Expressions, Equations and Function Families in Secondary Mathematics 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. Revised 9/4/2014 Expressions, Equations and Function Families in Secondary Mathematics Expressions Middle School Mathematics 6th Grade 7th Grade Numeric and Algebraic Expressions 8th Grade • Evaluating numerical expressions • Writing and interpreting algebraic expressions • Identifying and generating equivalent expressions th
Students extend the work from 5 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. 6-­‐EE.2a,b 6-­‐EE.2c 7-­‐EE.1 6-­‐EE.3 7-­‐EE.2 6-­‐EE.4 6-­‐EE.6 Revised 9/4/2014 High School Mathematics 8-­‐EE.1 Math I Math II Complex Algebraic Expressions Math III • 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 • Simplify, rewrite and interpret polynomial expressions expressions. As the complexity of o Add/subtract expressions expressions increases, students see them o Multiply expressions resulting in at most a cubic expression building from basic operations: they see • Simplify, rewrite and interpret exponential expressions including expressions expressions as sums of terms and products with non-­‐integer exponents. 2
of factors. Students begin to see how the The introduction of rational exponents • Simplify, rewrite, and interpret structure of an expression reveals various and systematic practice with the polynomial expressions (including characteristics in terms of the context of a properties of exponents in high school the completing the square for problem. This in turn allows them to use widens the field of operations for quadratic expressions) equivalent expressions in problem solving. manipulating expressions. • Simplify and rewrite rational Students should apply previous work with expressions the laws of exponents to rewrite expressions 2x
2 x
Students should develop fluency with manipulating expressions with a variety of algebraic expression types. such as (1.05) as (1.05 ) or to simplify 2 3 3
6 9
expressions such as (4x y ) = 64x y . N-­‐RN.2 A-­‐SSE.3a A-­‐APR.1 A-­‐SSE.1a,b A-­‐SSE.3c F-­‐IF.8a A-­‐SSE.2 A-­‐SSE.3b A-­‐APR.6; A-­‐APR.7 Expressions, Equations and Function Families in Secondary Mathematics Equations/Inequalities 6th Grade Middle School Mathematics 7th Grade 8th Grade High School Mathematics Math I Math II Math III 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. th
The work in 6 grade mathematics should • Write, solve and interpret multi-­‐step equations based on real-­‐world situations. In middle school mathematics, one-­‐
• Solve quadratic equations (by inspection, square roots, factoring, quadratic really connect expressions to equations. variable equations/inequalities focused on formula*, and graphing) • Multi-­‐step equations posed with positive and negative rational numbers. Students should view equations as primarily linear forms. The focus in Math I • Solve exponential equations using tables, graphs and common logs potentially equivalent expressions (may should be on solving more intricate one-­‐
• Use right triangle trigonometry to solve problems (to include trigonometric be true for some values of the variable.) variable equations/inequalities (still ratios and the Pythagorean Theorem) primarily linear). Students should be able Equations work at this level should The focus in Math II expands to algebraic • Analyze and solve multi-­‐step equations • Solve quadratic equations to justify their reasoning. Methods should evolve to more than one-­‐step and methods o
f s
olving q
uadratic e
quations given different solutions (one, none or • Solve exponential equations include, but not limited to, algebraic, connect to the work with rational (with the exception of completing the many) • Solve simple rational* and radical graphical and tabular methods. While numbers. square) and solving exponential equations equations. there is strong emphasis on linear • Write and solve one-­‐step equations • One-­‐step equations with nonnegative rational numbers 6-­‐EE.5 6-­‐EE.7 7-­‐EE.4a Revised 9/4/2014 using common logs. Recognizing the existence of non-­‐real solutions is expected at this level; however students will not be expected to work within the complex number system. In addition, right triangle trigonometry is used to solve problems and rational equations are introduced via inverse variation. equations/inequalities; an introduction to exponential and quadratic equations is a part of the Math I standards. Algebraic methods are not yet explored in Math I for quadratic and exponential equations. The work in high school mathematics extends to other one-­‐variable equations, for example quadratic, exponential, rational, and radical equations to name a few. A-­‐CED.1, A-­‐SSE.2 A-­‐REI.4b A-­‐REI.11 A-­‐SSE.3a 8-­‐EE.7a, b • Solve systems of equations by graphing and substitution. A-­‐CED.2; G-­‐SRT.8 Simultaneous Equations • Create and solve systems of linear • Create and solve systems of linear equations and inequalities by equations and inequalities by algebraic and graphical methods
algebraic and graphical methods At this level, the work with one-­‐variable equations extends to more complex equations. Completing the square is added as a method of solving quadratic equations along with the development of the quadratic formula. The work with exponential equations will expand to the use of natural logs in Math III. A-­‐REI.4a; N-­‐CN.7 A-­‐APR.4 • Create and solve systems of linear equations and inequalities by algebraic and graphical methods th
By 8 grade, students have done an The work in Math I will extend to other Students will continue to apply what was Math III continues the work with extensive amount of work with equations. algebraic methods of solving systems of learned in MS and/or Math 1 with systems systems with no limit to the combination They will apply those same principles to linear equations. Additionally solving of equations expanding to systems of of types of equations that constitute the solving simultaneous equations in two-­‐
systems of linear inequalities is introduced. equations comprised of linear and quadratic system. variables. This is the first occurrence of work equations and/or linear and inverse with systems of equations. Graphical variation equations (for example, 𝑦! = 𝑥 +
!
methods of solving systems should be given 2 and 𝑦! = ) !
great consideration with a connection to how algebraic methods help us to find accurate solutions without technology. A-­‐REI.5 8-­‐EE.8a,b,c A-­‐REI.6 A-­‐REI.7 A-­‐REI.11 A-­‐CED.3 Expressions, Equations and Function Families in Secondary Mathematics Functions Middle School Mathematics th
6 Grade th
High School Mathematics th
7 Grade 8 Grade Two-­‐variable Equations th
• Linear functions • Representing two quantities in • The work in 7 grade is with ratio, • Determine a function based on a real-­‐world problems. proportion and unit rate which table, graph, equation, verbal • Analyzing the relationship will eventually lead to work with description, etc. between dependent and linear equations and direct • Generate, interpret and compare independent variables. variation. functions. • (This is part of a different learning In e
lementary grades, students describe trajectory) patterns and express relationships between quantities. These ideas become th
semiformal in 8 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.) 6-­‐EE.9 8-­‐F.1 8-­‐F.5 Math I •
•
•
•
Math II Math III Functions and Function Families Linear functions, exponential functions (integer inputs) and quadratic functions (standard form) Writing functions to describe relationships between quantities; this includes combining functions, and explicit and recursive (NOW⇒NEXT) forms of a function Transformations of functions Comparison of functions 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 8-­‐F.4 8-­‐F.2 Revised 9/4/2014 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 • Polynomial functions and exponential functions is continued and • Logarithmic functions extends into more types of functions and • Trigonometric functions (specifically combinations of functions to create other sine and cosine functions) functions. For example, f(x) + g(x) where At this level, sine and cosine functions f(x) is a quadratic function and g(x) is a should be the focus with the caveat that linear function. Additionally, relationships tangent functions do not model natural that are not limited to a specific model of a phenomena; therefore are rarely used for common function family can be included in modeling. Transformation work extends Math II. The work of transformations of to add horizontal stretches and functions expands to include vertical compressions. Formal recursion function stretches and compressions. notation occurs at the Math III level. F-­‐IF.2 F-­‐IF.5 F-­‐IF.4 F-­‐IF.7e F-­‐BF.1a,b F-­‐BF.3 F-­‐TF.2 Expressions, Equations and Function Families in Secondary Mathematics 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: http://www.turnonccmath.net/index.php Cooney, T. J., Beckmann, S., & Lloyd, G. M. (2010). Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9-­‐12. 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. The Common Core Standards Writing Team. (2014, March 31). Progression Documents for the Common Core Math Standards (6-­‐8 Expressions and Equations). Retrieved from Institute for Mathematics and Education: http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_ee_2011_04_25.pdf, pg. 11 The Common Core Standards Writing Team. (2014, March 31). Progression Documents for the Common Core Math Standards (HS Algebra). Retrieved from Institute for Mathematics and Education: http://commoncoretools.me/wp-­‐content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf, pg. 4 The Common Core Standards Writing Team. (2014, March 31). Progression Documents for the Common Core Math Standards (HS Functions). Retrieved from Institute for Mathematics Education: http://commoncoretools.me/wp-­‐
content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf Revised 9/4/2014