Mathematics - 6th Grade Core

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	Mathematics - 6th Grade
	Course Preface Printable Version (pdf)

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 problem-solving 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 six, students have mastered the four arithmetic operations with whole numbers, positive rational numbers, positive decimals, and positive and negative integers; they accurately compute and solve problems. They find prime factorizations, least common multiples, and greatest common factors. They create, evaluate, and simplify expressions, and solve equations involving two operations and a single variable. They solve problems involving an unknown angle in a triangle or quadrilateral, and use properties of complementary and supplementary angles. Students know about π as the ratio between the circumference and the diameter of a circle and solve problems using the formulas for the circumference and area of a circle. Students analyze, draw conclusions, and make predictions based upon data and apply basic concepts of probability.

$Math 6 Reference Sheet$ Math 6 Reference Sheet (pdf)

Core Standards of the Course

Standard 1
Students will expand number sense to include operations with rational numbers.

Objective 1
Represent rational numbers in a variety of ways.

Recognize a rational number as a ratio of two integers, a to b, where b is not equal to zero.

Change whole numbers with exponents to standard form (e.g., 2⁴ = 16) and recognize that any non-zero whole number to the zero power equals 1 (e.g., 9⁰ = 1).

Write a whole number in expanded form using exponents (e.g., 876,539 = 8 x 10⁵ + 7 x 10⁴ + 6 x 10³ + 5 x 10² + 3 x 10¹ + 9 x 10⁰).

Express numbers in scientific notation using positive powers of ten.

Objective 2
Explain relationships and equivalencies among rational numbers.

Place rational numbers on the number line.

Compare and order rational numbers, including positive and negative mixed fractions and decimals, using a variety of methods and symbols, including the number line and finding common denominators.

Find equivalent forms for common fractions, decimals, percents, and ratios, including repeating or terminating decimals.

Relate percents less than 1% or greater than 100% to equivalent fractions, decimals, whole numbers, and mixed numbers.

Recognize that the sum of an integer and its additive inverse is zero.

Objective 3
Use number theory concepts to find prime factorizations, least common multiples, and greatest common factors.

Determine whether whole numbers to 100 are prime, composite, or neither.

Find the prime factorization of composite numbers to 100.

Find the greatest common factor and least common multiple for two numbers using a variety of methods (e.g., list of multiples, prime factorization).

Objective 4
Model and illustrate meanings of operations and describe how they relate.

Relate fractions to multiplication and division and use this relationship to explain procedures for multiplying and dividing fractions.

Recognize that ratios derive from pairs of rows in the multiplication table and connect with equivalent fractions.

Give mixed number and decimal solutions to division problems with whole numbers.

Objective 5
Solve problems involving multiple steps.

Select appropriate methods to solve a multi-step problem involving multiplication and division of fractions and decimals.

Use estimation to determine whether results obtained using a calculator are reasonable.

Use estimation or calculation to compute results, depending on the context and numbers involved in the problem.

Solve problems involving ratios and proportions.

Objective 6
Demonstrate proficiency with the four operations, with positive rational numbers, and with addition and subtraction of integers.

Multiply and divide a multi-digit number by a two-digit number, including decimals.

Add, subtract, multiply, and divide fractions and mixed numbers.

Add and subtract integers.

Mathematical Language and Symbols Students Should Use: prime, composite, exponent, least common multiple, least common denominator, greatest common factor, decimals, percents, divisible, divisibility, equivalent fractions, integer, dividend, quotient, divisor, factor, simplest terms, mixed numeral, improper fraction

Exploratory Concepts and Skills:

Explore the addition and subtraction of positive and negative fractions.
Investigate the concepts of ratio and proportion.
Investigate the distributive property of multiplication over addition of double-digit multipliers.

Standard 2
Students will use patterns, relations, and algebraic expressions to represent and analyze mathematical problems and number relationships.

Objective 1
Analyze algebraic expressions, tables, and graphs to determine patterns, relations, and rules.

Describe simple relationships by creating and analyzing tables, equations, and expressions.

Draw a graph and write an equation from a table of values.

Draw a graph and create a table of values from an equation.

Objective 2
Write, interpret, and use mathematical expressions, equations, and formulas to represent and solve problems that correspond to given situations.

Solve single variable linear equations using a variety of strategies.

Recognize that expressions in different forms can be equivalent and rewrite an expression to represent a quantity in a different way.

Evaluate and simplify expressions and formulas, substituting given values for the variables (e.g., 2x + 4; x = 2; therefore, 2 (2) + 4 = 8).

Mathematical Language and Symbols Students Should Use: order of operations, sequence, function, pattern, algebraic expression, approximately equal, notation for exponents: 4³ or 4^3, a number in front of a variable indicates multiplication (e.g., 3y means 3 times the quantity y), formula, generalization

Exploratory Concepts and Skills:

Use physical models to investigate and describe how a change in one variable affects a second variable.
Use models to develop understanding of slope as constant rate of change.
Model situations with proportional relationships and solve problems.

Standard 3
Students will use spatial and logical reasoning to recognize, describe, and analyze geometric shapes and principles.

Objective 1
Identify and analyze attributes and properties of geometric shapes to solve problems.

Identify the midpoint of a line segment and the center and circumference of a circle.

Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.

Develop and use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle in a triangle or quadrilateral.

Objective 2
Visualize and identify geometric shapes after applying transformations on a coordinate plane.

Rotate a polygon about the origin by a multiple of 90° and identify the location of the new vertices.

Translate a polygon either horizontally or vertically on a coordinate grid and identify the location of the new vertices.

Reflect a polygon across either the x- or y-axis and identify the location of the new vertices.

Mathematical Language and Symbols Students Should Use: midpoint, circumference, complementary and supplementary angles, rotate, translate, reflect, transformation

Exploratory Concepts and Skills:

Use manipulatives and technology to model geometric shapes.
Investigate tessellations.
Explore the angles formed by intersecting lines.
Identify and draw shapes and figures from different views/perspectives.

Standard 4
Students will understand and apply measurement tools and techniques and find the circumference and area of a circle.

Objective 1
Describe and find the circumference and area of a circle.

Explore the relationship between the radius and diameter of a circle to the circle's circumference to develop the formula for circumference.

Find the circumference of a circle using a formula.

Describe pi as the ratio of the circumference to the diameter of a circle.

Decompose a circle into a number of wedges and rearrange the wedges into a shape that approximates a parallelogram to develop the formula for the area of a circle.

Find the area of a circle using a formula.

Objective 2
Identify and describe measurable attributes of objects and units of measurement, and solve problems involving measurement.

Recognize that measurements are approximations and describe how the size of the unit used in measuring affects the precision.

Convert units of measurement within the metric system and convert units of measurement within the customary system.

Compare a meter to a yard, a liter to a quart, and a kilometer to a mile.

Determine when it is appropriate to estimate or use precise measurement when solving problems.

Derive and use the formula to determine the surface area and volume of a cylinder.

Standard 5
Students will analyze, draw conclusions, and make predictions based upon data and apply basic concepts of probability.

Objective 1
Design investigations to reach conclusions using statistical methods to make inferences based on data.

Design investigations to answer questions.

Extend data display and comparisons to include scatter plots and circle graphs.

Compare two similar sets of data on the same graph and compare two graphs representing the same set of data.

Recognize that changing the scale influences the appearance of a display of data.

Propose and justify inferences and predictions based on data.

Objective 2
Apply basic concepts of probability and justify outcomes.

Write the results of a probability experiment as a fraction between zero and one, or an equivalent percent.

Compare experimental results with theoretical results (e.g., experimental: 7 out of 10 tails; whereas, theoretical 5 out of 10 tails).

Compare individual, small group, and large group results of a probability experiment in order to more accurately estimate the actual probabilities.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Office of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Office of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.

For more information about this core curriculum, contact the USOE Specialist, DAVID SMITH or visit the Mathematics - Elementary Home Page. For general questions about Utah's Core Curriculum, contact the USOE Curriculum Director, Sydnee Dickson . UEN Contact Info: 801-581-2999 | 800-866-5852 | Contact Us