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Understand patterns, relations, and functions.
Kindergarten
1. Identify the attributes of objects (e.g., the ability to identify attributes
is a foundational skill for sorting and classifying).
2. Sort, classify, and order objects by size, number, and other properties.
3. Recognize, reproduce, describe, extend, and create repeating patterns (e.g.,
color, shape, size, sound, movement, simple numbers).
1st Grade
1. Recognize, reproduce, describe, extend, and create repeating patterns
(e.g., color, shape, size, sound, movement, simple numbers) and translate
from one
representation to another (e.g., red, red, blue, blue to step, step, clap,
clap).
2. Skip-count on a hundreds chart (e.g., by 2s up to 20 and 5s and 10s up
to 100) to identify, describe, and predict number patterns.
3. Identify number patterns on the hundreds chart.
2nd Grade
1. Recognize, reproduce, describe, extend, and create repeating and growing
patterns, and translate from one representation to another.
2. Skip-count using calculators or a hundreds chart to identify, describe,
predict, and make generalizations about number patterns to differentiate rote
counting versus the meaning of the numbers.
3. Construct and solve open sentences that have variables (e.g., 10 = [] +
7).
4. Relate everyday problem situations to number sentences involving addition
and subtraction (e.g., 25 students are going to the store. Five students can
ride in a car. How many cars will be needed?).
3rd Grade
1. Represent relationships of quantities in the form of mathematical
expressions, equations, or inequalities.
2. Solve problems involving numeric equations.
3. Select appropriate operational and relational symbols to make an expression
true (e.g., 'If 4 [] 3 = 12, what operational symbol goes in the box?').
4. Use models of feet and inches to express simple unit conversions in symbolic
form (e.g., 36 inches = [] feet x 12) that develop conceptual understanding
versus procedural skills.
5. Recognize and use the commutative property of multiplication (e.g., if
5 x 7 = 35, then what is 7 x 5?).
6. Create, describe, and extend numeric and geometric patterns including multiplication
patterns.
7. Represent simple functional relationships:
- solve simple problems involving a functional relationship between two
quantities (e.g., find the total cost of multiple items given the cost
per unit)
- extend
and recognize a linear pattern by its rules (e.g., the number of
legs on a given number of horses may be calculated by counting by 4s, by
multiplying
the number of horses by 4, or through the use of tables)
4th Grade
1. Represent and analyze patterns and simple functions using words,
tables, and graphs.
2. Create and describe numeric and geometric patterns including multiplication
and division patterns.
3.Express mathematical relationships using equations.
4. Use and interpret
variables, mathematical symbols, and properties to write and simplify expressions
and sentences:
- use letters, boxes, or other symbols
to stand for any number in simple expressions or equations (e.g.,
demonstrate an understanding of the concept of a variable)
- interpret and evaluate mathematical expressions using parentheses
- use and interpret
formulas (e.g., Area = Length x Width or A = L x W) to answer questions
about quantities and their relationships
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Represent and analyze mathematical situations and structures using algebraic symbols.
Kindergarten
1. Use concrete, pictorial, and verbal representation to develop an understanding
of invented and conventional symbols.
1st Grade
1. Write number sentences that use concrete objects, pictorial, and
verbal representations to express mathematical situations using invented
and conventional
symbols (e.g., +, -, =).
2. Demonstrate and describe the concept of equal (e.g., using objects, balance
scales).
3. Solve open number sentences that have variables representing numbers up
to 10 (e.g., 10 = [] + 2).
2nd Grade
1. Use mathematical language to describe a variety of representations and
mathematical ideas and situations.
2. Explain the concept of equal (e.g., quantities on both sides of equation
are the same) by using objects or giving examples.
3. Construct and solve open number sentences that have variables representing
numbers up to 20 (e.g. 20 = [] + 6).
4. Use objects, words, and symbols to explain the concept of addition.
3rd Grade
1. Determine the value of variables in missing part problems (e.g.,
139 + [] = 189).
2. Recognize and use the commutative and associative properties of addition
and multiplication (e.g., 'If 5 x 7 = 35, then what is 7 x 5? And if 5 x 7
x 3 = 105, then what is 7 x 3 x 5?').
3. Explore the ways that commutative, distributive, identity, and zero properties
are useful in computing with numbers.
4th Grade
1. Identify symbols and letters that represent the concept of a variable as
an unknown quantity.
2. Explore the uses of properties (commutative, distributive, associative)
in the computation of whole numbers.
3. Express mathematical relationships using equations.
4. Determine the value of variables in simple equations (e.g., 80 x 15 = 40
x []).
5. Develop simple formulas in exploring quantities and their relationships
(e.g., A = L x W).
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Use mathematical models to represent and understand quantitative relationships.
Kindergarten
1. Model situations that involve whole numbers using objects or pictures.
1st Grade
1. Represent equivalent forms of the same number through the use of physical
models, diagrams, and number expressions to 20 (e.g., 3 + 5 = 8, 2 + 6 = 8).
2. Describe situations that involve addition and subtraction of whole numbers
including objects, pictures, and symbols (e.g., Robert has four apples, Maria
has five more).
2nd Grade
1. Model situations of addition and subtraction of whole numbers using
objects, pictures, and symbols.
2. Solve problems related to trading (e.g., coin trading, measurement trading).
3. Solve addition and subtraction problems by using data from simple charts,
picture graphs, and number sentences.
3rd Grade
1. Model problem situations with objects and use representations such as pictures,
graphs, tables, and equations to draw conclusions.
2. Solve problems involving proportional relationships including unit pricing
(e.g., four apples cost 80 cents; therefore, one apple costs 20 cents).
3. Describe relationships of quantities in the form of mathematical expressions,
equations, or inequalities.
4. Select appropriate operational and relational symbols to make an expression
true (e.g.,' If 4 [] 3 = 12), what operational symbol goes in the box?').
4th Grade
1. Solve problems involving proportional relationships (including unit pricing
and map interpretations; e.g., one inch = five miles; therefore, five inches
= [] miles).
2. Model problem situations and use graphs, tables, pictures, and equations
to draw conclusions (e.g., different patterns of change).
3. Use and interpret formulas (e.g., Area = Length x Width or A = L x W) to
answer questions about quantities and their relationships.
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Analyze changes in various contexts.
Kindergarten
1. Verbally describe changes in various contexts (e.g., plants or animals
growing over time).
1st Grade
1. Describe qualitative change (e.g., a student growing taller, trees getting
bigger, ice melting).
2nd Grade
1. Describe quantitative change (e.g., a student growing two inches in one
year, water heating up to boil).
3rd Grade
1. Demonstrate how change in one variable can relate to a change in a second
variable (e.g., input-output machines, data tables).
4th Grade
1. Identify and describe situations with constant or varying rates of change
and compare them.
2. Determine how a change in one variable relates to a change in a second
variable (e.g., data tables, input-output machines).
3. Find and analyze patterns using data tables (e.g., T tables).
4. Demonstrate and describe varying rates of change in relation to real-world
situations (e.g., plant growth, students? heights).
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Understand patterns, relations, and functions.
5th Grade
1. Identify and graph ordered pairs in the first quadrant of the coordinate
plane.
2. Describe, represent, and analyze patterns and relationships.
3. Identify, describe, and continue patterns presented in a variety of formats
(e.g., numeric, visual, oral, written, kinesthetic, pictorial).
4. Generate a pattern using a written description.
6th Grade
1. Solve problems involving proportional relationships.
2. Graph ordered pairs in the coordinate plane.
3. Explain and use symbols to represent unknown quantities and variable relationships.
4. Explain and use the relationships among ratios, proportions, and percents.
5. Make generalizations based on observed patterns and relationships.
7th Grade
1. Identify and continue patterns presented in a variety of formats.
2. Represent a variety of relationships using tables, graphs, verbal rules,
and possible symbolic notation, and recognize the same general pattern presented
in different representations.
3. Simplify numerical expressions by applying properties of rational numbers,
and justify the process used.
4. Interpret and evaluate expressions involving integer powers and simple
roots.
5. Graph and interpret linear functions.
6. Solve problems involving rate, average speed, distance, and time.
8th Grade
1. Move between numerical, tabular, and graphical representations of linear
relationships.
2. Use variables to generalize patterns and information presented in tables,
charts, and graphs:
graph linear functions noting that the vertical change per
unit of horizontal change (the slope of the graph) is always the same
plot the values of quantities
whose ratios are always the same, fit a line to the plot, and understand
that the slope of the line equals the quantities
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Represent and analyze mathematical situations and structures using algebraic symbols.
5th Grade
1. Compute the value of the expression for specific numerical values of the
variable.
2. Use a letter to represent an unknown number.
3. Understand the differences between the symbols for 'less than', 'less than
or equal to', 'greater than', and 'greater than or equal to'.
6th Grade
1. Solve problems involving proportional relationships.
2. Use letters to represent an unknown in an equation.
3. Solve one-step linear equations and inequalities in one variable with positive
whole-number solutions.
4. Demonstrate that a variable can represent a single quantity that changes.
5. Demonstrate how changes in one variable affect other variables.
7th Grade
1. Write verbal expressions and sentences as algebraic expressions and equations:
- evaluate
algebraic expressions
- solve simple linear equations
- graph and interpret results
2. Use variables and appropriate operations to write an expression, an equation,
or an inequality that represents a verbal description.
3. Use the order of operations to evaluate algebraic expressions.
4. Simplify numerical expressions by applying properties of rational numbers.
5. Graph linear functions and identify slope as positive or negative.
6. Use letters as variables in mathematical expressions to describe how one
quantity changes when a related quantity changes.
8th Grade
1. Demonstrate the difference between an equation and an expression.
2. Solve two-step linear equations and inequalities in one variable with rational
solutions.
3. Evaluate formulas using substitution.
4. Demonstrate understanding of the relationships between ratios, proportions,
and percents and solve for a missing term in a proportion.
5. Graph solution sets of linear equations in two variables on the coordinate
plane.
6. Formulate and solve problems involving simple linear relationships, find
percents of a given number, variable situations, and unknown quantities.
7. Use symbols, variables, expressions, inequalities, equations, and simple
systems of equations to represent problem situations that involve variables
or unknown quantities.
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Use mathematical models to represent and understand quantitative relationships.
5th Grade
1. Use mathematical models to represent and explain mathematical concepts
and procedures.
2. Understand and use mathematical models such as:
- the number line to model the relationship between rational numbers and
rational number operations
- pictorial representation of addition and subtraction of rational
numbers with regrouping
- manipulatives or pictures to model computational procedures
- graphs, tables,
and charts to describe data
- diagrams or pictures to model problem
situations
3. Demonstrate how a situation can be represented in more than one way.
6th Grade
1.Develop and use mathematical models to represent and justify mathematical
relationships found in a variety of situations.
2.Create, explain, and use mathematical
models such as:
- Venn diagrams to show
the relationships between the characteristics of two or more sets
- equations and inequalities to model numerical relationships
- three-dimensional
geometric models
- graphs, tables, and charts to interpret
and analyze data
7th Grade
1. Create scale models and use them for dimensional drawings.
2. Understand and use the coordinate plane to graph ordered pairs and linear
equations.
3. Select and use an appropriate model for a particular situation.
8th Grade
1. Generate different representations to model a specific numerical relationship
given one representation of data (e.g., a table, a graph, an equation, a verbal
description).
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Analyze changes in various contexts.
5th Grade
1. Recognize and create patterns of change from everyday life using numerical
or pictorial representations.
2. Generalize patterns of change and recognize the same general patterns presented
in different representations.
6th Grade
1. Represent and explain changes using one-step equations with one variable.
2. Solve problems that involve change using proportional relationships.
3. Use ratios to predict changes in proportional situations.
4. Use tables and symbols to represent and describe proportional and other
relationships involving conversions, sequences, and perimeter.
5. Generate formulas to represent relationships involving changes in perimeter.
7th Grade
1. Use variables and appropriate operations to write an expression, an equation,
and/or an inequality that represents a verbal description involving change.
2. Interpret and evaluate expressions involving integer powers and simple
roots as they relate to change.
3. Graph and interpret linear functions as they are used to solve problems.
4. Solve two-step equations and inequalities with one variable over the rational
numbers, interpret the solution or solutions in the context from which they
arose, and verify the reasonableness of the results.
8th Grade
1. Use graphs, tables, and algebraic representations to make predictions and
solve problems that involve change.
2. Estimate, find, and justify solutions to problems that involve change using
tables, graphs, and algebraic expressions.
3. Use appropriate problem-solving strategies (e.g., drawing a picture, looking
for a pattern, systematic guessing and checking, acting it out, making a table
or graph, working a simpler problem, writing an algebraic expression or working
backward) to solve problems that involve change.
4. Solve multi-step problems that involve changes in rate, average speed,
distance, and time.
5. Analyze problems that involve change by identifying relationships, distinguishing
relevant from irrelevant information, identifying missing information, sequencing,
and observing patterns.
6. Generalize a pattern of change using algebra and show the relationship
among the equation, graph, and table of values.
7. Recognize the same general pattern of change presented in different representations.
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Represent and analyze mathematical situations and structures using algebraic symbols.
1. Classify numbers and members of the following sets:
- natural
- whole
- integers
- rationals
- irrationals
2. Simplify numerical expressions using the order of operations,
including
exponents.
3. Evaluate the numerical value of expressions of one or more variables that
are:
- polynomial
- rational
- radical
4. Simplify algebraic monomial expressions raised to a power (e.g.,
[5 x y2 ]3) and algebraic binomial (e.g., [5x2 + y]2) expressions raised
to a power.
5. Compare and order polynomial expressions by degree.
6. Represent and analyze relationships using written and verbal expressions,
tables, equations, and graphs, and describe the connections among those representations:
- translate
from verbal expression to algebraic formulae (e.g., 'Set
up the equations that represent the data in the following equation: John?s
father
is 23 years
older than John. John is 4 years older than his sister Jane.
John's mother is 3 years younger than John?s father. John?s mother is 9
times as
old as
Jane. How old are John, Jane, John?s mother, and John?s father?')
- given data in a
table, construct a function that represents these data (linear
only)
- given a graph, construct a function that represents the graph (linear
only)
7. Know, explain, and use equivalent representations for the same real number
including:
- integers
- decimals
- percents
- ratios
- scientific notation
- numbers with integer exponents
- inverses (reciprocal)
- prime factoring
8. Simplify algebraic expressions using the distributive property.
9. Explain and use the concept of absolute value.
10. Know, explain, and use equivalent representations for algebraic expressions.
11. Simplify square roots and cube roots with monomial radicands that are
perfect squares or perfect cubes (e.g., 9a^2x^4).
12. Calculate powers and roots of real numbers, both rational and irrational.
13.
Solve:
- formulas for specified variables
- radical equations involving one radical
14. Factor polynomials, difference
of squares and perfect square trinomials,
and the sum and difference of cubes.
15. Simplify fractions with polynomials in the numerator and denominator by
factoring both and reducing them to the lowest terms.
16. Manipulate simple expressions with + and - exponents.
17. Use the four basic operations (+, -, x, ?ith:
- linear expressions
- polynomial expressions
- rational expressions
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Understand patterns, relations, functions, and graphs.
B. Understand patterns, relations, functions, and graphs.
1. Distinguish between the concept of a relation and a function.
2. Determine
whether a relation defined by a graph, a set of ordered pairs, a table of
values, an equation, or a rule is a function.
3. Describe the concept
of a graph of a function.
4. Translate among tabular,
symbolic, and graphical representations of functions.
5.
Explain and use function notation.
6. Determine the domain of independent variables
and the range of dependent variables defined by a graph, a set of ordered
pairs, or a symbolic expression.
7.
Identify the independent and dependent variables from an application problem
(e.g., height of a child).
8. Describe the concept of a graph of an equation.
9. Understand symmetry of graphs.
10. Analyze and describe middle and end (asymptotic)
behavior of linear, quadratic, and exponential functions, and sketch the
graphs of functions.
11. Work with composition
of functions (e.g., find f of g when f(x) = 2x - 3 and g(x) = 3x - 2),
and find the domain, range, intercepts, zeros, and
local maxima or minima of the final function.
12. Use the quadratic formula and
factoring techniques to determine whether the graph of a quadratic function
will intersect the x-axis in zero, one,
or two points.
13. Apply quadratic equations to physical phenomena (e.g., the
motion of an
object under the force of gravity).
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Use mathematical models to represent and understand quantitative relationships.
1.
Model real-world phenomena using linear and quadratic equations and linear
inequalities (e.g., apply algebraic techniques to solve rate problems, work
problems, and percent mixture problems; solve problems that involve discounts,
markups, commissions, and profit and compute simple and compound interest;
apply quadratic equations to model throwing a baseball in the air ).
2. Use
a variety of computational methods (e.g., mental arithmetic, paper and
pencil, technological tools).
3. Express the relationship between two variables
using a table with a finite
set of values and graph the relationship.
4. Express the relationship between
two variables using an equation and a graph:
- graph
a linear equation and linear inequality in two variables
- solve linear
inequalities and equations in one variable
- solve systems of linear
equations in two variables and graph the solutions
- use the graph of a system of equations in two variables to help determine
the solution
5. Solve applications involving systems of equations.
6. Evaluate numerical and
algebraic absolute value expressions.
7. Create a linear
equation from a table of values containing co-linear data.
8.
Determine the solution to a system of equations in two variables from a given
graph.
9. Generate an algebraic sentence to model real-life situations.
10. Write an equation of the line that passes through two given points.
11. Understand
and use:
- such operations as taking the inverse, finding the reciprocal, taking
a root, and raising to a fractional power
- the rules of exponents
12. Verify that a point lies on a line, given an equation
of the line, and be able to derive linear equations by using the point-slope
formula.
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Analyze changes in various contexts.
1. Analyze the effects of parameter changes on these functions:
- linear (e.g.,
changes in slope or coefficients)
- quadratic (e.g., f[x-a] changes
coefficients and constants)
- exponential (e.g.,
changes caused by increasing x[x + c] or [ax])
- polynomial
(e.g., changes caused by positive or negative values of a, or in
a constant c)
2. Solve routine two- and three-step problems relating to change
using concepts such as:
exponents
- factoring
- ratio
- proportion
- average
- percent
3. Calculate the percentage of increase and decrease of a quantity.
4. Analyze the general shape of polynomial expressions and equations for different
degree polynomials (e.g., positive and negative general shapes for third-,
fourth-, and fifth-degree polynomials).
5. Estimate the rate of change of a function or equation by finding the slope
between two points on the graph.
6. Evaluate the estimated rate of change in the context of the problem.
7. Know Pascal?s triangle and use it to expand binomial expressions that are
raised to positive integer powers.
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