Standard 3 : Algebra (Mathematics)

Print


View this standard as a learning progression
Conceptual Strand
The Algebra Standard emphasizes relationships among quantities, including functions, patterns, ways of representing mathematical relationships, and the analysis of change.
Guiding Question
How does algebra encompass the relationships among quantities, the use of symbols, the modeling of phenomena, and the mathematical study of change?
 
Grade K
Grade Learning Expectations
GLE 0006.3.1  Identify, duplicate, and extend simple number patterns and sequential and growing patterns.
GLE 0006.3.2  Recognize attributes (such as color, shape, size) and patterns (such as repeated pairs, bilateral symmetry).
GLE 0006.3.3  Describe qualitative change.
Checks for Understanding
0006.3.1  Use a variety of manipulatives (such as connecting cubes, number cards, shapes) to create patterns.
0006.3.2  Name, copy, and extend patterns.
0006.3.3  Translate simple patterns into rules.
0006.3.4  Sort, order and classify objects by attribute and identify objects that do not belong in a particular group.
0006.3.5  Describe change in attributes according to qualitative criteria such as longer/shorter, colder/warmer, heavier/lighter.
State Performance Indicators
 
Grade 1
Grade Learning Expectations
GLE 0106.3.1  Identify, describe, and extend simple number patterns to develop strategies for adding and subtracting whole numbers.
GLE 0106.3.2  Understand that addition and subtraction are inverse operations.
GLE 0106.3.3  Extend the strategies for basic facts to include other properties of number and operations.
Checks for Understanding
0106.3.1  Find repeating patterns on the number line, addition table, and hundreds chart.
0106.3.2  Determine a reasonable next term in a given sequence and describe the rule.
0106.3.3  Use objects to illustrate the commutative property with basic facts and show that subtraction is not commutative.
0106.3.4  Demonstrate understanding of the basic equation a + b = c by using objects to illustrate the number sentences (fact families) associated with any particular sum.
0106.3.5  Use various strategies to find unknowns in problems involving addition and subtraction.
0106.3.6  Use objects to demonstrate the inverse relationship between addition and subtraction.
0106.3.7  Use the inverse relation between addition and subtraction to check arithmetic problems.
0106.3.8  Determine whether a number is odd or even by pairing objects.
0106.3.9  Recognize that zero is the identity element for addition.
State Performance Indicators
 
Grade 2
Grade Learning Expectations
GLE 0206.3.1  Develop pattern recognition.
GLE 0206.3.2  Extend knowledge of the properties of numbers and operations to multiplication.
GLE 0206.3.3  Solve simple arithmetic problems using various methods.
GLE 0206.3.4  Describe quantitative change.
Checks for Understanding
0206.3.1  Given rules, complete tables to reveal both arithmetic and geometric patterns.
0206.3.2  Given a description, extend or find a missing term in a pattern or sequence.
0206.3.3  Record and study patterns in lists of numbers created by repeated addition or subtraction.
0206.3.4  Generalize the patterns resulting from the addition, subtraction and multiplication of combinations of odd and even numbers.
0206.3.5  Understand and use the commutative and associative properties of addition and multiplication.
0206.3.6  Relate repeated addition to multiplication.
0206.3.7  Find unknowns in number sentences and problems involving addition, subtraction and multiplication.
0206.3.8  Describe change in measures according to quantitative criteria such as growing 2 inches in one year.
State Performance Indicators
 
Grade 3
Grade Learning Expectations
GLE 0306.3.1  Develop meaning for and apply the commutative, associative, and distributive properties using various representations.
GLE 0306.3.2  Develop understanding that a letter or a symbol can represent an unknown quantity in a simple mathematical expression/equation.
GLE 0306.3.3  Describe and analyze patterns and relationships in contexts.
GLE 0306.3.4  Create and represent patterns using words, tables, graphs, and symbols.
Checks for Understanding
0306.3.1  Show that addition and multiplication are commutative operations.
0306.3.2  Show that subtraction and division are not commutative operations.
0306.3.3  Use commutative, associative, and distributive properties to multiply whole numbers.
0306.3.4  Solve problems using the commutative, associative, and distributive properties.
0306.3.5  Find unknowns in number sentences and problems involving addition, subtraction, multiplication, or division.
0306.3.6  Analyze patterns in words, tables, and graphs to draw conclusions.
0306.3.7  Create different representations of a pattern given a verbal description.
0306.3.8  Analyze patterns in quantitative change resulting from computation.
State Performance Indicators
SPI 0306.3.1  Verify a conclusion using algebraic properties.
SPI 0306.3.2  Express mathematical relationships using number sentences/equations.
SPI 0306.3.3  Find the missing values in simple multiplication and division equations.
SPI 0306.3.4  Describe or extend (including finding missing terms) geometric and numeric patterns.
 
Grade 4
Grade Learning Expectations
GLE 0406.3.1  Extend understanding of a variable to equations involving whole numbers, fractions, decimals, and/or mixed numbers.
GLE 0406.3.2  Use mathematical language and modeling to develop descriptions, rules and extensions of patterns.
GLE 0406.3.3  Translate between different forms of representations of whole number relationships.
Checks for Understanding
0406.3.1  Find an unknown quantity in simple equations using whole numbers, fractions, decimals, and mixed numbers.
0406.3.2  Translate between symbols and words to represent quantities in expressions or equations.
0406.3.3  Create, explain and use a rule to generate terms of a pattern or sequence.
0406.3.4  Translate between symbolic, numerical, verbal, or pictorial representations of a whole number pattern or relationship.
State Performance Indicators
SPI 0406.3.1  Use letters and symbols to represent an unknown quantity and write a simple mathematical expression
SPI 0406.3.2  Make generalizations about geometric and numeric patterns.
SPI 0406.3.3  Represent and analyze patterns using words, function tables, and graphs.
 
Grade 5
Grade Learning Expectations
GLE 0506.3.1  Understand and use order of operations.
GLE 0506.3.2  Develop and apply the concept of variable.
GLE 0506.3.3  Understand and apply the substitution property.
GLE 0506.3.4  Solve single-step linear equations and inequalities.
Checks for Understanding
0506.3.1  Evaluate an expression by substituting non-negative rational number values for letter variables in the expression.
0506.3.2  Use variables appropriately to represent numbers whose values are not yet known.
0506.3.3  Solve single-step linear equations using inverse operations.
0506.3.4  Solve single-step linear inequalities and graph solutions on a number line.
0506.3.5  Determine if a given value is a solution to a linear equation/inequality.
0506.3.6  Recognize there are many numbers between any two whole numbers on the number line.
State Performance Indicators
SPI 0506.3.1  Evaluate algebraic expressions involving decimals and fractions using order of operations.
SPI 0306.3.2  Evaluate multi-step numerical expressions involving fractions using order of operations.
SPI 0506.3.3  Find the unknown in single-step equations involving fractions and mixed numbers.
SPI 0506.3.4  Given a set of values, identify those that make an inequality a true statement.
 
Grade 6
Grade Learning Expectations
GLE 0606.3.1  Write and solve two-step equations and inequalities.
GLE 0606.3.2  Interpret and represent algebraic relationships with variables in expressions, simple equations and inequalities.
GLE 0606.3.3  Extend order of operations to include grouping symbols and exponents.
GLE 0606.3.4  Use expressions, equations and formulas to solve problems.
GLE 0606.3.5  Use multiple representations including symbolic algebra to model and/or solve contextual problems that involve linear relationships.
GLE 0606.3.6  Understand and use the Cartesian coordinate system.
Checks for Understanding
0606.3.1  Write and solve two-step linear equations corresponding to given situations (non-negative numbers only).
0606.3.2  Write and solve one-step inequalities corresponding to given situations (non-negative numbers only).
0606.3.3  Recognize the use of juxtaposition (such as 3x, ab) to stand for multiplication, and the convention in these cases of writing numbers before letters.
0606.3.4  Generate data and graph relationships concerning measurement of length, area, volume, weight, time, temperature, money, and information.
0606.3.5  Use the commutative, associative and distributive properties to show that two expressions are equivalent.
0606.3.6  Use equations to describe simple relationships shown in a table or graph.
0606.3.7  Move fluently between different representations (such as verbal, tabular, numerical, algebraic, and graphical) of equations and expressions.
0606.3.8  Represent patterns using words, graphs, and simple symbolic notation.
0606.3.9  Write a contextual story modeled by a given graph.
0606.3.10  Understand that in an ordered pair (x, y), the x represents horizontal location and y represents vertical location.
0606.3.11  Identify the quadrant of the coordinate system in which a point lies.
State Performance Indicators
SPI 0606.3.1  Represent on a number line the solution of a linear inequality.
SPI 0606.3.2  Use order of operations and parentheses to simplify expressions and solve problems.
SPI 0606.3.3  Write equations that correspond to given situations or represent a given mathematical relationship.
SPI 0606.3.4  Rewrite expressions to represent quantities in different ways.
SPI 0606.3.5  Translate between verbal expressions/sentences and algebraic expressions/equations.
SPI 0606.3.6  Solve two-step linear equations using number sense, properties, and inverse operations.
SPI 0606.3.7  Use algebraic expressions and properties to analyze numeric and geometric patterns.
SPI 0606.3.8  Select the qualitative graph that models a contextual situation (e.g., water filling then draining from a bathtub).
SPI 0606.3.9  Graph ordered pairs of integers in all four quadrants of the Cartesian coordinate system.
 
Grade 7
Grade Learning Expectations
GLE 0706.3.1  Recognize and generate equivalent forms for simple algebraic expressions.
GLE 0706.3.2  Understand and compare various representations of relations and functions.
GLE 0706.3.3  Understand the concept of function as a rule that assigns to a given input one and only one number (the output).
GLE 0706.3.4  Use function notation where f(x) represents the output that the function f assigns to the input x.
GLE 0706.3.5  Understand and graph proportional relationships.
GLE 0706.3.6  Conceptualize the meanings of slope using various interpretations, representations, and contexts.
GLE 0706.3.7  Use mathematical models involving linear equations to analyze real-world phenomena.
GLE 0706.3.8  Use a variety of strategies to efficiently solve linear equations and inequalities.
Checks for Understanding
0706.3.1  Perform basic operations on linear expressions (including grouping, order of operations, exponents, simplifying and expanding).
0706.3.2  Represent and analyze mathematical situations using algebraic symbols.
0706.3.3  Identify a function from a written description, table, graph, rule, set of ordered pairs, and/or mapping.
0706.3.4  Make tables of inputs x and outputs f(x) for a variety of rules that include rational numbers (including negative numbers) as inputs.
0706.3.5  Plot points to represent tables of linear function values.
0706.3.6  Understand that the graph of a linear function f is the set of points on a line representing the ordered pairs (x, f(x)).
0706.3.7  Distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).
0706.3.8  Understand slope as the ratio of vertical change to horizontal change.
0706.3.9  Identify a function exhibiting a constant rate of change as a linear function and identify the slope as a unit rate.
0706.3.10  Solve problems involving unit rates (e.g., miles per hour, words per minute).
0706.3.11  Relate the features of a linear equation to a table and/or graph of the equation.
0706.3.12  Use linear equations to solve problems and interpret the meaning of slope, m, and the y-intercept, b, in f(x)= mx + b in terms of the context.
0706.3.13  Given a graph that exhibits the intersection of a line and the y-axis, write a linear function in slope-intercept form: y = mx + b.
0706.3.14  Understand that when solving linear inequalities, multiplication or division by a negative reverses the inequality symbol.
State Performance Indicators
SPI 0706.3.1  Evaluate algebraic expressions involving rational values for coefficients and/or variables.
SPI 0706.3.2  Determine whether a relation (represented in various ways) is a function.
SPI 0706.3.3  Given a table of inputs x and outputs f(x), identify the function rule and continue the pattern.
SPI 0706.3.4  Interpret the slope of a line as a unit rate given the graph of a proportional relationship.
SPI 0706.3.5  Represent proportional relationships with equations, tables and graphs.
SPI 0706.3.6  Solve linear equations with rational coefficients symbolically or graphically.
SPI 0706.3.7  Translate between verbal and symbolic representations of real-world phenomena involving linear equations.
SPI 0706.3.8  Solve contextual problems involving two-step linear equations.
SPI 0706.3.9  Solve linear inequalities in one variable with rational coefficients symbolically or graphically.
 
Grade 8
Grade Learning Expectations
GLE 0806.3.1  Recognize and generate equivalent forms for algebraic expressions.
GLE 0806.3.2  Represent, analyze, and solve problems involving linear equations and inequalities in one and two variables.
GLE 0806.3.3  Solve systems of linear equations in two variables.
GLE 0806.3.4  Translate among verbal, tabular, graphical and algebraic representations of linear functions.
GLE 0806.3.5  Use slope to analyze situations and solve problems.
GLE 0806.3.6  Compare and contrast linear and nonlinear functions.
Checks for Understanding
0806.3.1  Perform basic operations on algebraic expressions (including grouping, order of operations, exponents, square/cube roots, simplifying and expanding).
0806.3.2  Represent algebraic relationships with equations and inequalities.
0806.3.3  Solve systems of linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line.
0806.3.4  Understand the relationship between the graph of a linear inequality and its solutions.
0806.3.5  Solve linear inequalities in two variables (including those whose solutions require multiplication or division by a negative number).
0806.3.6  Identify x- and y-intercepts and slope of linear equations from an equation, graph or table.
0806.3.7  Analyze situations and solve problems involving constant rate of change.
0806.3.8  Recognize a proportion as a special case of a linear equation and understand that the constant of proportionality is the slope, and the resulting graph is a line through the origin.
0806.3.9  Given a function rule, create tables of values for x and y, and plot graphs of nonlinear functions.
0806.3.10  Distinguish quadratic and exponential functions as nonlinear using a graph and/or a table of values.
0806.3.11  Distinguish between the equations of linear, quadratic, and exponential functions (e.g. function families such as y=x2, y=2x, and y=2x).
0806.3.12  Understand how rates of change of nonlinear functions contrast with constant rates of change of linear functions.
0806.3.13  Represent situations and solve real-world problems using symbolic algebra.
State Performance Indicators
SPI 0806.3.1  Find solutions to systems of two linear equations in two variables.
SPI 0806.3.2  Solve the linear equation f(x) = g(x).
SPI 0806.3.3  Solve and graph linear inequalities in two variables.
SPI 0806.3.4  Translate between various representations of a linear function.
SPI 0806.3.5  Determine the slope of a line from an equation, two given points, a table or a graph.
SPI 0806.3.6  Analyze the graph of a linear function to find solutions and intercepts.
SPI 0806.3.7  Identify, compare and contrast functions as linear or nonlinear.
 
Algebra I
Course Learning Expectations
CLE 3102.3.1  Use algebraic thinking to analyze and generalize patterns.
CLE 3102.3.2  Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials.
CLE 3102.3.3  Understand and apply operations with rational expressions and equations.
CLE 3102.3.4  Solve problems involving linear equations and linear inequalities.
CLE 3102.3.5  Manipulate formulas and solve literal equations.
CLE 3102.3.6  Understand and use relations and functions in various representations to solve contextual problems.
CLE 3102.3.7  Construct and solve systems of linear equations and inequalities in two variables by various methods.
CLE 3102.3.8  Solve and understand solutions of quadratic equations with real roots.
CLE 3102.3.9  Understand and use exponential functions to solve contextual problems.
Checks for Understanding
3102.3.1  Recognize and extend arithmetic and geometric sequences.
3102.3.2  Explore patterns including Pascal’s Triangle and the Fibonacci sequence.
3102.3.3  Justify correct results of algebraic procedures using extension of properties of real numbers to algebraic expressions.
3102.3.4  Simplify expressions using exponent rules including negative exponents and zero exponents.
3102.3.5  Add, subtract, and multiply polynomials including squaring a binomial.
3102.3.6  Find the quotient of a polynomial and a monomial.
3102.3.7  Use various models (including area models) to represent products of polynomials.
3102.3.8  Find the GCF of the terms in a polynomial.
3102.3.9  Find two binomial factors of a quadratic expression.
3102.3.10  Add, subtract, multiply, and divide rational expressions and simplify results.
3102.3.11  Solve multi-step linear equations with one variable.
3102.3.12  Recognize and articulate when an equation has no solution, a single solution, or all real numbers as solutions.
3102.3.13  Solve multi-step linear inequalities with one variable and graph the solution on a number line.
3102.3.14  Solve absolute value equations and inequalities (including compound inequalities) with one variable and graph their solutions on a number line.
3102.3.15  Determine domain and range of a relation and articulate restrictions imposed either by the operations or by the real life situation that the function represents.
3102.3.16  Determine if a relation is a function from its graph or from a set of ordered pairs.
3102.3.17  Recognize “families” of functions.
3102.3.18  Analyze the characteristics of graphs of basic linear relations and linear functions including constant function, direct variation, identity function, vertical lines, absolute value of linear functions. Use technology where appropriate.
3102.3.19  Explore the characteristics of graphs of various nonlinear relations and functions including inverse variation, quadratic, and square root function. Use technology where appropriate.
3102.3.20  Understand that a linear equation has a constant rate of change called slope and represent slope in various forms.
3102.3.21  Determine the equation of a line using given information including a point and slope, two points, a point and a line parallel or perpendicular, graph, intercepts.
3102.3.22  Express the equation of a line in standard form, slope-intercept, and point-slope form.
3102.2.23  Determine the graph of a linear equation including those that depict contextual situations.
3102.2.24  Interpret the changes in the slope-intercept form and graph of a linear equation by looking at different values of the parameters, m and b.
3102.3.25  Find function values using f(x) notation or graphs.
3102.3.26  Graph linear inequalities on the coordinate plane and identify regions of the graph containing ordered pairs in the solution.
3102.3.27  Determine the number of solutions for a system of linear equations (0, 1, or infinitely many solutions).
3102.3.28  Solve systems of linear equations graphically, algebraically, and with technology.
3102.3.29  Solve contextual problems involving systems of linear equations or inequalities and interpret solutions in context.
3102.3.30  Solve quadratic equations using multiple methods: factoring, graphing, quadratic formula, or square root principle.
3102.3.31  Determine the number of real solutions for a quadratic equation including using the discriminant and its graph.
3102.3.32  Recognize the connection among factors, solutions (roots), zeros of related functions, and x-intercepts in equations that arise from quadratic functions.
3102.3.33  Recognize data that can be modeled by an exponential function.
3102.3.34  Graph exponential functions in the form y = a(bx) where b ? 0.
3102.3.35  Apply growth/decay and simple/compound interest formulas to solve contextual problems.
State Performance Indicators
SPI 3102.3.1  Express a generalization of a pattern in various representations including algebraic and function notation.
SPI 3102.3.2  Operate with polynomials and simplify results.
SPI 3102.3.3  Factor polynomials.
SPI 3102.3.4  Operate with, evaluate, and simplify rational expressions including determining restrictions on the domain of the variables.
SPI 3102.3.5  Write and/or solve linear equations, inequalities, and compound inequalities including those containing absolute value.
SPI 3102.3.6  Interpret various relations in multiple representations.
SPI 3102.3.7  Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
SPI 3102.3.8  Determine the equation of a line and/or graph a linear equation.
SPI 3102.3.9  Solve systems of linear equation/inequalities in two variables.
SPI 3102.3.10  Find the solution of a quadratic equation and/or zeros of a quadratic function.
SPI 3102.3.11  Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situation.
 
Geometry
Course Learning Expectations
CLE 3108.3.1  Use analytic geometry tools to explore geometric problems involving parallel and perpendicular lines, circles, and special points of polygons.
CLE 3108.3.2  Explore the effect of transformations on geometric figures and shapes in the coordinate plane.
Checks for Understanding
3108.3.1  Prove two lines are parallel, perpendicular, or oblique using coordinate geometry.
3108.3.2  Connect coordinate geometry to geometric figures in the plane (e.g. midpoints, distance formula, slope, and polygons).
3108.3.3  Find the equation of a circle given its center and radius and vice versa.
3108.3.4  Apply the midpoint and distance formulas to points and segments to find midpoints, distances, and missing information in two and three dimensions.
3108.3.5  Use mapping notation to identify the image of a transformation given the coordinates of the pre-image.
3108.3.6  Identify a transformation given its mapping notation.
State Performance Indicators
SPI 3108.3.1  Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles).
SPI 3108.3.2  Use coordinate geometry to prove characteristics of polygonal figures.
SPI 3108.3.3  Describe algebraically the effect of a single transformation (reflections in the x- or y-axis, rotations, translations, and dilations) on two-dimensional geometric shapes in the coordinate plane.
 
Algebra II
Course Learning Expectations
CLE 3103.3.1  Understand and apply properties of rational exponents and perform basic operations to simplify algebraic expressions.
CLE 3103.3.2  Understand, analyze, transform and generalize mathematical patterns, relations and functions using properties and various representations.
CLE 3103.3.3  Analyze and apply various methods to solve equations, absolute values, inequalities, and systems of equations over complex numbers.
CLE 3103.3.4  Graph and compare equations and inequalities in two variables. Identify and understand the relationships between the algebraic and geometric properties of the graph.
CLE 3103.3.5  Use mathematical models involving equations and systems of equations to represent, interpret and analyze quantitative relationships, change in various contexts, and other real-world phenomena.
Checks for Understanding
3103.3.1  Perform operations on algebraic expressions and justify the procedures.
3103.3.2  Determine the domain of a function represented in either symbolic or graphical form.
3103.3.3  Determine and graph the inverse of a function with and without technology.
3103.3.4  Analyze the effect of changing various parameters on functions and their graphs.
3103.3.5  Graph piece-wise and step functions.
3103.3.6  Simplify expressions and solve equations containing radicals.
3103.3.7  Solve quadratic equations by factoring, graphing, completing the square, extracting square roots and using the quadratic formula.
3103.3.8  Solve a three by three system of linear equations algebraically and by using inverse matrices and determinants with and without technology.
3103.3.9  Find an equation for a parabola when given its graph or when given its roots.
3103.3.10  Given a quadratic equation use the discriminant to determine the nature of the roots.
3103.3.11  Describe and articulate the characteristics and parameters of a parent function.
3103.3.12  Understand the relationship between real zeros of a function and the x-intercepts of its graph.
3103.3.13  Solve problems using exponential functions requiring the use of logarithms for their solutions.
3103.3.14  Define and use arithmetic and geometric sequences and series including using sigma and pi notation.
3103.3.15  Find the sum of an geometric series whose common ratio, r, is in the interval (-1,1).
3103.3.16  Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
3103.3.17  Know that the logarithm and exponential functions are inverses and use this information to solve real-world problems.
3103.3.18  Solve compound inequalities involving disjunction and conjunction and linear inequalities containing absolute values.
3103.3.19  Solve linear programming problems.
3103.3.20  Understand the relationships between the equations of conic sections and their graphs.
3103.3.21  Factor polynomials using a variety of methods including the factor theorem, synthetic division, long division, sums and differences of cubes, and grouping.
3103.3.22  Determine the number and possible types of zeros for a polynomial function and find the rational roots.
3103.3.23  Understand the connection between the roots, zeros, x-intercepts, factors of polynomials, and solutions of polynomial equations.
State Performance Indicators
SPI 3103.3.1  Add, subtract and multiply polynomials; divide a polynomial by a lower degree polynomial.
SPI 3103.3.2  Solve quadratic equations and systems, and determine roots of a higher order polynomial.
SPI 3103.3.3  Add, subtract, multiply, divide and simplify rational expressions including those with rational and negative exponents.
SPI 3103.3.4  Use the formulas for the general term and summation of finite arithmetic and both finite and infinite geometric series.
SPI 3103.3.5  Describe the domain and range of functions and articulate restrictions imposed either by the operations or by the contextual situations which the functions represent.
SPI 3103.3.6  Combine functions (such as polynomial, rational, radical and absolute value expressions) by addition, subtraction, multiplication, division, or by composition and evaluate at specified values of their variables.
SPI 3103.3.7  Identify whether a function has an inverse, whether two functions are inverses of each other, and/or explain why their graphs are reflections over the line y = x.
SPI 3103.3.8  Solve systems of three linear equations in three variables.
SPI 3103.3.9  Graph the solution set of two or three linear or quadratic inequalities.
SPI 3103.3.10  Identify and/or graph a variety of functions and their transformations.
SPI 3103.3.11  Graph conic sections (circles, parabolas, ellipses and hyperbolas) and understand the relationship between the standard form and the key characteristics of the graph.
SPI 3103.3.12  Interpret graphs that depict real-world phenomena.
SPI 3103.3.13  Solve contextual problems using quadratic, rational, radical and exponential equations, finite geometric series or systems of equations.
SPI 3103.3.14  Solve problems involving the binomial theorem and its connection to Pascal’s Triangle, combinatorics, and probability.