Expressions, Equations and Function Families in Secondary Mathematics

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1 Purpose of this document: This document displays how standards from Early Equations and Expressions in Elementary School progress across 6-12 mathematics to Functions in High School. The need to differentiate between common standards in the high school courses is what prompted the development of this document. It is to help clarify how the learning should progress for students from the middle grades across Math I, II and III. The major work of high school comes from the Algebra and Functions conceptual categories. It is important to examine the standards across a larger spectrum to gain the full picture of the standards progression for teachers at all levels of secondary mathematics; therefore the progression is being examined across 6-12 mathematics. Note that this progression document represents only one possible progression and it does not fully represent the scope of the work done in high school. For example, the work of functions crosses over into the Geometry and Statistics & Probability conceptual categories and is not necessarily represented in this document. How to use this document: This resource is intended to help schools and districts construct curriculum for their students that follow the learning progression of the middle and high school standards. The document displays the standards progressions within and between grade levels and courses. This helps to support the idea of NOT teaching the standards in isolation and to show how standards can be grouped to maximize understanding and learning for students. Defining Progressions Progressions describe a sequence of increasing sophistication in understanding and skill within an area of study. The CCSS- M requires an understanding of three core shifts in mathematics teaching and learning; greater focus on fewer topics, coherence within and across grades by linking topics in and across grade bands and a rigorous pursuit of conceptual understanding, procedural fluency and application. Three types of progressions 1. Learning progressions A learning progression is a road or pathway that students travel as they progress toward mastery of the skills needed for career and college readiness. Each road follows a route composed of a collection of building blocks that are defined by the content standards for a subject. Learning progressions are based on research on student learning. 2. Standards progressions The standards progressions support analysis of standards across grades and conceptual categories. The progressions within domains are being elaborated by the writers of the common core state standards in narrative form to show how standards connect to each other within and across grade levels and conceptual categories. The progressions are built into the standards to build upon each other as students move up in mathematics. 3. Task progressions A rich mathematical task can be reframed or resized to serve different mathematical goals. As students are exposed to more mathematical ideas, tools and strategies their approaches to problem- solving should increase along with their understanding of the big ideas illustrated in these tasks. Consequently, students from different grade levels should be able to solve similar rich mathematical task based on their exposure to different mathematical ideas.

2 Expressions Numeric and Algebraic Expressions Complex Algebraic Expressions Evaluating numerical expressions Writing and interpreting algebraic expressions Identifying and generating equivalent expressions Students extend the work from 5 th grade (5.OA.2) where they learned to write and interpret numerical expressions. At this level students will use order of operations to evaluate and generate expressions. This extends into expressions using variables. Using and connecting properties of operations to simplify expressions. Interpreting equivalent expressions in context At this level, students focus more extensively with algebraic expressions (general linear expressions) with rational coefficients. Students extend their understanding of order of operations using linear expressions. Expressions with radicals, integers and rational exponents Scientific notation Students will add the properties of integer exponents to their repertoire of rules for transforming expressions. Simplify, rewrite and interpret polynomial expressions o Add/subtract linear and quadratic polynomial expressions o Factor quadratic expressions (to reveal the roots; which are to later be connected to the zeros of a quadratic function) o Multiply two linear expressions Simplify, rewrite and interpret exponential expressions with rational exponents (Note: Rational exponents are limited to a numerator of 1 in Math 1). Students work with more complex algebraic expressions. As the complexity of expressions increases, students see them building from basic operations: they see expressions as sums of terms and products of factors. 2 Students begin to see how the structure of an expression reveals various characteristics in terms of the context of a problem. This in turn allows them to use equivalent expressions in problem solving. Students should apply previous work with the laws of exponents to rewrite expressions such as (1.05) 2x as ( ) x or to simplify expressions such as (4x 2 y 3 ) 3 = 64x 6 y 9. Simplify, rewrite and interpret polynomial expressions o Add/subtract expressions o Multiply expressions resulting in at most a cubic expression Simplify, rewrite and interpret exponential expressions including expressions with non- integer exponents. The introduction of rational exponents and systematic practice with the properties of exponents in high school widens the field of operations for manipulating expressions. Simplify, rewrite, and interpret polynomial expressions (including the completing the square for quadratic expressions) Simplify and rewrite rational expressions Students should develop fluency with manipulating expressions with a variety of algebraic expression types. 6- EE.2a,b 8- EE.1 N- RN.2 6- EE.2c 7- EE.1 A- APR.1 A- SSE.1a,b 6- EE.3 7- EE.2 A- SSE.3a A- SSE.3c A- SSE.3b F- IF.8a 6- EE.4 A- SSE.2 6- EE.6 A- APR.6; A- APR.7

3 Equations/Inequalities Write and solve one- step equations One- step equations with nonnegative rational numbers The work in 6 th grade mathematics should really connect expressions to equations. Students should view equations as potentially equivalent expressions (may be true for some values of the variable.) 6- EE.5 6- EE.7 Write, solve and interpret multi- step equations based on real- world situations. Multi- step equations posed with positive and negative rational numbers. Equations work at this level should evolve to more than one- step and connect to the work with rational numbers. 7- EE.4a Analyze and solve multi- step equations given different solutions (one, none or many) 8- EE.7a, b Solve systems of equations by graphing and substitution. By 8 th grade, students have done an extensive amount of work with equations. They will apply those same principles to solving simultaneous equations in two- variables. This is the first occurrence of work with systems of equations. Graphical methods of solving systems should be given great consideration with a connection to how algebraic methods help us to find accurate solutions without technology. 8- EE.8a,b,c One- variable Equations Solve linear and exponential equations to include solving equations based on contextual situations and identifying parts of equations within the given context. Solve one- variable equations by graphing each side of the equation separately and looking for a point of intersection. Note: At this level quadratic equations are NOT solved; however students should be able to factor a quadratic expression to reveal the roots (factors) of that expression so that a connection can be made to the zeros of a function. In middle school mathematics, one- Solve quadratic equations (by inspection, square roots, factoring, quadratic variable equations/inequalities focused on formula*, and graphing) primarily linear forms. The focus in Math I Solve exponential equations using tables, graphs and common logs should be on solving more intricate one- Use right triangle trigonometry to solve problems (to include trigonometric variable equations/inequalities (still ratios and the Pythagorean Theorem) primarily linear). Students should be able The focus in Math II expands to algebraic Solve quadratic equations to justify their reasoning. Methods should methods of solving quadratic equations include, but not limited to, algebraic, Solve exponential equations (with the exception of completing the graphical and tabular methods. While Solve simple rational* and radical square) and solving exponential equations there is strong emphasis on linear equations. using common logs. Recognizing the equations/inequalities; an introduction to At this level, the work with one- variable existence of non- real solutions is expected exponential and quadratic equations is a equations extends to more complex at this level; however students will not be part of the Math I standards. Algebraic equations. Completing the square is expected to work within the complex methods are not yet explored in Math I for added as a method of solving quadratic number system. In addition, right triangle quadratic and exponential equations. The equations along with the development trigonometry is used to solve problems and work in high school mathematics extends of the quadratic formula. The work with rational equations are introduced via to other one- variable equations, for exponential equations will expand to the inverse variation. example quadratic, exponential, rational, use of natural logs in Math III. and radical equations to name a few. A- CED.1, A- SSE.2 A- SSE.3a A- REI.4b A- REI.4a; N- CN.7 A- REI.11 A- APR.4 A- CED.2; G- SRT.8 Simultaneous Equations The work in Math I will extend to other algebraic methods of solving systems of linear equations. Additionally solving systems of linear inequalities is introduced. A- REI.5 A- REI.6 Students will continue to apply what was learned in MS and/or Math 1 with systems of equations expanding to systems of equations comprised of linear and quadratic equations and/or linear and inverse variation equations (for example, y! = x + 2 and y! =!! ) A- REI.7 A- REI.11 A- CED.3 Math III continues the work with systems with no limit to the combination of types of equations that constitute the system.

4 Functions Representing two quantities in real- world problems. Analyzing the relationship between dependent and independent variables. Two- variable Equations Functions and Function Families The work in 7 th grade is with ratio, Linear functions Linear functions, exponential functions (integer inputs) and quadratic functions (standard form) proportion and unit rate which Determine a function based on a Writing functions to describe relationships between quantities; this includes combining functions, and explicit and recursive will eventually lead to work with table, graph, equation, verbal (NOW NEXT) forms of a function linear equations and direct description, etc. Transformations of functions variation. Generate, interpret and compare Comparison of functions (This is part of a different learning functions. trajectory) 6- EE.9 In elementary grades, students describe patterns and express relationships between quantities. These ideas become semiformal in 8 th grade with an introduction to the concept of function. While function notation and formal vocabulary (domain, range, etc.) is held off until high school, students will understand the concept of a function and distinguish between linear and non- linear functions. Thus they will compare functions using a variety of representations (table, graphs, verbal descriptions, etc.) 8- F.1 8- F.5 8- F.4 8- F.2 Function notation is formally introduced in Math I as well as the formal vocabulary associated with functions. Students will continue their understanding by making connections between verbal, written and graphical depictions of the same function. Combining basic linear functions is an expectation in Math I. Students will explore basic transformations: f (x + k), f (x) + k of linear and exponential functions. Transformations are limited to horizontal and vertical translations of a function. F- IF.1 F- LE.1a- c Simple Inverse variation function Power (simple) functions Square root and cube root functions Absolute value functions Exponential functions Piece- wise defined Trigonometric functions (degrees only) In Math II, the work of linear, quadratic and exponential functions is continued and extends into more types of functions and combinations of functions to create other functions. For example, f(x) + g(x) where f(x) is a quadratic function and g(x) is a linear function. Additionally, relationships that are not limited to a specific model of a common function family can be included in Math II. The work of transformations of functions expands to include vertical stretches and compressions. F- IF.2 F- IF.5 F- IF.4 F- IF.7e F- BF.1a,b F- BF.3 Polynomial functions Logarithmic functions Trigonometric functions (specifically sine and cosine functions) At this level, sine and cosine functions should be the focus with the caveat that tangent functions do not model natural phenomena; therefore are rarely used for modeling. Transformation work extends to add horizontal stretches and compressions. Formal recursion function notation occurs at the Math III level. F- TF.2

5 Works Cited Confrey, J., Maloney, A., Nguyen, K., Lee, K. S., Panorkou, N., & Corley, A. (n.d.). Retrieved from Turn- On Common Core Math: Learning Trajectories for the Common Core State Standards for Mathematics: Cooney, T. J., Beckmann, S., & Lloyd, G. M. (2010). Developing Essential Understanding of Functions for Teaching Mathematics in Grades Reston: The National Council of Teachers of Mathematics, Inc. Lloyd, G., Herbel- Eisenmann, B., & Star, J. R. (2011). Developing Essential Understanding of Expressions, Equations & Expressions for Teaching Mathematics in Grades 6-8. Reston: National Council of Teachers of Mathematics. Standards (6-8 Expressions and Equations). Retrieved from Institute for Mathematics and Education: pg. 11 Standards (HS Algebra). Retrieved from Institute for Mathematics and Education: content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf, pg. 4 Standards (HS Functions). Retrieved from Institute for Mathematics Education: content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf