Ohio's Academic Content Standards in Mathematics
Patterns, Functions and Algebra StandardStudents 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.
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Prekindergarten |
| 1. | Sort, order and classify objects by one attribute (e.g., size, color, shape, use). (ORC Resources) | | 2. | Identify, copy, extend and create simple patterns or sequences of sound, shapes and motions in the context of daily activities and play. (ORC Resources) | | 3. | Use play, physical materials or drawings to model a simple problem (e.g., There are 6 cookies to be shared by 3 children. How many cookies can each child receive?). (ORC Resources) | | 4. | Model a problem situation using physical materials. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Kindergarten |
| 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. (ORC Resources) | | 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. (ORC Resources) | | 3. | Describe orally the pattern of a given sequence. (ORC Resources) | | 4. | Model a problem situation using physical materials. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 1 |
| 1. | Sort, classify and order objects by two or more attributes, such as color and shape, and explain how objects were sorted. (ORC Resources) | | 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. (ORC Resources) | | 3. | Describe orally the basic unit or general plan of a repeating or growing pattern. (ORC Resources) | | 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). (ORC Resources) | | 5. | Describe orally and model a problem situation using words, objects or number phrase or sentence. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 2 |
| 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. (ORC Resources) | | 2. | Use patterns to make generalizations and predictions; e.g., determine a missing element in a pattern. (ORC Resources) | | 3. | Create new patterns with consistent rules or plans, and describe the rule or general plan of existing patterns. (ORC Resources) | | 4. | Use objects, pictures, numbers and other symbols to represent a problem situation. (ORC Resources) | | 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 =  +  ... (ORC Resources) | | 6. | Use symbols to represent unknown quantities and identify values for symbols in an expression or equation using addition and subtraction; e.g., +  = 10, - 2 = 4. (ORC Resources) | | 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. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 3 |
| 1. | Extend multiplicative and growing patterns, and describe the pattern or rule in words. (ORC Resources) | | 2. | Analyze and replicate arithmetic sequences with and without a calculator. (ORC Resources) | | 3. | Use patterns to make predictions, identify relationships, and solve problems. (ORC Resources) | | 4. | Model problem situations using objects, pictures, tables, numbers, letters and other symbols. (ORC Resources) | | 5. | Write, solve and explain simple mathematical statements, such as 7 + > 8 or + 8 = 10. (ORC Resources) | | 6. | Express mathematical relationships as equations and inequalities. (ORC Resources) | | 7. | Create tables to record, organize and analyze data to discover patterns and rules. (ORC Resources) | | 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. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 4 |
| 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. (ORC Resources) | | 2. | Represent and analyze patterns and functions using words, tables and graphs. (ORC Resources) | | 3. | Construct a table of values to solve problems associated with a mathematical relationship. (ORC Resources) | | 4. | Use rules and variables to describe patterns and other relationships. (ORC Resources) | | 5. | Represent mathematical relationships with equations or inequalities. (ORC Resources) | | 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. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 5 |
| 1. | Justify a general rule for a pattern or a function by using physical materials, visual representations, words, tables or graphs. (ORC Resources) | | 2. | Use calculators or computers to develop patterns, and generalize them using tables and graphs. (ORC Resources) | | 3. | Use variables as unknown quantities in general rules when describing patterns and other relationships. (ORC Resources) | | 4. | Create and interpret the meaning of equations and inequalities representing problem situations. (ORC Resources) | | 5. | Model problems with physical materials and visual representations, and use models, graphs and tables to draw conclusions and make predictions. (ORC Resources) | | 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. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 6 |
| 1. | Represent and analyze patterns, rules and functions, using physical materials, tables and graphs. (ORC Resources) | | 2. | Use words and symbols to describe numerical and geometric patterns, rules and functions. (ORC Resources) | | 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. (ORC Resources) | | 4. | Solve simple linear equations and inequalities using physical models, paper and pencil, tables and graphs. (ORC Resources) | | 5. | Produce and interpret graphs that represent the relationship between two variables. (ORC Resources) | | 6. | Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations. (ORC Resources) | | 7. | Identify and describe situations with constant or varying rates of change, and compare them. (ORC Resources) | | 8. | Use technology to analyze change; e.g., use computer applications or graphing calculators to display and interpret rate of change. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 7 |
| 1. | Represent and analyze patterns, rules and functions with words, tables, graphs and simple variable expressions. (ORC Resources) | | 2. | Generalize patterns by describing in words how to find the next term. (ORC Resources) | | 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. (ORC Resources) | | 4. | Create visual representations of equation-solving processes that model the use of inverse operations. (ORC Resources) | | 5. | Represent linear equations by plotting points in the coordinate plane. (ORC Resources) | | 6. | Represent inequalities on a number line or a coordinate plane. (ORC Resources) | | 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. (ORC Resources) | | 8. | Use formulas in problem-solving situations. (ORC Resources) | | 9. | Recognize a variety of uses for variables; e.g., placeholder for an unknown quantity in an equation, generalization for a pattern, formula. (ORC Resources) | | 10. | Analyze linear and simple nonlinear relationships to explain how a change in one variable results in the change of another. (ORC Resources) | | 11. | Use graphing calculators or computers to analyze change; e.g., distance-time relationships. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 8 |
| 1. | Relate the various representations of a relationship; i.e., relate a table to graph, description and symbolic form. (ORC Resources) | | 2. | Generalize patterns and sequences by describing how to find the nth term. (ORC Resources) | | 3. | Identify functions as linear or nonlinear based on information given in a table, graph or equation. (ORC Resources) | | 4. | Extend the uses of variables to include covariants where y depends on x. (ORC Resources) | | 5. | Use physical models to add and subtract monomials and polynomials, and to multiply a polynomial by a monomial. (ORC Resources) | | 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. (ORC Resources) | | 7. | Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems. (ORC Resources) | | 8. | Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems. (ORC Resources) | | 9. | Solve linear equations and inequalities graphically, symbolically and using technology. (ORC Resources) | | 10. | Solve 2 by 2 systems of linear equations graphically and by simple substitution. (ORC Resources) | | 11. | Interpret the meaning of the solution of a 2 by 2 system of equations; i.e., point, line, no solution. (ORC Resources) | | 12. | Solve simple quadratic equations graphically; e.g., y = x² - 16. (ORC Resources) | | 13. | Compute and interpret slope, midpoint and distance given a set of ordered pairs. (ORC Resources) | | 14. | Differentiate and explain types of changes in mathematical relationships, such as linear vs. nonlinear, continuous vs. noncontinuous, direct variation vs. inverse variation. (ORC Resources) | | 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. (ORC Resources) | | 16. | Use graphing calculators or computers to analyze change; e.g., interest compounded over time as a nonlinear growth pattern. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 9 |
| 1. | Define function with ordered pairs in which each domain element is assigned exactly one range element. (ORC Resources) | | 2. | Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations. (ORC Resources) | | 3. | Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations. (ORC Resources) | | 4. | Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words. (ORC Resources) | | 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. (ORC Resources) | | 6. | Write and use equivalent forms of equations and inequalities in problem situations; e.g., changing a linear equation to the slope-intercept form. (ORC Resources) | | 7. | Use formulas to solve problems involving exponential growth and decay. (ORC Resources) | | 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. (ORC Resources) | | 9. | Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology. (ORC Resources) | | 10. | Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology. (ORC Resources) | | 11. | Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only). (ORC Resources) | | 12. | Simplify rational expressions by eliminating common factors and applying properties of integer exponents. (ORC Resources) | | 13. | Model and solve problems involving direct and inverse variation using proportional reasoning. (ORC Resources) | | 14. | Describe the relationship between slope and the graph of a direct variation and inverse variation. (ORC Resources) | | 15. | Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 10 |
| 1. | Define function formally and with f(x) notation. (ORC Resources) | | 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. (ORC Resources) | | 3. | Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height. (ORC Resources) | | 4. | Use algebraic representations and functions to describe and generalize geometric properties and relationships. (ORC Resources) | | 5. | Solve simple linear and nonlinear equations and inequalities having square roots as coefficients and solutions. (ORC Resources) | | 6. | Solve equations and inequalities having rational expressions as coefficients and solutions. (ORC Resources) | | 7. | Solve systems of linear inequalities. (ORC Resources) | | 8. | Graph the quadratic relationship that defines circles. (ORC Resources) | | 9. | Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals. (ORC Resources) | | 10. | Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions. (ORC Resources) | | 11. | Solve real-world problems that can be modeled, using systems of linear equations and inequalities. (ORC Resources) | | 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. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 11 |
| 1. | Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest. (ORC Resources) | | 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. (ORC Resources) | | 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. (ORC Resources) | | 4. | Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology. (ORC Resources) | | 5. | Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis, x-axis or y = x. (ORC Resources) | | 6. | Represent the inverse of a function symbolically and graphically as a reflection about y = x. (ORC Resources) | | 7. | Model and solve problems with matrices and vectors. (ORC Resources) | | 8. | Solve equations involving radical expressions and complex roots. (ORC Resources) | | 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). (ORC Resources) | | 10. | Describe the characteristics of the graphs of conic sections. (ORC Resources) | | 11. | Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation. (ORC Resources) |
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| PreK | K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| Grade 12 |
| 1. | Analyze the behavior of arithmetic and geometric sequences and series as the number of terms increases. (ORC Resources) | | 2. | Translate between the numeric and symbolic form of a sequence or series. (ORC Resources) | | 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. (ORC Resources) | | 4. | Represent the inverse of a transcendental function symbolically. (ORC Resources) | | 5. | Set up and solve systems of equations using matrices and graphs, with and without technology. (ORC Resources) | | 6. | Make arguments about mathematical properties using mathematical induction. (ORC Resources) | | 7. | Make mathematical arguments using the concepts of limit. (ORC Resources) | | 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. (ORC Resources) | | 9. | Translate freely between polar and Cartesian coordinate systems. (ORC Resources) | | 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. (ORC Resources) |
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