K-12 Mathematics
Benchmarks and Indicators by Standard
Click on the grade level under the standard to jump to the indicators for that
grade.
Number, Number Sense and Operations Standard
Geometry and Spatial Sense Standard
Patterns, Functions and Algebra Standard
Data Analysis and Probability Standard
Mathematical
Processes Standard23
Benchmarks
Number, Number Sense and Operations Standard
Students
demonstrate number sense, including an understanding of number systems and
operations and how they relate to one another. Students compute fluently and
make reasonable estimates using paper and pencil, technology-supported and
mental methods.
|
Notes |
By
the end of the K-2 program: A. Use place value concepts to represent whole
numbers using numerals, words and physical models. B. Recognize, classify, compare and order
whole numbers. C. Represent commonly used fractions using
words and physical models. D. Determine the value of a collection of coins
and dollar bills. E. Make change using coins for values up to
one dollar. F. Count, using numerals and ordinal numbers. G. Model, represent and explain addition as
combining sets and counting on. H. Model, represent and explain subtraction as
comparison, take-away and part-to-whole. I. Model, represent and explain
multiplication as repeated addition, rectangular arrays and skip counting. J. Model, represent and explain division as
sharing equally, repeated subtraction and rectangular arrays. |
By the end of the 3-4 program: A. Use place value structure of the base-ten
number system to read, write, represent and compare whole numbers and
decimals. B. Recognize and generate equivalent
representations for whole numbers, fractions and decimals. C. Represent commonly used fractions and mixed
numbers using words and physical models. D. Use models, points of reference and
equivalent forms of commonly used fractions to judge the size of fractions
and to compare, describe and order them. E. Recognize and classify numbers as prime or
composite and list factors. F. Count money and make change using both
coins and paper bills. G. Model and use commutative and associative
properties for addition and multiplication. H. Use relationships between operations, such
as subtraction as the inverse of addition and division as the inverse of
multiplication. |
Number,
Number Sense and Operations Standard
|
By the end of the 5-7 program: A. Represent and compare numbers less than 0
through familiar applications and extending the number line. B. Compare, order and convert among fractions,
decimals and percents. C. Develop meaning for percents, including
percents greater than 100 and less than 1. D. Use models and pictures to relate concepts
of ratio, proportion and percent. E. Use order of operations, including use of
parenthesis and exponents to solve multi-step problems, and verify and
interpret the results. F. Apply number system properties when
performing computations. G. Apply and explain the use of prime
factorizations, common factors, and common multiples in problem situations. H. Use and analyze the steps in standard and
non-standard algorithms for computing with fractions, decimals and integers. I. Use a variety of strategies, including
proportional reasoning, to estimate, compute, solve and explain solutions to
problems involving integers, fractions, decimals and percents. |
By the end of the 8-10
program: A. Use scientific notation to express large
numbers and numbers less than one. B. Identify subsets of the real number system. C. Apply properties of operations and the real
number system, and justify when they hold for a set of numbers. D. Connect physical, verbal and symbolic
representations of integers, rational numbers and irrational numbers. E. Compare, order and determine equivalent
forms of real numbers. F. Explain the effects of operations on the
magnitude of quantities. G.
Estimate,
compute and solve problems involving real numbers, including ratio,
proportion and percent, and explain solutions. H.
Find the
square root of perfect squares, and
approximate the square root of non-perfect squares. I. Estimate, compute and solve problems
involving scientific notation, square roots and numbers with integer
exponents. |
By the end of the 11-12
program: A. Demonstrate that vectors and matrices are
systems having some of the same properties of the real number system. B. Develop an understanding of properties of
and representations for addition and multiplication of vectors and matrices. C. Apply factorials and exponents, including
fractional exponents, to solve practical problems. D. Demonstrate fluency in operations with real
numbers, vectors and matrices, using mental computation or paper and pencil
calculations for simple cases and technology for more complicated cases. E. Represent and compute with complex numbers. |
Number,
Number Sense and Operations Standard
|
Notes |
By
the end of the K-2 program: K. Demonstrate fluency in addition facts with
addends through 9 and corresponding subtractions. L. Demonstrate fluency in adding and
subtracting multiples of 10, and recognize combinations that make 10. M. Add and subtract two-digit numbers with and
without regrouping. |
By the end of the 3-4 program: I. Demonstrate fluency in multiplication
facts with factors through 10 and corresponding divisions. J. Estimate the results of whole number
computations using a variety of strategies, and judge the reasonableness. K. Analyze and solve multi-step problems
involving addition, subtraction, multiplication and division of whole
numbers. L. Use a variety of methods and appropriate
tools (mental math, paper and pencil, calculators) for computing with whole
numbers. M. Add and subtract commonly used fractions
with like denominators and decimals, using models and paper and pencil. |
Benchmarks
Measurement Standard
Students
estimate and measure to a required degree of accuracy and precision by
selecting and using appropriate units, tools and technologies.
|
Notes |
By the end of the K-2
program: A. Explain the need for standard units of
measure. B. Select appropriate units for length,
weight, volume (capacity) and time, using: • objects;
i.e., non-standard units; • U.S.
customary units: inch, foot, yard, ounce, pound, cup, quart, gallon, minute,
hour, day, week and year; • metric
units: centimeter, meter, gram and liter. C. Develop common referents for units of
measure for length, weight, volume (capacity) and time to make comparisons
and estimates. D. Apply measurement techniques to measure
length, weight and volume (capacity). E. Recognize that using different units of
measurement will yield different numbers for the same measurement. |
By the end
of the 3-4 program: A. Select appropriate units for perimeter,
area, weight, volume (capacity), time and temperature, using: • objects
of uniform size; • U.S.
customary units; e.g., mile, square inch, cubic inch, second, degree
Fahrenheit, and other units as appropriate; • metric
units; e.g., millimeter, kilometer, square centimeter, kilogram, cubic
centimeter, degree Celsius, and other units as appropriate. B. Know that the number of units is inversely
related to the size of the unit for any item being measured. C. Develop common referents for units of
measure for length, weight, volume (capacity) and time to make comparisons
and estimates. D.
Identify
appropriate tools and apply counting techniques for measuring side lengths,
perimeter and area of squares, rectangles, and simple irregular
two-dimensional shapes, volume of rectangular prisms, and time and
temperature. E.
Tell time
to the nearest minute. |
Measurement
Standard
|
By the end of the 5-7
program: A. Select appropriate units to measure angles,
circumference, surface area, mass and volume, using: • U.S.
customary units; e.g., degrees, square feet, pounds, and other units as
appropriate; • metric
units; e.g., square meters, kilograms and other units as appropriate. B. Convert units of length, area, volume, mass
and time within the same measurement system. C. Identify appropriate tools and apply
appropriate techniques for measuring angles, perimeter or circumference and
area of triangles, quadrilaterals, circles and composite shapes, and surface
area and volume of prisms and cylinders. F.
Select a
tool and measure accurately to a specified level of precision. E. Use problem solving techniques and
technology as needed to solve problems involving length, weight, perimeter,
area, volume, time and temperature. F. Analyze and explain what happens to area
and perimeter or surface area and volume when the dimensions of an object are
changed. |
By the end of the 8-10
program: A. Solve increasingly complex non-routine
measurement problems and check for reasonableness of results. B. Use formulas to find surface area and volume
for specified three-dimensional objects accurate to a specified level of
precision. C. Apply indirect measurement techniques,
tools and formulas, as appropriate, to find perimeter, circumference and area
of circles, triangles, quadrilaterals and composite shapes, and to find
volume of prisms, cylinders, and pyramids. G.
Use
proportional reasoning and apply indirect measurement techniques, including
right triangle trigonometry and properties of similar triangles, to solve
problems involving measurements and rates. E. Estimate and compute various attributes,
including length, angle measure, area, surface area and volume, to a
specified level of precision. F. Write and solve real-world, multi-step
problems involving money, elapsed time and temperature, and verify
reasonableness of solutions. |
By the end of the 11-12
program: A. Explain differences among accuracy,
precision and error, and describe how each of those can affect solutions in
measurement situations. B. Apply various measurement scales to
describe phenomena and solve problems. C.
Estimate
and compute areas and volume in increasingly complex problem situations. D.
Solve
problem situations involving derived measurements; e.g., density,
acceleration. |
Measurement
Standard
|
Notes |
By the end of the K-2 program: |
By the end of the 3-4
program: |
Measurement
Standard
|
By the end of the 5-7
program: G. Understand and demonstrate the independence of perimeter and area for two-dimensional shapes and of surface area and volume for three-dimensional shapes. |
By the end of the 8-10 program: |
By the end of the 11-12
program: |
Benchmarks
Geometry and Spatial Sense Standard
Students
identify, classify, compare and analyze characteristics, properties and
relationships of one-, two- and three-dimensional geometric figures and
objects. Students use spatial reasoning, properties of geometric objects, and
transformations to analyze mathematical situations and solve problems.
|
Notes |
By the end of the K-2
program: A. Describe and create plane figures: circle,
rectangle, square, triangle, hexagon, trapezoid, parallelogram and rhombus,
and identify them in the environment. B. Describe solid objects: cube, rectangular
prism, sphere, cylinder, cone and pyramid, and identify them in the
environment. C. Sort and compare two-dimensional figures
and three-dimensional objects according to their characteristics and
properties. D. Identify, explain and model (superposition,
copying) the concept of shapes being congruent and similar. E.
Recognize
two- and three-dimensional objects from different positions. F.
Describe
location, using comparative (before, after), directional (above, below), and
positional (first, last) words. G. Identify and draw figures with line
symmetry. |
By the end of the 3-4 program: A. Provide rationale for groupings and comparisons
of two-dimensional figures and three-dimensional objects. B. Describe and identify points, lines and
planes in the environment. C. Describe and identify intersecting,
parallel and perpendicular lines or segments in the environment. D. Identify and draw right, obtuse, acute and
straight angles. E. Use attributes to describe, classify and
sketch plane figures and build solid objects. E.
Develop definitions of classes of shapes. F. Find and name locations in coordinate systems. G.
Identify
and describe line and rotational symmetry in two-dimensional shapes and
designs. H.
Describe,
identify and model reflections, rotations and translations, using physical
materials. I.
Describe
a motion or series of transformations that show two shapes are congruent. |
Geometry
and Spatial Sense Standard
|
By the end of the 5-7
program: A. Identify and label angle parts and the
regions defined within the plane where the angle resides. B. Draw circles, and identify and determine
the relationships among the radius, diameter, center and circumference. C. Specify locations and plot ordered pairs on
a coordinate plane. D. Identify, describe and classify types of
line pairs, angles, two-dimensional figures and three-dimensional objects
using their properties. E. Use proportions to express relationships
among corresponding parts of similar figures. F. Describe and use the concepts of
congruence, similarity and symmetry to solve problems. G. Describe and use properties of triangles to
solve problems involving angle measures and side lengths of right triangles. |
By the end of the 8-10
program: A. Formally define geometric figures. B. Describe and apply the properties of
similar and congruent figures; and justify conjectures involving similarity
and congruence. C. Recognize and apply angle relationships in
situations involving intersecting lines, perpendicular lines and parallel
lines. D. Use coordinate geometry to represent and
examine the properties of geometric figures. E. Draw and construct representations of two-
and three-dimensional geometric objects using a variety of tools, such as
straightedge, compass and technology. F.
Represent
and model transformations in a coordinate plane and describe the results. G.
Prove or
disprove conjectures and solve problems involving two- and three-dimensional
objects represented within a coordinate system. H.
Establish
the validity of conjectures about geometric objects, their properties and
relationships by counter-example, inductive and deductive reasoning, and
critiquing arguments made by others. |
By the end of the 11-12
program: A. Use trigonometric relationships to verify
and determine solutions in problem situations. B. Represent transformations within a
coordinate system using vectors and matrices. |
Geometry
and Spatial Sense Standard
|
Notes |
By
the end of the K-2 program: |
By
the end of the 3-4 program: |
Geometry
and Spatial Sense Standard
|
By the end of the 5-7 program: H. Predict and describe results (size,
position, orientation) of transformations of two-dimensional figures. I. Identify and draw three-dimensional
objects from different views (top, side, front and perspective). J. Apply properties of equality and
proportionality to solve problems involving congruent or similar figures;
e.g., create a scale drawing. |
By
the end of the 8-10 program: I. Use right triangle trigonometric
relationships to determine lengths and angle measures. |
By
the end of the 11-12 program: |
Benchmarks
Patterns, Functions and Algebra Standard
Students
use patterns, relations and functions to model, represent and analyze problem
situations that involve variable quantities. Students analyze, model and solve
problems using various representations such as tables, graphs and equations.
|
Notes |
By the end of the K-2
program: A. Sort, classify and order objects by size,
number and other properties, and describe the attributes used. B. Extend sequences of sounds and shapes or
simple number patterns, and create and record similar patterns. C. Create and extend patterns, and describe
the rule in words. D. Model problem situations, using objects,
pictures, numbers and other symbols. E. Solve open sentences and explain
strategies. F.
Represent an unknown quantity as a variable using a symbol, such as □, Δ, G. Describe and compare qualitative and
quantitative changes. |
By the end of the 3-4
program: A. Analyze and extend patterns, and describe
the rule in words. B. Use patterns to make predictions, identify
relationships, and solve problems. C. Write and solve open sentences and explain
strategies. D. Represent an unknown quantity as a variable
using a symbol, including letters. E. Use variables to create and solve equations
representing problem situations. F. Construct and use a table of values to
solve problems associated with mathematical relationships. G. Describe how a change in one variable
affects the value of a related variable. |
.
Patterns,
Functions and Algebra Standard
|
By the end of the 5-7 program: A. Describe, extend and determine the rule for
patterns and relationships occurring in numeric patterns, computation,
geometry, graphs and other applications. B. Represent, analyze and generalize a variety
of patterns and functions with tables, graphs, words and symbolic rules. C. Use variables to create and solve equations
and inequalities representing problem situations. D. Use symbolic algebra to represent and
explain mathematical relationships. F.
Use rules
and variables to describe patterns, functions and other relationships. F. Use representations, such as tables, graphs
and equations, to model situations and to solve problems, especially those
that involve linear relationships. G.
Write,
simplify and evaluate algebraic expressions. H. Solve linear equations and inequalities symbolically,
graphically and numerically. |
By the end of the 8-10 program: A. Generalize and explain patterns and
sequences in order to find the next term and the nth term. B. Identify and classify functions as linear
or nonlinear, and contrast their properties using tables, graphs or
equations. C. Translate information from one
representation (words, table, graph or equation) to another representation of
a relation or function. D. Use algebraic representations, such as
tables, graphs, expressions, functions and inequalities, to model and solve
problem situations. E. Analyze and compare functions and their
graphs using attributes, such as rates of change, intercepts and zeros. F. Solve and graph linear equations and
inequalities. G. Solve quadratic equations with real roots
by graphing, formula and factoring. H. Solve systems of linear equations involving
two variables graphically and symbolically. |
By the end of the 11-12 program: A. Analyze functions by investigating rates of
change, intercepts, zeros, asymptotes, and local and global behavior. B. Use the quadratic formula to solve
quadratic equations that have complex roots. C. Use recursive functions to model and solve
problems; e.g., home mortgages, annuities. D. Apply algebraic methods to represent and generalize
problem situations involving vectors and matrices. |
Patterns,
Functions and Algebra Standard
|
Notes |
By
the end of the K-2 program: |
By
the end of the 3-4 program: |
Patterns,
Functions and Algebra Standard
|
By the end of the 5-7 program: I. Explain how inverse operations are used to
solve linear equations. J. Use formulas in problem-solving
situations. K. Graph linear equations and inequalities. L. Analyze functional relationships, and
explain how a change in one quantity results in a change in the other. M. Approximate and interpret rates of change
from graphical and numerical data. |
By the end of the 8-10 program: I. Model and solve problem situations
involving direct and inverse variation. J. Describe and interpret rates of change
from graphical and numerical data. |
By the end of the 11-12 program: |
Benchmarks
Data Analysis and Probability Standard
Students
pose questions and collect, organize, represent, interpret and analyze data to
answer those questions. Students develop and evaluate inferences, predictions
and arguments that are based on data.
|
Notes |
By the end of the K-2
program: A. Pose questions and gather data about
everyday situations and familiar objects. B. Sort and classify objects by attributes,
and organize data into categories in a simple table or chart. C. Represent data using objects, picture
graphs and bar graphs. D. Describe the probability of chance events as
more, less or equally likely to occur. |
By the end of the 3-4
program: A. Gather and organize data from surveys and
classroom experiments, including data collected over a period of time. B. Read and interpret tables, charts, graphs
(bar, picture, line, line plot), and timelines as sources of information,
identify main idea, draw conclusions, and make predictions. C. Construct charts, tables and graphs to
represent data, including picture graphs, bar graphs, line graphs, line plots
and Venn diagrams. D. Read, interpret and construct graphs in
which icons represent more than a single unit or intervals greater than one; e.g.,
each b = 10 bicycles or the intervals on an
axis are multiples of 10. E. Describe data using mode, median and range. F. Conduct a simple probability experiment and
draw conclusions about the likelihood of possible outcomes. G. Identify and represent possible outcomes,
such as arrangements of a set of up to four members and possible combinations
from several sets, each containing 2 or 3 members. |
Data
Analysis and Probablity Standard
|
By the end of the 5-7 program: A. Read, create and use line graphs,
histograms, circle graphs, box-and-whisker plots, stem-and-leaf plots, and
other representations when appropriate. B. Interpret data by looking for patterns and
relationships, draw and justify conclusions, and answer related questions. C. Evaluate interpretations and conclusions as
additional data are collected, modify conclusions and predictions, and
justify new findings. D. Compare increasingly complex displays of
data, such as multiple sets of data on the same graph. E. Collect, organize, display and interpret
data for a specific purpose or need. F. Determine and use the range, mean, median
and mode to analyze and compare data, and explain what each indicates about
the data. G. Evaluate conjectures and predictions based
upon data presented in tables and graphs, and identify misuses of statistical
data and displays. H. Find all possible outcomes of simple
experiments or problem situations, using methods such as lists, arrays and
tree diagrams. |
By the end of the 8-10 program: A. Create, interpret and use graphical displays
and statistical measures to describe data; e.g., box-and-whisker plots,
histograms, scatterplots, measures of center and variability. B. Evaluate different graphical
representations of the same data to determine which is the most appropriate
representation for an identified purpose. C. Compare the characteristics of the mean,
median and mode for a given set of data, and explain which measure of center
best represents the data. D. Find, use and interpret measures of center
and spread, such as mean and quartiles, and use those measures to compare and
draw conclusions about sets of data. E. Evaluate the validity of claims and
predictions that are based on data by examining the appropriateness of the
data collection and analysis. F. Construct convincing arguments based on
analysis of data and interpretation of graphs. |
By the end of the 11-12 program: A. Create and
analyze tabular and graphical displays of data using appropriate tools,
including spreadsheets and graphing calculators. B. Use descriptive statistics to analyze and
summarize data, including measures of center, dispersion, correlation and
variability. C. Design and perform a statistical
experiment, simulation or study; collect and interpret data; and use
descriptive statistics to communicate and support predictions and
conclusions. D. Connect statistical techniques to
applications in workplace and consumer situations. |
Data
Analysis and Probability Standard
|
Notes |
By the end of the K-2
program: |
By the end of the 3-4
program: H. Use the set of possible outcomes to describe
and predict events. |
Data
Analysis and Probability Standard
|
By the end of the 5-7
program: I. Describe the probability of an event using
ratios, including fractional notation. J. Compare experimental and theoretical
results for a variety of simple experiments. K. Make and justify predictions based on
experimental and theoretical probabilities. |
By the end of the 8-10
program: G. Describe sampling methods and analyze the
effects of method chosen on how well the resulting sample represents the
population. H. Use counting techniques, such as
permutations and combinations, to determine the total number of options and
possible outcomes. I. Design an experiment to test a theoretical
probability, and record and explain results. J. Compute probabilities of compound events,
independent events, and simple dependent events. K. Make predictions based on theoretical
probabilities and experimental results. |
By the end of the 11-12
program: |
Benchmarks
Mathematical Processes Standard
Students
use mathematical processes and knowledge to solve problems. Students apply problem-solving and
decision-making techniques, and communicate mathematical ideas.
|
Notes |
By the end of the K-2
program: A. Use a variety of strategies to understand
problem situations; e.g., discussing with peers, stating problems in own
words, modeling problems with diagrams or physical materials, identifying a
pattern. B. Identify and restate in own words the
question or problem and the information needed to solve the problem. C. Generate alternative strategies to solve
problems. D. Evaluate the reasonableness of predictions,
estimations and solutions. E. Explain to others how a problem was solved. F. Draw pictures and use physical models to
represent problem situations and solutions. G. Use invented and conventional symbols and
common language to describe a problem situation and solution. H. Recognize the mathematical meaning of common
words and phrases, and relate everyday language to mathematical language and
symbols. |
By the end of the 3-4
program: A. Apply and justify the use of a variety of
problem-solving strategies; e.g., make an organized list, guess and check. B. Use an organized approach and appropriate
strategies to solve multi-step problems. C. Interpret results in the context of the
problem being solved; e.g., the solution must be a whole number of buses when
determining the number of buses necessary to transport students. D. Use mathematical strategies to solve problems
that relate to other curriculum areas and the real world; e.g., use a
timeline to sequence events; use symmetry in artwork. E.
Link
concepts to procedures and to symbolic notation; e.g., model 3 x 4 with a
geometric array, represent one-third by dividing an object into three equal
parts. F. Recognize relationships among different
topics within mathematics; e.g., the length of an object can be represented
by a number. |
The benchmarks for mathematical
processes articulate what students should demonstrate in problem solving,
representation, communication, reasoning and connections at key points in their
mathematics program. Specific grade-level indicators have not been included for
the mathematical processes standard because content and processes should be interconnected
at the indicator level. Therefore, mathematical processes have been embedded
within the grade-level indicators for the five content standards.
|
By the end of the 5-7
program: A. Clarify problem-solving situation and
identify potential solution processes; e.g., consider different strategies
and approaches to a problem, restate problem from various perspectives. B. Apply and adapt
problem-solving strategies to solve a variety of problems, including
unfamiliar and non-routine problem situations. C. Use more than one strategy to solve a
problem, and recognize there are advantages associated with various methods. D.
Recognize
whether an estimate or an exact solution is appropriate for a given problem
situation. F. Use deductive thinking to construct
informal arguments to support reasoning and to justify solutions to problems. G.
Use inductive thinking to generalize a pattern of observations for
particular cases, make conjectures, and provide supporting arguments for
conjectures. |
By the end of the 8-10
program: A. Formulate a problem or mathematical model in
response to a specific need or situation, determine information required to
solve the problem, choose method for obtaining this information, and set
limits for acceptable solution. B. Apply mathematical knowledge and skills
routinely in other content areas and practical situations. C. Recognize and use connections between
equivalent representations and related procedures for a mathematical concept;
e.g., zero of a function and the x-intercept
of the graph of the function, apply proportional thinking when measuring,
describing functions, and comparing probabilities. D. Apply reasoning processes and skills to
construct logical verifications or counter-examples to test conjectures and
to justify and defend algorithms and solutions. |
By the end of the 11-12
program: A. Construct algorithms for multi-step and
non-routine problems. B. Construct logical verifications or
counter-examples to test conjectures and to justify or refute algorithms and
solutions to problems. C. Assess the adequacy and reliability of
information available to solve a problem. D. Select and use various types of reasoning
and methods of proof. E. Evaluate a mathematical argument and use
reasoning and logic to judge its validity. F. Present complete and convincing arguments
and justifications, using inductive and deductive reasoning, adapted to be
effective for various audiences. G. Understand the difference between a
statement that is verified by mathematical proof, such as a theorem, and one that
is verified empirically using examples or data. |
Mathematical
Processes Standard
|
Notes |
By the end of the K-2
program: I. Communicate mathematical thinking by using
everyday language and appropriate mathematical language. |
By the end of the 3-4 program: G. Use reasoning skills to determine and
explain the reasonableness of a solution with respect to the problem
situation. H. Recognize basic valid and invalid arguments,
and use examples and counter examples, models, number relationships, and logic
to support or refute. I. Represent problem situations in a variety
of forms (physical model, diagram, in words or symbols), and recognize when
some ways of representing a problem may be more helpful than others. J. Read, interpret, discuss and write about
mathematical ideas and concepts using both everyday and mathematical
language. K. Use mathematical language to explain and
justify mathematical ideas, strategies and solutions. |
Mathematical
Processes Standard
|
By the end of the 5-7 program: G. Relate mathematical ideas to one another
and to other content areas; e.g., use area models for adding fractions,
interpret graphs in reading, science and social studies. H. Use representations to organize and
communicate mathematical thinking and problem solutions. I. Select, apply, and translate among
mathematical representations to solve problems; e.g., representing a number
as a fraction, decimal or percent as appropriate for a problem. J. Communicate mathematical thinking to
others and analyze the mathematical thinking and strategies of others. K. Recognize and use mathematical language and
symbols when reading, writing and conversing with others. |
By the end of the 8-10
program: E. Use a variety of mathematical
representations flexibly and appropriately to organize, record and
communicate mathematical ideas. F. Use precise mathematical language and
notations to represent problem situations and mathematical ideas. G. Write clearly and coherently about
mathematical thinking and ideas. H. Locate and interpret mathematical
information accurately, and communicate ideas, processes and solutions in a
complete and easily understood manner. |
By the end of the 11-12
program: H. Use formal mathematical language and
notation to represent ideas, to demonstrate relationships within and among
representation systems, and to formulate generalizations. I. Communicate mathematical ideas orally and
in writing with a clear purpose and appropriate for a specific audience. J. Apply mathematical modeling to workplace
and consumer situations, including problem formulation, identification of a
mathematical model, interpretation of solution within the model, and
validation to original problem situation. |
K-12
Mathematics
Grade-Level Indicators
Students demonstrate number sense, including an
understanding of number systems and operations and how they relate to one
another. Students compute fluently and make reasonable estimates using
paper and pencil, technology-supported and mental methods.
Number, Number Sense and Operations Standard
Kindergarten
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Compare and order whole numbers up to 10. 3.
Count to twenty; e.g., in play situations or while reading number
books. 4.
Determine “how many” in sets (groups) of 10 or fewer objects. 5.
Relate, read and write numerals for single-digit numbers (0 to 9). 6.
Construct multiple sets of objects each containing the same number of
objects. 7.
Compare the number of objects in two or more sets when one set has
one or two more, or one or two fewer objects. 8.
Represent and use whole numbers in flexible ways, including relating,
composing and decomposing numbers; e.g., 5 marbles can be 2 red and 3 green
or 1 red and 4 green. 9.
Identify and state the value of a penny, nickel and dime. 10.
Model and represent addition as combining sets and counting on, and
subtraction as take-away and comparison.
For example: a.
Combine and separate small sets of objects in contextual situations;
e.g., add or subtract one, two, or another small amount. b.
Count on (forward) and count back (backward) on a number line between
0 and 10. 11.
Demonstrate joining multiple groups of objects, each containing the
same number of objects; e.g., combining 3 bags of candy, each containing 2
pieces. 12.
Partition or share a small set of objects into groups of equal size;
e.g., sharing 6 stickers equally among 3 children. 13.
Recognize the number or quantity of sets up to 5 without counting;
e.g., recognize without counting the dot arrangement on a domino as 5. |
Grade One
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1. Use ordinal numbers to order objects; e.g., first, second, third. 2. Recognize and generate equivalent forms for the same number using physical models, words and number expressions; e.g., concept of ten is described by “10 blocks”, full tens frame, numeral 10, 5 + 5, 15 - 5, one less than 11, my brother’s age. 3.
Read and write the numerals for numbers to 100. 4.
Count forward to 100, count backwards from 100, and count forward or
backward starting at any number between 1 and 100. 5.
Use place value concepts to represent whole numbers using numerals,
words, expanded notation and physical models with ones and tens. For example: a.
Develop a system to group and count by twos, fives and tens. b.
Identify patterns and groupings in a 100's chart and relate to place
value concepts. c.
Recognize the first digit of a two-digit number as the most important
to indicate size of a number and the nearness to 10 or 100. 6.
Identify and state the value of a penny, nickel, dime, quarter and
dollar. 7.
Determine the value of a small collection of coins (with a total
value up to one dollar) using 1 or 2 different type coins, including pennies,
nickels, dimes and quarters. 8.
Show different combinations of coins that have the same value. 9.
Represent commonly used fractions using words and physical models for
halves, thirds and fourths, recognizing fractions are represented by equal
size parts of a whole and of a set of objects. 10.
Model, represent and explain addition as combining sets (part + part
= whole) and counting on. For
example: a.
Model and explain addition using physical materials in contextual Situations. b.
Draw pictures to model addition. c.
Write number sentences to represent addition. d.
Explain that adding two whole numbers yields a larger whole number. 11.
Model, represent and explain subtraction as take-away and comparison. For example: a.
Model and explain subtraction using physical materials in contextual
situations. b.
Draw pictures to model subtraction. c.
Write number sentences to represent subtraction. d.
Explain that subtraction of whole numbers yields an answer smaller
than the
original number. 12.
Use conventional symbols to represent the operations of addition and
subtraction. 13.
Model and represent multiplication as repeated addition and
rectangular arrays in contextual situations; e.g., four people will be at my
party and if I want to give 3 balloons to each person, how many balloons will
I need to buy? 14.
Model and represent division as sharing equally in contextual situations; e.g., sharing cookies. 15.
Demonstrate that equal means “the same as” using visual
representations. 16. Develop strategies for basic addition facts, such as: a.
Counting all; b.
Counting on; c.
one more, two more; d.
doubles; e.
doubles plus or minus one; f.
make ten; g.
using tens frames; h.
identity property (adding zero). 17. Develop strategies for basic subtraction
facts, such as: a.
relating to addition (for example, think of 7 - 3 = ? as “3 plus?
equals 7"); b.
one less, two less; c.
all but one (for example, 8 - 7, 5 - 4); d.
using tens frames; e.
missing addends. |
Grade Two
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Use place value concepts to represent, compare and order whole
numbers using physical models, numerals and words, with ones, tens and
hundreds. For example: a.
Recognize 10 can mean “10 ones” or a single entity (1 ten) through
physical models and trading games. b.
Read and write 3-digit numerals (e.g., 243 as two hundred forty
three, 24 tens and 3 ones, or 2 hundreds and 43 ones, etc.) and construct
models to represent each. 2. Recognize and classify numbers as even
or odd. 3.
Count money and make change using coins and a dollar bill. 4.
Represent and write the value of money using the ¢ sign and in
decimal form when using the $ sign. 5. Represent fractions (halves, thirds,
fourths, sixths and eighths), using words, numerals and physical models. For example: a.
Recognize that a fractional part can mean different amounts depending
on the original quantity. b.
Recognize that a fractional part of a rectangle does not have to be
shaded with contiguous parts. c.
Identify and illustrate parts of a whole and parts of sets of
objects. d.
Compare and order physical models of halves, thirds and fourths in
relations to 0 and 1. 6.
Model, represent and explain subtraction as comparison, take-away and
part-to-whole; e.g., solve missing addend problems by counting up or
subtracting, such as “I had six baseball cards, my sister gave me more, and I
now have ten. How many did she give me?” can be represented as 6 + ? = 10
or 10 - 6 = ?. 7.
Model, represent and explain multiplication as repeated addition,
rectangular arrays and skip counting. 8.
Model, represent and explain division as sharing equally and repeated
subtraction. 9.
Model and use the commutative property for addition. 10.
Demonstrate fluency in addition facts with addends through 9 and
corresponding subtractions; e.g., 9 + 9 = 18, 18 – 9 = 9. 11.
Add and subtract multiples of 10. 12.
Demonstrate multiple strategies for adding and subtracting 2- or
3-digit whole numbers, such as: a. compatible numbers; b.
compensatory numbers; c.
informal use of commutative and associative properties of addition. 13.
Estimate the results of whole number addition and subtraction
problems using front-end estimation, and judge the reasonableness of the
answers. |
Grade Three
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Identify
and generate equivalent forms of whole numbers; e.g., 36, 30 + 6, 9 x 4, 46 - 10, number
of inches in a yard. 2.
Use place
value concepts to represent whole numbers and decimals using numerals, words,
expanded notation and physical models. For example: a.
Recognize
100 means “10 tens” as well as a single entity (1 hundred) through physical
models and trading games. b.
Describe
the multiplicative nature of the number system; e.g., the structure of 3205
as 3 x 1000 plus 2 x 100 plus 5 x 1. c.
Model the
size of 1000 in multiple ways; e.g., packaging 1000 objects into 10 boxes of
100, modeling a meter with centimeter and decimeter strips, or gathering 1000
pop-can tabs. d. Explain the concept of tenths and hundredths
using physical models, such as metric pieces, base ten blocks, decimal
squares or money. 3.
Use mathematical language and
symbols to compare and order; e.g., less than, greater than, at most, at
least, <, >, =, ≤, ≥. 4.
Count money and make change using coins and paper bills to ten
dollars. 5.
Represent fractions and mixed numbers using words, numerals and
physical models. 6.
Compare and order commonly used fractions and mixed numbers using
number lines, models (such as fraction circles or bars), points of reference
(such as more or less than 7.
Recognize
and use decimal and fraction concepts and notations as related ways of
representing parts of a whole or a set; e.g., 3 of 10 marbles are red can
also be described as 8.
Model,
represent and explain multiplication; e.g., repeated addition, skip counting,
rectangular arrays and area model. For example: a.
Use
conventional mathematical symbols to write equations for word problems
involving multiplication. b.
Understand
that, unlike addition and subtraction, the factors in multiplication and
division may have different units; e.g., 3 boxes of 5 cookies each. 9.
Model,
represent and explain division; e.g., sharing equally, repeated subtraction,
rectangular arrays and area model. For example: a.
Translate
contextual situations involving division into conventional mathematical
symbols. b.
Explain
how a remainder may impact an answer in a real-world situation; e.g., 14
cookies being shared by 4 children. 10.
Explain
and use relationships between operations, such as: a.
relate
addition and subtraction as inverse operations; b.
relate
multiplication and division as inverse operations; c.
relate
addition to multiplication (repeated addition); d.
relate
subtraction to division (repeated subtraction). 11.
Model and
use the commutative and associative properties for addition and
multiplication. 12. Add and subtract whole numbers with and
without regrouping. 13
Demonstrate fluency in multiplication facts through 10 and
corresponding
division facts. 14.
Divide 2- and 3-digit numbers by a single-digit number, without
remainders
for division. 15. Evaluate the
reasonableness of computations based upon operations and the
numbers involved; e.g., considering relative size, place value and
estimates. |
Grade Four
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Identify and generate equivalent forms of fractions and
decimals. For example: a.
Connect physical, verbal and symbolic representations of fractions,
decimals and whole numbers; e.g., b.
Understand and explain that ten tenths is the same as one whole in
both fraction and decimal form. 2.
Use place value structure of the base-ten number system to read,
write, represent and compare whole numbers through millions and decimals
through thousandths. 3.
Round whole numbers to a given place value. 4.
Identify and represent factors and multiples of whole numbers through
100, and classify numbers as prime or composite. 5.
Use models and points of reference to compare commonly used
fractions. 6.
Use associative and distributive properties to simplify and perform
computations; e.g., use left to right multiplication and the distributive
property to find an exact answer without paper and pencil, such as 5 x 47 = 5 x
40 + 5 x 7 = 200 + 35 = 235. 7.
Recognize that division may be used to solve different types of
problem situations and interpret the meaning of remainders; e.g., situations
involving measurement, money. 8.
Solve problems involving counting money and making change, using both
coins and paper bills. 10.
Use physical models, visual representations, and paper and pencil to
add and subtract decimals and commonly used fractions with like denominators. 11.
Develop and explain strategies for performing computations mentally. 13.
Use a variety of methods and appropriate tools for computing with
whole numbers; e.g., mental math, paper and pencil, and calculator. 14.
Demonstrate fluency in adding and subtracting whole numbers and in
multiplying and dividing whole numbers by 1- and 2-digit numbers and
multiples of ten. |
, 
, “five tenths,” 0.5, shaded rectangles with half, and five
tenths.
Grade Five
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Use models and visual representation to develop the concept of ratio
as part-to-part and part-to-whole, and the concept of percent as
part-to-whole. 2.
Use various forms of “one” to demonstrate the equivalence of
fractions; e.g., 3.
Identify and generate equivalent forms of fractions, decimals and
percents. 4.
Round decimals to a given place value and round fractions (including
mixed numbers) to the nearest half. 5.
Recognize and identify perfect squares and their roots. 6.
Represent and compare numbers less than 0 by extending the number
line and using familiar applications; e.g., temperature, owing money. 7.
Use commutative, associative, distributive, identity and inverse
properties to simplify and perform computations. 8.
Identify and use relationships between operations to solve problems. 9.
Use order of operations, including use of parentheses, to simplify
numerical expressions. 10.
Justify why fractions need common denominators to be added or
subtracted. 11.
Explain how place value is related to addition and subtraction of
decimals; e.g., 0.2 + 0.14; the two tenths is added to the one tenth because
they are both tenths. 12.
Use physical models, points of reference, and equivalent forms to add
and subtract commonly used fractions with like and unlike denominators and
decimals. 13. Estimate the results of computations
involving whole numbers, fractions and
decimals, using a variety of
strategies. |
x 
.
Grade Six
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Decompose and recompose whole numbers using factors and exponents
(e.g., 32 = 2 x 2 x 2 x 2 x 2 = 25 ), and explain why “squared”
means “second power” and “cubed” means “third power.” 2.
Find and use the prime factorization of composite numbers. For example: a.
Use the prime factorization to recognize the greatest common factor
(GCF). b.
Use the prime factorization to recognize the least common multiple
(LCM). c.
Apply the prime factorization to solve problems and explain
solutions. 3.
Explain why a number is referred to as being “rational,” and
recognize that the expression 4.
Describe what it means to find a specific percent of a number, using
real-life examples. 5.
Use models and pictures to relate concepts of ratio, proportion and percent,
including percents less than 1 and greater than 100. 6.
Use the order of operations, including the use of exponents, decimals
and rational numbers, to simplify numerical expressions. 7.
Use simple expressions involving integers to represent and solve
problems; e.g., if a running back loses 15 yards on the first carry but gains
8 yards on the second carry, what is the net gain/loss? 8.
Represent multiplication and division situations involving fractions
and decimals with models and visual representations; e.g., show with pattern
blocks what it means to take 9.
Give examples of how ratios are used to represent comparisons; e.g.,
part-to-part, part-to-whole, whole-to-part. 10.
Recognize that a quotient may be larger than the dividend when the
divisor is a fraction; e.g., 6 ¸ 11.
Perform fraction and decimal computations and justify their
solutions; e.g., using manipulatives, diagrams, mathematical reasoning. 12.
Develop and analyze algorithms for computing with fractions and
decimals, and demonstrate fluency in their use. 13.
Estimate reasonable solutions to problem situations involving
fractions and decimals; e.g., 14.
Use proportional reasoning, ratios and percents to represent problem
situations and determine the reasonableness of solutions. 15.
Determine the percent of a number and solve related problems; e.g.,
find the percent markdown if the original price was $140, and the sale price
is $100. |
.
= 12.
+ 
Grade Seven
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Demonstrate an understanding of place value using powers of 10 and
write large numbers in scientific notation. 2.
Explain the meaning of exponents that are negative or 0. 3.
Describe differences between rational and irrational numbers; e.g.,
use technology to show that some numbers (rational) can be expressed as
terminating or repeating decimals and others (irrational) as non-terminating
and non-repeating decimals. 4.
Use order of operations and properties to simplify numerical
expressions involving integers, fractions and decimals. 5.
Explain the meaning and effect of adding, subtracting, multiplying
and dividing integers; e.g., how adding two integers can result in a lesser
value. 6.
Simplify numerical expressions involving integers and use integers to
solve real-life problems. 7.
Solve problems using the appropriate form of a rational number
(fraction, decimal or percent). 8.
Develop and analyze algorithms for computing with percents and
integers, and demonstrate fluency in their use. 9.
Represent and solve problem situations that can be modeled by and
solved using concepts of absolute value, exponents and square roots (for
perfect squares). |
Grade Eight
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Use scientific notation to express large numbers and small numbers
between 0 and 1. 2.
Recognize that natural numbers, whole numbers, integers, rational
numbers and irrational numbers are subsets of the real number system. 3.
Apply order of operations to simplify expressions and perform
computations involving integer exponents and radicals. 4.
Explain and use the inverse and identity properties and use inverse
relationships (addition/subtraction, multiplication/division, squaring/square
roots) in problem solving situations. 5.
Determine when an estimate is sufficient and when an exact answer is
needed in problem situations, and evaluate estimates in relation to actual
answers; e.g., very close, less than, greater than. 6.
Estimate, compute and solve problems involving rational numbers,
including ratio, proportion and percent, and judge the reasonableness of
solutions. 7.
Find the square root of perfect squares, and approximate the square
root of non-perfect squares as consecutive integers between which the root
lies; e.g., 8.
Add, subtract, multiply, divide and compare numbers written in
scientific notation. |
Grade Nine
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Identify and justify whether properties (closure, identity, inverse,
commutative and associative) hold for a given set and operations; e.g., even
integers and multiplication. 2.
Compare, order and determine equivalent forms for rational and
irrational numbers. 3.
Explain the effects of operations such as multiplication or division,
and of computing powers and roots on the magnitude of quantities. 4.
Demonstrate fluency in computations using real numbers. 5.
Estimate the solutions for problem situations involving square and
cube roots. |
Grade Ten
|
Number and Number Systems Meaning
of Operations Computation and Estimation |
1.
Connect physical, verbal and symbolic representations of irrational
numbers; e.g., construct 2.
Explain the meaning of the nth
root. 3.
Use factorial notation and computations to represent and solve
problem situations involving arrangements. 4.
Approximate the nth root of
a given number greater than zero between consecutive integers when n is an integer; e.g., the 4th
root of 50 is between 2 and 3. |
Grade Eleven
|
Number and Number Systems Meaning of Operations Computation and Estimation |
1.
Determine what properties hold for matrix addition and matrix
multiplication; e.g., use examples to show addition is commutative and when
multiplication is not commutative. 2.
Determine what properties hold for vector addition and
multiplication, and for scalar multiplication. 3.
Represent complex numbers on the complex plane. 4.
Use matrices to represent given information in a problem situation. 5.
Model using the coordinate plane, vector addition and scalar
multiplication. 6.
Compute sums, differences and products of matrices using paper and pencil
calculations for simple cases, and technology for more complicated cases. 7.
Compute sums, differences, products and quotients of complex numbers. 8.
Use fractional and negative exponents as optional ways of
representing and finding solutions for problem situations; e.g., 272/3
= (271/3) 2 = 9. 9.
Use vector addition and scalar multiplication to solve problems. |
Grade Twelve
|
Number and Number Systems Computation and Estimation |
1.
Determine what properties (closure, identity, inverse, commutative
and associative) hold for operations with complex numbers. 2. Apply combinations as a method to create coefficients for the
Binomial Theorem, and make connections to everyday and workplace problem
situations. |
K-12
Mathematics
Grade-Level Indicators
Students estimate and measure to a required degree of
accuracy and precision by selecting and using appropriate units, tools and
technologies.
Measurement Standard
Kindergarten
|
Measurement Units Use Measurement Techniques and Tools |
1.
Identify units of time (day, week, month, year) and compare calendar
elements; e.g., weeks are longer than days. 2.
Compare and order objects of different lengths, areas, weights and
capacities; and use relative terms, such as longer, shorter, bigger, smaller,
heavier, lighter, more and less. 3.
Measure length and volume (capacity) using uniform objects in the
environment. For example, find: a.
how many paper clips long is a pencil; b.
how many small containers it takes to fill one big container using
sand, rice, beans. 4.
Order events based on time.
For example: a.
activities that take a long or short time; b.
review what we do first, next, last; c. recall what we did or plan to do yesterday, today, tomorrow. |
Grade One
|
Measurement Units Use Measurement Techniques and Tools |
1.
Recognize and explain the need for fixed units and tools for
measuring length and weight; e.g., rulers and balance scales. 2.
Tell time to the hour and half hour on digital and analog (dial)
timepieces. 3.
Order a sequence of events with respect to time; e.g., summer, fall,
winter and spring; morning, afternoon and night. 4.
Estimate and measure weight using non-standard units; e.g., blocks of
uniform size. 5.
Estimate and measure lengths using non-standard and standard units;
i.e., centimeters, inches and feet. |
Grade Two
|
Measurement Units Use Measurement Techniques and Tools |
1.
Identify and select appropriate units of measure for: a.
length – centimeters, meters, inches, feet, or yards; b.
volume (capacity) – liters, cups, pints, or quarts; c.
weight – grams, ounces, or pounds; d.
time – hours, half-hours, quarter-hours, or minutes and time
designations a.m. or p.m. 2.
Establish personal or common referents for units of measure to make
estimates and comparisons; e.g., the width of a finger is a centimeter, a
large bottle of soda pop is 2 liters, a small paper clip weighs about one
gram. 3.
Describe and compare the relationships among units of measure, such
as centimeters and meters; inches, feet and yards; cups, pints and quarts;
ounces and pounds; and hours, half-hours, and quarter-hours; e.g., how many
inches in a foot? 4.
Tell time to the nearest minute interval on digital and to the
nearest 5 minute interval on analog (dial) timepieces. 5.
Estimate and measure the length and weight of common objects, using
metric and U.S. customary units, accurate to the nearest unit. 6.
Select and use appropriate measurement tools; e.g., a ruler to draw a
segment 3 inches long, a measuring cup to place 2 cups of rice in a bowl, a
scale to weigh 50 grams of candy. 7.
Make and test predictions about measurements, using different units
to measure the same length or volume. |
Grade Three
|
Measurement Units Use Measurement Techniques and Tools |
1.
Identify and select appropriate units for measuring: a.
length – miles, kilometers
and other units of measure as appropriate. b.
volume (capacity) – gallons; c.
weight – ounces, pounds, grams, or kilograms; d.
temperature – degrees (Fahrenheit or Celsius). 2.
Establish personal or common referents to include additional units;
e.g., a gallon container of milk; a postage stamp is about a square inch. 3.
Tell time to the nearest minute and find elapsed time using a
calendar or a clock. 4.
Read thermometers in both Fahrenheit and Celsius scales. 5.
Estimate and measure length, weight and volume (capacity), using
metric and U.S. customary units, accurate to the nearest 6. Use
appropriate measurement tools and techniques to construct a figure or
approximate an amount of specified length, weight or volume (capacity); e.g.,
construct a rectangle with length 2 7. Make estimates for perimeter, area and volume using links, tiles, cubes and other models.
|
inches and width 3 inches, fill a measuring cup to the 
cup mark.
Grade Four
|
Measurement Units Use Measurement Techniques and Tools |
1.
Relate the number of units to the size of the units used to measure
an object; e.g., compare the number of cups to fill a pitcher to the number
of quarts to fill the same pitcher. 2.
Demonstrate and describe perimeter as surrounding and area as
covering a two-dimensional shape, and volume as filling a three-dimensional
object. 3.
Identify and select appropriate units to measure: a.
perimeter – string or links (inches or centimeters). b.
area – tiles (square inches or square centimeters). c.
Volume – cubes (cubic inches or cubic centimeters). 4.
Develop and use strategies to find perimeter using string or links,
area using tiles or a grid, and volume using cubes; e.g., count squares to
find area of regular or irregular shapes on a grid, layer cubes in a box to
find its volume. 5.
Make simple unit conversions within a measurement system; e.g.,
inches to feet, kilograms to grams, quarts to gallons. 6.
Write, solve and verify solutions to multi-step problems involving
measurement. |
Grade Five
|
Measurement Units Use Measurement Techniques and Tools |
1.
Identify and select appropriate units to measure angles; i.e.,
degrees. 2.
Identify paths between points on a grid or coordinate plane and
compare the lengths of the paths; e.g., shortest path, paths of equal length. 3.
Demonstrate and describe the differences between covering the faces
(surface area) and filling the interior (volume) of three-dimensional
objects. 4.
Demonstrate understanding of the differences among linear units,
square units and cubic units. 5.
Make conversions within the same measurement system while performing
computations. 6.
Use strategies to develop formulas for determining perimeter and area
of triangles, rectangles and parallelograms, and volume of rectangular
prisms. 7.
Use benchmark angles (e.g.;
45º, 90º, 120º) to estimate the measure |
Grade Six
|
Measurement Units Use Measurement Techniques and Tools |
1.
Understand and describe the difference between surface area and
volume. 2.
Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of
sectors; e.g., 3.
Estimate perimeter or circumference and area for circles, triangles
and quadrilaterals, and surface area and volume for prisms and cylinders by: a.
estimating lengths using string or links, areas using tiles or grid,
and volumes using cubes; b.
measuring attributes (diameter, side lengths, or heights) and using
established formulas for circles, triangles, rectangles, parallelograms and
rectangular prisms. 4.
Determine which measure (perimeter, area, surface area, volume)
matches the context for a problem situation; e.g., perimeter is the context
for fencing a garden, surface area is the context for painting a room. 5.
Understand the difference between perimeter and area, and demonstrate
that two shapes may have the same perimeter, but different areas or may have
the same area, but different perimeters. 6.
Describe what happens to the perimeter and area of a two-dimensional
shape when the measurements of the shape are changed; e.g. length of sides
are doubled. |
circle, 
Grade Seven
|
Measurement Units Use Measurement Techniques and Tools |
1.
Select appropriate units for measuring derived measurements; e.g.,
miles per hour, revolutions per minute. 2.
Convert units of area and volume within the same measurement system
using proportional reasoning and a reference table when appropriate; e.g.,
square feet to square yards, cubic meters to cubic centimeters. 3.
Estimate a measurement to a greater degree of precision than the tool
provides. 4.
Solve problems involving proportional relationships and scale
factors; e.g., scale models that require unit conversions within the same
measurement system. 5.
Analyze problem situations involving measurement concepts, select
appropriate strategies, and use an organized approach to solve narrative and
increasingly complex problems. 6. Use strategies to develop formulas for finding area of trapezoids and volume of cylinders and prisms. 7.
Develop strategies to find the area of composite shapes using the
areas of triangles, parallelograms, circles and sectors. 8.
Understand the difference between surface area and volume and
demonstrate that two objects may have the same surface area, but different
volumes or may have the same volume, but different surface areas. 9.
Describe what happens to the surface area and volume of a
three-dimensional object when the measurements of the object are changed;
e.g., length of sides are doubled. |
Grade Eight
|
Measurement Units Use Measurement Techniques and Tools |
1.
Compare and order the relative size of common U.S. customary units
and metric units; e.g., mile and kilometer, gallon and liter, pound and
kilogram. 2.
Use proportional relationships and formulas to convert units from one
measurement system to another; e.g., degrees Fahrenheit to degrees Celsius. 3.
Use appropriate levels of precision when calculating with
measurements. 4.
Derive formulas for surface area and volume and justify them using
geometric models and common materials.
For example, find: a.
the surface area of a cylinder as a function of its height and
radius; b.
that the volume of a pyramid (or cone) is one-third of the volume of
a prism (or cylinder) with the same base area and height. 5.
Determine surface area for pyramids by analyzing their parts. 6. Solve and determine the reasonableness of the results for problems involving rates and derived measurements, such as velocity and density, using formulas, models and graphs. 7.
Apply proportional reasoning to solve problems involving indirect
measurements or rates. 8.
Find the sum of the interior and exterior angles of regular convex
polygons with and without measuring the angles with a protractor. 9.
Demonstrate understanding of the concepts of perimeter, circumference
and area by using established formula for triangles, quadrilaterals, and
circles to determine the surface area and volume of prisms, pyramids,
cylinders, spheres and cones. (Note: Only volume should be calculated for
spheres and cones.) 10.
Use conventional formulas to find the surface area and volume of
prisms, pyramids and cylinders and the volume of spheres and cones to a
specified level of precision. |
Grade Nine
|
Measurement Units Use Measurement Techniques and Tools |
1.
Convert rates within the
same measurement system; e.g., miles per hour to feet per second; kilometers
per hour to meters per second. 2.
Use unit analysis to check computations involving measurement. 3.
Use the ratio of lengths in similar two-dimensional figures or
three-dimensional objects to calculate the ratio of their areas or volumes
respectively. 4.
Use scale drawings and right triangle trigonometry to solve problems
that include unknown distances and angle measures. 5. Solve problems involving unit conversion for situations involving distances, areas, volumes and rates within the same measurement system. |
Grade Ten
|
Use Measurement Techniques and Tools |
1.
Explain how a small error in measurement may lead to a large error in
calculated results. 2.
Calculate relative error. 3.
Explain the difference between absolute error and relative error in
measurement. 4.
Give examples of how the same absolute error can be problematic in
one situation but not in another; e.g., compare “accurate to the nearest
foot” when measuring the height of a person versus when measuring the height
of a mountain. 5. Determine the measures of central and inscribed angles and their associated major and minor arcs. |
Grade Eleven
|
Measurement Units Use Measurement Techniques and Tools |
1.
Determine the number of significant digits in a measurement. 2.
Use radian and degree angle measures to solve problems and perform
conversions as needed. 3.
Derive a formula for the surface area of a cone as a function of its
slant height and the circumference of its base. 4.
Calculate distances, areas, surface areas and volumes of composite
three-dimensional objects to a specified number of significant digits. 5.
Solve real-world problems involving area, surface area, volume and
density to a specified degree of precision. |
Grade Twelve
|
Use Measurement Techniques and Tools |
1.
Solve problems involving derived measurements; e.g., acceleration and
pressure. 2.
Use radian measures in the solution of problems involving angular
velocity and acceleration. 3.
Apply informal concepts of successive approximation, upper and lower
bounds, and limits in measurement situations; e.g., measurement of some
quantities, such as volume of a cone, can be determined by sequences of
increasingly accurate approximations. |
K-12
Mathematics
Grade-Level Indicators
Geometry and Spatial Sense Standard
Kindergarten
|
Characteristics and Properties Spatial Relationships |
1.
Identify and sort two- dimensional
shapes and three- dimensional objects. For example: a.
Identify and describe two-dimensional figures and three-dimensional
objects from the environment using the child’s own vocabulary. b.
Sort shapes and objects into groups based on student-defined
categories. c.
Select all shapes or objects of one type from a group. d.
Build two-dimensional figures using paper shapes or tangrams; build
simple three-dimensional objects using blocks. 2.
Name and demonstrate the relative position of objects as follows: a.
place objects over, under, inside, outside, on, beside, between,
above, below, on top of, upside-down, behind, in back of, in front of; b.
describe placement of objects with terms such as on, inside, outside,
above, below, over, under, beside, between, in front of, behind. |
Grade One
|
Characteristics and Properties Spatial Relationships |
1.
Identify, compare, and sort two-dimensional shapes; i.e., square,
circle, ellipse, triangle, rectangle, rhombus, trapezoid, parallelogram,
pentagon, and hexagon. For example: a.
Recognize and identify triangles and rhombuses independent of
position, shape or size; b.
Describe two-dimensional shapes using attributes such as number of
sides and number of vertices (corners, or angles). 2.
Create new shapes by combining or cutting apart existing shapes. 3.
Identify the shapes of the faces of three-dimensional objects. 4.
Extend the use of location words to include distance (near, far,
close to) and directional words (left, right). 5.
Copy figures and draw simple two-dimensional shapes from memory. |
Grade Two
|
Characteristics and
Properties Spatial Relationships Transformations and Symmetry |
1.
Identify, describe, compare, and sort three-dimensional objects
(i.e., cubes, spheres, prisms, cones, cylinders and pyramids) according to
the shape of the faces or the numbers of faces, edges, or vertices. 2.
Predict what new shapes will be formed by combining or cutting apart
existing shapes. 3.
Recognize two-dimensional shapes and three-dimensional objects from
different positions. 4.
Identify and determine whether two-dimensional shapes are congruent
(same shape and size) or similar (same shape different size) by copying or
using superposition (lay one thing on top of another). 5.
Create and identify two-dimensional figures with line symmetry; e.g.,
what letter shapes, logos, polygons are symmetrical? |
Grade Three
|
Characteristics and Properties Spatial Relationships Transformations and Symmetry Visualization and Geometric Models |
1.
Analyze and describe properties of two-dimensional shapes and
three-dimensional objects using terms such as vertex, edge, angle, side and
face. 2. Identify and describe the relative size
of angles with respect to right angles as follows: a.
Use physical models, like straws, to make different sized angles by
opening and closing the sides, not by changing the side lengths. b.
Identify, classify and draw right, acute, obtuse and straight angles. 3.
Find and name locations on a labeled grid or coordinate system; e.g.,
a map or graph. 4.
Draw lines of symmetry to verify symmetrical two-dimensional shapes. 5.
Build a three-dimensional model of an object composed of cubes; e.g.,
construct a model based on an illustration or actual object. |
Grade Four
|
Characteristics and
Properties Spatial Relationships Transformations and Symmetry Visualization and Geometric Models |
1.
Identify, describe and model intersecting, parallel and perpendicular
lines and line segments; e.g., use straws or other material to model lines. 2.
Describe, classify, compare and model two- and three-dimensional
objects using their attributes. 3.
Identify similarities and differences of quadrilaterals; e.g.,
squares, rectangles, parallelograms and trapezoids. 4.
Identify and define triangles based on angle measures (equiangular,
right, acute and obtuse triangles) and side lengths (isosceles, equilateral
and scalene triangles). 5.
Describe points, lines and planes, and identify models in the
environment. 6.
Specify locations and plot ordered pairs on a coordinate plane, using
first quadrant points. 7.
Identify, describe and use reflections (flips), rotations (turns),
and translations (slides) in solving geometric problems; e.g., use
transformations to determine if 2 shapes are congruent. 8. Use geometric models to solve problems in other areas of
mathematics, such as number (multiplication/division) and measurement (area,
perimeter, border). |
Grade Five
|
Characteristics and
Properties Spatial Relationships Visualization and Geometric Models |
1.
Draw circles, and identify and determine relationships among the
radius, diameter, center and circumference; e.g., radius is half the
diameter, the ratio of the circumference of a circle to its diameter is an
approximation of π. 2.
Use standard language to describe line, segment, ray, angle, skew,
parallel and perpendicular. 3.
Label vertex, rays, interior and exterior for an angle. 4.
Describe and use properties of congruent figures to solve problems. 5. Use physical models to determine the sum
of the interior angles of triangles and quadrilaterals. 6. Extend understanding of coordinate system to include points
whose x or y values may be negative numbers. 7. Understand that the measure of an angle is determined by the
degree of rotation of an angle side rather than the length of either side. 8.
Predict what three-dimensional object will result from folding a
two-dimensional net, then confirm the prediction by folding the net. |
Grade Six
|
Characteristics and
Properties Spatial Relationships Transformation and Symmetry Visualization and Geometric Models |
1.
Classify and describe two-dimensional and three-dimensional geometric
figures and objects by using their properties; e.g., interior angle measures,
perpendicular/parallel sides, congruent angles/sides. 2.
Use standard language to define geometric vocabulary: vertex, face,
altitude, diagonal, isosceles, equilateral, acute, obtuse, etc. 3.
Use multiple classification criteria to classify triangles; e.g.,
right scalene triangle. 4.
Identify and define relationships between planes; i.e., parallel,
perpendicular and intersecting. 5. Predict and describe sizes, positions and orientations of
two-dimensional shapes after transformations such as reflections, rotations,
translations and dilations. 6.
Draw similar figures that model proportional relationships; e.g.,
model similar figures with a 1 to 2 relationship by sketching two of the same
figure, one with corresponding sides twice the length of the other. 7. Build three-dimensional objects built
with cubes and sketch the two- dimensional representations of each
side; i.e., projection sets. |
Grade Seven
|
Characteristics and
Properties Spatial Relationships Transformation and Symmetry Visualization and Geometric Models |
1.
Use proportional reasoning to describe and express relationships
between parts and attributes of similar and congruent figures. 2.
Determine sufficient (not necessarily minimal) properties that define
a specific two-dimensional figure or three-dimensional object. For example: a.
Determine when one set of figures is a subset of another; e.g., all
squares are rectangles. b.
Develop a set of properties that eliminates all but the desired
figure; e.g., only squares are quadrilaterals with all sides congruent and
all angles congruent. 3. Use and demonstrate understanding of
the properties of triangles. For
example: a.
Use Pythagorean Theorem to solve problems involving right triangles. b.
Use triangle angle sum relationships to solve problems. 4.
Determine necessary conditions for congruence of triangles. 5.
Apply properties of congruent or similar triangles to solve problems
involving missing lengths and angle measures. 6.
Determine and use scale factors for similar figures to solve problems
using proportional reasoning. 7. Identify the line and rotation symmetries of two-dimensional figures to solve problems. 8. Perform translations, reflections, rotations and dilations of two-dimensional figures using a variety of methods (paper folding, tracing, graph paper). 9. Draw representations of three-dimensional geometric objects from different views. |
Grade Eight
|
Characteristics and Properties Spatial Relationships Transformation and Symmetry Visualization and Geometric Models |
1.
Make and test conjectures about characteristics and properties (e.g.,
sides, angles, symmetry) of two-dimensional figures and three-dimensional
objects. 2. Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal. 3. Use proportions in several forms to solve problems involving similar figures (part-to-part, part-to-whole, corresponding sides between figures). 4. Represent and analyze shapes using coordinate geometry; e.g., given three vertices and the type of quadrilateral, find the coordinates of the fourth vertex. 5. Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations. 6.
Draw nets for a variety of prisms, pyramids, cylinders and cones. |
Grade Nine
|
Characteristics and Properties Visualization and Geometric Models |
1.
Define the basic trigonometric ratios in right triangles: sine,
cosine and tangent. 2.
Apply proportions and right triangle trigonometric ratios to solve
problems involving missing lengths and angle sizes in similar figures. 3. Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. |
Grade Ten
|
Characteristics and Properties Spatial Relationships Transformation and Symmetry Visualization and Geometric Models |
1.
Formally define and explain key aspects of geometric figures,
including: a.
interior and exterior angles of polygons; b.
segments related to triangles (median, altitude, midsegment); c.
points of concurrency related to triangles (centroid, incenter,
orthocenter, and circumcenter); d.
circles (radius, diameter, chord, circumference, major arc, minor
arc, sector, segment, inscribed angle). 2.
Recognize and explain the necessity for certain terms to remain
undefined, such as point, line and plane. 3.
Make, test and establish the validity of
conjectures about geometric properties and relationships using
counterexample, inductive and deductive reasoning, and paragraph or
two-column proof, including: a. prove the Pythagorean Theorem; b.
prove theorems involving triangle similarity and congruence; c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; d. test a conjecture using basic constructions made with a compass and straightedge or technology. 4.
Construct right triangles, equilateral triangles, parallelograms,
trapezoids, rectangles, rhombuses, squares and kites, using compass and
straightedge or dynamic geometry software. 5.
Construct congruent or similar figures using tools, such as compass,
straightedge, and protractor or dynamic geometry software. 6.
Identify the reflection and rotation symmetries of two- and
three-dimensional figures. 7.
Perform reflections and rotations using compass and straightedge
constructions and dynamic geometry software. 8.
Derive coordinate rules for translations, reflections and rotations
of geometric figures in the coordinate plane. 9.
Show and describe the results of combinations of translations,
reflections and rotations (compositions); e.g., perform compositions and
specify the result of a composition as the outcome of a single motion, when
applicable. 10.
Solve problems involving chords, radii, and arcs within the same
circle. |
Grade Eleven
|
Spatial Relationships Transformation and Symmetry Visualization and Geometric Models |
1.
Use polar coordinates to specify locations on a plane. 2.
Represent translations using vectors. 3.
Describe multiplication of a vector and a scalar graphically and
algebraically, and apply to problem situations. 4.
Use trigonometric relationships to determine lengths and angle
measures; i.e., Law of Sines and Law of Cosines. 5.
Identify, sketch and classify the cross sections of three-dimensional
objects. |
Grade Twelve
|
Transformation and Symmetry Visualization and Geometric Models |
1.
Use matrices to represent translations, reflections, rotations,
dilations and their compositions. 2.
Derive and apply the basic trigonometric identities; i.e., angle
addition, angle subtraction, and double angle. 3. Relate
graphical and algebraic representations of lines, simple curves and conic
sections. 4. Recognize and compare specific shapes and properties in multiple geometries; e.g., plane, spherical and hyperbolic. |
K-12
Mathematics
Grade-Level Indicators
Students
use patterns, relations and functions to model, represent and analyze
problem situations that involve variable quantities. Students analyze,
model and solve problems using various representations such as tables,
graphs and equations.
Patterns, Functions and Algebra Standard
Kindergarten
|
Use Patterns,
Relations and Functions Use Algebraic
Representations |
1.
Sort, classify and order
objects by size, number and other properties. For example: a. Identify how objects are alike and
different. b. Order three events or objects according to
a given attribute, such as time or size. c. Recognize and explain how objects can be
classified in more than one way. d. Identify what attribute was used to sort
groups of objects that have already been sorted. 2. Identify,
create, extend and copy sequences of sounds (such as musical notes), shapes
(such as buttons, leaves or blocks), motions (such as hops or skips), and
numbers from 1 to 10. 3.
Describe orally the pattern of a given sequence. 4.
Model a problem situation
using physical materials. |
Grade One
|
Use Patterns, Relations
and Functions Use Algebraic
Representations |
1. Sort, classify
and order objects by two or more attributes, such as color and shape, and
explain how objects were sorted. 2.
Extend sequences of sounds,
shapes or simple number patterns, and create and record similar
patterns. For example: a. Analyze and
describe patterns with multiple attributes using numbers and shapes; e.g.,
AA, B, aa, b, AA, B, aa, b,… b. Continue
repeating and growing patterns with materials, pictures and geometric items;
e.g., XO, XOO, XOOO, XOOOO. 3. Describe orally the basic unit or general
plan of a repeating or growing pattern. 4.
Solve open sentences by representing
an expression in more than one way using the commutative property; e.g., 4 +
5 = 5 + 4 or the number of blue balls plus red balls is the same as the
number of red balls plus blue balls (R+B=B+R). 5. Describe orally and model a problem situation using words, objects or number phrase or sentence. |
Grade Two
|
Use Patterns,
Relations and Functions Use Algebraic
Representations Analyze Change |
1.
Extend simple number patterns (both repeating and growing
patterns), and create similar patterns using different objects, such as using
physical materials or shapes to represent numerical patterns. 2.
Use
patterns to make generalizations and predictions; e.g., determine a missing
element in a pattern. 3.
Create
new patterns with consistent rules or plans, and describe the rule or general
plan of existing patterns. 4.
Use
objects, pictures, numbers and other symbols to represent a problem
situation. 5. Understand
equivalence and extend the concept to situations involving symbols; e.g., 4 +
5 = 9 and 9 = 4 + 5 and 4 + 5 = 3 + 6 = Δ + □ … 6.
Use symbols to represent unknown quantities and identify values
for symbols in an expression or equation using addition and subtraction;
e.g., □ + 7.
Describe
qualitative and quantitative changes, especially those involving addition and
subtraction; e.g., a student growing taller versus a student growing two
inches in one year. |
Grade Three
|
Use Patterns,
Relations and Functions Use Algebraic
Representations Analyze Change |
1. Extend
multiplicative and growing patterns, and describe the pattern or rule in
words. 2.
Analyze and replicate arithmetic sequences with and
without a calculator. 3.
Use patterns to make predictions, identify relationships, and
solve problems. 4.
Model
problem situations using objects, pictures, tables, numbers, letters and
other symbols. 5.
Write, solve and explain simple mathematical statements, such as
7 + □ > 8 or ∆ + 8 = 10. 6. Express
mathematical relationships as equations and inequalities. 7.
Create tables to record, organize and analyze data to discover
patterns and rules. 8.
Identify and describe quantitative changes, especially
those involving addition and subtraction; e.g., the height of water in a
glass becoming 1 centimeter lower each week due to evaporation. |
Grade Four
|
Use Patterns,
Relations and Functions Use Algebraic
Representation Analyze Change |
1.
Use models and words to describe, extend and make generalizations of
patterns and relationships occurring in computation, numerical patterns,
geometry, graphs and other applications. 2.
Represent and analyze patterns and functions using words,
tables and graphs. 3.
Construct a table of values to solve problems associated
with a mathematical relationship. 4.
Use rules and variables to describe patterns and other
relationships. 5.
Represent mathematical relationships with equations or inequalities. 6.
Describe how a change in one variable affects the value
of a related variable; e.g., as one increases the other increases or as one
increases the other decreases. |
Grade Five
|
Use Patterns,
Relations and Functions Use Algebraic Representations Analyze Change |
1.
Justify a general rule for a pattern or a function by using physical
materials, visual representations, words, tables or graphs. 2.
Use calculators or computers to develop patterns, and generalize them
using tables and graphs. 3.
Use variables as unknown quantities in general rules when describing
patterns and other relationships. 4.
Create and interpret the meaning of equations and inequalities
representing problem situations. 5.
Model problems with physical materials and visual representations,
and use models, graphs and tables to draw conclusions and make predictions. 6. Describe how the quantitative change in a variable affects the value of a related variable; e.g., describe how the rate of growth varies over time, based upon data in a table or graph. |
Grade Six
|
Use Patterns,
Relations and Functions Use Algebraic Representations Analyze Change |
1.
Represent and analyze patterns, rules and functions, using physical
materials, tables and graphs. 2.
Use words and symbols to describe numerical and geometric patterns,
rules and functions. 3.
Recognize and generate equivalent forms of algebraic expressions, and
explain how the commutative, associative and distributive properties can be
used to generate equivalent forms; e.g., perimeter as 2(1 + w) or 21 + 2w. 4.
Solve simple linear equations and inequalities using physical models,
paper and pencil, tables and graphs. 5.
Produce and interpret graphs that represent the relationship between
two variables. 6.
Evaluate simple expressions by replacing variables with given values,
and use formulas in problem-solving situations. 7.
Identify and describe situations with constant or varying rates of
change, and compare them. 8.
Use technology to analyze change; e.g., use computer applications or
graphing calculators to display and interpret rate of change. |
Grade Seven
|
Use Patterns, Relations and Functions Use Algebraic Representations Analyze Change |
1.
Represent and analyze patterns, rules and functions with words,
tables, graphs and simple variable expressions. 2.
Generalize patterns by describing in words how to find the next term. 3.
Recognize and explain when numerical patterns are linear or nonlinear
progressions; e.g., 1,3,5,7... is linear and 1,3,4,8,16... is nonlinear. 4.
Create visual representations of equation-solving processes that
model the use of inverse operations. 5.
Represent linear equations by plotting points in the coordinate
plane. 6.
Represent inequalities on a number line or a coordinate plane. 7. Justify that two forms of
an algebraic expression are equivalent, and recognize when an expression is
simplified; e.g., 4m = m + m + m + m or a · 5 + 4 = 5a + 4. 8.
Use formulas in problem-solving situations. 9.
Recognize a variety of uses for variables; e.g., placeholder for an
unknown quantity in an equation, generalization for a pattern, formula. 10.
Analyze linear and simple nonlinear relationships to explain how a
change in one variable results in the change of another. 11.
Use graphing calculators or computers to analyze change; e.g.,
distance-time relationships. |
Grade Eight
|
Use Patterns,
Relations and Functions Use Algebraic Representations Analyze Change |
1.
Relate the various representations of a relationship; i.e., relate a
table to graph, description and symbolic form. 2.
Generalize patterns and sequences by describing how to find the nth term. 3.
Identify functions as linear or nonlinear based on information given
in a table, graph or equation. 4.
Extend the uses of variables to include covariants where y depends on x. 5.
Use physical models to add and subtract monomials and polynomials,
and to multiply a polynomial by a monomial. 6.
Describe the relationship between the graph of a line and its
equation, including being able to explain the meaning of slope as a constant
rate of change and y-intercept in
real-world problems. 7.
Use symbolic algebra (equations and inequalities), graphs and tables
to represent situations and solve problems. 8.
Write, simplify and evaluate algebraic expressions (including
formulas) to generalize situations and solve problems. 9.
Solve linear equations and inequalities graphically, symbolically and
using technology. 10. Solve 2 by 2 systems of linear
equations graphically and by simple substitution. 11.
Interpret the meaning of the solution of a 2 by 2 system of
equations; i.e., point, line, no solution. 12.
Solve simple quadratic equations graphically; e.g., y = x2 – 16. 13.
Compute and interpret slope, midpoint and distance given a set of
ordered pairs. 14.
Differentiate and explain types of changes in mathematical
relationships, such as linear vs. nonlinear, continuous vs. noncontinuous,
direct variation vs. inverse variation.
15.
Describe and compare how changes in an equation affects the related
graphs; e.g., for a linear equation changing the coefficient of x affects the slope and changing the
constant affects the intercepts. 16.
Use graphing calculators or computers to analyze change; e.g.,
interest compounded over time as a nonlinear growth pattern. |
Grade Nine
|
Use Patterns,
Relations and Functions Use Algebraic
Representations Analyze Change |
1.
Define function with ordered pairs in which each domain element is
assigned exactly one range element. 2.
Generalize patterns using functions or relationships (linear,
quadratic and exponential), and freely translate among tabular, graphical and
symbolic representations. 3.
Describe problem situations (linear, quadratic and exponential) by
using tabular, graphical and symbolic representations. 4.
Demonstrate the relationship among zeros of a function, roots of
equations, and solutions of equations graphically and in words. 5.
Describe and compare characteristics of the following families of
functions: linear, quadratic and exponential functions; e.g., general shape,
number of roots, domain, range, rate of change, maximum or minimum. 6.
Write and use equivalent forms of equations and inequalities in
problem situations; e.g., changing a linear equation to the slope-intercept
form. 7.
Use formulas to solve problems involving exponential growth and
decay. 8.
Find linear equations that represent lines that pass through a given
set of ordered pairs, and find linear equations that represent lines parallel
or perpendicular to a given line through a specific point. 9.
Solve and interpret the meaning of 2 by 2 systems of linear equations
graphically, by substitution and by elimination, with and without
technology. 10.
Solve quadratic equations with real roots by factoring, graphing,
using the quadratic formula and with technology. 11.
Add, subtract, multiply and divide monomials and polynomials
(division of polynomials by monomials only). 12.
Simplify rational expressions by eliminating common factors and
applying properties of integer exponents. 13.
Model and solve problems involving direct and inverse variation using
proportional reasoning. 14.
Describe the relationship between slope and the graph of a direct
variation and inverse variation. 15.
Describe how a change in the value of a constant in a linear or
quadratic equation affects the related graphs. |
Grade Ten
|
Use Patterns,
Relations and Functions Use Algebraic Representations Analyze Change |
1.
Define function formally and with f(x) notation. 2.
Describe and compare characteristics of the following families of functions:
square root, cubic, absolute value and basic trigonometric functions; e.g.,
general shape, possible number of roots, domain and range. 3.
Solve equations and formulas for a specified variable; e.g., express
the base of a triangle in terms of the area and height. 4.
Use algebraic representations and functions to describe and
generalize geometric properties and relationships. 5.
Solve simple linear and nonlinear equations and inequalities having
square roots as coefficients and solutions. 6.
Solve equations and inequalities having rational expressions as
coefficients and solutions. 7.
Solve systems of linear inequalities. 8.
Graph the quadratic relationship that defines circles. 9.
Recognize and explain that the slopes of parallel lines are equal and
the slopes of perpendicular lines are negative reciprocals. 10.
Solve real-world problems that can be modeled using linear,
quadratic, exponential or square root functions. 11.
Solve real-world problems that can be modeled, using systems of
linear equations and inequalities. 12.
Describe the relationship between slope of a line through the origin
and the tangent function of the angle created by the line and the positive x-axis. |
Grade Eleven
|
Use Patterns,
Relations and Functions Use Algebraic Representations Analyze Change |
1.
Identify and describe problem situations involving an iterative
process that can be represented as a recursive function; e.g., compound
interest. 2.
Translate a recursive function into a closed form expression or
formula for the nth term to solve a
problem situation involving an iterative process; e.g., find the value of an
annuity after 7 years. 3.
Describe and compare the characteristics of the following families of
functions: quadratics with complex roots, polynomials of any degree,
logarithms, and rational functions; e.g., general shape, number of roots,
domain and range, asymptotic behavior. 4.
Identify the maximum and minimum points of polynomial, rational and
trigonometric functions graphically and with technology. 5.
Identify families of functions with graphs that
have rotation symmetry or reflection symmetry about the y-axis, x-axis or y = x. 6.
Represent the inverse of a function symbolically and graphically as a
reflection about y = x. 7.
Model and solve problems with matrices and vectors. 8.
Solve equations involving radical expressions and complex roots. 9.
Solve 3 by 3 systems of linear equations by elimination and using
technology, and interpret graphically what the solution means (a point, line,
plane, or no solution). 10.
Describe the characteristics of the graphs of conic sections. 11.
Describe how a change in the value of a constant in an exponential,
logarithmic or radical equation affects the graph of the equation. |
Grade Twelve
|
Use Patterns,
Relations and Functions Use Algebraic
Representations Analyze Change |
Analyze the behavior of arithmetic and
geometric sequences and series as the number of terms increases. 2. Translate between the numeric and symbolic
form of a sequence or series. 3. Describe
and compare the characteristics of transcendental and periodic functions;
e.g., general shape, number of roots, domain and range, asymptotic behavior,
extrema, local and global behavior. 4.
Represent the inverse of a transcendental function symbolically. 5. Set
up and solve systems of equations using matrices and graphs, with and without
technology. 6. Make
arguments about mathematical properties using mathematical induction. 7. Make
mathematical arguments using the concepts of limit. 8. Compare
estimates of the area under a curve over a bounded interval by partitioning
the region with rectangles; e.g., make successive estimates using
progressively smaller rectangles. 9. Translate
freely between polar and Cartesian coordinate systems. 10. Use the concept of
limit to find instantaneous rate of change for a point on a graph as the
slope of a tangent at a point. |
K-12
Mathematics
Grade-Level Indicators
Students pose questions and collect,
organize, represent, interpret and analyze data to answer those questions.
Students develop and evaluate inferences, predictions and arguments that
are based on data.
Data Analysis and Probability Standard
Kindergarten
|
Data Collection Statistical Methods |
Gather
and sort data in response to questions posed by teacher and students; e.g.,
how many sisters and brothers, what color shoes. Arrange objects
in a floor or table graph according to attributes, such as use, size, color,
or shape. Select
the category or categories that have the most or fewest objects in a floor or
table graph. |
Grade One
|
Data Collection Statistical Methods Probability |
1.
Identify multiple categories for sorting data. 2.
Collect and organize data into charts using tally marks. 3.
Display data in picture graphs with units of 1 and bar graphs with
intervals of 1. 4.
Read and interpret charts, picture graphs and bar graphs as sources
of information to identify main ideas, draw conclusions, and make
predictions. 5.
Construct a question that can be answered by using information from a
graph. 6.
Arrange five objects by an attribute, such as size or weight, and
identify the ordinal position of each object. 7.
Answer questions about the number of objects represented in a picture
graph, bar graph or table graph; e.g., category with most, how many more in a
category compared to another, how many altogether in two categories. 8.
Describe the likelihood of simple events as possible/impossible and
more likely/less likely; e.g., when using spinners or number cubes in
classroom activities. |
Grade Two
|
Data Collection Statistical Methods Probability |
1.
Pose questions, use observations, interviews and surveys to collect
data, and organize data in charts, picture graphs and bar graphs. 2.
Read, interpret and make comparisons and predictions from data
represented in charts, line plots, picture graphs and bar graphs. 3.
Read and construct simple timelines to sequence events. 4.
Write a few sentences to describe and compare categories of data
represented in a chart or graph, and make statements about the data as a
whole. 5.
Identify untrue or inappropriate statements about a given set of
data. 6.
Recognize that data may vary from one population to another; e.g.,
favorite TV shows of students and of parents. 7.
List some of the possible outcomes of a simple experiment, and predict
whether given outcomes are more, less or equally likely to occur. 8.
Use physical models and pictures to represent possible arrangements
of 2 or 3 objects. |
Grade Three
|
Data Collection Statistical Methods Probability |
1.
Collect and organize data from an experiment, such as recording and
classifying observations or measurements, in response to a question posed. 2.
Draw and interpret picture graphs in which a symbol or picture
represents more than one object. 3.
Read, interpret and construct bar graphs with intervals greater than
one. 4.
Support a conclusion or prediction orally and in writing, using
information in a table or graph. 5.
Match a set of data with a graphical representation of the data. 6.
Translate information freely among charts, tables, line plots,
picture graphs and bar graphs; e.g., create a bar graph from the information
in a chart. 7.
Analyze and interpret information represented on a timeline. 8.
Identify the mode of a data set and describe the information it gives
about a data set. 9.
Conduct a simple experiment or simulation of a simple event, record
the results in a chart, table or graph, and use the results to draw
conclusions about the likelihood of possible outcomes. 10.
Use physical models, pictures, diagrams and lists to solve problems
involving possible arrangements or combinations of two to four objects. |
Grade Four
|
Data Collection Statistical Methods Probability |
1.
Create a plan for collecting data for a specific purpose. 2.
Represent and interpret data using tables, bar graphs, line plots and
line graphs. 3.
Interpret and construct Venn diagrams to sort and describe data. 4.
Compare different representations of the same data to evaluate how
well each representation shows important aspects of the data, and identify
appropriate ways to display the data. 5.
Propose and explain interpretations and predictions based on data
displayed in tables, charts and graphs. 6.
Describe the characteristics of a set of data based on a graphical
representation, such as range of the data, clumps of data, and holes in the
data. 7.
Identify the median of a set of data and describe what it indicates
about the data. 8.
Use range, median and mode to make comparisons among related sets of
data. 9.
Conduct simple probability experiments and draw conclusions from the
results; e.g., rolling number cubes or drawing marbles from a bag. 10.
Represent the likelihood of possible outcomes for chance situations;
e.g., probability of selecting a red marble from a bag containing 3 red and 5
white marbles. 11.
Relate the concepts of impossible and certain-to-happen events to the
numerical values of 0 (impossible) and 1 (certain). 12.
Place events in order of likelihood and use a diagram or appropriate language
to compare the chance of each event occurring; e.g. impossible, unlikely,
equal, likely, certain. |
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13.
List and count all possible combinations using one member from each of
several sets, each containing 2 or 3 members; e.g., the number of possible
outfits from 3 shirts, 2 shorts and 2 pair of shoes. |
Grade Five
|
Data Collection Statistical Methods Probability |
1.
Read, construct and interpret frequency tables, circle graphs and
line graphs. 2.
Select and use a graph that is appropriate for the type of data to be
displayed; e.g., numerical vs. categorical data, discrete vs. continuous
data. 3.
Read and interpret increasingly complex displays of data, such as
double bar graphs. 4.
Determine appropriate data to be collected to answer questions posed
by students or teacher, collect and display data, and clearly communicate
findings. 5.
Modify initial conclusions, propose and justify new interpretations
and predictions as additional data are collected. 6.
Determine and use the range, mean, median and mode, and explain what
each does and does not indicate about the set of data. 7.
List and explain all possible outcomes in a given situation. 8.
Identify the probability of events within a simple experiment, such
as three chances out of eight. 9.
Use 0,1 and ratios between 0 and 1 to represent the probability of
outcomes for an event, and associate the ratio with the likelihood of the
outcome. 10.
Compare what should happen (theoretical/expected results) with what
did happen (experimental/actual results) in a simple experiment. 11.
Make predictions based on experimental and theoretical probabilities. |
Grade Six
|
Data Collection Statistical Methods Probability |
1.
Read, construct and interpret line graphs, circle graphs and
histograms. 2.
Select, create and use graphical representations that are appropriate
for the type of data collected. 3.
Compare representations of the same data in different types of
graphs, such as a bar graph and circle graph. 4.
Understand the different information provided by measures of center
(mean, mode and median) and measures of spread (range). 5.
Describe the frequency distribution of a set of data, as shown in a
histogram or frequency table, by general appearance or shape; e.g., number of modes, middle of data and
level of symmetry, outliers. 6.
Make logical inferences from statistical data. 7.
Design an experiment to test a theoretical probability and explain
how the results may vary. |
Grade Seven
|
Data Collection Statistical Methods Probability |
1.
Read, create and interpret box-and-whisker plots, stem-and-leaf
plots, and other types of graphs, when appropriate. 2.
Analyze how decisions about graphing affect the graphical
representation; e.g., scale, size of classes in a histogram, number of
categories in a circle graph. 3.
Analyze a set of data by using and comparing combinations of measures
of center (mean, mode, median) and measures of spread (range, quartile,
interquartile range), and describe how the inclusion or exclusion of outliers
affects those measures. 4.
Construct opposing arguments based on analysis of the same data,
using different graphical representations. 5.
Compare data from two or more samples to determine how sample
selection can influence results. 6.
Identify misuses of statistical data in articles, advertisements, and
other media. 7.
Compute probabilities of compound events; e.g., multiple coin tosses
or multiple rolls of number cubes, using such methods as organized lists,
tree diagrams and area models. 8.
Make predictions based on theoretical probabilities, design and
conduct an experiment to test the predictions, compare actual results to
predicted results, and explain differences. |
Grade Eight
|
Data Collection Statistical Methods Probability |
1.
Use, create and interpret scatterplots and other types of graphs as
appropriate. 2.
Evaluate different graphical representations of the same data to
determine which is the most appropriate representation for an identified
purpose; e.g., line graph for change over time, circle graph for
part-to-whole comparison, scatterplot for relationship between two variants. 3.
Differentiate between discrete and continuous data and appropriate
ways to represent each. 4.
Compare two sets of data using measures of center (mean, mode,
median) and measures of spread (range, quartiles, interquartile range, percentiles).
5.
Explain the mean’s sensitivity to extremes and its use in comparison
with the median and mode. 6.
Make conjectures about possible relationship in a scatterplot and
approximate line of best fit. 7.
Identify different ways of selecting samples, such as survey
response, random sample, representative sample and convenience sample. 8.
Describe how the relative size of a sample compared to the target
population affects the validity of predictions. 9.
Construct convincing arguments based on analysis of data and
interpretation of graphs. 10.
Calculate the number of possible outcomes for a situation,
recognizing and accounting for when items may occur more than once or when
order is important. 11.
Demonstrate an understanding that the probability of either of two
disjoint events occurring can be found by adding the probabilities for each
and that the probability of one independent event following another can be
found by multiplying the probabilities. |
Grade Nine
|
Data Collection Statistical Methods Probability |
1.
Classify data as univariate (single variable) or bivariate (two
variables) and as quantitative (measurement) or qualitative (categorical)
data. 2.
Create a scatterplot for a set of bivariate data, sketch the line of
best fit, and interpret the slope of the line of best fit. 3.
Analyze and interpret frequency distributions based on spread,
symmetry, skewness, clusters and outliers. 4.
Describe and compare various types of studies (survey, observation,
experiment), and identify possible misuses of statistical data. 5.
Describe characteristics and limitations of sampling methods, and
analyze the effects of random versus biased sampling; e.g., determine and
justify whether the sample is likely to be representative of the population. 6.
Make inferences about relationships in bivariate data, and recognize
the difference between evidence of relationship (correlation) and causation. 7.
Use counting techniques and the Fundamental Counting principle to
determine the total number of possible outcomes for mathematical situations. 8.
Describe, create and analyze a sample space and use it to calculate
probability. 9.
Identify situations involving independent and dependent events, and
explain differences between and common misconceptions about probabilities
associated with those events. 10.
Use theoretical and experimental probability, including simulations
or random numbers, to estimate probabilities and to solve problems dealing
with uncertainty; e.g., compound events, independent events, simple dependent
events. |
Grade Ten
|
Data Collection Statistical Methods Probability |
1.
Describe measures of center and the range verbally, graphically and
algebraically. 2.
Represent and analyze bivariate data using appropriate graphical
displays (scatterplots, parallel box-and-whisker plots, histograms with more
than one set of data, tables, charts, spreadsheets) with and without
technology. 3.
Display bivariate data where at least one variable is categorical. 4.
Identify outliers on a data display; e.g., use the interquartile
range to identify outliers on a box-and-whisker plot. 5.
Provide examples and explain how a statistic may or may not be an
attribute of the entire population; e.g., intentional or unintentional bias
may be present. 6.
Interpret the relationship between two variables using multiple
graphical displays and statistical measures; e.g., scatterplots, parallel
box-and-whisker plots, and measures of center and spread. 7.
Model problems dealing with uncertainty with area models (geometric
probability). 8.
Differentiate and explain the relationships between the probability
of an event and the odds of an event, and compute one given the other. |
Grade Eleven
|
Data Collection Statistical Methods Probability |
1.
Design a statistical experiment, survey or study for a problem;
collect data for the problem; and interpret the data with appropriate
graphical displays, descriptive statistics, concepts of variability,
causation, correlation and standard deviation. 2.
Describe the role of randomization in a well-designed study,
especially as compared to a convenience sample, and the generalization of
results from each. 3.
Describe how a linear transformation of univariate data affects
range, mean, mode, and median. 4.
Create a scatterplot of bivariate data, identify trends, and find a
function to model the data. 5.
Use technology to find the Least Squares Regression Line, the
regression coefficient, and the correlation coefficient for bivariate data
with a linear trend, and interpret each of these statistics in the context of
the problem situation. 6.
Use technology to compute the standard deviation for a set of data,
and interpret standard deviation in relation to the context or problem
situation. 7.
Describe the standard normal curve and its general properties, and
answer questions dealing with data assumed to be normal. 8.
Analyze and interpret univariate and bivariate data to identify
patterns, note trends, draw conclusions, and make predictions. 9.
Evaluate validity of results of a study based on characteristics of
the study design, including sampling method, summary statistics and data
analysis techniques. 10.
Understand and use the concept of random variable, and compute and
interpret the expected value for a random variable in simple cases. 11.
Examine statements and decisions involving risk; e.g., insurance
rates and medical decisions. |
Grade Twelve
|
Data Collection Statistical Methods |
1.
Identify and use various sampling methods (voluntary response,
convenience sample, random sample, stratified random sample, census) in a
study. 2.
Transform bivariate data so it can be modeled by a function; e.g.,
use logarithms to allow nonlinear relationship to be modeled by linear
function. 3.
Describe the shape and find all summary statistics for a set of
univariate data, and describe how a linear transformation affects shape,
center and spread. 4.
Apply the concept of a random variable to generate and interpret
probability distributions, including binomial, normal and uniform. 5.
Use sampling distributions as the basis for informal inference. 6.
Use theoretical or experimental probability, including simulations,
to determine probabilities in real-world problem situations involving
uncertainty, such as mutually exclusive events, complementary events and
conditional probability. |
Note:
The benchmarks for mathematical processes articulate what students should
demonstrate in problem solving, representation, communication, reasoning
and connections at key points in their mathematics program. Specific
grade-level indicators have not been included for the mathematical
processes standard because content and processes should be interconnected
at the indicator level. Therefore, mathematical processes have been
embedded within the grade-level indicators for the five content standards.