Teacher Road Map for Lesson 23: Solving Complicated Radical Equations
|
|
- Georgiana Bennett
- 6 years ago
- Views:
Transcription
1 eacher Road Map for Solving Complicated Radical Equations Objective in Student Friendly Language: oday I am: comparing two methods for solving complicated radical equations So that I can: determine the steps to solve radical equations. I ll know I have it when I can: apply this skill to a geometry problem. Flow: o Students compare two methods for solving complicated radical equations. o Students solve four equations and use them to write the steps for solving radical equations. o Students look at situations where the radical cannot be isolated. o Students apply their skills to a geometry problem. Big Ideas: o Radical equations become polynomial equations through exponentiation. CCS Standards: o A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, startingfrom the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. o A-REI. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 33
2 Student Outcomes Students develop facility in solving radical equations. Lesson Notes In the previous lesson, students were introduced to the notion of solving radical equations and checking for extraneous solutions (A-REI.). Students continue this work by looking at radical equations that contain variables on both sides. he main point to stress to students is that radical equations become polynomial equations through exponentiation. So we really have not left the notion of polynomials that have been studied throughout this module. his lesson also provides opportunities to emphasize MP.7 (look for and make use of structure). In the last lesson you solved equations with radicals but there was only one variable term. In this lesson, you ll expand your equation-solving skills to include equations with radical and non-radical variable terms. Opening Exercise (15 minutes) 1. Carlos and Andrea were solving the equation x + x= 0. Andrea says that there are two solutions, 0 and -. Carlos says the only solution is - because he divided both sides by x and got x + = 0. Who is correct and why? Andrea is correct. If we replace x with either 0 or, the answer is 0. When Carlos divided both sides of the equation by x, he changed the solutions from 0 and to simply. He lost one solution to the equation. Performing algebraic steps may alter the set of solutions to the original equation.. Carlos and Andrea are solving a different equation x = 3. Andrea says the solution is 9 because she squared both sides and got x = 9. Carlos says there is no solution. Who is correct? Why? Andrea is correct when she says to eliminate a radical from an equation, we raise both sides to an exponent, but in this case her answer is extraneous. If we let x = 9, then we get 9 = 3, and 3 3, so 9 is not a solution. he danger in squaring both sides of an equation is that it sometimes produces an equation whose solution set is not equivalent to that of the original equation. If both sides of x = 3 are Scaffolding Use several examples to illustrate that if aa > 0, then an equation of the form xx = aa will not have a solution (e.g., xx = 4, xx = 5). Extension: Write an equation that has an extraneous solution of xx =
3 squared, the equation x = 9 is produced, but 9 is not a solution to the original equation. he original equation has no solution. Because of this danger, the final essential step of solving a radical equation is to check the solution or solutions to ensure that an extraneous solution was not produced by the step of squaring both sides. We could have predicted that the equation would have no solution since the square root of a number is never equal to a negative value, so there is no x-value so that x = 3. While this problem is difficult, students should attempt to solve it on their own first, by applying their understandings of radicals. Students should be asked to verify the solution they come up with and describe their solution method. Discuss Exercise 3 as a class once they have worked on it individually. 3. Carlos and Andrea tried one more problem, 6 = x+ x. hey both agreed that you need to square both sides, but Andrea wants to isolate the x first and Carlos thinks they should isolate the x. Finish both methods and then decide which you like best. Do they both give the same answer? ( 6 x) Carlos Method 6 = x+ x 6 x = x 36 1 x + x= x = x = ( 1 x) = ( x x 36) 1 x x x x= x x 71x + 7x = x x 71x 7x+ 196 (Note: at this point Carlos needs to know a couple of the factors of this polynomial to continue. 9 and 4 are both factors and can be used to rewrite this equation in factored form.) Andrea s Method 6 = x+ 6 x= x ( 6 x) = ( x) 36 1x+ x = x x 13x+ 36 = 0 ( x 9)( x 4) = 0 x= 9; x= 4 Check: 6-9=-3 9 = 3 Check: 6-4= = 4 here is one extraneous solution (x = 9) and one valid solution (x = 4). x 35
4 Discussion Questions How does this equation differ from the ones from yesterday s lesson? here are two xx s; one inside and one outside of the radical. Explain how you were able to determine the solution to the equation above. Isolate the radical and square both sides. Solve the resulting equation. Did that change the way in which the equation was solved? Not really; we still eliminated the radical by squaring both sides. What type of equation were we left with after squaring both sides? A quadratic polynomial equation Why did 9 fail to work as a solution? Exercise 4 (13 minutes) he square root of 9 takes only the positive value of 3. Allow students time to work the problems independently and then pair up to compare solutions. Use this time to informally assess student understanding by examining their work. Display student responses, making sure that students checked for extraneous solutions. he steps students write may vary. Discuss as a class. 4. Solve each radical equation and then write the steps needed to solve radical equations. A. 33xx = 11 + xx he only solution is 1. Note that 1 is an extraneous solution. 9 B. 33 = 44 xx xx he two solutions are 9 and 1. C. xx + 55 = xx 11 he only solution is 44. Note that 11 is an extraneous solution. D xx 88 = 00 Steps to Solve Radical Equations 1. If there are two radicals, then have one on each side of the equation. If there is only one radical, then isolate it as much as possible.. Square both sides of the equation. 3. If there were two radicals in the original equation, you ll still have one radical after squaring. Isolate that radical and square both sides again. 4. You should now have a polynomial equation. Get all the terms on one side of the equation so that your equation is equal to Factor then solve. 6. Check for extraneous solutions. here are no solutions. 36
5 Exercise 5 (5 minutes) What if there is no way to isolate the radical? What is going to be the easiest way to square both sides? 5. Solve the equation xx + xx + 33 = 33. MP.7 Give students time to work on Exercise 5 independently. Point out that even though we had to square both sides twice, we were still able to rewrite the equation as a polynomial. Exercise 5 Solve the equation xx + xx + 33 = 33. Scaffolding: What if we had squared both sides of the equation as it was presented? his is similar to Carlos method in the Opening Exercise. xx + 33 = 33 xx xx + 33 = (33 xx) xx + 33 = xx + xx 11 = xx 11 = xx Check: = 11 + = 33 So the solution is 11. Exercise 6 (7 minutes) Allow students time to work the problems independently and then pair up to compare solutions. Circulate to assess understanding. Consider targeted instruction with a small group of students while others work independently. Display student responses, making sure that students check for extraneous solutions. 6. Solve the following equations. A. xx 3 + xx + 5 = 4 x = 4 B. 3 + xx = xx + 81 x =
6 7. Consider the triangle AAAAAA shown to the right where AAAA = DDDD, and BBBB is the altitude of the triangle. A. If the length of BBBB is xx cccc, and the length of AAAA is 1111 cccc, how long is AD and DC? 9 cm each B. Write an expression for the lengths of AAAA and BBBB in terms of xx. (Hint: Use the Pythagorean heorem.) AAAA = BBBB = 81 + xx cm C. Write an expression for the perimeter of AAAAAA in terms of xx xx + 18 cm D. Find the value of xx for which the perimeter of AAAAAA is equal to 3333 cccc. 19 cm Lesson Summary If a = b and n is an integer, then a n = b n. However, the converse is not necessarily true. he statement a n = b n does not imply that a = b. herefore, it is necessary to check for extraneous solutions when both sides of an equation are raised to an exponent. Example: (-3) = 3 but
7 Closing (5 minutes) Ask students to respond to these prompts in writing or with a partner. Use these responses to informally assess their understanding of the lesson. How did these equations differ from the equations seen in the previous lesson? Most of them contained variables on both sides of the equation or a variable outside of the radical. How were they similar to the equations from the previous lesson? hey were solved using the same process of squaring both sides. Even though they were more complicated, the equations could still be rewritten as a polynomial equation and solved using the same process seen throughout this module. Give an example where aa nn = bb nn but aa bb. We know that ( 3) = 3 but 3 3. Exit icket (5 minutes) 39
8 Name Date Solving Radical Equations Exit icket 1. Solve xx + 15 = xx + 6. Verify the solution(s).. Explain why it is necessary to check the solutions to a radical equation. 330
9 Exit icket Sample Solutions 1. Solve xx = xx Verify the solution(s). he solutions are 33 and 77. Check xx = 33: ( 33) = 99 = = 33 So, 33 is a valid solution = xx = xx = (xx + 33)(xx + 77) Check xx = 77: herefore, the only solution to the original equation is 33. ( 77) = 11 = = 11 Since 11 11, we see that 11 is an extraneous solution.. Explain why it is necessary to check the solutions to a radical equation. Raising both sides of an equation to a power can produce an equation whose solution set is not equivalent to that of the original equation. In the problem above, xx = 77 does not satisfy the equation. Homework Problem Set Sample Solutions 1. Solve. A. xx 55 xx + 66 = B. xx 55 + xx + 66 = 00 No solution C. xx 55 xx + 66 = No solution D. xx 55 xx + 66 = 4444 E. xx + 44 = 33 xx 3333 F. xx + 44 = 33 + xx No solution 331
10 G. xx + 33 = 55xx H. xx + 11 = xx I. xx xx = K. xx = 44xx J. xx = 11 44xx L. 44xx 11 = xx 11 M. xx + = 44 xx 66 N. xx xx 1111 = O. xx = xx , 88 P. xx = 99xx , 88. Consider the right triangle AAAAAA shown to the right, with AAAA = 88 and BBBB = xx. A. Write an expression for the length of the hypotenuse in terms of xx. AAAA = 64 + xx B. Find the value of xx for which AAAA AAAA = 99. he solutions to the mathematical equation 64 + x 8 = 9 are 15 and 15. Since lengths must be positive, 15 is an extraneous solution, and x =
Lesson 29: Solving Radical Equations
Lesson 29: Solving Radical Equations Student Outcomes Students develop facility in solving radical equations. Lesson Notes In the previous lesson, students were introduced to the notion of solving radical
More informationLesson 28: A Focus on Square Roots
now Lesson 28: A Focus on Square Roots Student Outcomes Students solve simple radical equations and understand the possibility of extraneous solutions. They understand that care must be taken with the
More informationLesson 18: Recognizing Equations of Circles
Student Outcomes Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in xx and yy is the equation for a circle. Lesson Notes
More informationLesson 9: Radicals and Conjugates
Lesson 9: Radicals and Conjugates Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert
More informationLesson 9: Law of Cosines
Student Outcomes Students prove the law of cosines and use it to solve problems (G-SRT.D.10). Lesson Notes In this lesson, students continue the study of oblique triangles. In the previous lesson, students
More informationUnit 3 Day 4. Solving Equations with Rational Exponents and Radicals
Unit Day 4 Solving Equations with Rational Exponents and Radicals Day 4 Warm Up You know a lot about inverses in mathematics we use them every time we solve equations. Write down the inverse operation
More informationTeacher Road Map for Lesson 10: True and False Equations
Teacher Road Map for Objective in Student Friendly Language: Today I am: sorting equation cards So that I can: determine when an equation is true or false. I ll know I have it when I can: write my own
More informationLesson 3: Advanced Factoring Strategies for Quadratic Expressions
Advanced Factoring Strategies for Quadratic Expressions Student Outcomes Students develop strategies for factoring quadratic expressions that are not easily factorable, making use of the structure of the
More informationLesson 8: Complex Number Division
Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers. Lesson Notes This is the second day of a two-day lesson
More informationLesson 26: Solving Rational Equations
Lesson 2: Solving Rational Equations Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to
More informationLesson 8: Why Stay with Whole Numbers?
Student Outcomes Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Students create functions that
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More informationLesson 5: Criterion for Perpendicularity
Student Outcomes Students explain the connection between the Pythagorean theorem and the criterion for perpendicularity. Lesson Notes It is the goal of this lesson to justify and prove the following: Theorem:
More informationEureka Math. Algebra II Module 1 Student File_A. Student Workbook. This file contains Alg II-M1 Classwork Alg II-M1 Problem Sets
Eureka Math Algebra II Module 1 Student File_A Student Workbook This file contains Alg II- Classwork Alg II- Problem Sets Published by the non-profit GREAT MINDS. Copyright 2015 Great Minds. No part of
More informationLesson 23: Deriving the Quadratic Formula
: Deriving the Quadratic Formula Opening Exercise 1. Solve for xx. xx 2 + 2xx = 8 7xx 2 12xx + 4 = 0 Discussion 2. Which of these problems makes more sense to solve by completing the square? Which makes
More informationNever leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
More informationTABLE OF CONTENTS. Introduction to Finish Line Indiana Math 10. UNIT 1: Number Sense, Expressions, and Computation. Real Numbers
TABLE OF CONTENTS Introduction to Finish Line Indiana Math 10 UNIT 1: Number Sense, Expressions, and Computation LESSON 1 8.NS.1, 8.NS.2, A1.RNE.1, A1.RNE.2 LESSON 2 8.NS.3, 8.NS.4 LESSON 3 A1.RNE.3 LESSON
More informationTransition to College Math and Statistics
Transition to College Math and Statistics Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear College Algebra Students, This assignment
More informationCCSS Math- Algebra. Domain: Algebra Seeing Structure in Expressions A-SSE. Pacing Guide. Standard: Interpret the structure of expressions.
1 Domain: Algebra Seeing Structure in Expressions A-SSE Standard: Interpret the structure of expressions. H.S. A-SSE.1a. Interpret expressions that represent a quantity in terms of its context. Content:
More informationEureka Math. Grade 8, Module 7. Teacher Edition
A Story of Ratios Eureka Math Grade 8, Module 7 Teacher Edition Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, sold, or commercialized, in
More informationLesson 12: Overcoming Obstacles in Factoring
Lesson 1: Overcoming Obstacles in Factoring Student Outcomes Students factor certain forms of polynomial expressions by using the structure of the polynomials. Lesson Notes Students have factored polynomial
More informationEquations and Inequalities. College Algebra
Equations and Inequalities College Algebra Radical Equations Radical Equations: are equations that contain variables in the radicand How to Solve a Radical Equation: 1. Isolate the radical expression on
More informationStudent Outcomes. Lesson Notes. Classwork. Example 1 (5 minutes) Students apply knowledge of geometry to writing and solving linear equations.
Student Outcomes Students apply knowledge of geometry to writing and solving linear equations. Lesson Notes All of the problems in this lesson relate to what students have learned about geometry in recent
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 informationLesson 10: The Power of Algebra Finding Pythagorean Triples
Lesson 10: The Power of Algebra Finding Pythagorean Triples Student Outcomes Students explore the difference of two squares identity x " y " = (x y)(x + y) in the context of finding Pythagorean triples.
More informationLesson 7: Algebraic Expressions The Commutative and Associative Properties
Algebraic Expressions The Commutative and Associative Properties Classwork Exercise 1 Suzy draws the following picture to represent the sum 3 + 4: Ben looks at this picture from the opposite side of the
More informationLesson 24: True and False Number Sentences
NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values
More informationLESSON 13.1 NONLINEAR EQUATIONS
LESSON. NONLINEAR EQUATIONS LESSON. NONLINEAR EQUATIONS 58 OVERVIEW Here's what you'll learn in this lesson: Solving Equations a. Solving polynomial equations by factoring b. Solving quadratic type equations
More informationVOYAGER INSIDE ALGEBRA CORRELATED TO THE NEW JERSEY STUDENT LEARNING OBJECTIVES AND CCSS.
We NJ Can STUDENT Early Learning LEARNING Curriculum OBJECTIVES PreK Grades 8 12 VOYAGER INSIDE ALGEBRA CORRELATED TO THE NEW JERSEY STUDENT LEARNING OBJECTIVES AND CCSS www.voyagersopris.com/insidealgebra
More informationSpring 2018 Math Week Week 1 Task List
Spring 2018 Math 143 - Week 1 25 Week 1 Task List This week we will cover Sections 1.1 1.4 in your e-book. Work through each of the following tasks, carefully filling in the following pages in your notebook.
More informationLesson 18: Equations Involving a Variable Expression in the Denominator
: Equations Involving a Variable Expression in the Denominator Student Outcomes Students interpret equations like 3 as two equations 3 and 0 joined by and. Students find the solution set for this new system
More informationAdding and Subtracting Polynomials
Exploratory Exercise Kim was working on a problem in math when she ran across this problem. Distribute and simplify if possible. 2(3x + 5) Kim s dad said, I remember doing something like this in school.
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationLesson 23: Complicated Quadratics
: Complicated Quadratics Opening Discussion 1. The quadratic expression 2x 2 + 4x + 3 can be modeled with the algebra tiles as shown below. Discuss with your group a method to complete the square with
More informationWest Windsor-Plainsboro Regional School District Algebra Grade 8
West Windsor-Plainsboro Regional School District Algebra Grade 8 Content Area: Mathematics Unit 1: Foundations of Algebra This unit involves the study of real numbers and the language of algebra. Using
More information2. Similarly, 8 following generalization: The denominator of the rational exponent is the index of the radical.
RD. Rational Exponents Rational Exponents In sections P and RT, we reviewed properties of powers with natural and integral exponents. All of these properties hold for real exponents as well. In this section,
More informationLesson 25: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise Over the years, many students and teachers have thought of ways to help us all remember the quadratic formula. Below is the YouTube link to a video created by two teachers
More information11.3 Solving Radical Equations
Locker LESSON 11. Solving Radical Equations Common Core Math Standards The student is expected to: A-REI. Solve simple rational and radical equations in one variable, and give examples showing how extraneous
More informationa factors The exponential 0 is a special case. If b is any nonzero real number, then
0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the
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 informationLesson 24: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise 1. Examine the two equation below and discuss what is the most efficient way to solve each one. A. 4xx + 5xx + 3 = xx 3xx B. cc 14 = 5cc. Solve each equation with the
More informationHonors Math 2 Unit 5 Exponential Functions. *Quiz* Common Logs Solving for Exponents Review and Practice
Honors Math 2 Unit 5 Exponential Functions Notes and Activities Name: Date: Pd: Unit Objectives: Objectives: N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of
More informationdue date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish)
Honors PreCalculus Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear Honors PreCalculus Students, This assignment is designed
More informationMathematics Curriculum
8 GRADE Mathematics Curriculum GRADE 8 MODULE 1 Table of Contents 1 Integer Exponents and Scientific Notation Module Overview... 2 Topic A: Exponential Notation and Properties of Integer Exponents (8.EE.A.1)...
More informationMathematics Curriculum
New York State Common Core Mathematics Curriculum MODULE 1 Table of Contents 1 Relationships Between Quantities and Reasoning with Equations and... 3 Topic A: Introduction to Functions Studied this Year
More informationMathematics High School Algebra
Mathematics High School Algebra Expressions. An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels,
More informationLesson 23: The Defining Equation of a Line
Student Outcomes Students know that two equations in the form of and graph as the same line when and at least one of or is nonzero. Students know that the graph of a linear equation, where,, and are constants
More informationLesson 11: The Special Role of Zero in Factoring
Lesson 11: The Special Role of Zero in Factoring Student Outcomes Students find solutions to polynomial equations where the polynomial expression is not factored into linear factors. Students construct
More informationF.4 Solving Polynomial Equations and Applications of Factoring
section F4 243 F.4 Zero-Product Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or first-degree, equations. Solving higher
More informationAlgebra 2 and Mathematics 3 Critical Areas of Focus
Critical Areas of Focus Ohio s Learning Standards for Mathematics include descriptions of the Conceptual Categories. These descriptions have been used to develop critical areas for each of the courses
More informationAlgebra Summer Review Packet
Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills
More informationName Period Date. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 2: Solving Quadratic Equations 1
Name Period Date QUAD2.1 QUAD2.2 QUAD2.3 The Square Root Property Solve quadratic equations using the square root property Understand that if a quadratic function is set equal to zero, then the result
More informationTeacher s Copy. Algebra 2 Unit 1. Polynomial, Rational, and Radical Relationships. Eureka Math. Eureka Math
Teacher s Copy Algebra 2 Unit 1 Polynomial, Rational, and Radical Relationships Eureka Math Eureka Math Mathematics Curriculum MODULE 1 Table of Contents 1 Polynomial, Rational, and Radical Relationships
More informationTennessee s State Mathematics Standards - Algebra II
Domain Cluster Standard Scope and Clarifications The Real Number System (N-RN) Extend the properties of exponents to rational exponents 1. Explain how the definition of the meaning of rational exponents
More informationSection 2: Equations and Inequalities
Topic 1: Equations: True or False?... 29 Topic 2: Identifying Properties When Solving Equations... 31 Topic 3: Solving Equations... 34 Topic 4: Solving Equations Using the Zero Product Property... 36 Topic
More informationLesson 30: Linear Systems in Three Variables
Lesson 30: Linear Systems in Three Variables Student Outcomes Students solve linear systems in three variables algebraically. Lesson Notes Students solved systems of linear equations in two variables using
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 informationEquations and Inequalities
Equations and Inequalities 2 Figure 1 CHAPTER OUTLINE 2.1 The Rectangular Coordinate Systems and Graphs 2.2 Linear Equations in One Variable 2.3 Models and Applications 2.4 Complex Numbers 2.5 Quadratic
More informationLesson 8: Graphs of Simple Non Linear Functions
Student Outcomes Students examine the average rate of change for non linear functions and learn that, unlike linear functions, non linear functions do not have a constant rate of change. Students determine
More informationUnit 3 Radical and Rational Functions Algebra 2
Number of Days: 29 11/27/17 1/19/18 Unit Goals Stage 1 Unit Description: A theme of Unit 3 is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers.
More informationLesson 22: Equivalent Rational Expressions
0 Lesson 22: Equivalent Rational Expressions Student Outcomes Students define rational expressions and write them in equivalent forms. Lesson Notes In this module, students have been working with polynomial
More informationBig Ideas Chapter 6: Exponential Functions and Sequences
Big Ideas Chapter 6: Exponential Functions and Sequences We are in the middle of the year, having finished work with linear equations. The work that follows this chapter involves polynomials and work with
More informationA Correlation of. To the. North Carolina Standard Course of Study for Mathematics High School Math 2
A Correlation of 2018 To the North Carolina Standard Course of Study for Mathematics High School Math 2 Table of Contents Standards for Mathematical Practice... 1 Number and Quantity... 8 Algebra... 9
More informationAlgebra I PBA/MYA. solutions of the equation t = 4t. equation t = 4t
A-SSE.-1 A-SSE.3a Statement Text Clarifications MP Calculator Use the structure of numerical expressions and polynomial expressions in one variable to identify ways to rewrite it. Choose and produce an
More informationAlgebra II. A2.1.1 Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
Standard 1: Relations and Functions Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
More informationAlgebra II Syllabus CHS Mathematics Department
1 Algebra II Syllabus CHS Mathematics Department Contact Information: Parents may contact me by phone, email or visiting the school. Teacher: Mrs. Tara Nicely Email Address: tara.nicely@ccsd.us Phone Number:
More informationManipulating Radicals
Lesson 40 Mathematics Assessment Project Formative Assessment Lesson Materials Manipulating Radicals MARS Shell Center University of Nottingham & UC Berkeley Alpha Version Please Note: These materials
More informationA. Graph the parabola. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say?
Hart Interactive Honors Algebra 1 Lesson 6 M4+ Opening Exercises 1. Watch the YouTube video Imaginary Numbers Are Real [Part1: Introduction] by Welch Labs (https://www.youtube.com/watch?v=t647cgsuovu).
More informationLesson 2. Homework Problem Set Sample Solutions S.19
Homework Problem Set Sample Solutions S.9. Below are formulas Britney Gallivan created when she was doing her paper-folding extra credit assignment. his formula determines the minimum width, WW, of a square
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 information12.2 Simplifying Radical Expressions
x n a a m 1 1 1 1 Locker LESSON 1. Simplifying Radical Expressions Texas Math Standards The student is expected to: A.7.G Rewrite radical expressions that contain variables to equivalent forms. Mathematical
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 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 informationLESSON 10.1 QUADRATIC EQUATIONS I
LESSON 10.1 QUADRATIC EQUATIONS I LESSON 10.1 QUADRATIC EQUATIONS I 409 OVERVIEW Here s what you ll learn in this lesson: Solving by Factoring a. The standard form of a quadratic equation b. Putting a
More informationOverview (90 Days) Properties of Equality Properties of Inequality Solve Linear Function
Pre- Requisites Skills: Overview (90 Days) Students need to know the meanings of exponents (2 4 = 2 2 2 2) Students need to know how to graph a simple inequality on a number line (i.e. x > 4) Students
More informationLesson 4: Numbers Raised to the Zeroth Power
Student Outcomes Students know that a number raised to the zeroth power is equal to one. Students recognize the need for the definition to preserve the properties of exponents. Classwork Concept Development
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 informationCommon Core State Standards for Mathematics - High School PARRC Model Content Frameworks Mathematics Algebra 2
A Correlation of CME Project Algebra 2 Common Core 2013 to the Common Core State Standards for , Common Core Correlated to the Number and Quantity The Real Number System N RN Extend the properties of exponents
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 informationWest Windsor-Plainsboro Regional School District Math A&E Grade 7
West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale
More informationUnit 1: Equations and Inequalities
Math I Name: Unit 1: Equations and Inequalities Day 1 Date A DAY B Day Aug. 29 Aug. 30 Topic General Class Procedures and Rules PRACTICE: #1-10 Grade (Teacher fills in) 2 3 4 5 6 Aug. 31 Sept. 2 Sept.
More informationAlgebra II Assessment. Eligible Texas Essential Knowledge and Skills
Algebra II Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra II Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.
More informationTopic: Solving systems of equations with linear and quadratic inequalities
Subject & Grade: Mathematics, 9 th Grade Topic: Solving systems of equations with linear and quadratic inequalities Aim: How would you find the solution set of a linear and quadratic inequality? Materials:.
More informationALGEBRA II CURRICULUM MAP
2017-2018 MATHEMATICS ALGEBRA II CURRICULUM MAP Department of Curriculum and Instruction RCCSD Common Core Major Emphasis Clusters The Real Number System Extend the properties of exponents to rational
More informationNorth Carolina MATH I (Traditional) Pacing Guide
North Carolina MATH I (Traditional) 2018-2019 Pacing Guide Note: The eight Standards for Mathematical Practice describe the varieties of expertise that mathematics educators should seek to develop in their
More informationSequenced Units for Arizona s College and Career Ready Standards MA40 Algebra II
Sequenced Units for Arizona s College and Career Ready Standards MA40 Algebra II Year at a Glance Semester 1 Semester 2 Unit 1: Linear Functions (10 days) Unit 2: Quadratic Functions (10 days) Unit 3:
More informationEquations Involving Factored Expressions
Exploratory Exercise 1. A. Jenna said the product of two numbers is 20. Would the factors have to be 4 and 5? Why? B. Julie said the product of two numbers is 20. Would both factors have to be less than
More informationEureka Math. Grade 8, Module 7. Student File_B. Contains Exit Ticket and Assessment Materials
A Story of Ratios Eureka Math Grade 8, Module 7 Student File_B Contains and Assessment Materials Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced,
More informationLesson 13: More Factoring Strategies for Quadratic Equations & Expressions
: More Factoring Strategies for Quadratic Equations & Expressions Opening Exploration Looking for Signs In the last lesson, we focused on quadratic equations where all the terms were positive. Juan s examples
More informationExample: x 10-2 = ( since 10 2 = 100 and [ 10 2 ] -1 = 1 which 100 means divided by 100)
Scientific Notation When we use 10 as a factor 2 times, the product is 100. 10 2 = 10 x 10 = 100 second power of 10 When we use 10 as a factor 3 times, the product is 1000. 10 3 = 10 x 10 x 10 = 1000 third
More informationLESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II
1 LESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarify concepts and remove ambiguity from the analysis of problems.
More informationVARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II
Name: Date: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarify concepts and remove ambiguity from the analysis of problems.
More informationEureka Lessons for 6th Grade Unit FIVE ~ Equations & Inequalities
Eureka Lessons for 6th Grade Unit FIVE ~ Equations & Inequalities These 2 lessons can easily be taught in 2 class periods. If you like these lessons, please consider using other Eureka lessons as well.
More informationWhy It s Important. What You ll Learn
How could you solve this problem? Denali and Mahala weed the borders on the north and south sides of their rectangular yard. Denali starts first and has weeded m on the south side when Mahala says he should
More informationExtending the Number System
Analytical Geometry Extending the Number System Extending the Number System Remember how you learned numbers? You probably started counting objects in your house as a toddler. You learned to count to ten
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 informationPearson Mathematics Algebra 2 Common Core 2015
A Correlation of Pearson Mathematics Algebra 2 Common Core 2015 to the Common Core State Standards for Bid Category 13-050-10 A Correlation of Pearson Common Core Pearson Number and Quantity The Real Number
More informationAlgebra 1. Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots.
Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots. A1.1.1 Compare real number expressions. A1.1.2 Simplify square
More informationWHCSD Grade Content Area
Course Overview and Timing This section is to help you see the flow of the unit/topics across the entire school year. Quarter Unit Description Unit Length Early First Quarter Unit 1: Investigations and
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 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