Functions. Essential Understandings and How These Important Ideas Relate to One Another
|
|
- Alfred Richardson
- 6 years ago
- Views:
Transcription
1 Functions Essential Understandings and How These Important Ideas Relate to One Another
2 Page 2
3 Opening Exercises 1. The function f has a domain of {9, 11, 13, 15} and a range of {8, 10, 12}. Could f be represented by {(9, 8), (11, 10), (13, 12)}? Justify your reasoning. 2. The graph below shows the circulation of newspapers in a town for integer years. According to the graph, is the circulation of newspapers a function of the years shown? Justify your reasoning. Page 3
4 Self-Assessment 5 Exemplary (I have exceeded that target.) 4 Accomplished (I have met the target.) 3 Emerging (I am more than half-way there.) 2 Beginning (I have started making progress.) 1 No Understanding (I have not yet started making progress.) Pre-assessment Learning Target Post-Assessment 1. I can articulate essential understandings about functions and how important mathematical ideas about functions relate to one another. 2. I can identify problems that align to the standards within the Functions conceptual category. Page 4
5 Norms of Collaboration (Garmston & Wellman, 1999) The following norms support a community of adult learning: 1. Promoting a Spirit of Inquiry 2. Pausing 3. Paraphrasing 4. Probing 5. Putting ideas on the table 6. Paying attention to self and others 7. Presuming positive intentions Page 5
6 Page 6
7 Part 1 The Concept of Function Page 7
8 Page 8
9 The concept of function is intentionally broad and flexible, allowing it to apply to a wide range of situations. The notion of concept encompasses many types of mathematical entities in addition to classical functions that describe quantities that vary continuously. For example, matrices and arithmetic and geometric sequences can be viewed as functions. ~Developing Essential Understanding of Functions, NCTM, 2010 Consider all possible circular animal pens enclosed by a length of fencing. For each length of fencing, there is a corresponding area that the fencing will enclose when it is formed into a circle. Describe the relationship between the area of a circular pen and the length of fencing that it takes to enclose the pen. Page 9
10 The following are quotes taken from different sources. Select one quote that strikes you as the most interesting or surprising. Share your thoughts with a partner. 1. A function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). ~Standard F-IF.1 2. A function is a correspondence between two sets, X and Y, in which each element of X is matched to one and only one element of Y. The set X is called the domain of the function. ~Algebra 1 Module 3 Lesson 9 3. By definition, functions are single-valued. In other words, for each element of the domain, there is exactly one element of the range of the function. ~Developing Essential Understanding of Functions, NCTM, Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of continuous variation. For example, sequences are functions. ~Developing Essential Understanding of Functions, NCTM, The domain and range of functions do not have to be numbers. ~Developing Essential Understanding of Functions, NCTM, Functions describe situations where one quantity determines another. Because we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models. ~ New York State P-12 Common Core Learning Standards for Mathematics 7. These ideas become semi-formal in Grade 8 with the introduction of the concept of function: a rule that assigns to each input exactly one output. ~Standard 8.F.1 8. Traditional pattern activities, where students are asked to continue a pattern through observation, are not a mathematical topic, and do not appear in the Standards in their own right. ~ Progressions for the Common Core State Standards, Time normally spent on exercises involving the vertical line test, or searching lists of ordered pairs to find two with the same x-coordinate and different y-coordinate, can be reallocated elsewhere. Indeed, the vertical line test is problematic. ~ Progressions for the Common Core State Standards, 2013 Page 10
11 Exercises 1. The table below shows the cost of parking in a 24-hour garage for a given number of hours 0 < t < 24. Does this correspondence represent a function? Justify your reasoning. 2. Is the relation y = g(x) = 2 x a function from the Real numbers to the Real numbers? Justify your reasoning. Page 11
12 3. Is the relation below a function from the Real numbers to the Real numbers? Justify your reasoning. x + 2 x 2 y = p(x) = { x x 0 4. Is y a function of x? Justify your reasoning. x x < 0 f(x) = { x x 0 5. Does each element of the domain correspond to exactly one element in the range in the following sequence? Justify your reasoning. f(1) = 2 f(n) = f(n 1) + 2n for all natural numbers n > 1 6. If f(x) = 1 x + 9, which statement is always true? 3 (1) f(x) < 0 (3) If x < 0, then f(x) < 0. (2) f(x) > 0 (4) If x > 0, then f(x) > 0. Page 12
13 Definition Facts/Characteristics Examples FUNCTION Nonexamples
14
15 Part 2 Interpreting Functions Page 15
16 Page 16
17 Interpreting Functions F-IF Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. Use the graph (for example, by marking specific points) to illustrate the statements in (a) (d). If possible, label the coordinates of any points you draw. a. f(0) = 2 b. f( 3) = f(3) = f(9) = 0 c. f(2) = g(2) d. g(x) > f(x) for x > 2 Page 17
18 Interpreting Functions F-IF Understand the concept of a function and use function notation. 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. This task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context. Let f(t) be the number of people, in millions, who own cell phones years after Explain the meaning of the following statements. a. f(10) = b. f(a) = 20 c. f(20) = b d. n = f(t) Page 18
19 Interpreting Functions F-IF Understand the concept of a function and use function notation. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. A sequence is simply a function whose domain is restricted. For example, arithmetic sequences are restrictions of linear functions to the positive integers, and geometric sequences are restrictions of exponential functions to the positive integers. These are two examples of sequences expressed using function notation. August 2014 Algebra 1 Regents Exam #24 If f(1) = 3 and f(n) = 2f(n 1) + 1, then f(5) = (1) 5 (3) 21 (2) 11 (4) 43 June 2014 Algebra 1 Regents Exam #21 A sunflower is 3 inches tall at week 0 and grows 2 inches each week. Which function(s) shown below can be used to determine the height, f(n), of the sunflower in n weeks? I. f(n) = 2n + 3 II. f(n) = 2n + 3(n 1) III. f(n) = f(n 1) + 2 where f(0) = 3 (1) I and II (3) III, only (2) II, only (4) I and III Page 19
20 Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. This task helps reinforce the idea that when a variable represents time t = 0, is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that. The figure shows the graph of T, the temperature (in degrees Fahrenheit) over one particular 20-hour period in Santa Elena as a function of time t. a. Estimate (14). b. If t = 0 corresponds to midnight, interpret what we mean by T(14) in words. c. Estimate the highest temperature during this period from the graph. d. When was the temperature decreasing? e. If Anya wants to go for a two-hour hike and return before the temperature gets over 80 degrees, when should she leave? Page 20
21 Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. June 2014 Algebra 1 Regents Exam #2 Officials in a town use a function, C, to analyze traffic patterns. C(n) represents the rate of traffic through an intersection where n is the number of observed vehicles in a specified time interval. What would be the most appropriate domain for the function? (1) { 2, 1, 0, 1, 2, 3, } (3) {0, 1 2, 1, 1 1 2, 2, } (2) { 2, 1, 0, 1, 2, 3} (4) {0, 1, 2, 3, } August 2014 Algebra 1 Regents Exam #24 The function h(t) = 16t represents the height, h(t), in feet, of an object from the ground at t seconds after it is dropped. A realistic domain for this function is (1) 3 t 3 (3) 0 h(t) 144 (2) 0 t 3 (4) all real numbers Algebra 1 Module 3 End of Module Assessment #3a A boy bought 6 guppies at the beginning of the month. One month later the number of guppies in his tank had doubled. His guppy population continued to grow in this same manner. His sister bought some tetras at the same time. The table below shows the number of tetras, t, after n months have passed since they bought the fish. Create a function g to model the growth of the boy s guppy population, where g(n) is the number of guppies at the beginning of each month and n is the number of months that have passed since he bought the 6 guppies. What is a reasonable domain for g in this situation? Page 21
22 Functions provide a means to describe how related quantities vary together. We can classify, predict, and characterize various kinds of relationships by attending to the rate at which one quantity varies with respect to the other. ~Developing Essential Understanding of Functions, NCTM, 2010 Consider the table of the function f below. What do you notice? What stands out? x f(x) Page 22
23 Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Algebra 1 Module 4 Lesson 10 Example 2e-f The table below represents the value of Andrew s stock portfolio, with V representing the value of the portfolio, in hundreds of dollars, and t is the time, in months, since he started investing. How fast is Andrew s stock value decreasing between [10, 12]? Find another two-month interval where the average rate of change is faster than [10, 12] and explain why. Are there other two-month intervals where the rate of change is same as [10, 12]? Explain your answer. t (months) V(t) (hundreds of dollars) Algebra 1 Module 4 End of Module Assessment #3d An arrow is shot into the air. A function representing the relationship between the number of seconds it is in the air, t, and the height of the arrow in meters, h, is given by: What is the average rate of change for the interval from t = 1 to t = 2 seconds? Compare your answer to the average rate of change for the interval from t = 2 to t = 3 seconds and explain the difference in the context of the problem. Page 23
24 Functions can be represented in multiple ways, including algebraic (symbolic), graphical, verbal, and tabular representations. Links among these different representations are important to studying relationships and change. ~Developing Essential Understanding of Functions, NCTM, 2010 Consider the relationship represented in four different ways below. When might each representation be more useful than the others? a. A movie theater has operating costs of $1025 per day. Tickets cost $7.50 each. The movie theater s profit each day depends on the number of tickets sold. b. P = 7.5T 1025 c. d. T P 0 $ $ $ $ $ $ $1225 Page 24
25 Interpreting Functions F-IF Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. August 2014 Algebra 1 Regents Exam #8 The value of the x-intercept for the graph of 4x 5y = 40 is (1) 10 (3) 4 5 (2) 4 5 (4) 8 June 2014 Algebra 1 Regents Exam #25 Draw the graph of y = x 1 on the set of axes below. Page 25
26 Interpreting Functions F-IF Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Suppose h(t) = 5t t + 3 is an expression giving the height of a diver above the water (in meters), t seconds after the diver leaves the springboard. A. How high above the water is the springboard? Explain how you know. B. When does the diver hit the water? C. At what time on the diver's descent toward the water is the diver again at the same height as the springboard? D. When does the diver reach the peak of the dive? Page 26
27 Interpreting Functions F-IF Analyze functions using different representations. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Algebra 1 Module 4 Lesson 22 Exit Ticket Two people, in two different apartment buildings, have buzzers that don t work. They both must throw the keys to their apartments out of the window to their guests, who will then use the keys to enter. Tenant one throws the keys such that the relationship between the height of the keys (in feet), and the time that has passed (in seconds) can be modeled by h(t) = 16t t + 9. Tenant two throws the keys such that the height/time relationship can be modeled by the graph below. Whose window is higher? Explain how you know. Page 27
28 Page 28
29 Part 3 Building Functions Page 29
30 Page 30
31 Building Functions F-BF Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. On June 1, a fast growing species of algae is accidentally introduced into a lake in a city park. It starts to grow and cover the surface of the lake in such a way that the area covered by the algae doubles every day. If it continues to grow unabated, the lake will be totally covered and the fish in the lake will suffocate. At the rate it is growing, this will happen on June 30. (a) When will the lake be covered half-way? (b) On June 26, a pedestrian who walks by the lake every day warns that the lake will be completely covered soon. Her friend just laughs. Why might her friend be skeptical of the warning? (c) On June 29, a clean-up crew arrives at the lake and removes almost all of the algae. When they are done, only 1% of the surface is covered with algae. How well does this solve the problem of the algae in the lake? Page 31
32 Functions can be combined by adding, subtracting, multiplying, dividing, and composing them. Functions sometimes have inverses. Functions can often be analyzed by viewing them as made from other functions. ~Developing Essential Understanding of Functions, NCTM, 2010 Consider the following question: What is the effect (on the graph of a function) of adding a constant function? Page 32
33 Building Functions F-BF Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Consider the functions listed below. What does x need to be in order for the argument to be 0? f(x) = x = 0 x = f(x + 1) = x + 1 = 0 x = f(x + 2) = x + 2 = 0 x = f(x + 3) = x + 3 = 0 x = f(x 1) = x 1 = 0 x = f(x 2) = x 2 = 0 x = f(x 3) = x 3 = 0 x = The graph of f(x + 4) is a shift. It translates the graph of f(x) units. The graph of f(x 5) is a shift. It translates the graph of f(x) units. In general, f(x + a) translates f(x). If a > 0, the graph slides and if a < 0, the graph slides. Can you justify why this is true? Page 33
34 Page 34
35 Part 4 Linear and Exponential Models Page 35
36 Page 36
37 Functions can be classified into different families of functions, each with its own unique characteristics. Different families can be used to model different real-world phenomena. ~Developing Essential Understanding of Functions, NCTM, 2010 Consider which function would best model each of the following: a. Gregory plans to purchase a video game player. He has $500 in his savings account, and plans to save $20 per week from his allowance until he has enough money to buy the player. He needs to figure out how long it will take. b. c. d. Margie got $1000 from her grandmother to start her college fund. She is opening a new savings account and finds out that her bank offers a 2% annual interest rate, compounded monthly. She wants to calculate the amount of money in the bank compounded monthly. e. The function that associates to the input x the output y given by y = 5(x + 5)(x 6). Page 37
38 Linear, Quadratic, & Exponential Models F-LE Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. August 2014 Algebra 1 Regents Exam #10 A population that initially has 20 birds approximately doubles every 10 years. Which graph represents this population growth? August 2014 Algebra 1 Regents Exam #12 Which situation could be modeled by using a linear function? (1) a bank account balance that grows at a rate of 5% per year, compounded annually (2) a population of bacteria that doubles every 4.5 hours (3) the cost of cell phone service that charges a base amount plus20 cents per minute (4) the concentration of medicine in a person s body that decays by a factor of one-third every hour Page 38
39 Linear, Quadratic, & Exponential Models F-LE 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). August 2014 Algebra 1 Regents Exam #16 The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is a 1, which is an equation for the nth term of this sequence? (1) a n = 8n + 10 (3) a n = 16n + 10 (2) a n = 8n 14 (4) a n = 16n 38 June 2014 Algebra 1 Regents Exam #15 The table below represents the function F. The equation that represents this function is (1) F(x) = 3 x (3) F(x) = 2 x + 1 (2) F(x) = 3x (4) F(x) = 2x + 3 Page 39
40 Linear, Quadratic, & Exponential Models F-LE 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Algebra 1 Module 3 Lesson 5 Opening Exercise Two equipment rental companies have different penalty policies for returning a piece of equipment late: Company 1: On day 1, the penalty is $5. On day 2, the penalty is $10. On day 3, the penalty is $15. On day 4, the penalty is $20 and so on, increasing by $5 each day the equipment is late. Company 2: On day 1, the penalty is $0.01. On day 2, the penalty is $0.02. On day 3, the penalty is $0.04. On day 4, the penalty is $0.08 and so on, doubling in amount each additional day late. Jim rented a digger from Company 2 because he thought it had the better late return policy. The job he was doing with the digger took longer than he expected, but it did not concern him because the late penalty seemed so reasonable. When he returned the digger 15 days late, he was shocked by the penalty fee. What did he pay, and what would he have paid if he had used Company 1 instead? Company 1 Company 2 Day Penalty Day Penalty Page 40
41 1. Which company has a greater 15 day late charge? 2. Describe how the amount of the late charge changes from any given day to the next successive day in both companies 1 and How much would the late charge have been after 20 days under Company 2? Page 41
42 Linear, Quadratic, & Exponential Models F-LE Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context. June 2014 Algebra 1 Regents Exam #7 A company that manufactures radios first pays a start-up cost, and then spends a certain amount of money to manufacture each radio. If the cost of manufacturing r radios is given by the function c(r) = 5.25r + 125, then the value 5.25 best represents (1) the start-up cost (2) the profit earned from the sale of one radio (3) the amount spent to manufacture each radio (4) the average number of radios manufactured June 2014 Algebra 1 Regents Exam #26 The breakdown of a sample of a chemical compound is represented by the function p(t) = 300(0.5) t, where p(t) represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent. Page 42
43 Exit Ticket: How have these sessions helped to deepen your understanding of functions? We appreciate your feedback, as it will help us reflect on our practice. What worked well? What needs improvement? What are you walking away with? What more do you need? Page 43
A Story of Functions Curriculum Overview
Rationale for Module Sequence in Algebra I Module 1: By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze
More informationALGEBRA I CCR MATH STANDARDS
RELATIONSHIPS BETWEEN QUANTITIES AND REASONING WITH EQUATIONS M.A1HS.1 M.A1HS.2 M.A1HS.3 M.A1HS.4 M.A1HS.5 M.A1HS.6 M.A1HS.7 M.A1HS.8 M.A1HS.9 M.A1HS.10 Reason quantitatively and use units to solve problems.
More informationAlgebra I Number and Quantity The Real Number System (N-RN)
Number and Quantity The Real Number System (N-RN) Use properties of rational and irrational numbers N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational
More informationMathematics Standards for High School Algebra I
Mathematics Standards for High School Algebra I Algebra I is a course required for graduation and course is aligned with the College and Career Ready Standards for Mathematics in High School. Throughout
More informationMathematics High School Functions
Mathematics High School Functions Functions describe situations where one quantity determines another. For example, the return on $10,000 invested at an annualized percentage rate of 4.25% is a function
More informationAlgebra I, Common Core Correlation Document
Resource Title: Publisher: 1 st Year Algebra (MTHH031060 and MTHH032060) University of Nebraska High School Algebra I, Common Core Correlation Document Indicates a modeling standard linking mathematics
More informationMathematics. Number and Quantity The Real Number System
Number and Quantity The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties
More informationMATHEMATICS COURSE SYLLABUS
Course Title: Algebra 1 Honors Department: Mathematics MATHEMATICS COURSE SYLLABUS Primary Course Materials: Big Ideas Math Algebra I Book Authors: Ron Larson & Laurie Boswell Algebra I Student Workbooks
More informationModel Traditional Pathway: Model Algebra I Content Standards [AI]
Model Traditional Pathway: Model Algebra I Content Standards [AI] Number and Quantity The Real Number System AI.N-RN A. Extend the properties of exponents to rational exponents. 1. Explain how the definition
More informationDublin City Schools Mathematics Graded Course of Study Algebra I Philosophy
Philosophy The Dublin City Schools Mathematics Program is designed to set clear and consistent expectations in order to help support children with the development of mathematical understanding. We believe
More informationAlgebra I. 60 Higher Mathematics Courses Algebra I
The fundamental purpose of the course is to formalize and extend the mathematics that students learned in the middle grades. This course includes standards from the conceptual categories of Number and
More informationMathematics High School Algebra I
Mathematics High School Algebra I All West Virginia teachers are responsible for classroom instruction that integrates content standards and mathematical habits of mind. Students in this course will focus
More informationTennessee s State Mathematics Standards - Algebra I
Domain Cluster Standards Scope and Clarifications Number and Quantity Quantities The Real (N Q) Number System (N-RN) Use properties of rational and irrational numbers Reason quantitatively and use units
More informationAlgebra 1. Mathematics Course Syllabus
Mathematics Algebra 1 2017 2018 Course Syllabus Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit The fundamental purpose of this course is to formalize
More informationSolving Quadratic Equations Using Multiple Methods and Solving Systems of Linear and Quadratic Equations
Algebra 1, Quarter 4, Unit 4.1 Solving Quadratic Equations Using Multiple Methods and Solving Systems of Linear and Quadratic Equations Overview Number of instructional days: 13 (1 day = 45 minutes) Content
More informationN-Q2. Define appropriate quantities for the purpose of descriptive modeling.
Unit 1 Expressions Use properties of rational and irrational numbers. N-RN3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number
More informationComparison of Virginia s College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics
Comparison of Virginia s College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics February 17, 2010 1 Number and Quantity The Real Number System
More informationLesson 3.notebook May 17, Lesson 2 Problem Set Solutions
Lesson 2 Problem Set Solutions Student Outcomes Lesson 3: Analyzing a Verbal Description > Students make sense of a contextual situation that can be modeled with linear, quadratic, and exponential functions
More informationBeal City High School Algebra 2A Curriculum and Alignment
Beal City High School Algebra 2A Curriculum and Alignment UNIT 1 Linear Functions (Chapters 1-3) 1. Combine like terms, solve equations, solve inequalities, evaluate expressions(1-2,3,4) 2. Solve an equation
More informationAlgebra I Curriculum Crosswalk
Algebra I Curriculum Crosswalk The following document is to be used to compare the 2003 North Carolina Mathematics Course of Study for Algebra I and the State s for Mathematics Algebra I course. As noted
More informationUnit 5: Representations of Linear Relations
Time Frame: Approximately 3-5 weeks Connections to Previous Learning: Students build upon previous understandings of linear equations and functions and apply them to various representations of linear relationships,
More informationALGEBRA I. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. (N-RN2)
ALGEBRA I The Algebra I course builds on foundational mathematical content learned by students in Grades K-8 by expanding mathematics understanding to provide students with a strong mathematics education.
More information3. If 4x = 0, the roots of the equation are (1) 25 and 25 (2) 25, only (3) 5 and 5 (4) 5, only 3
ALGEBRA 1 Part I Answer all 24 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. For each statement or question, choose the word or expression that,
More informationSequence of Algebra 1 Units Aligned with the California Standards
Sequence of Algebra 1 Units Aligned with the California Standards Year at a Glance Unit Big Ideas Math Algebra 1 Textbook Chapters Dates 1. Equations and Inequalities Ch. 1 Solving Linear Equations MS
More informationHigh School Programs. Math 2 II UNIT 2 OVERVIEW: Modeling with Quadratic Functions Parent Guide
Unit Outcomes At the end of this unit, your student should be able to: Determine whether an expression is a polynomial. Add and subtracting polynomials. Multiply up to three linear expressions. Create
More informationCommon Core State Standards: Algebra 1
Common Core State Standards: Number and Quantity Standards The Real Number System Extend the properties of exponents to rational exponents. N-RN.1 Explain how the definition of the meaning of rational
More informationIntegrated CME Project Mathematics I-III 2013
A Correlation of -III To the North Carolina High School Mathematics Math I A Correlation of, -III, Introduction This document demonstrates how, -III meets the standards of the Math I. Correlation references
More informationAlgebra I. 60 Higher Mathematics Courses Algebra I
The fundamental purpose of the course is to formalize and extend the mathematics that students learned in the middle grades. This course includes standards from the conceptual categories of Number and
More informationCalifornia Common Core State Standards for Mathematics Standards Map Algebra I
A Correlation of Pearson CME Project Algebra 1 Common Core 2013 to the California Common Core State s for Mathematics s Map Algebra I California Common Core State s for Mathematics s Map Algebra I Indicates
More informationThe School District of Palm Beach County Algebra 1 Honors Unit A: Data Analysis
Unit A: Data Analysis MAFS.912.S ID.1.1 MAFS.912.S ID.1.2 MAFS.912.S ID.1.3 MAFS.912.S ID.2.5 Calculator: Yes Mathematics Florida Represent data with plots on the real number line (dot plots, histograms,
More informationComplete Week 18 Package
Complete Week 18 Package Jeanette Stein Table of Contents Unit 4 Pacing Chart -------------------------------------------------------------------------------------------- 1 Day 86 Bellringer --------------------------------------------------------------------------------------------
More informationObservations Homework Checkpoint quizzes Chapter assessments (Possibly Projects) Blocks of Algebra
September The Building Blocks of Algebra Rates, Patterns and Problem Solving Variables and Expressions The Commutative and Associative Properties The Distributive Property Equivalent Expressions Seeing
More informationAlgebra I Sample Unit Outline
Algebra I Sample Unit Outline Organizing Theme Topic Unit 1: Intro. to Topic 2 Topic 3 Topic 4 Topic 5 Topic 6 Topic 7 Topic 8 Topic 9 Build functions that model situations Unit 1: Intro. to Data- Summarize,
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Modeling & Problem Solving Common Core Standard: A-REI.4a: Solve quadratic equations in one
More informationAlgebra , Martin-Gay
A Correlation of Algebra 1 2016, to the Common Core State Standards for Mathematics - Algebra I Introduction This document demonstrates how Pearson s High School Series by Elayn, 2016, meets the standards
More information0814AI Common Core State Standards
0814AI Common Core State Standards 1 Which statement is not always true? 1) The product of two irrational numbers is irrational. 2) The product of two rational numbers is rational. 3) The sum of two rational
More informationSequence of Algebra AB SDC Units Aligned with the California Standards
Sequence of Algebra AB SDC Units Aligned with the California Standards Year at a Glance Unit Big Ideas Math Algebra 1 Textbook Chapters Dates 1. Equations and Inequalities Ch. 1 Solving Linear Equations
More informationAlgebra 1 Standards Curriculum Map Bourbon County Schools. Days Unit/Topic Standards Activities Learning Targets ( I Can Statements) 1-19 Unit 1
Algebra 1 Standards Curriculum Map Bourbon County Schools Level: Grade and/or Course: Updated: e.g. = Example only Days Unit/Topic Standards Activities Learning Targets ( I 1-19 Unit 1 A.SSE.1 Interpret
More informationUnit 4. Exponential Function
Unit 4. Exponential Function In mathematics, an exponential function is a function of the form, f(x) = a(b) x + c + d, where b is a base, c and d are the constants, x is the independent variable, and f(x)
More informationStandard Description Agile Mind Lesson / Activity Page / Link to Resource
Publisher: Agile Mind, Inc Date: 19-May-14 Course and/or Algebra I Grade Level: TN Core Standard Standard Description Agile Mind Lesson / Activity Page / Link to Resource Create equations that describe
More informationThroughout Algebra I, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice:
In the three years prior to Algebra I, students have already begun their study of algebraic concepts. They have investigated variables and expressions, solved equations, constructed and analyzed tables,
More informationAlgebra 1 Syllabus
Algebra 1 Syllabus 2017-18 dennis_jenkins@crpusd.org Welcome to algebra, a course designed to prepare students for geometry and any other courses taken after it. Students are required by the state of California
More informationAlgebra 1 Yearlong Curriculum Plan. Last modified: June 2014
Algebra 1 Yearlong Curriculum Plan Last modified: June 2014 SUMMARY This curriculum plan is divided into four academic quarters. In Quarter 1, students first dive deeper into the real number system before
More informationCommon Core State Standards with California Additions 1 Standards Map. Algebra I
Common Core State s with California Additions 1 s Map Algebra I *Indicates a modeling standard linking mathematics to everyday life, work, and decision-making N-RN 1. N-RN 2. Publisher Language 2 Primary
More informationFUNCTIONS Families of Functions Common Core Standards F-LE.A.1 Distinguish between situations that can be
M Functions, Lesson 5, Families of Functions (r. 2018) FUNCTIONS Families of Functions Common Core Standards F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential
More informationSubject Algebra 1 Unit 1 Relationships between Quantities and Reasoning with Equations
Subject Algebra 1 Unit 1 Relationships between Quantities and Reasoning with Equations Time Frame: Description: Work with expressions and equations through understanding quantities and the relationships
More informationGSE Algebra 1. Unit Two Information. Curriculum Map: Reasoning with Linear Equations & Inequalities
GSE Algebra 1 Unit Two Information EOCT Domain & Weight: Equations 30% Curriculum Map: Reasoning with Linear Equations & Inequalities Content Descriptors: Concept 1: Create equations that describe numbers
More informationAlgebra I. Time Frame Standard Resources Notes. Page 1 of 22
Page 1 of 22 Module 1 4. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and
More informationMathematics Standards for High School Financial Algebra A and Financial Algebra B
Mathematics Standards for High School Financial Algebra A and Financial Algebra B Financial Algebra A and B are two semester courses that may be taken in either order or one taken without the other; both
More informationLesson 1: Multiplying and Factoring Polynomial Expressions
Lesson 1 Lesson 1: Multiplying and Factoring Polynomial Expressions When you multiply two terms by two terms you should get four terms. Why is the final result when you multiply two binomials sometimes
More informationMathematics. Algebra Course Syllabus
Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit Mathematics Algebra 1 2018 2019 Course Syllabus Algebra I formalizes and extends the mathematics students
More informationHonors Algebra I
emath Instruction Unit 3 emath Instruction emath Instruction Unit 1 Term 1 The Building Blocks of Algebra A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4
More informationMATHEMATICS Math I. Number and Quantity The Real Number System
MATHEMATICS Math I The high school mathematics curriculum is designed to develop deep understanding of foundational math ideas. In order to allow time for such understanding, each level focuses on concepts
More informationLINEAR EQUATIONS Modeling Linear Equations Common Core Standards
E Linear Equations, Lesson 1, Modeling Linear Functions (r. 2018) LINEAR EQUATIONS Modeling Linear Equations Common Core Standards F-BF.A.1 Write a function that describes a relationship between two quantities.
More informationUnit 3: Linear and Exponential Functions
Unit 3: Linear and Exponential Functions In Unit 3, students will learn function notation and develop the concepts of domain and range. They will discover that functions can be combined in ways similar
More informationHigh School Algebra I Scope and Sequence by Timothy D. Kanold
High School Algebra I Scope and Sequence by Timothy D. Kanold First Semester 77 Instructional days Unit 1: Understanding Quantities and Expressions (10 Instructional days) N-Q Quantities Reason quantitatively
More informationCurriculum Map Algebra I Quarter 1
Quarter 1 How can algebra describe the relationship between sets of numbers? Algebra Creating Equations AI.A.CED.1 * Create equations and inequalities in one variable and use them to solve problems. Include
More informationSequenced Units for Arizona s College and Career Ready Standards MA27 Algebra I
Sequenced Units for Arizona s College and Career Ready Standards MA27 Algebra I Year at a Glance Semester 1 Semester 2 Unit 1: Solving Linear Equations (12 days) Unit 2: Solving Linear Inequalities (12
More informationEighth Grade Algebra I Mathematics
Description The Appleton Area School District middle school mathematics program provides students opportunities to develop mathematical skills in thinking and applying problem-solving strategies. The framework
More informationSequenced Units for the Common Core State Standards in Mathematics High School Algebra I
In the three years prior to Algebra I, students have already begun their study of algebraic concepts. They have investigated variables and expressions, solved equations, constructed and analyzed tables,
More informationAlgebra 1. Functions and Modeling Day 2
Algebra 1 Functions and Modeling Day 2 MAFS.912. F-BF.2.3 Which statement BEST describes the graph of f x 6? A. The graph of f(x) is shifted up 6 units. B. The graph of f(x) is shifted left 6 units. C.
More informationAlgebra 2 Standards. Essential Standards:
Benchmark 1: Essential Standards: 1. Alg2.M.F.LE.A.02 (linear): I can create linear functions if provided either a graph, relationship description or input-output tables. - 15 Days 2. Alg2.M.A.APR.B.02a
More informationAlgebra 1 Mathematics: to Hoover City Schools
Jump to Scope and Sequence Map Units of Study Correlation of Standards Special Notes Scope and Sequence Map Conceptual Categories, Domains, Content Clusters, & Standard Numbers NUMBER AND QUANTITY (N)
More informationUNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS
UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.
More informationWA State Common Core Standards - Mathematics
Number & Quantity The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties
More informationAlgebra 1R/H Regents Review 7. x 1, for which value of x is
Algebra 1R/H Regents Review 7 NAME Date ( ) = x 2 2x 8 and g ( x) = 1 4 f ( x) = g ( x)? (Use 2 nd calc intersect on the graph.) 152) If f x (1) 1.75 and 1.438 (3) 1.438 and 0 (2) 1.75 and 4 (4) 4 and
More informationAlgebra I Remediation Guide
Algebra I Remediation Guide Focused remediation helps target the skills students need to more quickly access and practice on-grade level content. This chart is a reference guide for teachers to help them
More informationAlgebra I High School Math Solution West Virginia Correlation
M.A1HS.1 M.A1HS.2 M.A1HS.4a M.A1HS.4b Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret
More informationSection 8 Topic 1 Comparing Linear, Quadratic, and Exponential Functions Part 1
Section 8: Summary of Functions Section 8 Topic 1 Comparing Linear, Quadratic, and Exponential Functions Part 1 Complete the table below to describe the characteristics of linear functions. Linear Functions
More informationCurriculum Scope & Sequence. Subject/Grade Level: MATHEMATICS/HIGH SCHOOL (GRADE 7, GRADE 8, COLLEGE PREP)
BOE APPROVED 9/27/11 Curriculum Scope & Sequence Subject/Grade Level: MATHEMATICS/HIGH SCHOOL Course: ALGEBRA I (GRADE 7, GRADE 8, COLLEGE PREP) Unit Duration Common Core Standards / Unit Goals Transfer
More informationUnit 2 Linear Functions and Systems of Linear Functions Algebra 1
Number of Days: MS 44 10/16/17 12/22/17 HS 44 10/16/17 12/22/17 Unit Goals Stage 1 Unit Description: Unit 2 builds upon students prior knowledge of linear models. Students learn function notation and develop
More informationFLORIDA STANDARDS TO BOOK CORRELATION
FLORIDA STANDARDS TO BOOK CORRELATION Florida Standards (MAFS.912) Conceptual Category: Number and Quantity Domain: The Real Number System After a standard is introduced, it is revisited many times in
More informationCalifornia Common Core State Standards for Mathematics Standards Map Mathematics I
A Correlation of Pearson Integrated High School Mathematics Mathematics I Common Core, 2014 to the California Common Core State s for Mathematics s Map Mathematics I Copyright 2017 Pearson Education, Inc.
More informationDESK Secondary Math II
Mathematical Practices The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically
More informationUnit 3: Linear and Exponential Functions
Unit 3: Linear and Exponential Functions In Unit 3, students will learn function notation and develop the concepts of domain and range. They will discover that functions can be combined in ways similar
More informationCurriculum Summary 8 th Grade Algebra I
Curriculum Summary 8 th Grade Algebra I Students should know and be able to demonstrate mastery in the following skills by the end of Eighth Grade: The Number System Extend the properties of exponents
More informationThe Common Core Georgia Performance Standards (CCGPS) for Grades K-12 Mathematics may be accessed on-line at:
FORMAT FOR CORRELATION TO THE COMMON CORE GEORGIA PERFORMANCE STANDARDS (CCGPS) Subject Area: Mathematics State-Funded Course: 27.09710 Coordinate Algebra I Textbook Title: Publisher: and Agile Mind The
More informationSubject Area Algebra I Grade Level 9_
MVNTA COMMON CORE TEMPLATE Subject Area Algebra I Grade Level 9_ BUCKET ONE BIG ROCKS Reason quantitatively and use units to solve problems. Understand the concept of a function and use function notation.
More informationMathematics Standards for High School Algebra II
Mathematics Standards for High School Algebra II Algebra II is a course required for graduation and is aligned with the College and Career Ready Standards for Mathematics in High School. Throughout the
More informationMilford Public Schools Curriculum. Department: Mathematics Course Name: Algebra 1 Level 2
Milford Public Schools Curriculum Department: Mathematics Course Name: Algebra 1 Level 2 UNIT 1 Unit Title: Intro to Functions and Exponential Expressions Unit Description: Students explore the main functions
More informationAlgebra 1 Pacing Guide First Nine Weeks
Multi-Variable Categorical Data (Fractions, Decimals, and Percents) One-Variable Data Distributions (Mean, Median, Mode, and Range) Quantitative Reasoning Algebraic Models Algebra 1 Pacing Guide First
More informationThe steps in Raya s solution to 2.5 (6.25x + 0.5) = 11 are shown. Select the correct reason for line 4 of Raya s solution.
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. Unit 2: Reasoning with Linear Equations and Inequalities The perimeter
More informationGuide Assessment Structure Algebra I
Guide Assessment Structure Algebra I The Common Core State Standards for Mathematics are organized into Content Standards which define what students should understand and be able to do. Related standards
More informationHouston County School System
NUMBER AND QUANTITY The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties
More informationSECONDARY MATHEMATICS I
SECONDARY MATHEMATICS I The fundamental purpose of SECONDARY Mathematics I is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, organized into units,
More informationFoundations of Algebra/Algebra/Math I Curriculum Map
*Standards N-Q.1, N-Q.2, N-Q.3 are not listed. These standards represent number sense and should be integrated throughout the units. *For each specific unit, learning targets are coded as F for Foundations
More informationIntermediate Algebra Final Exam Review
Intermediate Algebra Final Exam Review Note to students: The final exam for MAT10, MAT 11 and MAT1 will consist of 30 multiple-choice questions and a few open-ended questions. The exam itself will cover
More information, Algebra I, Quarter 1
2017.18, Algebra I, Quarter 1 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1.
More informationNRSD Curriculum - Algebra 1
NUMBER AND QUANTITY The Real Number System NRSD Curriculum - Algebra 1 Extend the properties of exponents to rational exponents. 9-12.N-RN.1 Explain how the definition of the meaning of rational exponents
More informationContinuing Quadratic/Polynomial Real-World Problems
Algebra 1, Quarter 3, Unit 3.1 Continuing Quadratic/Polynomial Real-World Problems Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Understand closed operations.
More informationSchool District of Marshfield Course Syllabus
School District of Marshfield Course Syllabus Course Name: Algebra I Length of Course: 1 Year Credit: 1 Program Goal(s): The School District of Marshfield Mathematics Program will prepare students for
More informationA Story of Functions: A Curriculum Overview for Grades 9-12
Common Core 9-12 Mathematics Curriculum GRADE A Story of Functions: A Curriculum Overview for Grades 9-12 Table of Contents Introduction... 2 Curriculum Map... 4 Standards of Mathematical Practice... 5
More information4. Based on the table below, what is the joint relative frequency of the people surveyed who do not have a job and have a savings account?
Name: Period: Date: Algebra 1 Common Semester 1 Final Review 1. How many surveyed do not like PS4 and do not like X-Box? 2. What percent of people surveyed like the X-Box, but not the PS4? 3. What is the
More informationCumberland County Schools
Cumberland County Schools MATHEMATICS Algebra II The high school mathematics curriculum is designed to develop deep understanding of foundational math ideas. In order to allow time for such understanding,
More informationAlgebra II/Math III Curriculum Map
6 weeks Unit Unit Focus Common Core Math Standards 1 Simplify and perform operations with one variable involving rational, exponential and quadratic functions. 2 Graph and evaluate functions to solve problems.
More informationFunction: State whether the following examples are functions. Then state the domain and range. Use interval notation.
Name Period Date MIDTERM REVIEW Algebra 31 1. What is the definition of a function? Functions 2. How can you determine whether a GRAPH is a function? State whether the following examples are functions.
More informationUnit 0. Unit 1. OUHSD Mathematics Pacing Guide. OUHSD Mathematics Pacing Guide. Total Days 15. Total Days 23. Unit 0: Bridge to Math 1.
OUHSD Mathematics Pacing Guide Quarter 1 Unit 0: Bridge to Unit 0 Total Days 15 Cluster Heading Standard MVP "REVIEW: Use own Resources Solving Equations: One- step; Two- step; Multi- step; Variables on
More information4. Based on the table below, what is the joint relative frequency of the people surveyed who do not have a job and have a savings account?
Name: Period: Date: Algebra 1 Common Semester 1 Final Review Like PS4 1. How many surveyed do not like PS4 and do not like X-Box? 2. What percent of people surveyed like the X-Box, but not the PS4? 3.
More informationSchool District of Marshfield Course Syllabus
School District of Marshfield Course Syllabus Course Name: Algebra II Length of Course: 1 Year Credit: 1 Program Goal: The School District of Marshfield Mathematics Program will prepare students for college
More informationBridge to Algebra II Standards for Mathematical Practice
Bridge to Algebra II Standards for Mathematical Practice The Standards for Mathematical Practices are to be interwoven and should be addressed throughout the year in as many different units and tasks as
More information