Learning Progression

Algebraic Concepts: Functional Representations

Grade 2	Grade 3	Grade 4	Grade 5	Grade 6	Grade 7	Grade 8	Algebra I - Module 1 - Operations and Linear Functions & Inequalities	Algebra I - Module 2 - Linear Functions and Data Organizations	Algebra II - Module 1 - Numbers Systems and Data Analysis	Algebra II - Module 2 - Non-Linear Expressions and Equations
M03.B-O.3.1.5 Identify arithmetic patterns (including patterns in the addition table or multiplication table) and/or explain them using properties of operations. Example 1: Observe that 4 times a number is always even. Example 2: Explain why 6 times a number can be decomposed into three equal addends. When grade appropriate instruction pertaining to an eligible content should begin	M03.B-O.3.1.5 Identify arithmetic patterns (including patterns in the addition table or multiplication table) and/or explain them using properties of operations. Example 1: Observe that 4 times a number is always even. Example 2: Explain why 6 times a number can be decomposed into three equal addends. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
	M04.B-O.3.1.1 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Example 1: Given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms alternate between odd and even numbers. Example 2: Given the rule “increase the number of sides by 1” and starting with a triangle, observe that the tops of the shapes alternate between a side and a vertex. When grade appropriate instruction pertaining to an eligible content should begin	M04.B-O.3.1.1 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Example 1: Given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms alternate between odd and even numbers. Example 2: Given the rule “increase the number of sides by 1” and starting with a triangle, observe that the tops of the shapes alternate between a side and a vertex. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
		M04.B-O.3.1.2 Determine the missing elements in a function table (limit to +, –, or × and to whole numbers or money). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
		M04.B-O.3.1.3 Determine the rule for a function given a table (limit to +, –, or × and to whole numbers). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
			M05.B-O.2.1.1 Generate two numerical patterns using two given rules. Example: Given the rule “add 3” and the starting number 0 and given the rule “add 6” and the starting number 0, generate terms in the resulting sequences. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
			M05.B-O.2.1.2 Identify apparent relationships between corresponding terms of two patterns with the same starting numbers that follow different rules. Example: Given two patterns in which the first pattern follows the rule “add 8” and the second pattern follows the rule “add 2,” observe that the terms in the first pattern are 4 times the size of the terms in the second pattern. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
			M06.B-E.3.1.2 Analyze the relationship between the dependent and independent variables using graphs and tables and/or relate these to an equation. When grade appropriate instruction pertaining to an eligible content should begin	M06.B-E.3.1.2 Analyze the relationship between the dependent and independent variables using graphs and tables and/or relate these to an equation. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						M08.B-E.2.1.1 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						M08.B-E.2.1.2 Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
					M08.B-E.2.1.3 Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. When grade appropriate instruction pertaining to an eligible content should begin	M08.B-E.2.1.3 Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						M08.B-F.1.1.1 Determine whether a relation is a function. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
					M08.B-F.1.1.2 Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. When grade appropriate instruction pertaining to an eligible content should begin	M08.B-F.1.1.2 Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						M08.B-F.1.1.3 Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
				M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. When grade appropriate instruction pertaining to an eligible content should begin	M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. When grade appropriate instruction pertaining to an eligible content should begin	M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A1.2.1.2.1 Create, interpret and/or use the equation, graph or table of a linear function. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						A1.2.1.2.2 Translate from one representation of a linear function to another (graph, table and equation). When grade appropriate instruction pertaining to an eligible content should begin		A1.2.1.2.2 Translate from one representation of a linear function to another (graph, table and equation). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A1.2.2.1.1 Identify, describe and/or use constant rates of change. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						A1.2.2.1.2 Apply the concept of linear rate of change (slope) to solve problems. When grade appropriate instruction pertaining to an eligible content should begin		A1.2.2.1.2 Apply the concept of linear rate of change (slope) to solve problems. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A1.2.2.1.3 Write or identify a linear equation when given the graph of the line 2 points on the line, or the slope and a point on a line, (Linear equation may be in point-slope, standard and/or slope-intercept form). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A1.2.2.1.4 Determine the slope and/or y-intercept represented by a linear equation or graph. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
					M08.B-F.2.1.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally. When grade appropriate instruction pertaining to an eligible content should begin	M08.B-F.2.1.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
						A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically. When grade appropriate instruction pertaining to an eligible content should begin		A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A1.2.1.1.2 Determine if a relation is a function given a set of points or a graph. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
							A2.1.3.2.1 Determine how a change in one variable relates to a change in a second variable (e.g., y=4/x, if x doubles, what happens to y?). When grade appropriate instruction pertaining to an eligible content should begin	A2.1.3.2.1 Determine how a change in one variable relates to a change in a second variable (e.g., y=4/x, if x doubles, what happens to y?). When grade appropriate instruction pertaining to an eligible content should begin	A2.1.3.2.1 Determine how a change in one variable relates to a change in a second variable (e.g., y=4/x, if x doubles, what happens to y?). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A2.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern with a rule algebraically and/or graphically. When grade appropriate instruction pertaining to an eligible content should begin		A2.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern with a rule algebraically and/or graphically. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.1.1.2 Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, find the 20^th term). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.1.1.3 Determine the domain, range or inverse of a relation. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.1.1.4 Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increasing/decreasing, intercepts, zeros, and asymptotes). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.2.1.1 Create, interpret and/or use the equation, graph or table of a polynomial function (including quadratics). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.2.1.2 Create, interpret and/or use the equation, graph or table of an exponential or logarithmic function (including common and natural logarithms). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.2.1.3 Determine, use and/or interpret minimum and maximum values over a specified interval of a graph of a polynomial, exponential or logarithmic function. When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
										A2.2.2.1.4 Translate a polynomial, exponential or logarithmic function from one representation to another (graph, table and equation). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content
								A2.2.2.2.1 Identify or describe the effect of changing parameters within a family of functions (e.g., y = x² and y = x² + 3, or y = x²and y = 3x²). When grade appropriate instruction pertaining to an eligible content should begin		A2.2.2.2.1 Identify or describe the effect of changing parameters within a family of functions (e.g., y = x² and y = x² + 3, or y = x²and y = 3x²). When students are expected to demonstrate the knowledge, skills, and abilities described by an eligible content

Learning Progression

Algebraic Concepts: Functional Representations

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