing the cubes—for example as a
stack

of three rectangular layers, each three by four cubes. Students deepen their
understanding

of the relationship between volume and the linear dimensions of length, width,
and

height. Once students have developed viable strategies for finding the volume of

rectangular prisms, they extend their understanding of volume to other solids
such as

pyramids, cylinders, and cones, measured in cubic units.

**Emphases**

Features of Shape

• Describing and classifying 2-D figures

• Describing and measuring angles

• Creating and describing similar shapes

• Translating between 2-D and 3-D shapes

Linear and Area Measurement

• Finding the perimeter and area of rectangles

Volume

• Structuring rectangular prisms and determining their volume

• Structuring prisms, pyramids, cylinders, and cones and determining their
volume

**Benchmarks**

• Identify different quadrilaterals by attribute, and know that some
quadrilaterals

can be classified in more than one way

• Use known angle sizes to determine the sizes of other angles (30 degrees, 45

degrees, 60 degrees, 90 degrees, 120 degrees, and 150 degrees)

• Determine the perimeter and area of rectangles

• Identify mathematically similar polygons

• Find the volume of rectangular prisms

• Use standard units to measure volume

• Identify how the dimensions of a box change when the volume is changed

• Explain the relationship between the volumes of prisms and pyramids with the

same base and height

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