Background. A Transformation from Traditional College Algebra to Freudenthal College Algebra. Background. Traditional
|
|
- Dwight Park
- 5 years ago
- Views:
Transcription
1 A Transformation from Traditional College Algebra to Freudenthal College Algebra Monica Geist, Ph.D. Front Range Community College Background Started teaching College Algebra in 1997 Very traditional/formal algebra Logarithms were a disaster November of 005 Attended a 4 hour Realistic Mathematics Education workshop at CU Henk Van der Kooij presented on logarithms Background Spring 006 asked Henk for his help with logarithms Summer 006 Henkwrote a replacement chapter for Exponential and Logarithms Fall 006 Pilot tested this chapter Spring 007 Changed entire course modeling after the Freudenthal philosophy Traditional No education courses required State mandated curriculum Book dependent As books changed, so did we Followed the order the topics were presented in the book No learning trajectories built into these book Freudenthal Traditional Must know your pre requisites don t have time to re teach these.1 Lines and Slope. Distance and Midpoint Formulas; Circles.3 Basic of Functions.4 Graphs of Functions.5 Transformations of Functions.6 Combinations of Functions; Composite Functions.7 Inverse Functions
2 Traditional Day One Point-Slope Form of the Equation of a Line The point-slope equation of a nonvertical line of slope m that passes through the point (x 1, y 1 ) is y y 1 = m(x x 1 ). Traditional Formal Algebra Equations of Lines Point-slope form: y y 1 = m(x x 1 ) Slope-intercept form: y = m x + b Horizontal line: y = b Vertical line: x = a General form: Ax + By + C = 0 Freudenthal Day One LINEAR GROWTH Suppose gasoline costs $.50 per gallon. Draw a graph that models putting g gallons of gas into a car. Let the horizontal axis be number of gallons and the vertical axis be the total cost, in dollars. Let C be the total cost for the gasoline. Let g be the number of gallons. Create an equation that models the scenario described above. Traditional Polynomials The Standard Form of a Quadratic Function The quadratic function f (x) a(x h) k, a 0 is in standard form. The graph of f is a parabola whose vertex is the point (h, k). The parabola is symmetric to the line x h. If a 0, the parabola opens upward; if a 0, the parabola opens downward. Freudenthal Polynomials Calculator Exploration What do you get if you multiply a line times a line? What do you get if you multiply three lines? What about four lines? Freudenthal Polynomials Linear function times linear function results in: Quadratic function (graphically) Line times line results in parabola Linear function times linear function times linear function results in: Cubic function (graphically) Line times line times line results in a cubic graph
3 Freudenthal Polynomials Linear Factorization Theorem All polynomials, f(x) = a n x n + a n 1 x n a 1 x+a 0, can be written as a product of linear factors f(x) = a n (x c 1 )(x c ) (x c n ) Traditional Exponential Functions Definition of the Exponential Function The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than 1 ( b 0 and b 1) and x is any real number Here are some examples of exponential functions. f (x) = x g(x) = 10 x h(x) = 3 x+1 Base is. Base is 10. Base is 3. On the same day two friends both buy a foal. Both foals have a weight of 50 kilogram. After one month they compare the weights. Fred says, "My foal grew 10 kg." Andy answers, "My foal grew 0 %. After another month they meet again and compare the weights. Fred: "another 10 kg!"; Andy: "another 0%." 1. What are the weights of the foals two months after they were bought?. Compare the weight of Andy's foal after two months to the weight on the day it was bought. What is the percentage increment (percent increase)? # months after purchase Weight of Fred s foal = x 50 = = x 60 = Weight of Andy s foal # months after purchase Weight of Fred s foal = x 50 = = x 60 = = x 7 = 86.4 Weight of Andy s foal = x 86.4 = = x = = x = = x = = x = The growth of Andy's foal deserves a closer look! Looking at the weights in the table, you can find the following relationship: weight this month 1. weigth last month
4 This relationship can also be written as: weight this month = 1. weight last month We call the consecutive weights of Andy's foal: W A (t) with t = 0, 1,, 3,... With this function we can write the statement as: W A (1) = 1. W A (0) W A () = 1. W A (1) = W A (0) = 1. W A (0) W A (3) = 1. W A () = W A (0) = 1. 3 W A (0) W A (4) = 1. W A (3) = W A (0) = 1. 4 W A (0) Bacteria: Escherichia coli (E. coli), is one of the main species of bacteria that live in the lower intestines of mammals including us. They are necessary for the proper digestion of food and are part of the intestinal flora. The human body has the same number of human cells and E.coli cells: In real life, E.coli duplicate every two hours. Notice what E. coli looks like magnified to 10,000x. E. coli magnified to 10,000x. Suppose a culture of E.coli is grown in a lab. We put 18 of them in an ideal environment and they duplicate every two hours. How many are there after one day? And how many after one week? The number of E.coli is E(t) and t indicates the time (in units of hours) after the start. What formula describes the number of E.coli as a function of time t? How many are there after one day? 5488 And how many after one week?.48 x 10 7 The number of E.coli is E(t) and t indicates the time (in units of hours) after the start. What formula describes the number of E.coli as a function of time t? E(t) = 18*() t A growth process that can be described with a function like G(t) = c(g) t is called an exponential growth process. Linear growth is characterized by: every fixed time step results in a fixed addition Exponential growth is characterized by: every fixed time step results in a fixed multiplication.
5 Traditional Logarithmic Functions Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log b x is the logarithmic function with base b. Location of Base and Exponent in Exponential and Logarithmic Forms Exponent Exponent Logarithmic form: y = log b x Exponential Form: b y = x. Base Base Traditional Logarithmic Functions Properties for Expanding Logarithmic Expressions For M > 0 and N > 0: 1. log b (MN) log b M log b N M. log b log N b M log b N 3. log b M p plog b M Let s return to exponential growth. Duckweed (or lemna minor) is a type of plant that multiplies fast. The number doubles every day. A pond that has this plant can get completely covered in a short time. The surface then looks like green grass. Because the area that is covered is important we will look at the area (in m ) covered by the plant instead of thinking about numbers of individual plants. Beginning with 1 m of plants on one day (starting time t = 0), after 1 day (t = 1) there are m of it. The formula that describes this process is a = t The graph shows this process for the first four days. Use this graph to answer the following questions. 1. At what moment will there be 3 m of duckweed in the pond? And 6 m? And 1 m? Now without the help of the graph: at what moment are there 4 m of duckweed?. The same question for 5 m, 10 m and 0 m area (in m ) time (in days) 3. Complete the empty cells in the following table using the information from problem 1 and. 3. Complete the empty cells in the following table using the information from problem 1 and. Area (m ) Time (days) 0 1 Area (m ) Time (days) 1.6 Area (m ) Time (days).3 Area (m ) Time (days) 0 Area (m ) Time (days) Area (m ) Time (days) Area (m ) Time (days) Area (m ) Time (days)
6 a t a t a t a t t() t(3) t(6) t(4) t(6) t(4) log log 3 log 6 log A log B log ( AB) a t a t a t a t t(0) t(5) t(4) t(15) t(3) t(5) log 0 log log log B log A log 4 A B a t a t a t a t t() t() t() 3* t() t(8) 1 log log log A B B log A
7 Traditional Systems of Equations Solving Linear Systems by Addition If necessary, rewrite both equations in the form Ax + By = C. If necessary, multiply either equation or both equations by appropriate nonzero numbers so that the sum of the x-coefficients or the sum of the y- coefficients is 0. Add the equations in step. The sum is an equation in one variable. Solve the equation from step 3. Back-substitute the value obtained in step 4 into either of the given equations and solve for the other variable. Check the solution in both of the original equations. Freudenthal Systems of Equations How many bananas do you need to balance the third scale? Explain your reasoning. Traditional End of Term Students felt rushed through the course No depth of knowledge Even A students complained of not understanding what logarithm were Rates of A, B, C s ranged from 5 40% Rates of D, F, W s ranged from 60 75% Freudenthal Students come alive and are more engaged Students ask deeper questions Students can explain what they are doing and have a deeper understanding of concepts 95% of students can explain the meaning of a logarithm Rates of A, B, C s ranged from 55 65% Rates of D, F, W s ranged from 35 45% Comments from Students I understood the main idea. It wasn t just performing problems. It was understanding why behind the problem. It used to be if you put a problem in front of me, I couldn t do it unless you told me what you wanted me to do here I understand everything and can just do it. Comments from Students My big math moment was when I realized the relationship between logarithms and exponential equations. My worst math experience was one day, while I was driving home, I realized I was trying to model the car in front of me with a graph.
8 Teaching Algebra: An Autoethnographic Poem algebra what is it good for when will we ever use this algebra skills do not transfer to life algebra skills do not transfer to critical thinking am I wasting my life by teaching algebra disillusionment disenchantment disappointment I must change my career do something useful contribute to society in a better way While returning to school with new career plans in the works Realistic Mathematics Education entered my life sure, I ll try it I ll try anything that might help teaching algebra is just temporary I still need this job for now different focus different style slower pace deeper learning more engagement increased participation class is turned upside down high achieving students are frustrated can t memorize their way out of this unit low achieving students finally wake up no memorizing needed just thinking thinking about the problem what it means student responses it was surprisingly enjoyable I understood it very well we talked a lot about exponential/logarithms in many different settings seeing it over and hearing it over helps a lot I came out of this with some understanding the visual charts and graphs were very helpful I did feel a different type of learning going on inside my head I enjoyed the graph discussions can you teach like this every day I had a good day I finally understood math today
9 teacher response I had a good day, too I finally taught math today
Multiple Representations: Equations to Tables and Graphs Transcript
Algebra l Teacher: It s good to see you again. Last time we talked about multiple representations. If we could, I would like to continue and discuss the subtle differences of multiple representations between
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 informationPreCalculus Summer Assignment (2018/2019)
PreCalculus Summer Assignment (2018/2019) We are thrilled to have you join the Pre-Calculus family next year, and we want you to get a jump-start over the summer! You have learned so much valuable information
More informationA Study Guide for. Students PREPARING FOR GRADE. Nova Scotia Examinations in Mathematics
A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics For more information,
More informationChapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills...
Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... identifying and graphing quadratic functions transforming quadratic equations solving quadratic equations using factoring
More informationInstructor Notes for Chapters 3 & 4
Algebra for Calculus Fall 0 Section 3. Complex Numbers Goal for students: Instructor Notes for Chapters 3 & 4 perform computations involving complex numbers You might want to review the quadratic formula
More informationLooking Ahead to Chapter 10
Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.
More informationALGEBRA II: A. Survey of Functions. Jeff Shriner
ALGEBRA II: A Survey of Functions Jeff Shriner J S Created for University of Colorado PCDP, Summer 2014 1 C O N T E N T S 0 0.1: pre-requisite review 5 0 0.2: functions 7 0 0.3: inverse functions 11 0
More informationCh. 7.6 Squares, Squaring & Parabolas
Ch. 7.6 Squares, Squaring & Parabolas Learning Intentions: Learn about the squaring & square root function. Graph parabolas. Compare the squaring function with other functions. Relate the squaring function
More informationMath 3C Midterm 1 Study Guide
Math 3C Midterm 1 Study Guide October 23, 2014 Acknowledgement I want to say thanks to Mark Kempton for letting me update this study guide for my class. General Information: The test will be held Thursday,
More informationINSTRUCTIONS USEFUL FORMULAS
MATH 1100 College Algebra Spring 18 Exam 1 February 15, 2018 Name Student ID Instructor Class time INSTRUCTIONS 1. Do not open until you are told to do so. 2. Do not ask questions during the exam. 3. CAREFULLY
More informationBUILT YOU. ACT Pathway. for
BUILT for YOU 2016 2017 Think Through Math s is built to equip students with the skills and conceptual understandings of high school level mathematics necessary for success in college. This pathway progresses
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationCalculus Honors and Introduction to Calculus
Calculus Honors and Introduction to Calculus Name: This summer packet is for students entering Calculus Honors or Introduction to Calculus in the Fall of 015. The material represents concepts and skills
More informationWhat students need to know for PRE-CALCULUS Students expecting to take Pre-Calculus should demonstrate the ability to:
What students need to know for PRE-CALCULUS 2014-2015 Students expecting to take Pre-Calculus should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every
More informationAmarillo ISD Math Curriculum
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationSUMMER REVIEW PACKET. Name:
Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the
More informationUMUC MATH-107 Final Exam Information
UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationAmarillo ISD Algebra II Standards
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationName: Geometry & Intermediate Algebra Summer Assignment
Name: Geometry & Intermediate Algebra Summer Assignment Instructions: This packet contains material that you have seen in your previous math courses (Pre- Algebra and/or Algebra 1). We understand that
More informationAmarillo ISD Math Curriculum
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationSummer Packet A Math Refresher For Students Entering IB Mathematics SL
Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school
More informationINTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2
INTERNET MAT 117 Solution for the Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (i) Group
More informationFinal Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i
Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add
More informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationBasic Fraction and Integer Operations (No calculators please!)
P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.
More informationTest 2 Review Math 1111 College Algebra
Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.
More informationCALCULUS AB SUMMER ASSIGNMENT
CALCULUS AB SUMMER ASSIGNMENT Dear Prospective Calculus Students, Welcome to AP Calculus. This is a rigorous, yet rewarding, math course. Most of the students who have taken Calculus in the past are amazed
More informationFoundations of Math II Unit 5: Solving Equations
Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following
More informationGUIDED NOTES 4.1 LINEAR FUNCTIONS
GUIDED NOTES 4.1 LINEAR FUNCTIONS LEARNING OBJECTIVES In this section, you will: Represent a linear function. Determine whether a linear function is increasing, decreasing, or constant. Interpret slope
More informationAlgebra 2B Semester Credit by Exam Information
Algebra B Semester Credit by Exam Information Plano ISD Now uses the Texas Tech University ISD Credit by Exam Test for Algebra Format: second semester consists of 5 multiple choice questions and 5 open
More informationAmarillo ISD Math Curriculum
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
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 informationCALCULUS Differential and Integral The Domain and the Range, Algebraic of functions
University of Al-Qadisiyh College of Education Department of Mathematics Department of physics Class: First CALCULUS Differential and Integral The Domain and the Range, Algebraic of functions Fieras Joad
More informationINTERNET MAT 117 Review Problems. (1) Let us consider the circle with equation. (b) Find the center and the radius of the circle given above.
INTERNET MAT 117 Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (b) Find the center and
More informationAmarillo ISD Math Curriculum
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationFinite Differences TEACHER NOTES MATH NSPIRED
Math Objectives Students will be able to recognize that the first set of finite differences for a linear function will be constant. Students will be able to recognize that the second set of finite differences
More informationParabolas and lines
Parabolas and lines Study the diagram at the right. I have drawn the graph y = x. The vertical line x = 1 is drawn and a number of secants to the parabola are drawn, all centred at x=1. By this I mean
More informationQuarter 2 400, , , , , , ,000 50,000
Algebra 2 Quarter 2 Quadratic Functions Introduction to Polynomial Functions Hybrid Electric Vehicles Since 1999, there has been a growing trend in the sales of hybrid electric vehicles. These data show
More informationAmarillo ISD Algebra I Standards
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationCurriculum Map: Mathematics
Curriculum Map: Mathematics Course: Honors Algebra II Grade(s): 9/10 Unit 1: Expressions, Equations, and Inequalities In this unit, students review basics concepts and skills of algebra studied in previous
More informationDear Future CALCULUS Student,
Dear Future CALCULUS Student, I am looking forward to teaching the AP Calculus AB class this coming year and hope that you are looking forward to the class as well. Here a few things you need to know prior
More informationImportant Math 125 Definitions/Formulas/Properties
Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient
More informationCALCULUS. Department of Mathematical Sciences Rensselaer Polytechnic Institute. May 8, 2013
Department of Mathematical Sciences Ready... Set... CALCULUS 3 y 2 1 0 3 2 1 0 1 x 2 3 1 2 3 May 8, 2013 ii iii Ready... Set... Calculus This document was prepared by the faculty of the Department of Mathematical
More informationMATH 1130 Exam 1 Review Sheet
MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This
More informationAP Calculus Summer Assignment Summer 2017 Expectations for Summer Assignment on the first day of the school year.
Summer 07 Expectations for Summer Assignment This packet is to be submitted to your Calculus BC teacher on the first day of the school year. All work must be shown in the packet OR on separate paper attached
More information2012 Texas Essential Knowledge and Skills for Algebra II in Pearson Texas Algebra II
2012 Texas Essential Knowledge and Skills for Algebra II in Pearson Texas Algebra II The following table shows where each of the from the 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra II
More informationBooker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website.
BTW Math Packet Advanced Math Name Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. Go to the BTW website
More informationDepartment of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections 3.1, 3.3, and 3.5
Department of Mathematics, University of Wisconsin-Madison Math 11 Worksheet Sections 3.1, 3.3, and 3.5 1. For f(x) = 5x + (a) Determine the slope and the y-intercept. f(x) = 5x + is of the form y = mx
More informationSUMMER MATH PACKET ALGEBRA TWO COURSE 229
SUMMER MATH PACKET ALGEBRA TWO COURSE 9 MATH SUMMER PACKET INSTRUCTIONS MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2014 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationPrentice Hall CME Project, Algebra
Prentice Hall CME Project, Algebra 2 2009 Grades 9-12 C O R R E L A T E D T O Grades 9-12 CME Project Algebra 2 2009 Introduction The Correlation of CME Project, Algebra 2 2009 to the Idaho Content Standards,
More informationAP CALCULUS AB. Summer Assignment. Page 1
AP CALCULUS AB Summer Assignment Page 1 Welcome to AP Calculus AB. This will be the toughest class yet in your mathematical careers but the benefit you will receive by having this experience in high school
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More informationInstructors Manual Algebra and Trigonometry, 2e Cynthia Y. Young
Dear Instructor, Instructors Manual Algebra and Trigonometry, 2e Cynthia Y. Young I hope that when you read my Algebra and Trigonometry book it will be seamless with how you teach. My goal when writing
More information3. Solve the following inequalities and express your answer in interval notation.
Youngstown State University College Algebra Final Exam Review (Math 50). Find all Real solutions for the following: a) x 2 + 5x = 6 b) 9 x2 x 8 = 0 c) (x 2) 2 = 6 d) 4x = 8 x 2 e) x 2 + 4x = 5 f) 36x 3
More informationStudents expecting to take Advanced Qualitative Reasoning should demonstrate the ability to:
What students need to know for AQR Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every
More informationGeneral Form: y = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0
Families of Functions Prepared by: Sa diyya Hendrickson Name: Date: Definition: function A function f is a rule that relates two sets by assigning to some element (e.g. x) in a set A exactly one element
More informationLesson 9 Exploring Graphs of Quadratic Functions
Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point
More informationMath ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying
Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More information(4) Statements that contain the word "including" reference content that must be
111.40. Algebra II, Adopted 2012 (One-Half to One Credit). (a) General requirements. Students shall be awarded one-half to one credit for successful completion of this course. Prerequisite: Algebra I.
More informationMy website:
Honors Algebra 2 Summer Assignment To: All Students Taking Honors Algebra 2 for the school year 2017-2018 From: Mrs. Murphy Contact Information: michele.murphy@carlisleindians.org Dear Students and Parents,
More informationSCIE 4101 Spring Math Review Packet #2 Notes Algebra I
SCIE 4101 Spring 011 Math Review Packet # Notes Algebra I I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first characteristic of
More informationMath Fundamentals for Statistics I (Math 52) Unit 7: Connections (Graphs, Equations and Inequalities)
Math Fundamentals for Statistics I (Math 52) Unit 7: Connections (Graphs, Equations and Inequalities) By Scott Fallstrom and Brent Pickett The How and Whys Guys This work is licensed under a Creative Commons
More informationDear Future CALCULUS Student,
Dear Future CALCULUS Student, I am looking forward to teaching the AP Calculus AB class this coming year and hope that you are looking forward to the class as well. Here a few things you need to know prior
More informationCheck boxes of Edited Copy of Sp Topics (was 261-pilot)
Check boxes of Edited Copy of 10023 Sp 11 253 Topics (was 261-pilot) Intermediate Algebra (2011), 3rd Ed. [open all close all] R-Review of Basic Algebraic Concepts Section R.2 Ordering integers Plotting
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 information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationAP Calculus AB Summer Packet 2018
AP Calculus AB Summer Packet 2018 Dear Calculus Student: To be successful in AP Calculus AB, you must be proficient at solving and simplifying each type of problem in this packet. This is a review of Algebra
More informationPearson Georgia High School Mathematics Advanced Algebra
A Correlation of Pearson Georgia High School Mathematics Advanced Algebra 2014 to the Gwinnett County Academic Knowledge and Skills (AKS) Mathematics Table of Contents A - Process Skills... 1 C - Geometry...
More informationA Level Maths summer preparation work
A Level Maths summer preparation work Welcome to A Level Maths! We hope you are looking forward to two years of challenging and rewarding learning. You must make sure that you are prepared to study A Level
More informationAlgebra 3-4 Unit 1 Absolute Value Functions and Equations
Name Period Algebra 3-4 Unit 1 Absolute Value Functions and Equations 1.1 I can write domain and range in interval notation when given a graph or an equation. 1.1 I can write a function given a real world
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More information(MATH 1203, 1204, 1204R)
College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related
More information1.) Suppose the graph of f(x) looks like this (each tick mark denotes 1 unit). x y
College Algebra Summer 2014 Exam File Exam #1 1.) Suppose the graph of f(x) looks like this (each tick mark denotes 1 unit). Graph g(x) = -0.5 f(x + 1) - 3 2.) Consider the following table of values. x
More informationAbsolute Value Inequalities (Advanced Only)
Unit 1: Linear Functions and Inequalities Time Frame: 5 weeks August 16 to September 20, 2011 Unit Description This unit focuses on the development of concepts of functions that was begun in Algebra I
More informationSemester Review Packet
MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree
More informationUnderstanding Functions. Elizabeth Lancellotti
Understanding Functions Elizabeth Lancellotti My unit for the Delaware Teachers Institute will focus on understanding and interpreting the characteristics of functions and function notation. In our curriculum,
More informationAlgebra II Vocabulary Word Wall Cards
Algebra II Vocabulary Word Wall Cards Mathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development. The cards should
More informationare topics that you have not covered yet. Just do the best you can.
Summer assignment for Honors Algebra II 1 Honors Algebra II 010 Summer Assignment Dear student, Welcome to Honors Algebra II! You have signed up for a rigorous course that will challenge your minds, get
More informationC. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2
AP Calculus Students: Welcome to AP Calculus. Class begins in approimately - months. In this packet, you will find numerous topics that were covered in your Algebra and Pre-Calculus courses. These are
More informationSummer Work for students entering PreCalculus
Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate
More informationMath 111: Final Review
Math 111: Final Review Suggested Directions: Start by reviewing the new material with the first portion of the review sheet. Then take every quiz again as if it were a test. No book. No notes. Limit yourself
More informationAlgebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher
Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3-hole punch into your MATH BINDER. Read and work
More informationChapter 1 Homework Problems
Chapter 1 Homework Problems Lesson 1.1.1 1-4. Angelica is working with function machines. She has the two machines shown at right. She wants to put them in order so that the output of the first machine
More informationInstructional Materials for the WCSD Math Common Finals
201-2014 Algebra 2 Semester 1 Instructional Materials for the WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Math Common Final blueprint for
More informationAre you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*
Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course
More informationChapter 8B - Trigonometric Functions (the first part)
Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of
More informationAccel Alg E. L. E. Notes Solving Quadratic Equations. Warm-up
Accel Alg E. L. E. Notes Solving Quadratic Equations Warm-up Solve for x. Factor. 1. 12x 36 = 0 2. x 2 8x Factor. Factor. 3. 2x 2 + 5x 7 4. x 2 121 Solving Quadratic Equations Methods: (1. By Inspection)
More informationAlgebra 2A Semester Credit by Exam Information
Algebra 2A Semester Credit by Exam Information Plano ISD Now uses the Texas Tech University ISD Credit by Exam Test for Algebra 2 Format: first semester consists of 35 multiple choice questions and 5 open
More informationFinal Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14
Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)
More informationA VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES ALGEBRA I
A VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES ALGEBRA I Revised TEKS (2012): Building to Algebra I Linear Functions, Equations, and Inequalities A Vertical Look at Key Concepts and Procedures Determine
More informationSUMMER MATH PACKET. Geometry A COURSE 227
SUMMER MATH PACKET Geometry A COURSE 7 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of the summer packet is
More informationMATHEMATICS TIPS (MARCH 2018) Leo Ramirez Sr. (The Wizard Maker)
MATHEMATICS TIPS (MARCH 08) Leo Ramirez Sr. (The Wizard Maker) www.rammaterials.com For more Mathematics tips, visit the STORE and look for instructional Mathematics workbooks.. Tickets were sold for a
More informationSCIE 4101 Fall Math Review Packet #2 Notes Patterns and Algebra I Topics
SCIE 4101 Fall 014 Math Review Packet # Notes Patterns and Algebra I Topics I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
More informationAmarillo ISD Math Curriculum
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More information