Number Properties and Operations

PROGRAM OF STUDIES

Big Idea: Number Properties and Operations
Middle grades students understand fractions, decimals, percents and integers, compare them and locate
their relative positions on a number line. They develop and use proportional reasoning to solve problems.
They work with large numbers and small numbers. They use factors, multiples and prime factorizations.
They perform arithmetic operations with fractions, decimals and integers, use properties in computation,
develop fluency and develop strategies to estimate the result of operations on rational numbers.
Academic Expectations
2.7 Students understand number concepts and use numbers appropriately and accurately.
2.8 Students understand various mathematical procedures and use them appropriately and
accurately.
Grade 6 Enduring Knowledge – Understandings
Students will understand that
• numbers, ways of representing numbers, relationships among numbers and number systems are
means of representing real-world quantities.
• meanings of and relationships among operations provide tools necessary to solve realistic
problems encountered in everyday life.
• computing fluently and making reasonable estimates with fractions, decimals and whole numbers
increases the ability to solve realistic problems encountered in everyday life.
• proportional reasoning is a tool for modeling and solving problems encountered in everyday
situations.
Grade 6 Skills and Concepts – Number Sense
Students will
• continue to develop number sense using fractions, decimals and percents, including percents
greater than 100% and improper fractions
• extend applications of operations (+,-,×,÷ ) to include fractions and decimals
• develop place value of large and small numbers, including decimals
• explore positive integral exponents (e.g. squares, cubes)
• compare, order and convert between whole numbers, fractions, decimals and percents using
concrete materials, drawings or pictures and mathematical symbols (e.g., <,=,>,=,=,. , order on
a number line)
Grade 6 Skills and Concepts – Estimation
Students will
• estimate and mentally compute to solve real-world and/or mathematical problems with whole
numbers, fractions, decimals and percents, checking for reasonable and appropriate
computational results
• estimate large and small quantities of objects

Measurement

PROGRAM OF STUDIES

Big Idea: Measurement
Students continue to measure and estimate measurements including fractions and decimals. They use
formulas to find perimeter, area, circumference and volume. They use rulers and protractors. They use
U.S. Customary and metric units of measurement.
Academic Expectations
2.10 Students understand measurement concepts and use measurements appropriately and
accurately.
2.11 Students understand mathematical change concepts and use them appropriately and accurately.
Grade 6 Enduring Knowledge – Understandings
Students will understand that
• there are two major measurement systems (U.S. Customary and metric) and either may be used
to solve problems.
• measurable attributes of objects and the units, systems and processes of measurement are
powerful tools for making sense of the world around them.
• measurements are determined by using appropriate techniques, tools, formulas and degree of
accuracy needed for the situation.
Grade 6 Skills and Concepts – Measuring Physical Attributes
Students will
• find perimeter of regular and irregular polygons in metric and U.S. customary units
• read and use measurement tools (e.g., rulers, scales, protractors, angle rulers)
• find area of plane figures composed of triangles, squares and rectangles by subdividing and
measuring; use square units appropriately
• estimate and find angle measures and segment measures
• estimate measurements in standard units, including fractions and decimals
• explain how measurements and measurement formulas are related or different (e.g., compare the
perimeter with the area of a rectangle)
Grade 6 Skills and Concepts – Systems of Measurement
Students will
• describe and provide examples of U.S. Customary and metric units of measurement; use these
units to solve real-world and/or mathematical problems
• estimate, compare and convert (meaning to make ballpark comparisons/not memorize
conversion factors between U.S. and metric) units of measurement for length, weight/mass and
volume/capacity within the U.S. customary system and within the metric system:
o length (e.g., parts of an inch, inches, feet, yards, miles, millimeters, centimeters, meters,
kilometers);
o weight/mass (e.g., pounds, tons, grams, kilograms);
o volume/capacity (e.g., cups, pints, quarts, gallons, milliliters, liters)

Geometry

PROGRAM OF STUDIES

Big Idea: Geometry
Middle grade students expand analysis of two-dimensional shapes and three-dimensional shapes. They
translate shapes in a coordinate plane. They extend work with congruent and similar figures, including
proportionality.
Academic Expectation
2.8 Students understand various mathematical procedures and use them appropriately and
accurately.
2.9 Students understand space and dimensionality concepts and use them appropriately and
accurately.
Grade 6 Enduring Knowledge – Understandings
Students will understand that
• characteristics and properties of two-dimensional figures and three-dimensional objects describe
the world and are used to develop mathematical arguments about geometric relationships and to
evaluate the arguments of others.
• representational systems, including coordinate geometry, are means for specifying locations and
describing spatial relationships and are organizers for making sense of the world around them.
• transformations and symmetry are used to analyze real-world situations (e.g., art, nature,
construction and scientific exploration).
• shape and area are conserved during mathematical transformations (flips, slides and turns).
Scale conserves shape, but changes size.
• visualization, spatial reasoning and geometric relationships model real-world situations.
Grade 6 Skills and Concepts – Shapes and Relationships
Students will
• formulate and use the rules for the sum of angle measures in a triangle (180°) and in a
quadrilateral (360°)
• identify and use relationships among lines (e.g., parallel, perpendicular)
• identify, describe and provide examples of the basic geometric elements (points, rays, lines,
segments, angles [acute, right, obtuse], planes, radius, diameter, circumference)
• identify, describe and provide examples and properties of two-dimensional figures (circles,
triangles [acute, right, obtuse, scalene, isosceles, equilateral], quadrilaterals, regular polygons);
apply these properties and figures to solve real-world problems
• describe, provide examples of and identify properties (e.g., vertices, angles, faces, edges,
congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms and
pyramids)
• describe and provide examples of congruent and similar plane figures; apply congruent and
similar plane figures to solve real-world problems
Grade 6 Skills and Concepts – Transformations of Shapes
Students will
• determine lines of symmetry for a plane figure, sketch plane figures with multiple lines of
symmetry and apply line symmetry to real-world and/or mathematical situations
• transform (translate and reflect across a horizontal or vertical line) figures in the first quadrant of
the coordinate plane and determine new coordinates of the shape after transformation
• explore the rotation of a figure in a plane in the first quadrant, with and without manipulatives
Grade 6 Skills and Concepts – Coordinate Geometry
Students will
• identify and graph ordered pairs on a positive coordinate system, identifying the origin, axes and
ordered pairs
• apply graphing in the positive coordinate system to solve real-world and mathematical problems

Data Analysis and Probability

PROGRAM OF STUDIES

Big Idea: Data Analysis and Probability
Middle grades students extend the early development of data representations and examine the
appropriateness of graphs and representations of data. They examine central tendencies and dispersion.
They develop organized approaches to counting and use experimental and theoretical probabilities.
Academic Expectations
2.7 Students understand number concepts and use numbers appropriately and accurately.
2.8 Students understand various mathematical procedures and use them appropriately and
accurately.
2.13 Students understand and appropriately use statistics and probability.
Grade 6 Enduring Knowledge – Understandings
Students will understand that
• quantitative literacy is a necessary tool to be an intelligent consumer and citizen.
• the collection, organization, interpretation and display of data can be used to answer questions.
• the choice of data display can affect the visual message communicated.
• inferences and predictions from data are used to make critical and informed decisions.
• for a given set of data or a graph, statistical measures (mean, median, mode, range) can be used
to describe the distribution of the data.
• probability can be used to make decisions or predictions or to draw conclusions.
Grade 6 Skills and Concepts – Data Representations
Students will
• select an appropriate graph to represent given data and justify the selection
• collect, organize, construct, analyze and interpret data in a variety of graphical methods,
including line plots, line graphs, circle graphs, bar graphs and stem-and-leaf plots
• compare data from various types of graphs
• relate different representations of data (e.g., tables, graphs, diagrams, plots)
Grade 6 Skills and Concepts – Characteristics of Data
Students will
• make predictions, draw conclusions and verify results from statistical data and probability
experiments
• determine and apply measures of distribution (mean, median, mode, range)
Grade 6 Skills and Concepts – Experiments and Samples
• pose questions; collect, organize and display data
• explore how sample size affects the reliability of the outcome
Grade 6 Skills and Concepts – Probability
Students will
• describe or determine (e.g., tables, tree diagrams) the sample space of an event
• investigate solutions to probability problems using counting techniques, tree diagrams, charts and
tables
• make predictions, draw conclusions and verify results from statistical data and probability
experiments
• determine simple probabilities based on the results of an experiment and make inferences based
on the data
• explore the role of probability in decision making

Algebraic Thinking

PROGRAM OF STUDIES

Big Idea: Algebraic Thinking
Middle grade students extend pattern work to include arithmetic sequences. They use linear functions
and linear equations. They plot rational number pairs in the Cartesian plane. They simplify algebraic and
numeric expressions. They explore the effects of change on related variables. They use and solve twostep
single variable equations and inequalities.
Academic Expectations
2.8 Students understand various mathematical procedures and use them appropriately and
accurately.
2.11 Students understand mathematical change concepts and use them appropriately and accurately.
2.12 Students understand mathematical structure concepts including the properties and logic of
various mathematical systems.
Grade 6 Enduring Knowledge – Understandings
Students will understand that
• patterns, relations and functions are tools that help explain or predict real-world phenomena.
• numerical patterns can be written as rules that generate the pattern.
• algebra represents mathematical situations and structures for analysis and problem solving.
• real-world situations can be represented using mathematical models to analyze quantitative
relationships.
• functions are used to analyze change in various contexts and model real-world phenomena.
Functions can be written in words, in a symbolic sentence or in a table.
Grade 6 Skills and Concepts – Patterns, Relations and Functions
Students will
• recognize, create and extend patterns (give an informal description of the continuation of a
pattern and/or generalize a pattern through a verbal rule)
• represent, interpret and describe function relationships through tables, graphs and verbal rules
• organize input-output coordinate pairs into tables and plot points in the first quadrant of a
coordinate (Cartesian) system/grid
• explain how the change in one quantity affects change in another quantity (e.g., in tables or
graphs, input/output tables)
Grade 6 Skills and Concepts – Variables, Expressions and Operations
Students will
• explore the use of variables in expressions and equations
• substitute numerical values for variables and evaluate algebraic expressions
• describe, define and provide examples of algebraic expressions based on real-world and/or
mathematical situations
Grade 6 Skills and Concepts – Equations and Inequalities
Students will
• use concrete and/or informal methods to solve equations with one variable that model real-world
situations
• solve problems involving simple formulas (e.g., A=lw, D=rt)
• model and solve real-world problems with one variable equations and inequalities (e.g., 8x=4,
x+2>5)