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 three, students develop understandings of multiplication and division of
whole numbers. They use properties to develop increasingly more sophisticated strategies to
solve problems involving basic multiplication and division facts. They relate division to multiplication. Students understand fraction equivalence for simple fractions;
they recognize that the size of a fractional part is relative to the size of the whole. They
understand meanings of fractions to represent parts of a whole, parts of a set, or distances on a
number line. They compare and order simple fractions by using models, benchmark fractions, or
common denominators.
Students investigate, analyze, and classify twodimensional shapes by their sides and angles.
They decompose, combine, and transform polygons to understand properties of twodimensional
space and use those properties to solve problems. Students construct and analyze frequency
tables, bar graphs, picture graphs, and line plots and use them to solve problems.
Core Standards of the Course
Standard 1
Students will understand the baseten numeration system, place value concepts, simple fractions and perform operations with whole numbers.
Objective 1
Represent whole numbers up to 10,000, comprehend place value concepts, and identify relationships among whole numbers using baseten models and symbolic notation.

Read, write, and represent whole numbers using standard and expanded form.

Demonstrate multiple ways to represent numbers using models and symbolic representations (e.g., fifty is the same as two groups of 25, the number of pennies in five dimes, or 75  25).

Identify the place and the value of a given digit in a fourdigit numeral and round numbers to the nearest ten, hundred, and thousand.

Order and compare whole numbers on a number line and use the inequality symbols <, >, ≠, and = when comparing whole numbers.

Identify factors and multiples of whole numbers.
Objective 2
Use fractions to communicate and compare parts of the whole.

Identify the denominator of a fraction as the number of equal parts of the unit whole and the numerator of a fraction as the number of equal parts being considered.

Define regions and sets of objects as a whole and divide the whole into equal parts using a variety of objects, models, and illustrations.

Name and write a fraction to represent a portion of a unit whole for halves, thirds, fourths, sixths, and eighths.

Place fractions on the number line and compare and order fractions using models, pictures, the number line, and symbols.

Find equivalent fractions using concrete and pictorial representations.
Objective 3
Model problems involving addition, subtraction, multiplication, and division.

Demonstrate the meaning of multiplication and division of whole numbers through the use of a variety of representations (e.g., equalsized groups, arrays, area models, and equal jumps on a number line for multiplication, partitioning and sharing for division).

Use a variety of strategies and tools, such as repeated addition or subtraction, equal jumps on the number line, and counters arranged in arrays to model multiplication and division problems.

Demonstrate, using objects, that multiplication and division by the same numbers are inverse operations (e.g., 3 x = 12 is the same as 12 ÷ 3 = and = 4).

Demonstrate the effect of place value when multiplying whole numbers by 10.

Write a story problem that relates to a given addition, subtraction, or multiplication equation, and write a number sentence to solve a problem related to the students' environment.
Objective 4
Compute and solve problems involving addition and subtraction of 3 and 4 digit numbers and basic facts of multiplication and division.

Use a variety of methods to facilitate computation (e.g., estimation, mental math strategies, paper and pencil).

Find the sum or difference of numbers, including monetary amounts, using models and strategies such as expanded form, compensation, partial sums, and the standard algorithm.

Compute basic multiplication facts (010) and related division facts using a variety of strategies based on properties of addition and multiplication (i.e., commutative, associative, identity, zero, and the distributive properties).
Standard 2
Students will use patterns, symbols, operations, and properties of addition and multiplication to represent and describe simple number relationships.
Objective 1
Create, represent, and analyze growing patterns.

Create and extend growing patterns using objects, numbers, and tables.

Describe how patterns are extended using manipulatives, pictures, and numerical representations.
Objective 2
Recognize, represent, and simplify simple number relationships using symbols, operations, and properties.

Represent numerical relationships as expressions, equations, and inequalities.

b. Solve equations involving equivalent expressions (e.g., 6 + 4 = + 7).

Use the >, <, and = symbols to compare two expressions involving addition and subtraction (e.g., 4 + 6 3 + 2; 3 + 5 16  9).

Recognize and use the commutative, associative, distributive, and identity properties of addition and multiplication, and the zero property of multiplication.
Standard 3
Students will describe and analyze attributes of twodimensional shapes.
Objective 1
Describe and compare attributes of twodimensional shapes.

Identify, describe, and classify polygons (e.g., pentagons, hexagons, octagons).

Identify attributes for classifying triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle).

Identify attributes for classifying quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square).

Identify right angles in geometric figures, or in appropriate objects, and determine whether other angles are greater or less than a right angle.
Objective 2
Demonstrate the meaning of congruence through applying transformations.

Demonstrate the effect of reflection, translation, or rotation using objects.

Determine whether two polygons are congruent by reflecting, translating, or rotating one polygon to physically fit on top of the other.
Standard 4
Students will select and use appropriate units and measurement tools to solve problems.
Objective 1
Select and use appropriate tools and units to estimate and measure length, weight, capacity, time, and perimeter of twodimensional figures.

Describe the partwhole relationships (e.g., 3 feet in a yard, a foot is 1/3 of a yard) between metric units of length (i.e., centimeter, meter), and among customary units of length (i.e., inch, foot, yard), capacity (i.e., cup, quart), and weight (i.e., pound, ounce).

Measure the length of objects to the nearest centimeter, meter, half and quarterinch, foot, and yard.

Measure capacity using cups and quarts, and measure weight using pounds and ounces.

Identify the number of minutes in an hour, the number of hours in a day, the number of days in a year, and the number of weeks in a year.

Describe perimeter as a measurable attribute of twodimensional figures, and estimate and measure perimeter with metric and customary units.
Objective 2
Solve problems involving measurements.

Determine simple equivalences of measurements (e.g., 30 inches = 2 feet and 6 inches; 6 cups = 1 1/2 quarts; 90 min. = 1 hr. 30 min.).

Compare given objects according to measurable attributes (i.e., length, weight, capacity).

Solve problems involving perimeter.

Determine elapsed time in hours (e.g., 7:00 a.m. to 2:00 p.m.).
Standard 5
Students will collect and organize data to make predictions and identify basic concepts of probability.
Objective 1
Collect, organize, and display data to make predictions.

Collect, read, represent, and interpret data using tables, graphs, and charts, including keys (e.g., pictographs, bar graphs, frequency tables, line plots).

Make predictions based on a data display.
Objective 2
Identify basic concepts of probability.

Describe the results of events using the terms "certain," "likely," "unlikely," and "impossible."

Conduct simple probability experiments, record possible outcomes systematically, and display results in an organized way (e.g., chart, graph).

Use results of simple probability experiments to describe the likelihood of a specific outcome in the future.