Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals

Size: px
Start display at page:

Download "Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals"

Transcription

1 Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius In partnership with the Utah State Office of Education "2013"Mathematics"Vision"Project" "MVP" In"partnership"with"the"Utah"State"Office"of"Education""" Licensed(under(the(Creative(Commons(Attribution4NonCommercial4ShareAlike(3.0(Unported(license." 1

2 Task 3.1 Maximum Power In this task, we will be experimenting with some useful properties of exponents. To start, let s make sure we remember exactly how exponents work. In much the same way that multiplication is simply repeated addition, exponents are a tool for expressing repeated multiplication. So for example = = = = a. Rewrite the following expression in its expanded form: b. Now use your expanded form to rewrite the expression as a single base raised to a single power: 2. Repeat the process above to simplify the following expressions: Can you identify a shortcut that would have allowed you to simplify these expressions more quickly? 4. Use your answer to question 3 to complete the following mathematical property: The Product of Powers Rule: bb xx bb yy = bb 5. Would this property help you simplify the following expression? Why or why not? Use this new property to simplify each of the following expressions: xx 3 xx 6 mm 2 mm 3 mm 2

3 There are more exponent properties we can discover by thinking about expanded form. For example, we can consider what happens when we raise an exponential expression to a further power, like in this example: (44 22 ) 33 = = Repeat the process above to simplify the following expressions. (2 3 ) 4 (7 4 ) 5 9. Can you identify a shortcut that would have allowed you to simplify these expressions more quickly? 10. Use your answer to question 9 to complete the following mathematical property: The Power of Powers Rule: (bb xx ) yy = bb 11. Use this new property to simplify each of the following expressions: (5 5 ) 12 (xx 7 ) 3 (xx 6 yy 2 ) 4 Here s another use for expanded form: = = Repeat the process above to simplify the following expressions:

4 13. Can you identify a shortcut that would have allowed you to simplify these expressions more quickly? 14. Use your answer to question 13 to complete the following mathematical property: The Quotient of Powers Rule: bb xx bbyy = bb 15. Use this new property to simplify each of the following expressions: xx 5 xx 10 yy 5 xx 2 xx Use all of the rules you have learned in this module to simplify the following expression: xx9 bb 2 3 xx 4 bb 5 4

5 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals READY, SET, GO Name Period Date READY Topic: Evaluating Functions Graphically 1. Sketch a graph of each of the following functions: a. f(x) = 2 x b. g(x) = 3 x 2. Use the above graphs to evaluate each of the following. If you are unsure of an exact answer, estimate. a. f(0) = e. g(1) = b. f(1) = f. g(2) = c. f(2) = g. g(1.5) = d. f(1.5) = h. g(0.5) = Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 5

6 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals SET Topic: Properties of Exponents Rewrite each of the following expressions in its simplest exponential form using the Product of Powers, Power of Powers, and Quotient of Powers Rules. GO Topic: Imaginary Numbers Simplify each of the following expressions. 6

7 Task 3.2 Be Rational We are very familiar with the idea of exponents, which tell us how many copies of a particular number to multiply together (or if the exponent is negative, how many times to divide by that number). Today, we will be exploring what happens when our exponent isn t a whole number, but instead a rational number (i.e. a fraction). 1. Use the provided graph in order to calculate the following values. If you cannot determine the exact value from the graph, estimate. ff(xx) = 2 xx ff(0) = ff(1) = ff(2) = ff 1 2 = ff 3 2 = 2. Using a calculator and the above function, calculate each of the following values (round to the 2 nd decimal place): ff 1 2 = ff 3 2 = ff 9 4 = Your calculator has no trouble providing you with answers for the above problems, but you would probably struggle to explain why the answers make sense. If exponents tell us how many copies of a number to multiply together, how exactly do we multiply together 0.5, 1.5, or 2.25 copies of a number together? Clearly, exponents are more complicated than we thought, so let s try to think about this problem a different way. 3. Assuming that each of the following patterns are geometric (they change by multiplying), find the common ratio and fill in the missing values

8 4. Now we ll try working with some related exponential equations. How might you fill out the missing columns in the following table? xx What s Happening? What s Really Happening? ff(xx) = 1 9 xx From the last two problems, we can see that the following two things are true: = 3 and 9 = 3 Based on these two statements, what conclusion might you make? 6. Try filling out the following table the same way: xx What s Happening? What s Really Happening? ff(xx) = 1 8 xx 7. Based on the above table, how might you finish the following statement? = In this task, we have been working with rational exponents, where in the past all of our exponents have been integers. When faced with a rational exponent, we can handle it the following way: bb xx yy = = x tell us y tells us 8

9 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals READY, SET, GO Name Period Date READY Topic: Finding a Prime Factorization In past math classes, you may have built a factor tree to help you find the prime factorization of a number. For instance, if you wanted the prime factorization of the number 54, you might build a tree like this one: Each number in a box is a composite number, and each circled number is a prime number (meaning it can only be divided by itself or by 1). Once every "branch" of your tree ends with a circle, you know you are finished. Using this tree, we can see that the prime factorization of 54 is......because there was one 2 and three 3's. 54 = 2 3³ Build factor trees and write the prime factorization of each of the following numbers: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 9

10 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals SET Fill in the tables of values and find the factor used to move between whole number values, F w, as well as the factor, F c, used to move between each column of the table. 7. x y 4 16 d. F w = e. F c = F w F w 8. x y 5 15 d. F w = e. F c = F w F w Rewrite each of the following as the product of whole numbers: Ex: GO Topic: Square Numbers Make a list of the first 15 square numbers. The list has been started for you: 12. 1, 4, 9, 10

11 Task 3.3 I m Fluent in Exponents Today s goal is to practice with rational exponents and strengthen your mathematical fluency, the same way you might grow more fluent in a spoken language. 1. Without using a calculator, determine the value of each of the following (your answers should be whole numbers): = = = = 2. It is often useful to be able to think of mathematical expressions in different forms. Rewrite each of the following exponential expressions in their matching radical forms: EEEE: oooo ( 7) = xx 7 2 = (7yy) 4 9 = 5 3bb 2dd = 3. Now try going the other way: 11 aa 3 h = = 2xx = 4. How might you change these to exponential form? 4 aa 3 bb 8 2 = xx 1 yy 7 zz 3 = 5. All of the exponent rules we learned apply to rational exponents the same way they applied to integer exponents. Use exponent properties to simplify each of the following: xx 1 7 xx 3 7 = xx 1 2 xx 2 3 = (xx 1 7) 3 2 = (xx 2 5) 5 2 = 11

12 SECONDARY MATH II // MODULE Imaginary Numbers, Exponents and Radicals 3.3 READY, SET, GO Name Period Date READY Topic: Square Factors Rewrite each number as the product of two factors, one of which is a square number. Try to use the largest square factor possible: Ex: 48 = x³ SET Topic: Radical notation and radical exponents Each of the expressions below can be written using either radical notation, or rational exponents. Rewrite each of the given expressions in the form that is missing. Radical Form 9. Exponential Form " "

13 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals GO Topic: Exponent Rules Simplify the following expressions using the product, power, and quotient rules: 16) 2 7 x x 17) x ) 5 2x 19) ) 0 8x 21) ) 3 2x 8x 4 23) (-3) 4 24) 7 xy xy ) x 3x x 26) 3st ) 2 7 3m n m 5 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 13

14 Task 3.4 Simple Is as Simple Does We have often seen that multiple equations can represent the same function. This is because these equations are really just different forms of the same equation. For example yy = and yy = 33(xx 22) are just slope-intercept and point-slope form equations of the same line. The same idea can work for numbers, like how 1111, , and are all just alternative ways of writing the number 4. Today we will look for a similar relationship in radicals. 1. Use a calculator to approximate the value of each of the following radicals. Round your answers to the first decimal place: = = 8800 = A quick check with a calculator shows us that even though the numbers looked different, they are in fact the same. Today s goal is to figure out exactly why these values are the same, and then to extend that idea to include more complicated radicals ( radical is a fancy word for square roots, cube roots, etc.). 2. See if you can simplify each of the following expressions in any way: = = aa xx bb xx = 4 1/2 5 1/2 = Rational exponents are equivalent to radicals, so a rule that works in exponential form, like 44 11/ /22 = /22, must also work in radical form, like = This creates a brand new rule for dealing with radicals: aa bb = aa bb Using this same property, we can demonstrate that the above answers must be equal. 3. Rewrite 28 as the product of two factors, one of which is a square number. Then use the above property to simplify 28: 4. Try simplifying each of the following square roots the same way we simplified = 75 = 72 = 24 = 14

15 Another way you might simplify radicals is by using prime factorization. For example, if I wanted to calculate 2222, I would first build a factor tree and find the prime factorization. Then I would rewrite and solve my problem using that prime factorization: 2222 = = = Try simplifying the same square roots using this new method. You might find the last one a little trickier: 18 = 75 = 72 = 24 = 6. By using the prime factorization, we are able to simplify more complex radicals. Try simplifying each of the following. Remember, the inverse of a 3 rd root is a third power, the inverse of a 4 th root is a 4 th power, etc = 80 3 = 162 = 15

16 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals READY, SET, GO Name Period Date READY Topic: Combining Like Terms Simplify each of the radicals SET Topic: Simplifying radicals, imaginary numbers When simplifying expressions, we can only combine like terms. Simplify each of the expressions below. You may need to simplify some radicals in order to identify like terms. 10. x + 2x 11. 2x + 3x + 5y Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 16

17 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals GO Topic: Solve Quadratic Equations Identify a pattern and then find the next three terms of the sequence: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 17

Integrated Math 1 Honors Module 11H Exponents Ready, Set, Go! Homework Solutions

Integrated Math 1 Honors Module 11H Exponents Ready, Set, Go! Homework Solutions 1 Integrated Math 1 Honors Module 11H Exponents Ready, Set, Go! Homework Solutions Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

More information

Integrated Math 3 Module 5 Honors Modeling with Functions Ready, Set, Go! Homework Solutions

Integrated Math 3 Module 5 Honors Modeling with Functions Ready, Set, Go! Homework Solutions 1 Integrated Math 3 Module 5 Honors Modeling with Functions Ready, Set, Go! Homework Solutions Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon,

More information

SECONDARY MATH THREE. An Integrated Approach. MODULE 2 Logarithmic Functions

SECONDARY MATH THREE. An Integrated Approach. MODULE 2 Logarithmic Functions SECONDARY MATH THREE An Integrated Approach MODULE 2 Logarithmic Functions The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2018 Original work 2013 in partnership with

More information

due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish)

due 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 information

Integrated Math 1 Honors Module 9H Quadratic Functions Ready, Set, Go Homework Solutions

Integrated Math 1 Honors Module 9H Quadratic Functions Ready, Set, Go Homework Solutions 1 Integrated Math 1 Honors Module 9H Quadratic Functions Ready, Set, Go Homework Solutions Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet

More information

ALGEBRA II. Standard Teacher Notes. An Integrated Approach. MODULE 4 Polynomial Functions

ALGEBRA II. Standard Teacher Notes. An Integrated Approach. MODULE 4 Polynomial Functions ALGEBRA II An Integrated Approach Standard Teacher Notes MODULE 4 Polynomial Functions The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2018 Original work 2013 in partnership

More information

Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics

Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics 1 Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org

More information

SECONDARY MATH THREE. An Integrated Approach. MODULE 3 Polynomial Functions

SECONDARY MATH THREE. An Integrated Approach. MODULE 3 Polynomial Functions SECONDARY MATH THREE An Integrated Approach MODULE 3 Polynomial Functions The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2018 Original work 2013 in partnership with the

More information

Solving Quadratic Equations Review

Solving Quadratic Equations Review Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic

More information

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

More information

Integrated Math 3 Module 8 Honors Limits & Introduction to Derivatives Ready, Set Go Homework Solutions

Integrated Math 3 Module 8 Honors Limits & Introduction to Derivatives Ready, Set Go Homework Solutions 1 Integrated Math 3 Module 8 Honors Limits & Introduction to Derivatives Ready, Set Go Homework Solutions Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis

More information

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, and Janet Sutorius. Mathematics, Algebra II

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, and Janet Sutorius. Mathematics, Algebra II Resource Title: Algebra II Mathematics Student Edition Publisher: Mathematics Vision Project ISBN: This is an e-book located at http://www.mathematicsvisionproject.org Media: Authors: Copyright: internet

More information

LESSON 9.1 ROOTS AND RADICALS

LESSON 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 information

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, and Janet Sutorius. Mathematics, Algebra I

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, and Janet Sutorius. Mathematics, Algebra I Resource Title: Algebra One Mathematics Student Edition Publisher: Mathematics Vision Project ISBN: This is an e-book located at http://www.mathematicsvisionproject.org Media: Authors: internet pdf Scott

More information

Math 5a Reading Assignments for Sections

Math 5a Reading Assignments for Sections Math 5a Reading Assignments for Sections 4.1 4.5 Due Dates for Reading Assignments Note: There will be a very short online reading quiz (WebWork) on each reading assignment due one hour before class on

More information

Section 4.6 Negative Exponents

Section 4.6 Negative Exponents Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals

More information

Transition to College Math and Statistics

Transition 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 information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

Summer Work for students entering PreCalculus

Summer 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 information

Bridge to Algebra II Standards for Mathematical Practice

Bridge 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

Solving Quadratic & Higher Degree Equations

Solving 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

Algebra & Trig Review

Algebra & 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 information

Summer Work for students entering PreCalculus

Summer 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 information

SECONDARY MATH THREE. Standard Teacher Notes. An Integrated Approach. MODULE 3 Polynomial Functions

SECONDARY MATH THREE. Standard Teacher Notes. An Integrated Approach. MODULE 3 Polynomial Functions SECONDARY MATH THREE An Integrated Approach Standard Teacher Notes MODULE 3 Polynomial Functions The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2018 Original work 2013

More information

a factors The exponential 0 is a special case. If b is any nonzero real number, then

a 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 information

Math Lecture 23 Notes

Math Lecture 23 Notes Math 1010 - Lecture 23 Notes Dylan Zwick Fall 2009 In today s lecture we ll expand upon the concept of radicals and radical expressions, and discuss how we can deal with equations involving these radical

More information

Instructor Quick Check: Question Block 12

Instructor Quick Check: Question Block 12 Instructor Quick Check: Question Block 2 How to Administer the Quick Check: The Quick Check consists of two parts: an Instructor portion which includes solutions and a Student portion with problems for

More information

Chapter REVIEW ANSWER KEY

Chapter REVIEW ANSWER KEY TEXTBOOK HELP Pg. 313 Chapter 3.2-3.4 REVIEW ANSWER KEY 1. What qualifies a function as a polynomial? Powers = non-negative integers Polynomial functions of degree 2 or higher have graphs that are smooth

More information

UNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base

UNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base UNIT 4: RATIONAL AND RADICAL EXPRESSIONS M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 4.1 Product Rule Objective I will be able to multiply powers when they have the same base, including simplifying algebraic expressions

More information

POLYNOMIAL EXPRESSIONS PART 1

POLYNOMIAL EXPRESSIONS PART 1 POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so

More information

Lesson 23: The Defining Equation of a Line

Lesson 23: The Defining Equation of a Line Classwork Exploratory Challenge/Exercises 1 3 1. Sketch the graph of the equation 9xx +3yy = 18 using intercepts. Then, answer parts (a) (f) that follow. a. Sketch the graph of the equation yy = 3xx +6

More information

WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations

WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations SECONDARY MATH TWO An Integrated Approach WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2017 Original

More information

Solving Quadratic & Higher Degree Equations

Solving 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

Lesson #33 Solving Incomplete Quadratics

Lesson #33 Solving Incomplete Quadratics Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

More information

George Ranch High School Pre-Calculus PAP Su Summer Packet Problems, 2017

George Ranch High School Pre-Calculus PAP Su Summer Packet Problems, 2017 Name Date Due 4 th day of class (Thursday, August 1st, 017) George Ranch High School Pre-Calculus PAP Su Summer Packet Problems, 017 Pre-Calculus is a very exciting class. Because we will need to hit the

More information

A. Graph the parabola. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say?

A. 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 information

Chapter 7: Exponents

Chapter 7: Exponents Chapter : Exponents Algebra Chapter Notes Name: Notes #: Sections.. Section.: Review Simplify; leave all answers in positive exponents:.) m -.) y -.) m 0.) -.) -.) - -.) (m ) 0.) 0 x y Evaluate if a =

More information

P.1 Prerequisite skills Basic Algebra Skills

P.1 Prerequisite skills Basic Algebra Skills P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable

More information

Lesson 24: True and False Number Sentences

Lesson 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 information

Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics

Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics 1 Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org

More information

Math 096--Quadratic Formula page 1

Math 096--Quadratic Formula page 1 Math 096--Quadratic Formula page 1 A Quadratic Formula. Use the quadratic formula to solve quadratic equations ax + bx + c = 0 when the equations can t be factored. To use the quadratic formula, the equation

More information

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.). College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite

More information

5.5 Special Rights. A Solidify Understanding Task

5.5 Special Rights. A Solidify Understanding Task SECONDARY MATH III // MODULE 5 MODELING WITH GEOMETRY 5.5 In previous courses you have studied the Pythagorean theorem and right triangle trigonometry. Both of these mathematical tools are useful when

More information

LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 253

LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 253 LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 5 OVERVIEW Here's what you'll learn in this lesson: Properties of Exponents Definition of exponent, power, and base b. Multiplication Property c. Division Property

More information

Algebra I Polynomials

Algebra I Polynomials Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

More information

Common Core Algebra 2. Chapter 5: Rational Exponents & Radical Functions

Common Core Algebra 2. Chapter 5: Rational Exponents & Radical Functions Common Core Algebra 2 Chapter 5: Rational Exponents & Radical Functions 1 Chapter Summary This first part of this chapter introduces radicals and nth roots and how these may be written as rational exponents.

More information

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step

More information

Ch. 7.6 Squares, Squaring & Parabolas

Ch. 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 information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never 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 information

CP Algebra 2. Unit 2-1 Factoring and Solving Quadratics

CP Algebra 2. Unit 2-1 Factoring and Solving Quadratics CP Algebra Unit -1 Factoring and Solving Quadratics Name: Period: 1 Unit -1 Factoring and Solving Quadratics Learning Targets: 1. I can factor using GCF.. I can factor by grouping. Factoring Quadratic

More information

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

More information

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2 5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate

More information

Solving Quadratic & Higher Degree Equations

Solving 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

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... 7 Introduction... 7 Integer Exponents... 8 Rational Exponents...5 Radicals... Polynomials...30 Factoring Polynomials...36

More information

Stephen F Austin. Exponents and Logarithms. chapter 3

Stephen F Austin. Exponents and Logarithms. chapter 3 chapter 3 Starry Night was painted by Vincent Van Gogh in 1889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Exponents and Logarithms This chapter focuses on understanding

More information

Extending the Number System

Extending 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 information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions

More information

Algebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Algebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is

More information

Are you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*

Are 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 information

DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II

DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II Y X MODULE 1 Quadratic Equations BUREAU OF SECONDARY EDUCATION

More information

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There

More information

Lesson 24: Using the Quadratic Formula,

Lesson 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 information

NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4

NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4 Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as

More information

8th Grade. Equations with Roots and Radicals.

8th Grade. Equations with Roots and Radicals. 1 8th Grade Equations with Roots and Radicals 2015 12 17 www.njctl.org 2 Table of Contents Radical Expressions Containing Variables Click on topic to go to that section. Simplifying Non Perfect Square

More information

Math 0320 Final Exam Review

Math 0320 Final Exam Review Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

More information

Spring 2018 Math Week Week 1 Task List

Spring 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 information

Elementary Algebra

Elementary Algebra Elementary Algebra 978-1-63545-008-8 To learn more about all our offerings Visit Knewton.com/highered Source Author(s) (Text or Video) Title(s) Link (where applicable) Flatworld Text John Redden Elementary

More information

Positive exponents indicate a repeated product 25n Negative exponents indicate a division by a repeated product

Positive exponents indicate a repeated product 25n Negative exponents indicate a division by a repeated product Lesson.x Understanding Rational Exponents Sample Lesson, Algebraic Literacy Earlier, we used integer exponents for a number or variable base, like these: x n Positive exponents indicate a repeated product

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 2 nd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 2 nd Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

Intermediate Algebra Textbook for Skyline College

Intermediate Algebra Textbook for Skyline College Intermediate Algebra Textbook for Skyline College Andrew Gloag Anne Gloag Mara Landers Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org To access a customizable

More information

Finite Mathematics : A Business Approach

Finite Mathematics : A Business Approach Finite Mathematics : A Business Approach Dr. Brian Travers and Prof. James Lampes Second Edition Cover Art by Stephanie Oxenford Additional Editing by John Gambino Contents What You Should Already Know

More information

MA094 Part 2 - Beginning Algebra Summary

MA094 Part 2 - Beginning Algebra Summary MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

More information

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer. 2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution

More information

Part 2 - Beginning Algebra Summary

Part 2 - Beginning Algebra Summary Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian

More information

Lesson 25: Using the Quadratic Formula,

Lesson 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 information

Algebra I. Polynomials.

Algebra I. Polynomials. 1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

More information

Elementary Algebra Study Guide Some Basic Facts This section will cover the following topics

Elementary Algebra Study Guide Some Basic Facts This section will cover the following topics Elementary Algebra Study Guide Some Basic Facts This section will cover the following topics Notation Order of Operations Notation Math is a language of its own. It has vocabulary and punctuation (notation)

More information

Developed in Consultation with Virginia Educators

Developed in Consultation with Virginia Educators Developed in Consultation with Virginia Educators Table of Contents Virginia Standards of Learning Correlation Chart.............. 6 Chapter 1 Expressions and Operations.................... Lesson 1 Square

More information

College Algebra Through Problem Solving (2018 Edition)

College Algebra Through Problem Solving (2018 Edition) City University of New York (CUNY) CUNY Academic Works Open Educational Resources Queensborough Community College Winter 1-25-2018 College Algebra Through Problem Solving (2018 Edition) Danielle Cifone

More information

Eureka 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-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 information

5.3 Other Algebraic Functions

5.3 Other Algebraic Functions 5.3 Other Algebraic Functions 397 5.3 Other Algebraic Functions This section serves as a watershed for functions which are combinations of polynomial, and more generally, rational functions, with the operations

More information

When they compared their results, they had an interesting discussion:

When they compared their results, they had an interesting discussion: 27 2.5 Making My Point A Solidify Understanding Task Zac and Sione were working on predicting the number of quilt blocks in this pattern: CC BY Camille King https://flic.kr/p/hrfp When they compared their

More information

Chapter 4: Radicals and Complex Numbers

Chapter 4: Radicals and Complex Numbers Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y

More information

Important Math 125 Definitions/Formulas/Properties

Important 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 information

9.2 Multiplication Properties of Radicals

9.2 Multiplication Properties of Radicals Section 9.2 Multiplication Properties of Radicals 885 9.2 Multiplication Properties of Radicals Recall that the equation x 2 = a, where a is a positive real number, has two solutions, as indicated in Figure

More information

Math ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying

Math ~ 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 information

Chapter 5 Simplifying Formulas and Solving Equations

Chapter 5 Simplifying Formulas and Solving Equations Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify

More information

2.2 Radical Expressions I

2.2 Radical Expressions I 2.2 Radical Expressions I Learning objectives Use the product and quotient properties of radicals to simplify radicals. Add and subtract radical expressions. Solve real-world problems using square root

More information

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 McDougal Littell Algebra 1 2007 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 The main goal of Algebra is to

More information

Math 119 Main Points of Discussion

Math 119 Main Points of Discussion Math 119 Main Points of Discussion 1. Solving equations: When you have an equation like y = 3 + 5, you should see a relationship between two variables, and y. The graph of y = 3 + 5 is the picture of this

More information

POWER ALGEBRA NOTES: QUICK & EASY

POWER ALGEBRA NOTES: QUICK & EASY POWER ALGEBRA NOTES: QUICK & EASY 1 Table of Contents Basic Algebra Terms and Concepts... 5 Number Operations... 5 Variables... 5 Order of Operation... 6 Translating Verbal and Algebraic Phrases... 7 Definition

More information

Algebra 2 Honors: Final Exam Review

Algebra 2 Honors: Final Exam Review Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt

More information

Practical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software

Practical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software Practical Algebra A Step-by-step Approach Brought to you by Softmath, producers of Algebrator Software 2 Algebra e-book Table of Contents Chapter 1 Algebraic expressions 5 1 Collecting... like terms 5

More information

Lesson 6b Rational Exponents & Radical Functions

Lesson 6b Rational Exponents & Radical Functions Lesson 6b Rational Exponents & Radical Functions In this lesson, we will continue our review of Properties of Exponents and will learn some new properties including those dealing with Rational and Radical

More information

Definition of a Logarithm

Definition of a Logarithm Chapter 17 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents

More information

Before we do that, I need to show you another way of writing an exponential. We all know 5² = 25. Another way of writing that is: log

Before we do that, I need to show you another way of writing an exponential. We all know 5² = 25. Another way of writing that is: log Chapter 13 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents

More information

LESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II

LESSON #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 information

When they compared their results, they had an interesting discussion:

When they compared their results, they had an interesting discussion: 27 2.5 Making My Point A Solidify Understanding Task Zac and Sione were working on predicting the number of quilt blocks in this pattern: CC BY Camille King https://flic.kr/p/hrfp When they compared their

More information

More Polynomial Equations Section 6.4

More Polynomial Equations Section 6.4 MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division

More information