Mathematics Cumulative Progress Indicators (CPIs) for the end of the designated grade span

Place a "+" for an expectation that represents a strength & a "-" for a weakness

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STANDARD 4.3     (PATTERNS AND ALGEBRA)     ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES  AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.

A.     Patterns

By the end of Grade 2, students will:

 1.    Recognize, describe, extend, and create patterns.

·        Using concrete materials (manipulatives), pictures, rhythms, & whole numbers

·        Descriptions using words and symbols (e.g., “add two” or “+ 2”)

·        Repeating patterns

·        Whole number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 1.         Recognize, describe, extend, and create patterns.

·        Descriptions using words and number sentences/expressions

·        Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 1.         Recognize, describe, extend, and create patterns.

·        Descriptions using words, number sentences/expressions, graphs, tables, variables (e.g., shape, blank, or letter)

·        Sequences that stop or that continue infinitely

·        Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)

·        Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4, . . . could be 8, or 7, or … )

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Recognize, describe, extend, and create patterns involving whole numbers.

·        Descriptions using tables, verbal rules, simple equations, and graphs

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.         Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.

·        Descriptions using tables, verbal rules, simple equations, and graphs

·        Formal iterative formulas (e.g., NEXT = NOW * 3)

·        Recursive patterns, including Pascal’s Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1,  1,  2,  3,  5,  8, . . .    (where NEXT = NOW + PREVIOUS)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 1.         Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

·        Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

·        Finite and infinite sequences

·        Generating sequences by using calculators to repeatedly apply a formula

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 1.         Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

·        Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

·        Finite and infinite sequences

·        Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

·        Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)

·        Generating sequences by using calculators to repeatedly apply a formula

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.         Use models and algebraic formulas to represent and analyze sequences and series.

·        Explicit formulas for n^th terms

·        Sums of finite arithmetic series

·        Sums of finite and infinite geometric series

 2.       Develop an informal notion of limit.

 3.       Use inductive reasoning to form generalizations.

B.     Functions and Relationships

By the end of Grade 2, students will:

 1.    Use concrete and pictorial models of function machines to explore the basic concept of a function.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 1.         Use concrete and pictorial models to explore the basic concept of a function.

·        Input/output tables, T-charts

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 1.         Use concrete and pictorial models to explore the basic concept of a function.

·        Input/output tables, T-charts

·        Combining two function machines

·        Reversing a function machine

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Describe arithmetic operations as functions, including combining operations and reversing them.

 2.         Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.         Describe the general behavior of functions given by formulas or verbal rules (e.g., graph to determine whether increasing or decreasing, linear or not).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 1.         Graph functions, and understand and describe their general behavior.

·        Equations involving two variables

 1.         Graph functions, and understand and describe their general behavior.

·        Equations involving two variables

·        Rates of change (informal notion of slope)

 2.         Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.       Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.

 2.       Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

·        Slope of a line or curve

·        Domain and range

·        Intercepts

·        Continuity

·        Maximum/minimum

·        Estimating roots of equations

·        Intersecting points as solutions of systems of equations

·        Rates of change

 3.       Understand and perform transformations on commonly-used functions.

·        Translations, reflections, dilations

·        Effects on linear and quadratic graphs of parameter changes in equations

·        Using graphing calculators or computers for more complex functions

 4.       Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.

·        Linear vs. non-linear

·        Symmetry

·        Increasing/decreasing on an interval

C.     Modeling

By the end of Grade 2, students will:

 1.    Recognize and describe changes over time (e.g., temperature, height).

 2.    Construct and solve simple open sentences involving addition or subtraction.

·        Result unknown (e.g., 6 – 2  = __  or  n = 3 + 5)

·        Part unknown (e.g., 3 + ÿ = 8)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 1.         Recognize and describe change in quantities.

·        Graphs representing change over time (e.g., temperature, height)

2.         Construct and solve simple open sentences involving addition or subtraction (e.g., 3 + 6 = __,  n = 15 – 3,  3 + __ = 3,  16 – c = 7).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 1.         Recognize and describe change in quantities.

·        Graphs representing change over time (e.g., temperature, height)

·        How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration)

 2.         Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = __, n = 15 ¸ 3,  3 x __ = 0,  16 – c = 7).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Use number sentences to model situations.

·        Using variables to represent unknown quantities

·        Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations

 2.         Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

·        Changes over time

·        Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.         Use patterns, relations, and linear functions to model situations.

·        Using variables to represent unknown quantities

·        Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

 2.         Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

·        Changes over time

·        Relations between quantities

·        Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

1.      Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

  2.         Use patterns, relations, symbolic algebra, and linear functions to model situations.

·        Using manipulatives, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

·        Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 1.         Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

 2.         Use patterns, relations, symbolic algebra, and linear functions to model situations.

·        Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities

·        Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.       Use functions to model real-world phenomena and solve problems that involve varying quantities.

·        Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)

·        Direct and inverse variation

·        Absolute value

·        Expressions, equations and inequalities

·        Same function can model variety of phenomena

·        Growth/decay and change in the natural world

·        Applications in mathematics, biology, and economics (including compound interest)

 2.       Analyze and describe how a change in an independent variable leads to change in a dependent one.

 3.       Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).

D.    Procedures

By the end of Grade 2, students will:

 1.   Understand and apply (but don’t name) the following properties of addition:

·        Commutative (e.g., 5 + 3 = 3 + 5)

·        Zero as the identity element (e.g., 7 + 0 = 7)

·        Associative (e.g., 7 + 3 + 2 can be found by first adding either 7 + 3 or 3 + 2)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 1.         Understand and apply the properties of operations and numbers.

·        Commutative (e.g., 3 x 7 = 7 x 3)

·        Identity element for multiplication is 1 (e.g., 1 x 8 = 8)

·        Any number multiplied by zero is zero

 2.         Understand and use the concepts of equals, less than, and greater than to describe relations between numbers.

·        Symbols ( = , < , > )

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 1.         Understand, name, and apply the properties of operations and numbers.

·        Commutative (e.g., 3 x 7 = 7 x 3)

·        Identity element for multiplication is 1 (e.g., 1 x 8 = 8)

·        Associative (e.g., 2 x 4 x 25 can be found by first multiplying either 2 x 4 or 4 x 25)

·        Division by zero is undefined 

·        Any number multiplied by zero is zero.

 2.         Understand and use the concepts of equals, less than, and greater than in simple number sentences.

·        Symbols ( = , < , > )

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Solve simple linear equations with manipulatives and informally

·        Whole-number coefficients only, answers also whole numbers

·        Variables on one side of equation

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.       Solve simple linear equations with manipulatives and informally.

·        Whole-number coefficients only, answers also whole numbers

·        Variables on one or both sides of equation

 2.       Understand and apply the properties of operations and numbers.

·        Distributive property

·        The product of a number and its reciprocal is 1

 3.       Evaluate numerical expressions.

 4.       Extend understanding and use of inequality.

·        Symbols ( ³ , ¹ , £ )

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 1.       Use graphing techniques on a number line.

·        Absolute value

·        Arithmetic operations represented by vectors (arrows) (e.g., “-3 + 6” is “left 3, right 6”)

 2.       Solve simple linear equations informally and graphically.

·        Multi-step, integer coefficients only (although answers may not be integers)

·        Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology

 3.       Create, evaluate, and simplify algebraic expressions involving variables.

·        Order of operations, including appropriate use of parentheses

·        Substitution of a number for a variable

 4.       Understand and apply the properties of operations, numbers, equations, and inequalities.

·        Additive inverse

·        Multiplicative inverse

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 1.         Use graphing techniques on a number line.

·        Absolute value

·        Arithmetic operations represented by vectors (arrows) (e.g., “-3 + 6” is “left 3, right 6”)

 2.       Solve simple linear equations informally, graphically, and using formal algebraic methods.

·        Multi-step, integer coefficients only (although answers may not be integers)

·        Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology

 3.       Solve simple linear inequalities.

 4.       Create, evaluate, and simplify algebraic expressions involving variables.

·        Order of operations, including appropriate use of parentheses

·        Distributive property

·        Substitution of a number for a variable

·        Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa

 5.       Understand and apply the properties of operations, numbers, equations, and inequalities.

·        Additive inverse

·        Multiplicative inverse

·        Addition and multiplication properties of equality

·        Addition and multiplication properties of inequalities

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.       Evaluate and simplify expressions.

·        Add and subtract polynomials

·        Multiply a polynomial by a monomial or binomial

·        Divide a polynomial by a monomial

 2.       Select and use appropriate methods to solve equations and inequalities.

·        Linear equations – algebraically

·        Quadratic equations – factoring (when the coefficient of x² is 1) and using the quadratic formula

·        All types of equations using graphing, computer, and graphing calculator techniques

 3.       Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.