Course Description
Intended Learning Outcomes for Third Through Sixth Grade Mathematics
The main intent of mathematics instruction is for students to value and use mathematics and
reasoning skills to investigate and understand the world.
The Intended Learning Outcomes (ILOs) describe the skills and attitudes students should acquire as
a result of successful mathematics instruction. They are an essential part of the Mathematics Core
Curriculum and provide teachers with a standard for student learning in mathematics.
ILOs for mathematics:
 Develop a positive learning attitude toward mathematics.
 Become effective problem solvers by selecting appropriate methods, employing a
variety of strategies, and exploring alternative approaches to solve problems.
 Reason logically, using inductive and deductive strategies and justify conclusions.
 Communicate mathematical ideas and arguments coherently to peers, teachers, and
others using the precise language and notation of mathematics.
 Connect mathematical ideas within mathematics, to other disciplines, and to everyday
experiences.
 Represent mathematical ideas in a variety of ways.
Significant mathematics understanding occurs when teachers incorporate ILOs in planning
mathematics instruction. The following are ideas to consider when planning instruction for students
to acquire the ILOs:
 Develop a positive learning attitude toward mathematics.
When students are confident in their mathematical abilities, they demonstrate persistence
in completing tasks. They pose mathematical questions about objects, events, and processes
while displaying a sense of curiosity about numbers and patterns. It is important to build on
students' innate problemsolving inclinations and to preserve and encourage a disposition
that values mathematics.
 Become effective problem solvers by selecting appropriate methods, employing a
variety of strategies, and exploring alternative approaches to solve problems.
Problem solving is the cornerstone of mathematics. Mathematical knowledge is
generated through problem solving as students explore mathematics. To become effective
problem solvers, students need many opportunities to formulate questions and model
problem situations in a variety of ways. They should generalize mathematical relationships
and solve problems in both mathematical and everyday contexts.
 Reason logically, using inductive and deductive strategies and justify conclusions.
Mathematical reasoning develops in classrooms where students are encouraged to put
forth their own ideas for examination. Students develop their reasoning skills by making
and testing mathematical conjectures, drawing logical conclusions, and justifying their
thinking in developmentally appropriate ways. Students use models, known facts, and
relationships to explain reasoning. As they advance through the grades, students' arguments
become more sophisticated.
 Communicate mathematical ideas and arguments coherently to peers, teachers, and
others using the precise language and notation of mathematics.
The ability to express mathematical ideas coherently to peers, teachers, and others
through oral and written language is an important skill in mathematics. Students develop
this skill and deepen their understanding of mathematics when they use accurate
mathematical language to talk and write about what they are doing. When students talk and
write about mathematics, they clarify their ideas and learn how to make convincing
arguments and represent mathematical ideas verbally, pictorially, and symbolically.
 Connect mathematical ideas within mathematics, to other disciplines, and to everyday
experiences.
Students develop a perspective of the mathematics field as an integrated whole by
understanding connections within mathematics. Students should be encouraged to explore
the connections that exist with other disciplines and between mathematics and their own
experiences.
 Represent mathematical ideas in a variety of ways.
Mathematics involves using various types of representations, including concrete,
pictorial, and symbolic models. Students use a variety of mathematical representations to
expand their capacity to think logically about mathematics.
By the end of grade five, students increase their facility with the four basic arithmetic operations
applied to whole numbers, fractions, and decimals. They locate integers on a number line and
ordered pairs of integers on the coordinate plane. They determine rules for numerical patterns,
work with expressions including order of operations, and solve singleoperation equations
involving a single variable. They classify angles, triangles, and quadrilaterals, and analyze
relationships among lines, triangles and quadrilaterals. They recognize and determine surface
area and volume of threedimensional shapes, including right prisms. Students understand the
concepts of mean, median, mode, and range of data sets and can calculate them. They use line
plots, bar graphs, and line graphs to record and analyze data.
Math 5 Reference Sheet (pdf)
Core Standards of the Course
Standard 1
Students will expand number sense to include integers and perform operations with whole numbers, simple fractions, and decimals.
Objective 1
Represent whole numbers and decimals from thousandths to one billion, fractions, percents, and integers.

Read and write numbers in standard and expanded form.

Demonstrate multiple ways to represent whole numbers, decimals, fractions, percents, and integers using models and symbolic representations (e.g., 108 = 2 x 50 + 8; 108 = 10^{2} + 8; 90% = 90 out of 100 squares on a hundred chart).

Identify, read, and locate fractions, mixed numbers, decimals, and integers on the number line.

Represent repeated factors using exponents.

Describe situations where integers could be used in the students' environment.
Objective 2
Explain relationships and equivalencies among integers, fractions, decimals, and percents.

Compare fractions by finding a common denominator.

Order integers, fractions (including mixed numbers), and decimals using a variety of methods, including the number line.

Rewrite mixed numbers and improper fractions from one form to the other and represent each using regions, sets of objects, or line segments.

Represent commonly used fractions as decimals and percents in a variety of ways (e.g., models, fraction strips, pictures, calculators, algorithms).

Model and calculate equivalent forms of a fraction (including simplest form).

Rename whole numbers as fractions with different denominators (e.g., 5 = 5/1, 3 = 6/2, 1 = 7/7).
Objective 3
Use number theory concepts to develop and use divisibility tests; classify whole numbers to 50 as prime, composite, or neither; and find common multiples and factors.

Identify patterns with skip counting and multiples to develop and use divisibility tests for determining whether a whole number is divisible by 2, 3, 5, 6, 9, and 10.

Use strategies for classifying whole numbers to 50 as prime, composite, or neither.

Rewrite a composite number between 2 and 50 as a product of only prime numbers.

Find common multiples and factors and apply to adding and subtracting fractions.
Objective 4
Model and illustrate meanings of multiplication and division.

Represent divisionwithremainder using whole numbers, decimals, or fractions.

Describe the effect of place value when multiplying and dividing whole numbers and decimals by 10, 100, and 1,000.

Model multiplication of fractions and decimals (e.g., tenths multiplied by tenths, a whole number multiplied by tenths, or a whole number with tenths multiplied by tenths) in a variety of ways (e.g., manipulatives, number line and area models, patterns).
Objective 5
Solve problems involving one or two operations.

Determine when it is appropriate to use estimation, mental math strategies, paper and pencil, and algorithms.

Make reasonable estimations of fraction and decimal sums, differences, and products, including knowing whether results obtained using a calculator are reasonable.

Write number sentences that can be used to solve a twostep problem.

Interpret divisionwithremainder problems as they apply to the environment (e.g., If there are 53 people, how many vans are needed if each van holds 8 people?).
Objective 6
Demonstrate proficiency with multiplication and division of whole numbers and compute problems involving addition, subtraction, and multiplication of decimals and fractions.

Multiply multidigit whole numbers by a twodigit whole number with fluency, using efficient procedures.

Divide multidigit dividends by a onedigit divisor with fluency, using efficient procedures.

Add and subtract decimals with fluency, using efficient procedures.

Add and subtract fractions with fluency.

Multiply fractions.
Standard 2
Students will use patterns and relations to represent and analyze mathematical problems and number relationships using algebraic symbols.
Objective 1
Identify, analyze and determine a rule for predicting and extending numerical patterns involving operations whole numbers, decimals, and fractions.

Analyze and make predictions about numeric patterns, including decimals and fractions.

Determine a rule for the pattern using organized lists, tables, objects, and variables.
Objective 2
Use algebraic expressions, inequalities, or equations to represent and solve simple realworld problems.

Use properties and the order of operations involving addition, subtraction, multiplication, division, and the use of parentheses to compute with whole numbers, decimals, and fractions.

Use patterns, models, and relationships as contexts for writing and solving simple equations and inequalities with whole number solutions (e.g., 6x = 54; x + 3 = 7).
Standard 3
Students will use spatial reasoning to recognize, describe, and analyze geometric shapes and principles.
Objective 1
Describe relationships between two and threedimensional shapes and analyze attributes and properties of geometric shapes.

Draw, label, and describe line segments, rays, lines, parallel lines, and perpendicular lines.

Draw, label, and define an angle as two rays sharing a common endpoint (vertex).

Classify triangles and quadrilaterals and analyze the relationships among the shapes in each classification (e.g., a square is a rectangle).

Relate pyramids and right prisms to the twodimensional shapes (nets) from which they were created.

Identify properties and attributes of solids (i.e., right prisms, pyramids, cylinders, cones) and describe them by the number of edges, faces, and vertices as well as the types of faces.
Objective 2
Specify locations in a coordinate plane.

Locate points defined by ordered pairs of integers.

Write an ordered pair for a point in a coordinate plane with integer coordinates.

Specify possible paths between locations on a coordinate plane and compare distances of the various paths.
Standard 4
Students will determine area of polygons and surface area and volume of threedimensional shapes.
Objective 1
Determine the area of polygons and apply to realworld problems.

Determine the area of a trapezoid by the composition and decomposition of rectangles, triangles, and parallelograms.

Determine the area of irregular and regular polygons by the composition and decomposition of rectangles, triangles, and parallelograms.

Compare areas of polygons using different units of measure within the same measurement system (e.g., square feet, square yards).
Objective 2
Recognize, describe, and determine surface area and volume of threedimensional shapes.

Quantify volume by finding the total number of samesized units of volume needed to fill the space without gaps or overlaps.

Recognize that a cube having a 1 unit edge is the standard unit for measuring volume expressed as a cubic unit.

Derive and use the formula to determine the volume of a right prism with a triangular or rectangular base.

Relate the formulas for the areas of triangles, rectangles, or parallelograms to the surface area of a right prism.

Derive and use the formula to determine the surface area of a right prism and express surface area in square units.
Standard 5
Students will construct, analyze, and construct reasonable conclusions from data and apply basic concepts of probability.
Objective 1
Formulate and answer questions using statistical methods to compare data, and propose and justify inferences based on data.

Construct, analyze, and display data using an appropriate format (e.g., line plots, bar graphs, line graphs).

Recognize the differences in representing categorical and numerical data.

Identify minimum and maximum values for a set of data.

Identify and calculate the mean, median, mode, and range.
Objective 2
Apply basic concepts of probability.

Describe the results of experiments involving random outcomes using a variety of notations (e.g., 4 out of 9, 4/9).

Recognize that probability is always a value between 0 and 1 (inclusively).

Express the likelihood of an outcome in a simple experiment as a value between 0 and 1 (inclusively).