Size: px
Start display at page:

Transcription

1 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 are still at the alpha stage and are not expected to be perfect. The revision process concentrated on addressing any major issues that came to light during the first round of school trials of these early attempts to introduce this style of lesson to US classrooms. In many cases, there have been very substantial changes from the first drafts and new, untried, material has been added. We suggest that you check with the Nottingham team before releasing any of this material outside of the core project team. If you encounter errors or other issues in this version, please send details to the MAP team c/o 2012 MARS University of Nottingham

4 Common issues: Student cannot get started Student uses only guess and check Student provides no, or poor explanation for conclusions Student makes mistakes with manipulation of the equations For example: The student equates expressions when she has only squared one side of the equation. Or: The student subtracts a value from one side of the equation but not the other. Student does not choose appropriate operations when manipulating the equations For example: The student does not square to remove square roots. Or: The student does not simplify the expressions on each side. Student misinterprets the meaning of algebraic equations and expressions For example: The student assumes that because the expressions on each side of an equation are different, the equation can never be true. Or: The student assumes that squaring an expression in x means that every x in the expression just becomes x 2. Student is confused by roots of negative values For example: The student says that the statement is sometimes true because you can t use negative values of x (Q1. or Q2.) Student correctly answers the questions The student needs an extension task. Suggested questions and prompts: Try some values to decide whether you think the equation can ever be true. How can you get rid of the square root? You have checked a few values of x. How can you manipulate the equation to check your answer is correct in all cases? How can you be sure there are no other values of x for which this equation is true/false? How can you use math to explain your answer? You have changed the equation. How can you be sure that the two sides of the equation are in fact equal? How could you manipulate the algebra to simplify the equation? What other operation could you use to change the equation? Why would that be a helpful move? What is the positive square root of an expression that is squared? How could squaring both sides of the equation help? Try some values for x. Are there any values for x that would make the equation true? When the square root of 3 is multiplied by itself, what is the result? Is the statement sometimes, always, or never true for positive x? Can you take the square root of a negative number? Look at this inequality: 2 x > 3x. Is it always, sometimes, or never true? Write an equation that is only true when x is a complex number MARS University of Nottingham 3

6 Now square the right hand side. You may wish to make a deliberate mistake to address the common misconception that squaring an expression involving square roots involves only removing the square root signs. Write this on the board: ( x " 5) 2 = x " 5 Is this correct? [No.] Explain. [Squaring an expression in the brackets does not just mean squaring each component within the brackets separately.] Ask students to help expand the brackets. Write this on the board: ( x " 5) 2 = ( x " 5) ( x " 5) = x " 2 x The two expressions (1) and (2) are equal if: (2) x " 5 = x " 2 x Prompt students to manipulate and simplify the resulting equation. Is there any way of simplifying this? Remember to perform the same operation on both sides when manipulating the equation. Help students to show that the expressions are equal, when x = 5: x " 5 = x " 2 x " 2 x 5 =10 " x 5 = 5! x = 5! x = 5 Collaborative Activity 1 (20 minutes) Organize the class into groups of two or three students. Give each group a large sheet of paper and Cards A - F from the sheet Always, Sometimes, or Never True? On the board, show students how to divide their large sheet of paper into three columns, with headings Always True, Sometimes True, Never True. Explain the activity to students. You have a set of cards with statements on them. Your task is to decide whether the statements are always, sometimes, or never true. In your group, take turns to place a card in a column. Then explain your choice to your partner or partners. If you think the statements is sometimes true, you will need to find the values of x for which it is true and values of x for which it is not true. You will need to explain why the equation is true for just those values of x. If you think the statement is always true, or never true, you will need to explain why you re sure of that. Remember, if you re showing it is always true or never true, substituting just a few values of x is not enough. Your partner then has to decide whether the card is in the correct place. When you agree, write down your explanations on paper. You may want to use the slide, Always, Sometimes, or Never True? to display the instructions for group work MARS University of Nottingham 5

9 Ask students to work first individually, and then in pairs to show you how to manipulate the equation, to show that: x " y = x " y Squaring the left side of the equation: ( x " y ) 2 = x " y Squaring the right side of the equation: ( x " y ) 2 = ( x " y ) x " y ( ) = x + y - 2 x y The two expressions are equal when: x " y = x + y - 2 x y 2 x y = 2y x y = y Squaring both sides : xy = y 2 ( ) = 0 y y " x y = 0 and y = x. As they work on the problem, support the students as in the first collaborative activity. If necessary, discuss the solution as a class. Collaborative Activity 2 (10 minutes) Once you've established how to replace the number with a variable: Choose one or two of the statements from your poster. Replace the numbers in the equations with a variable, say, y. Check to see if the statement should still remain in the same category. For the statements you place in the sometimes true category, you should figure out exactly which values of x and y make the equation true. They can write their equations and answers on the posters. Support the students as in the first collaborative activity. Collaborative Activity 3 (10 minutes) (optional) If you want to work on complex numbers in the lesson, give each group Cards G and H. These two cards introduce the students to complex numbers. Now consider whether these new statements are always, sometimes, or never true. What values of x make this statement true? Support student reasoning: What is the result of substituting x = -1 into the statement? How can you check the statement is correct for just this value / these values of x? 2012 MARS University of Nottingham 8

12 Manipulating Radicals Teacher Guide Alpha Version January 2012 Solutions Assessment Task: Operations With Radicals x 1. The statement is always true, for any value of x. 7 = x 7 Whatever the value of x substituted into the equation, it remains true. Statements that are always true are identities. Some students might justify this claim only by substituting a few values of x and showing that the equation is true for those values. This is not sufficient justification. Others may just appeal directly to the laws for the " a % manipulation of indices: \$ ' = a # b & b x ( ) x ( ). x Whilst this shows that the statement is true, it does not help explain why the statement is always true. Squaring and taking the positive square root of the right hand side of the equation uses the laws for the manipulation of indices, but shows how this does not change the value of the fraction: " x % 2 x ( x \$ ' = # 7 & 7 ( 7 = x 7 This is clearly true for any value of x. Replacing 7 by any other numbers does not affect the truth of the statement. This can be shown by the same methods for a particular value, but that does not establish the generality. A full solution will involve replacing 7 with a variable, y, say, and manipulating the algebra to show equality MARS University of Nottingham 11

13 x " y = x " y. This statement is sometimes true, that is, true when either x = y or when y = 0. For a full solution, the student is asked not only to identify some values of x for which the statement is true and some for which it is false, but to state exactly which values of x make the statement true. Students may just notice that if x = y, both sides are zero. This is not a complete solution, as it does not provide any justification for the general statement and it misses the solution x = y. Consider the left hand side of the equation. Squaring: ( x " y ) 2 = x " y # x " y = x " y Squaring the right hand side of the equation: ( x " y ) 2 = x 2 " 2 x y + y 2 = x " 2 x y + y If the two sides of the equation are equal, then x " y = x " 2 x y + y " 2y = 2 x y " y = x y Squaring both sides: y 2 = xy Therefore x = y or y = 0. A common error is to divide x y = y by y, which hides the solution y = (1" 2x )(1+ 2x ) = "5. This statement is sometimes true, when x = 3. (1! 2x )(1+ 2x ) =!5 " 1! 2x =!5 " x = MARS University of Nottingham 12

14 Lesson Task For each card, students should identify values for x for which the equation is true, and explain why the equation is true (or is not true) for just those values. Card A x + 2 = x + 2 This equation is sometimes true, when x = 0. Squaring both sides of the equation: ( x + 2) 2 = x + 2 " x + 2 = x + 2 ( ) 2 ( )( x + 2) " x + 2 = x x 2 Subtracting x + 2 from each side of the equation: 2 x 2 = 0 " x 2 = 0 This is true only if x =0. For Collaborative Activity 2, we replace 2 with y, so the equation becomes x + y = x + y. Following the same method as before, find that the statement is true only if x = 0 or. Logically, this includes the possibility that both values are zero. Card B 3x = 3 " x This equation is always true. Squaring the right hand side of the equation, then taking the positive square root: ( 3 " x ) 2 = ( 3 " x ) " ( 3 " x ) = 3 " 3 " x " x = 3 " x = 3x Therefore this equation is an identity, true for any value of x. Card C x = x + 2 This equation is sometimes true. Some students may simply remove the square root and square notation, and assume that this equation is always true. Other students may try a few values of x and assume it is never true. Suppose that Then " \$ # x % 2 ' & x = x + 2 = ( x + 2) 2 " x = ( x + 2) ( x + 2) = x # 2x = x 2 + 4x + 4 Thus the equation is true just when 4x = 0 which occurs only when x = 0. For Collaborative Activity 2, we replace 2 with y, say MARS University of Nottingham 13

15 380 If we suppose that x 2 + y 2 = x + y Then Thus the equation is true just when 2xy = 0 in which case either x = 0 or y = 0. Card D x 2 " 2 2 = x " 2 This equation is sometimes true, when x = 2. As with Card C, an error that students sometimes make is to assume that you can simply remove the squares and root notation from x 2 " 2 2 and that the equation is thus always true. Suppose that x 2 " 2 2 = x " 2. Then So " \$ # x 2 + y 2 % 2 ' & Thus the equation is true just when 4x = 8. This occurs when x = 2. For Collaborative Activity 2, replace 2 with y. If x 2 " y 2 = x " y. = ( x + y) 2 " x 2 + y 2 = ( x + y) ( x + y) = x 2 + 2xy + y 2 = x 2 + 2xy + y 2 # % \$ x 2 " & ( = x " 2 ' ( ) 2 x 2 " 2 2 = ( x " 2) ( x " 2) = x 2 " 2 # 2x = x 2 " 4x Then # % \$ x 2 " y 2 2 & ( ' = ( x " y) So Thus the general equation is true just when "y 2 = " 2xy + y 2, that is, when 2y 2 = 2xy This occurs when A possible student error is to divide through by y at this point, which hides the solution y=0. Instead, factorize: x 2 " y 2 = ( x " y) ( x " y) = x 2 " 2xy + y 2 y 2 = xy 401 y 2 = xy " y 2 # xy = 0 " y( y # x) = 0 The equation is true just when y=x or y=0. Card E 406 x 2 " 3 2 = 3x This equation is always true. 407 Using the laws for the manipulation of indices, x 2 = x and 3 2 = 3. So x 2 " 3 2 = 3x 2012 MARS University of Nottingham 14

16 For Collaborative Activity 2, replace 3 with y. x 2 " y 2 = xy 410 Since x 2 = x for any value, x 2 " y 2 = xy Card F x = x 3 This equation is always true Using the laws for the manipulation of indices, x 2 = x and 3 2 = 3. Thus = x As with Card G, this is also true if 3 is replaced with y, except that we must then exclude the possibility that y=0. Card G ( 3 + x )( 3 " x ) = 10. This equation is sometimes true if x may take negative values (and the square root of a negative number is permitted). In this case, x = -1. If x is only considered to be zero or positive, then the statement is never true. To find the values of x for which it is true, first clear the brackets. ( 3 + x )( 3 " x ) = 10 " 9 # 3 x + 3 x # x = 10 " 9 # x = 10 Therefore the equation is true if x = "1 Substituting these values of x back into the equation uses i, the square root of -1. ( 3+ "1)( 3 " "1) =10 ( 3 + i) ( 3 " i) = 10 Thus if students look only at values of x that are positive or zero, this statement is never true. Since Card G and H are introduced after Collaborative Activity 2, the general case (substituting y for 3) is not considered by students. This could, though, be used as an extension activity. Card H x x x " 1 = 2x allowed, when x = -1. Consider x x " 1. Squaring this expression: This equation is sometimes true, when x = 1, and also if complex numbers are ( x x " 1) 2 = ( x x " 1)( x x " 1) = x x +1 x " 1 + x " 1 = 2(x + x + 1 x " 1) = 2(x + x 2 " 1) 2012 MARS University of Nottingham 15

17 437 ( ) 2 = 2x Squaring the right hand expression gives 2x. 438 Thus the two sides are equal if 2x = 2(x + x 2 " 1) This occurs if x 2 " 1 = 0, that is, if x 2 = 1. Thus x = 1 or x = -1. In the latter case, both sides of the expression are complex numbers. Operations With Radicals (revisited) These solutions are analogous to those in the initial assessment, so we will not repeat all the reasoning x! x = 2 x is sometimes true when x = 0 or x = 4. If the two expressions are equal x = 2 x " x 2 = 4x " x(x # 4) = 0 " x = 0 or x = 4. If 2 were replaced by y then the equation is sometimes true when x = 0 and when x = y x y = x y is always true (1! 4x )(1+ 4x ) =!15 is sometimes true when x = MARS University of Nottingham 16

18 Manipulating Radicals Student Materials Alpha Version January 2012 Operations With Radicals In these questions, the symbol means the positive square root. So 9 = +3. For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. If you choose sometimes true, state all values of x that make it true. 1. x 7 = x 7 Always True Sometimes True Never True It is true for Show your reasoning: If you change 7 to another number, is your answer still correct? Explain. 2. x " y = x " y Always True Sometimes True Never True! It is true for Show your reasoning: 3. (1" 2x )(1+ 2x ) = "5 Always True Sometimes True Never True! It is true for Show your reasoning: 2012 MARS University of Nottingham S-1

19 Manipulating Radicals Student Materials Alpha Version January 2012 Cards: Always, Sometimes, or Never True? A. B. x + 2 = x + 2 3x = 3 x C. D. x = x + 2 x 2 " 2 2 = x " 2 E. F. x = 3x x = x 3 G. H. ( 3 + x) ( 3 " x) = 10 x x " 1 = 2x 2012 MARS University of Nottingham S-2

20 Manipulating Radicals Student Materials Alpha Version January 2012 Operations With Radicals (revisited) In these questions, the symbol means the positive square root. So 9 = +3. For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. If you choose sometimes true, state all values of x that make it true. 1. x " x = 2 x Always True Sometimes True Never True It is true for Show your reasoning: If you change 2 to another number, is your answer still correct? Explain. 2. x y = x y Always True Sometimes True Never True! It is true for Show your reasoning: 3. (1! 4x )(1+ 4x ) =!15 Always True Sometimes True Never True It is true for Show your reasoning: 2012 MARS University of Nottingham S-3

21 Always, Sometimes, or Never True? 1. Take turns to place a card in a column. 2. If you think the equation is sometimes true: Find the values of x for which the equation is true, and the values of x for which it is not true. Explain why the equation is true for just those values of x. If you think the equation is always true or never true: Explain how you can be sure that this is the case. 3. Other members of the group then decide if the card is in the correct place. If you think it is correctly placed, explain why. If you think it is not correctly placed, move the card to a new position and explain your reasoning. 4. When you all agree, write your reasoning on the poster next to the card. Alpha Version January MARS, University of Nottingham Projector Resources: 1

Mathematical goals. Starting points. Materials required OHT 1 Graphs 1. Time needed. To enable learners to:

Level A14 of challenge: D A14 Exploring equations in parametric parametric form form Mathematical goals Starting points To enable learners to: Materials required OHT 1 Graphs 1 Time needed find stationary

Serena: I don t think that works because if n is 20 and you do 6 less than that you get 20 6 = 14. I think we should write! 6 > 4

24 4.6 Taking Sides A Practice Understanding Task Joaquin and Serena work together productively in their math class. They both contribute their thinking and when they disagree, they both give their reasons

Solutions of Linear Equations

Lesson 14 Part 1: Introduction Solutions of Linear Equations Develop Skills and Strategies CCSS 8.EE.C.7a You ve learned how to solve linear equations and how to check your solution. In this lesson, you

Unit 4 Patterns and Algebra

Unit 4 Patterns and Algebra In this unit, students will solve equations with integer coefficients using a variety of methods, and apply their reasoning skills to find mistakes in solutions of these equations.

Essential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving

Locker LESSON 3. Complex Numbers Name Class Date 3. Complex Numbers Common Core Math Standards The student is expected to: N-CN. Use the relation i = 1 and the commutative, associative, and distributive

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities (optional)

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities (optional) CCSD Teachers note: CCSD syllabus objectives (2.8)The student will solve multi-step inequalities and (2.9)The student will

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

Understanding and Using Variables

Algebra is a powerful tool for understanding the world. You can represent ideas and relationships using symbols, tables and graphs. In this section you will learn about Understanding and Using Variables

Part 2 Number and Quantity

Part Number and Quantity Copyright Corwin 08 Number and Quantity Conceptual Category Overview Students have studied number from the beginning of their schooling. They start with counting. Kindergarten

SFUSD Mathematics Core Curriculum Development Project

1 SFUSD Mathematics Core Curriculum Development Project 2014 2015 Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own

ACTIVITY 15 Continued Lesson 15-2

Continued PLAN Pacing: 1 class period Chunking the Lesson Examples A, B Try These A B #1 2 Example C Lesson Practice TEACH Bell-Ringer Activity Read the introduction with students and remind them of the

Class Assessment Checklist

Appendix A solving Linear Equations Grade 9 Mathematics, Applied (MFM1P) Class Assessment Checklist Categories/Mathematical Process/Criteria Thinking The student: Reasoning and Proving Communication Representing

Total=75 min. Materials BLM cut into cards BLM

Unit 2: Day 4: All together now! Math Learning Goals: Minds On: 15 Identify functions as polynomial functions. Consolidate understanding of properties of functions that include: linear, Action: 50 quadratic,

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.

Unit 6 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 6 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics

and Transitional Comprehensive Curriculum. Algebra I Part 2 Unit 7: Polynomials and Factoring

Algebra I Part Unit 7: Polynomials and Factoring Time Frame: Approximately four weeks Unit Description This unit focuses on the arithmetic operations on polynomial expressions as well as on basic factoring

SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS

SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS 6.EE.6 and 6.EE.7 CONTENTS The types of documents contained in the unit are listed below. Throughout the unit, the documents are arranged by lesson.

Chapter 1: Foundations for Algebra

Chapter 1: Foundations for Algebra 1 Unit 1: Vocabulary 1) Natural Numbers 2) Whole Numbers 3) Integers 4) Rational Numbers 5) Irrational Numbers 6) Real Numbers 7) Terminating Decimal 8) Repeating Decimal

Math 31 Lesson Plan. Day 2: Sets; Binary Operations. Elizabeth Gillaspy. September 23, 2011

Math 31 Lesson Plan Day 2: Sets; Binary Operations Elizabeth Gillaspy September 23, 2011 Supplies needed: 30 worksheets. Scratch paper? Sign in sheet Goals for myself: Tell them what you re going to tell

Algebra: Linear UNIT 16 Equations Lesson Plan 1

1A UNIT 16 Equations Lesson Plan 1 Introduction to coding T: Today we're going to code and decode letters and numbers. OS 16.1 T: What can you see on the OHP? (A circle with letters in it) T: How is it

Pre-enrolment task for 2014 entry Physics Why do I need to complete a pre-enrolment task? This bridging pack serves a number of purposes. It gives you practice in some of the important skills you will

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,

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,

Shenandoah University. (PowerPoint) LESSON PLAN *

Shenandoah University (PowerPoint) LESSON PLAN * NAME DATE 10/28/04 TIME REQUIRED 90 minutes SUBJECT Algebra I GRADE 6-9 OBJECTIVES AND PURPOSE (for each objective, show connection to SOL for your subject

Finding Complex Solutions of Quadratic Equations

COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the

Name 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

Mathematics Enhancement Programme

UNIT 9 Lesson Plan 1 Perimeters Perimeters 1 1A 1B Units of length T: We're going to measure the lengths of the sides of some shapes. Firstly we need to decide what units to use - what length-units do

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,

DATE: Algebra 2. Unit 1, Lesson 2: n th roots and when are n th roots real or not real?

Algebra 2 DATE: Unit 1, Lesson 2: n th roots and when are n th roots real or not real? Objectives - Students are able to evaluate perfect n th roots. - Students are able to estimate non-perfect n th roots

Solve Systems of Equations Algebraically

Part 1: Introduction Solve Systems of Equations Algebraically Develop Skills and Strategies CCSS 8.EE.C.8b You know that solutions to systems of linear equations can be shown in graphs. Now you will learn

Chapter 1: Climate and the Atmosphere

Chapter 1: Climate and the Atmosphere ECC: 1.2.1 WARM-UP Students complete and discuss their responses to prompts in an Anticipation Guide. (10 min) Anticipation Guide. The expectation is that you will

Indirect Measurement Technique: Using Trigonometric Ratios Grade Nine

Ohio Standards Connections Measurement Benchmark D Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve

ED 357/358 - FIELD EXPERIENCE - LD & EI LESSON DESIGN & DELIVERY LESSON PLAN #4

ED 357/358 - FIELD EXPERIENCE - LD & EI LESSON DESIGN & DELIVERY LESSON PLAN #4 Your Name: Sarah Lidgard School: Bentheim Elementary School Lesson: Telling Time Length: approx. 50 minutes Cooperating Teacher:

Objective: Recognize halves within a circular clock face and tell time to the half hour.

Lesson 13 1 5 Lesson 13 Objective: Recognize halves within a circular clock face and tell time to the half Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief

SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION

Intermediate Algebra TEST 1 Spring 014 NAME: Score /100 Please print SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION Course Average No Decimals No mixed numbers No complex fractions No boxed or circled

Polynomials. This booklet belongs to: Period

HW Mark: 10 9 8 7 6 RE-Submit Polynomials This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher

SOLVING LINEAR INEQUALITIES

Topic 15: Solving linear inequalities 65 SOLVING LINEAR INEQUALITIES Lesson 15.1 Inequalities on the number line 15.1 OPENER Consider the inequality x > 7. 1. List five numbers that make the inequality

Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and

and Transitional Comprehensive Curriculum. Algebra II Unit 4: Radicals and the Complex Number System

01-1 and 01-14 Transitional Comprehensive Curriculum Algebra II Unit 4: Radicals and the Complex Number System Time Frame: Approximately three weeks Unit Description This unit expands student understanding

Numbers. The aim of this lesson is to enable you to: describe and use the number system. use positive and negative numbers

Module One: Lesson One Aims The aim of this lesson is to enable you to: describe and use the number system use positive and negative numbers work with squares and square roots use the sign rule master

Lesson Plan by: Stephanie Miller

Lesson: Pythagorean Theorem and Distance Formula Length: 45 minutes Grade: Geometry Academic Standards: MA.G.1.1 2000 Find the lengths and midpoints of line segments in one- or two-dimensional coordinate

Factoring. Expressions and Operations Factoring Polynomials. c) factor polynomials completely in one or two variables.

Factoring Strand: Topic: Primary SOL: Related SOL: Expressions and Operations Factoring Polynomials AII.1 The student will AII.8 c) factor polynomials completely in one or two variables. Materials Finding

These Choice Boards cover the fourteen units for eighth grade Common Core math.

A Note to the Teacher: Thank you so much for purchasing this Choice Board BUNDLE from The Math Station on Teachers Pay Teachers. I hope you like it! I started using choice boards last year and LOVE them!

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.

Grade 5 Purpose: To use number relationships to solve problems and to learn number facts To use known facts and relationships to determine unknown facts To develop and test conjectures To make generalizations

GRADE 7 MATH LEARNING GUIDE. Lesson 26: Solving Linear Equations and Inequalities in One Variable Using

GRADE 7 MATH LEARNING GUIDE Lesson 26: Solving Linear Equations and Inequalities in One Variable Using Guess and Check Time: 1 hour Prerequisite Concepts: Evaluation of algebraic expressions given values

Name Period Date. polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2

Name Period Date POLYNOMIALS Student Packet 3: Factoring Polynomials POLY3 STUDENT PAGES POLY3.1 An Introduction to Factoring Polynomials Understand what it means to factor a polynomial Factor polynomials

Unit 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

Lesson 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

Math 90 Lecture Notes Chapter 1

Math 90 Lecture Notes Chapter 1 Section 1.1: Introduction to Algebra This textbook stresses Problem Solving! Solving problems is one of the main goals of mathematics. Think of mathematics as a language,

VARIABLES, 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.

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

Section 6: Polynomials

Foundations of Math 9 Updated September 2018 Section 6: Polynomials This book belongs to: Block: Section Due Date Questions I Find Difficult Marked Corrections Made and Understood Self-Assessment Rubric

Algebra 1 Scope and Sequence Standards Trajectory

Algebra 1 Scope and Sequence Standards Trajectory Course Name Algebra 1 Grade Level High School Conceptual Category Domain Clusters Number and Quantity Algebra Functions Statistics and Probability Modeling

8 th Grade Intensive Math Ready Florida MAFS Student Edition August-September 2014 Lesson 1 Part 1: Introduction Properties of Integer Exponents Develop Skills and Strategies MAFS 8.EE.1.1 In the past,

Mathematics Enhancement Programme

1A 1B UNIT 3 Theorem Lesson Plan 1 Introduction T: We looked at angles between 0 and 360 two weeks ago. Can you list the different types of angles? (Acute, right, reflex, obtuse angles; angles on straight

Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and

Constructing and solving linear equations

Key Stage 3 National Strategy Guidance Curriculum and Standards Interacting with mathematics in Key Stage 3 Constructing and solving linear equations Teachers of mathematics Status: Recommended Date of

Session 3. Journey to the Earth s Interior

Session 3. Journey to the Earth s Interior The theory of plate tectonics represents a unifying set of ideas that have great power in explaining and predicting major geologic events. In this session, we

Lesson 3: Solving Equations A Balancing Act

Opening Exercise Let s look back at the puzzle in Lesson 1 with the t-shape and the 100-chart. Jennie came up with a sum of 380 and through the lesson we found that the expression to represent the sum

Algebra 1. Mathematics Course Syllabus

Mathematics Algebra 1 2017 2018 Course Syllabus Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit The fundamental purpose of this course is to formalize

Powers, Algebra 1 Teacher Notes

Henri Picciotto Powers, Algebra 1 Teacher Notes Philosophy The basic philosophy of these lessons is to teach for understanding. Thus: - The lessons start by describing a situation without invoking new

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities A note to substitute teachers: pre-algebra teachers agree that all units of study are important, but understanding this unit seems to

Mathematics. Algebra Course Syllabus

Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit Mathematics Algebra 1 2018 2019 Course Syllabus Algebra I formalizes and extends the mathematics students

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

Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing

LECSS Physics 11 Introduction to Physics and Math Methods 1 Revised 8 September 2013 Don Bloomfield

LECSS Physics 11 Introduction to Physics and Math Methods 1 Physics 11 Introduction to Physics and Math Methods In this introduction, you will get a more in-depth overview of what Physics is, as well as

Unit 3, Lesson 2: Exploring Circles

Unit 3, Lesson 2: Exploring Circles Lesson Goals Describe the characteristics that make something a circle. Be introduced to the terms diameter, center, radius, and circumference. Required Materials rulers

Analysis of California Mathematics standards to Common Core standards Algebra I

Analysis of California Mathematics standards to Common Core standards Algebra I CA Math Standard Domain Common Core Standard () Alignment Comments in 1.0 Students identify and use the arithmetic properties

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

Algebra 1 Yearlong Curriculum Plan Last modified: June 2014 SUMMARY This curriculum plan is divided into four academic quarters. In Quarter 1, students first dive deeper into the real number system before

Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number

Egyptian Fractions: Part I

Egyptian Fractions: Part I Prepared by: Eli Jaffe October 8, 2017 1 Cutting Cakes 1. Imagine you are a teacher. Your class of 10 students is on a field trip to the bakery. At the end of the tour, the baker

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

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

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition Students have the options of either purchasing the loose-leaf

Objective: Recognize halves within a circular clock face and tell time to the half hour. (60 minutes) (2 minutes) (5 minutes)

Lesson 11 1 Lesson 11 Objective: Recognize halves within a circular clock face and tell time to the half Suggested Lesson Structure Fluency Practice Application Problem Concept Development Student Debrief

WSMA Algebra - Expressions Lesson 14

Algebra Expressions Why study algebra? Because this topic provides the mathematical tools for any problem more complicated than just combining some given numbers together. Algebra lets you solve word problems

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,

1 Modular Arithmetic Grade Level/Prerequisites: Time: Materials: Preparation: Objectives: Navajo Nation Math Circle Connection

1 Modular Arithmetic Students will explore a type of arithmetic that results from arranging numbers in circles instead of on a number line. Students make and prove conjectures about patterns relating to

5( 4) 4 = x. Answers to Warm Up: Solve the equation and then graph your solution on the number line below.

Grade Level/Course: Grade 7, Grade 8, and Algebra 1 Lesson/Unit Plan Name: Introduction to Solving Linear Inequalities in One Variable Rationale/Lesson Abstract: This lesson is designed to introduce graphing

Instructors 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

Grades Algebra 1. Polynomial Arithmetic Equations and Identities Quadratics. By Henri Picciotto. 395 Main Street Rowley, MA

Grades 7 10 ALGEBRA LAB GEAR Algebra 1 Polynomial Arithmetic Equations and Identities Quadratics Factoring Graphing Connections By Henri Picciotto 395 Main Street Rowley, MA 01969 www.didax.com Contents

Chapter 9: Roots and Irrational Numbers

Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic

Lesson 5: Other Examples of Combustion

Lesson 5: Other Examples of Combustion Students learn to distinguish organic from inorganic materials and practice explanations of combustion for other organic materials. They also take the unit posttest.

A-Level Notes CORE 1

A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is

Algebra 1 Syllabus

Algebra 1 Syllabus 2017-18 dennis_jenkins@crpusd.org Welcome to algebra, a course designed to prepare students for geometry and any other courses taken after it. Students are required by the state of California

Park School Mathematics Curriculum Book 1, Lesson 1: Defining New Symbols

Park School Mathematics Curriculum Book 1, Lesson 1: Defining New Symbols We re providing this lesson as a sample of the curriculum we use at the Park School of Baltimore in grades 9-11. If you d like

Contents. To the Teacher... v

Katherine & Scott Robillard Contents To the Teacher........................................... v Linear Equations................................................ 1 Linear Inequalities..............................................

Factors, Zeros, and Roots

Factors, Zeros, and Roots Mathematical Goals Know and apply the Remainder Theorem Know and apply the Rational Root Theorem Know and apply the Factor Theorem Know and apply the Fundamental Theorem of Algebra

Gary School Community Corporation Mathematics Department Unit Document. Unit Name: Polynomial Operations (Add & Sub)

Gary School Community Corporation Mathematics Department Unit Document Unit Number: 1 Grade: Algebra 1 Unit Name: Polynomial Operations (Add & Sub) Duration of Unit: A1.RNE.7 Standards for Mathematical

Mathematics High School Algebra I

Mathematics High School Algebra I All West Virginia teachers are responsible for classroom instruction that integrates content standards and mathematical habits of mind. Students in this course will focus

Lesson 14: Solving Inequalities

Student Outcomes Students learn if-then moves using the addition and multiplication properties of inequality to solve inequalities and graph the solution sets on the number line. Classwork Exercise 1 (5

Algebra , Martin-Gay

A Correlation of Algebra 1 2016, to the Common Core State Standards for Mathematics - Algebra I Introduction This document demonstrates how Pearson s High School Series by Elayn, 2016, meets the standards

Egyptian Fractions: Part I

Egyptian Fractions: Part I Prepared by: Eli Jaffe October 8, 2017 1 Cutting Cakes 1. Imagine you are a teacher. Your class of 10 students is on a field trip to the bakery. At the end of the tour, the baker

Atoms. Grade Level: 4 6. Teacher Guidelines pages 1 2 Instructional Pages pages 3 5 Activity Pages pages 6 7 Homework Page page 8 Answer Key page 9

Atoms Grade Level: 4 6 Teacher Guidelines pages 1 2 Instructional Pages pages 3 5 Activity Pages pages 6 7 Homework Page page 8 Answer Key page 9 Classroom Procedure: 1. Display the different items collected

Lesson 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

Lesson 6: Algebra. Chapter 2, Video 1: "Variables"

Lesson 6: Algebra Chapter 2, Video 1: "Variables" Algebra 1, variables. In math, when the value of a number isn't known, a letter is used to represent the unknown number. This letter is called a variable.