Norwalk La Mirada Unified School District Common Core Math 1 Curriculum Map Semester One

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1 Norwalk La Mirada Unified School District Common Core Math 1 Curriculum Map Semester One Unit 1 - Relationships between Quantities and Review (Approximately 15 days) Overview: Students will understand and explain what the terms of an expression or equation mean in the context of a situation that it models, as linear equations and linear inequalities. Students build functions that model relationships between two quantities. This unit should also introduce students to sharing mathematical knowledge with a study team as they work together to solve problems. Essential Standards: N.Q.2; A.SSE.1a; A.REI.11 Supplementary Standards: N.Q.1,3; A.CED.1,4; A.REI.1,3 Understand how to pull out important information from given context (word problems) (N.Q.1) Interpret the meaning of each of the parts (terms, factors, coefficients, and constants) in terms of the context that they represent (A.SSE.1a) Write algebraic expressions and equations given a context (A.CED.1) Solve equations and inequalities for a variable (A.REI.3) Explain how to solve equations and inequalities for a variable (A.REI.1) Solve equations and inequalities from a given context (word problems) for a variable (N.Q.1-3) Rearrange a given formula in order to solve for a specific variable or quantity of interest (A.CED.4) Find and explain a solution from a graph. (A.REI.11) Solve equations and inequalities in one variable. Graph with technology and interpret the solutions in context. (A.REI.3.1,11 ) Understand and identify terms, factors, coefficients, constants, and expressions Equality Properties (Addition, Subtraction, Multiplication, Division) Operations with Integers Operations with fractions/decimals Textbook : 1.2, 1.3, 2.1, 2.2, 2.4, 2.5 Illustrative Mathematics Task: Ice Cream Van (N.Q.1) Mixing Candies (A.SSE.1a) The Bank Account (A.SSE.1a) Escalator (A.CED.2) Student Interactive Resource : Algebraic Expressions Millionaire Game Textbook : 1.1, 2.2, 2.3, 2.4, 2.5 Illustrative Mathematics Task: Reasoning with Linear Inequalities (A.REI.1 (3)) Students should be introduced to problem-based learning. Within this format, they will begin to use a team approach to learning mathematics. They will see that everyone is responsible to contribute to the completion of the problems. Team roles, and prompts for students to use them, should support this approach to learning.

2 Norms for working in study teams should be established starting with the first lesson. Development of effective team discussion and collaboration should be a focus throughout this unit. Students need to talk about the mathematics in order to learn it well, and lessons in this unit should help them become comfortable doing it. Solving is review for the students, so a there should be a focus in doing so in context and relating solutions to the x-values of intersections of the graphs of each side of the equation. Students are not expected to graph linear functions by hand yet. They may use technology to graph and check algebraic solutions. Technology in this Chapter: Students should be able use technology to create tables, draw graphs, and find intersections. Where is this Going? Students will focus on graphing linear functions by hand in Unit 3, and practice solving by graphing by hand. They will then return to intersections as solutions when they start graphing systems of equations by hand in Unit 4. Unit 2 - Functions (Approximately 15 days) Overview: In previous courses, students may have learned about relationships between two quantities that they could be graphed with a straight line. In this unit, they will explore nonlinear functions and learn how to describe functions completely. They will investigate the shapes and behaviors of several different nonlinear functions. Students will also perform simple operations on function Essential Standards: F.IF.1,2 Supplementary Standards: F.IF.4,5,9; F.BF.1b Find the domain and range from a given context (F.IF.1) Given a table or graph, determine whether or not it is a function (F.IF.1) Given a table of values or a graph of a function, identify the domain and range (F.IF.1) Evaluate a function with function notation over a given domain in order to determine the range (from an equation, graph, or context) (F.IF.2) Interpret key features from a graph (including intercepts, positive/ negative, increasing/ decreasing, max/min, and domain). (F.IF.4,5) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by descriptions) (F.IF.9) Perform the four basic operations on functions Understand relationships that can be graphed with a straight line Textbook : 3.2, 3.3, 3.4, 4.1, 4.2, 4.3 Illustrative Mathematics Task : Cell Phones Yam in the Oven Using Function Notation II Teacher Resource : Graphing Stories (Video + handout) A-CED Planes and Wheat (A.CED.2)

3 f(x) and g(x): o (f+g)(x) o (f g)(x) o (f g)(x) o (f g)(x) (F.BF.1b) Unit 2 is an introduction to the course on several levels. Students will focus on graphing and various functions. These lessons serve as a prelude to some of the mathematics in the course and use skills that the students bring to it from their previous studies. Students will be exposed very early in this course to real-life situations where the growth is not necessarily linear. In the next few lessons that follow, the main objective is for students to be able to fully describe the key elements of a nonlinear graph (shape, increasing/decreasing, intercepts, domain and range, maximum and minimum, endpoints, lines of symmetry, and so forth.) Students will also observe that different families of functions have graphs that are different shapes. By the end of this unit, students will not only see graphs for lines, parabolas, and exponentials, but they will also see square root, absolute-value, and cube root graphs. As they move into the latter part of the unit, students will determine whether a relationship defined by a graph, by a set of ordered pairs, or by a symbolic expression is a function. Then students will learn how to use function notation, and they will determine the domain and range of a function by looking at a graph. Finally, students will close the unit with simple function operations. Technology in this Chapter: Students should be able use technology to create tables, draw graphs, find intersections, and define functions using linear regression features. In this unit students will: model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning. Where is this Going? In Unit 3, students will expand on their knowledge of linear functions from their previous math course. Students will explore slope in more depth, investigate connections between multiple representations of a line, and write equations of lines from a variety of representations. In Unit 5 students will study arithmetic ( linear ) and geometric ( exponential ) sequences, which leads into a deeper investigation of exponential functions. Unit 3 - Linear Functions (Approximately 35 days) Overview: Unit 3 will focus on the starting value and growth of linear functions. You will look for connections between the multiple representations of linear functions: table, graph, equation, and situation. In this unit, you will come to a deeper understanding of slope than you may have had in previous courses, and you will explore the idea of slope as a rate of change. Essential Standards: A.REI.3.1,12; F.IF.7,7a; Supplementary Standards: A.CED.1,2,3,4; A.REI.10; F.BF.1a; F.IF.4,5,6 / Common Assessments Write an explicit linear function to represent Identifying Student Resources :

4 a given pattern from a context. (F.BF.1a) Create a table of values given an equation. (A.REI.10) Graph points from a table of values. (A.REI.10) Determine the approximate rate of change given a graph and interval(s) for linear functions. (F.IF.6) Calculate the rate of change for a linear function over an interval (from an equation or from a table of values), and explain what it means in terms of the context. (F.IF.6) Predict a rate of change for future intervals. (F.IF.6) Create an equation from context through use of tables and graphs. (A.CED.2) Rearrange a given formula in order to solve for a specific variable or quantity of interest (A.CED.4) Interpret key features from a graph (including intercepts, positive/ negative, increasing/ decreasing, and domain) for linear functions. (F.IF.4,5) Graph a linear equation in two variables (from the form y=mx+b) on a coordinate plane. (F.IF.7a) Graph linear functions, show and describe key features of the graph (F.IF.7, 7a) Create a graph that represents a given context and determine its key features for linear functions. (F.IF.4,5) Create and graph a linear equation in two variables from a given context (A.CED.2) Describe the appropriate units needed for a given situation/context. (N.Q.2) Interpret constraints within situations (Domain & Range) (A.CED.3) Identify constraints of a linear equation or inequality, within the context of a given situation (A.CED.3) Model a real-world context by writing linear equations and inequalities in one variable. Solve the equation and inequality for a quantity using appropriate units (A.CED.1) Interpret the solution of an inequality in terms of the context of the problem. (A.REI.3.1) Graph the solutions to a linear inequality in two variables (A.REI.12) domain/range, independent/depend ent variable Understand slope and y-intercept Know how to read a graph Understand how to solve for a variable Understand how to solve for a variable Graphing Linear Equations DESMOS Land the Plane (A.CED.2) DESMOS Polygraph Lines (A.CED.2) Game: Slope-Intercept Hoops (A.CED.2) Changing cost per minute (A.CED.2) DESMOS Marbleslides - Lines (A.CED.2) Illustrative Mathematics Tasks : Kitchen Floor Tiles (A.SSE.1) Graphing Stories + Video Clips (A.CED.2) Giving Raises (N.Q.2) Paying the Rent (A.CED.1) Bernardo and Sylvia play a game (A.CED.3) Teacher & Student Resource: (Lesson Plan using technology on Desmos (F.IF.6)): Charge! Teacher & Student Resource: (Lesson Plan using technology on Desmos): LEGO - Prices YummyMath 3-Act Task: The Biggest Lego Set Ever Teacher & Student Interactive Resource (Lesson Plan using technology on Geogebra): Graphing System of Linear Inequalities Owsiak Student Geogebra Interactive Resource: Graphing LInear Inequality Brezinski Teacher & Student Geogebra Interactive Resource (Lesson Plan using technology on Geogebra):

5 Graphing Linear Inequalities Beckwith Textbook : 5.1, 5.2, 5.3, 6.1, 6.2, 6.3, 6.4, 6.5, 7.1, 7.2, 7.3 In Unit 3, students will build on their study of linear functions from previous courses. The major theme of this unit is making connections between the multiple representations of linear functions: table, graph, equation, and situation. By the end of the unit, students will be able to convert readily between the multiple representations, indicating the starting value and growth in each representation. Students come to a deeper understanding of slope than they may have had in previous courses. Students start the unit understanding slope as the growth of geometric tile patterns. Then they move on to finding the slope using slope triangles on a graph. Students alternate between y = mx + b notation and function notation to write the equation of a line. They conclude that vertical lines are not functions. After learning how to find slope and build linear equations using multiple strategies, students will apply this knowledge in a variety of situations. Students investigate slope as speed. From there, they extend their knowledge of slopes to understand that it is a rate of change in several different real-life situations. Students will have developed algorithms, some of them algebraic, to find the equation of a line given any two pieces of information: the slope and y-intercept, the slope and a point on the line, or two points on the line. Students should be able to use technology to find the equations of lines from both graphs and tables. Students should practice finding the equation of a line through two points and recognize slope as a rate. Students should start getting comfortable with the graphing calculator technology that they will rely on later in the course. Technology in this Chapter: Students should be able use technology to create tables, draw graphs, and find intersections. In this unit students should: make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, look for and express regularity in repeated reasoning. Unit 4 will focus on algebraically solving problems using systems of equations. In Unit 5 students will study arithmetic ( linear ) and geometric ( exponential ) sequences, which leads into a deeper investigation of exponential functions, where students will complete the connections in a multiple representations web for exponential functions. Unit 4 - Systems of Linear Functions (Approximately 15 days) Overview: In Unit 3, you studied the connections between the multiple representations of linear equations and learned how to write equations from situations. In this chapter, you will learn how to solve word problems by writing a pair of equations, called a system of equations. Then you will solve the system of equations with the same multiple representations you used for solving linear equations: table, graph, and by manipulating the equations. Along the way, you will develop ways to solve different forms of systems, and will learn how to recognize when one method may be more efficient than another. By the end

6 of this unit, you will know multiple ways to find the point of intersection of two lines and will be able to solve systems that arise from different situations. Essential Standards: A.REI.6,11 Supplementary Standards: A.REI.12 Solve systems of linear equations exactly (algebraically, table, etc.) and approximately (graphs); focus on pairs of linear equations in two variables (A.REI.6) Explain why the x-coordinates of the points where the linear graphs of y = f ( x ) and y = g ( x ) intersect are the solutions of the equation f ( x ) = g ( x ). Find the solutions approximately: technology to graph, tables, successive approximations. *This is a modeling standard linking math to everyday life, work, and decision-making* (A.REI.11) Graph linear functions Interpret key features of a graph Textbook : 11.1,11.2,11.3,11.4 LearnZillion Video Lesson: Solve a system of equations using a table Lake Sonoma (F.IF.4) Tasks: Estimating a solution via Graphs (A.REI.6) Containers (F.IF.4) Fishing Adventures 3 (A.REI.12) Solution Sets (A.REI.12) Graph the solutions to a linear inequality in two variables as a half-plane; graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes (A.REI.12) Graph systems of linear equations Solve linear inequalities Textbook : 12.2 Unit 4 requires students to tie together their understanding of graphing, solving equations, systems of equations, and solutions. By the end of this unit, students will have three algebraic methods to solve systems of linear equations, as well as a better understanding of what a solution is and how to solve situational problems by writing and solving equations. This unit is divided into three sections: Section 1: In this section, students will learn how to write and solve one- and two-variable equations from word problems. They will also continue to solve systems of equations using the Equal Values Method from previous courses. When the equations are not in y = mx + b form, students will first solve both equations for the same variable (often y) using what they learned in Unit 3. This will motivate the need for the Substitution Method. Section 2: To avoid the messiness that can be caused by using the Equal Values Method for systems that are not in y = mx + b form, students will develop the Substitution Method. This method will help students solve systems when only one equation is solved for a variable. Likewise, the Elimination Method will be introduced as a way to solve systems when neither equation is solved for a variable. Each new method will be analyzed conceptually and will be applied to situational descriptions. Students will focus on which method is the most efficient and most likely to lead to an accurate solution for each type of system. Section 3: Review and extend content both from Units 1 through 3 and from content in the first sections of Unit 4. Require students to make important connections between equations, graphs, tables, and systems of

7 equations. In this unit students will: make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure. While the focus of this unit is limited to systems of linear equations, students will later study systems of non-linear equations in subsequent courses. Students will continue to write and solve linear equations in the context of arithmetic sequences in Unit 5 and do so to compare linear functions to exponential functions and geometric series. Semester Two Unit 5 - Exponential vs Linear (Approximately days) Overview: Unit 5 provides an opportunity to review and strengthen algebra skills while the students learn about arithmetic and geometric sequences. Early in the unit, students will find yourself using familiar strategies such as looking for patterns and making tables to write algebraic equations describing sequences of numbers. A little later in the chapter, students will develop shortcuts for writing equations for certain kinds of sequences. They will then learn more about the family of exponential functions. Students will also build more advanced algebra skills, such as solving for an indicated variable, simplifying or rewriting exponential expressions, and finding the exponential function that passes exactly through any pair of given points. Finally, students will learn about several important applications of exponential functions. Essential Standards: A.CED.1; A.REI.3; F.IF.3,4,7e Supplementary Standards: F.IF.2,5,6,9;F.BF.1a,1b,2; F.LE.1a,1b,1c,.2,3,5 Students will learn what sequences are and will become familiar with two important types of sequences: arithmetic and geometric (F.IF.3) Write and use a recursive formula from a given context (F.BF.1a) Find a specific term(s) or a missing term(s) in a given sequence (F.BF.2) Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms (F.BF.2) Use a recursive formula in order to create a table of values and a graph (F.IF.3) Calculate the rate of change for linear and exponential functions over an interval *This is a modeling standard linking math to everyday life, work, and decision-making* (F.IF.6) Exponent rules, especially zero and negative exponent simplification Textbook : 3.3, 3.4, 4.1,4.2,4.3, 14.1, 14.2, 14.3, 14.4, 14.5, 15.1, 15.2, 15.3, 15.4 Teacher Resource: (Lesson Plan with manipulatives) Teaching Sequences Illustrative Mathematics Task: Illegal Fish Population from A Saturating Exponential

8 Recognize situations where a quantity changes at a constant rate (linear) or percent rate (exponential) per unit interval *This is a modeling standard linking math to everyday life, work, and decision-making* (F.LE.1b,c) Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly *This is a modeling standard linking math to everyday life, work, and decision-making* (F.LE.3) Prove that over equal intervals, linear functions grow by equal differences while exponential functions grow by equal factors *This is a modeling standard linking math to everyday life, work, and decision-making* (F.LE.1a) Identify and interpret the parameters in a linear or exponential function in terms of a context *This is a modeling standard linking math to everyday life, work, and decision-making* (F.LE.5) Relate the domain of linear and exponential functions to their graphs *This is a modeling standard linking math to everyday life, work, and decision-making* (F.IF.5) Students will recognize the connections between arithmetic and geometric sequences and linear and exponential functions (F.IF.3) Given the equation of a linear or exponential function from a context, evaluate the function for a specific input value and interpret the results in terms of the context (F.IF.2) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by descriptions) (F.IF.9) Interpret key features from a graph (including intercepts, positive/negative, increasing/decreasing, and domain) Include linear and exponential functions *This is a modeling standard linking math to everyday life, work, and decision-making* (F.IF.4) Graph an exponential function from a table of values, identify intercepts and asymptotes (F.IF.7e) Write an exponential function that describes a relationship between two quantities (F.BF.1) Create linear and exponential functions from a graph or a table of values (F.LE.2) Create exponential equations in one variable and use them to solve problems (A.CED.1) Explain why the x-coordinates of the points where the linear graphs of y = f ( x ) and y = g ( x ) intersect are the solutions of the equation f ( x ) =

9 g ( x ). Find the solutions approximately: technology to graph, tables, successive approximations. (A.REI.11) Students will investigate exponential growth in the context and eventually compare linear growth with exponential growth. Students should start with an investigation of a mix of sequences which they are asked to categorize into different families by looking at the multiple representations of a sequence: the sequence itself, a table, and a graph. Students should learn the vocabulary of sequences as they write the equation for arithmetic sequences and represent arithmetic sequences with recursive equations. Students will make the connection between arithmetic sequences and the linear functions they studied in Unit 3. The equation t(n) = a + d n for the nth term of an arithmetic sequence is a direct analogy to the general equation of a linear function, f (x) = mx + b. However, sequences commonly start with n = 1 as the first term. That means students will have to work backwards in a sequence to find the initial value or starting point in the equation where n = 0. Students should compare the growth of a linear and exponential sequence, and conclude that exponential growth will always exceed the other two. Students will write the equations of exponential sequences, both explicitly and recursively. The domain of sequences should be compared to the domain of the corresponding function. This chapter has these main objectives. Students will: Enhance their understanding of exponential functions through multiple representations (tables, graphs, and equations) and applications. Distinguish between the growth in linear situations and exponential situations. Model situations using step functions. Begin to use the properties of exponents to transform expressions for exponential functions. Interpret negative exponents in an everyday situation. Learn how to graph exponential functions and use them to model everyday situations and solve problems. Learn how to find exponential equations when given two points. In this unit students will: make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, look for and express regularity in repeated reasoning. Overview: Unit 6 - Rigid Transformations and Congruence (Approximately days) Essential Standards: Supplementary Standards:

10 Overview: Unit 7 - Connecting Algebra with Geometry (Approximately days) Essential Standards: F.BF.3;G.CO.2,5 Supplementary Standards: Understand the concept of a parent function, particularly for linear and exponential functions. Relate rigid transformations of geometric figures to transformations of linear and exponential parent functions in the coordinate plane. (F.BF.3,G.CO.2) Graph linear and exponential functions by hand and then describe how the resulting graph has been transformed from the parent function (F.BF.3,G.CO.5) Overview: Unit 8 - Statistical Models (Approximately days) Essential Standards: Supplementary Standards: