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The California Mathematics Content Standards

Grade Five Mathematics Content Standards

By the end of grade five, students increase their facility with the four basic arithmetic
operations applied to fractions and decimals and learn to add and subtract positive and
negative numbers. They know and use common measuring units to determine length and
area and know and use formulas to determine the volume of simple geometric figures.
Students know the concept of angle measurement and use a protractor and compass to
solve problems . They use grids, tables, graphs, and charts to record and analyze data.

Note: The sample
problems illustrate
the standards and
are written to help
clarify them. Some
problems are written
in a form that can be
used directly with
students; others will
need to be modified,
particularly in the
primary grades,
before they are
used with students.

The symbol
identifies the key
standards for
grade five.



































Number Sense

1.0 Students compute with very large and very small numbers, positive integers,
decimals, and fractions and understand the relationship between decimals,
fractions, and percents. They understand the relative magnitudes of numbers:

1.1 Estimate, round, and manipulate very large (e.g., millions) and very small
(e.g., thousandths) numbers.

Interpret percents as a part of a hundred; find decimal and percent equivalents
for common fractions and explain why they represent the same value;
compute a given percent of a whole number.

A test had 48 problems. Joe got 42 correct.

1. What percent were correct?

2. What percent were wrong?

3. If Moe got 93.75% correct, how many problems did he get correct?

1.3 Understand and compute positive integer powers of nonnegative integers;
compute examples as repeated multiplication.

Extend the tables shown below:

Determine the prime factors of all numbers through 50 and write the
numbers as the product of their prime factors by using exponents to show
multiples of a factor (e.g., 24 = 2 × 2 × 2 × 3 = 23 × 3).

Find the prime factorization of 48 and use exponents where appropriate.

Identify and represent on a number line decimals, fractions, mixed numbers,
and positive and negative integers.

2.0 Students perform calculations and solve problems involving addition, subtraction,
and simple multiplication and division of fractions and decimals:

Add, subtract, multiply, and divide with decimals; add with negative
integers; subtract positive integers from negative integers; and verify the
reasonableness of the results.

Determine the following numbers:

1. 11 + (-23)

2. (-15) - 128

3. (-27) + (-45)

Demonstrate proficiency with division, including division with positive
decimals and long division with multidigit divisors.

Find the quotient:

6 divided by .025

Solve simple problems, including ones arising in concrete situations, involving
the addition and subtraction of fractions and mixed numbers ( like and
denominators of 20 or less ), and express answers in the simplest form.

Given the following three pairs of fractions (3/8 and 1/6, and , 16 and
), find for each pair its:

1. Sum

2. Difference

2.4 Understand the concept of multiplication and division of fractions.

2.5 Compute and perform simple multiplication and division of fractions and
apply these procedures to solving problems.

Given the following three pairs of fractions (3/8 and 1/6, and , 16 and
), find for each pair its:

1. Product

2. Quotient in simplest terms

Ericka has yards of cloth to make shirts. Each shirt requires 7/8 yard.
How many shirts can she make? How much cloth will she have left over?

Algebra and Functions

1.0 Students use variables in simple expressions, compute the value of the expression
for specific values of the variable, and plot and interpret the results:

1.1 Use information taken from a graph or equation to answer questions about a
problem situation.

Use a letter to represent an unknown number ; write and evaluate simple
algebraic expressions in one variable by substitution.

3x + 2 = 14. What is x?

1.3 Know and use the distributive property in equations and expressions with

Identify and graph ordered pairs in the four quadrants of the coordinate

Plot the points (1, 2), (-4, -3), (12, -1), (0, 4), (-4, 0).

Solve problems involving linear functions with integer values; write the
; and graph the resulting ordered pairs of integers on a grid.

Measurement and Geometry

1.0 Students understand and compute the volumes and areas of simple objects:

Derive and use the formula for the area of a triangle and of a parallelogram
by comparing it with the formula for the area of a rectangle (i.e., two of the
same triangles make a parallelogram with twice the area; a parallelogram is
compared with a rectangle of the same area by pasting and cutting a right
triangle on the parallelogram).

Find the area and perimeter.

Construct a cube and rectangular box from two-dimensional patterns and
use these patterns to compute the surface area for these objects.

Understand the concept of volume and use the appropriate units in common
measuring systems (i.e., cubic centimeter [cm 3], cubic meter [m3], cubic
inch [in. 3], cubic yard [yd.3]) to compute the volume of rectangular solids.

1.4 Differentiate between , and use appropriate units of measures for, two- and
three-dimensional objects (i.e., find the perimeter, area, volume).

2.0 Students identify, describe, and classify the properties of, and the relationships
between, plane and solid geometric figures:

Measure, identify, and draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate tools (e.g., straightedge,
ruler, compass, protractor, drawing software).

Know that the sum of the angles of any triangle is 180° and the sum
of the angles of any quadrilateral is 360° and use this information to solve

2.3 Visualize and draw two-dimensional views of three-dimensional objects
made from rectangular solids.

Statistics, Data Analysis, and Probability

1.0 Students display, analyze, compare, and interpret different data sets, including
data sets of different sizes:

1.1 Know the concepts of mean, median, and mode; compute and compare
simple examples to show that they may differ.

1.2 Organize and display single-variable data in appropriate graphs and representations
(e.g., histogram, circle graphs) and explain which types of graphs are
appropriate for various data sets.

1.3 Use fractions and percentages to compare data sets of different sizes.

Identify ordered pairs of data from a graph and interpret the meaning of the
data in terms of the situation depicted by the graph.

Know how to write ordered pairs correctly; for example, (x, y).

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant
from irrelevant information, sequencing and prioritizing information,
and observing patterns.

1.2 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex

2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs,
tables, diagrams, and models, to explain mathematical reasoning.

2.4 Express the solution clearly and logically by using the appropriate mathematical
notation and terms and clear language; support solutions with
evidence in both verbal and symbolic work .

2.5 Indicate the relative advantages of exact and approximate solutions to
problems and give answers to a specified degree of accuracy.

2.6 Make precise calculations and check the validity of the results from the
context of the problem.

3.0 Students move beyond a particular problem by generalizing to other

3.1 Evaluate the reasonableness of the solution in the context of the original

3.2 Note the method of deriving the solution and demonstrate a conceptual
understanding of the derivation by solving similar problems.

3.3 Develop generalizations of the results obtained and apply them in other
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