# Curriculum Alignment Project

Size: px
Start display at page:

Transcription

1 Curriculum Alignment Project Math Unit Date: Unit Details Title: Solving Linear Equations Level: Developmental Algebra Team Members: Michael Guy Mathematics, Queensborough Community College, CUNY Jonathan Cornick Mathematics, Queensborough Community College, CUNY Frank Gardella Mathematics Education, Hunter College, CUNY Brooke Nixon-Friedheim Long Island City High School, NYCDOE

2 Graduate NYC: Unit Plan Unit Title _Solving Linear Equations Teacher _ Cornick, Gardella, Guy, Nixon-Friedheim Grade Level _CUNY developmental alg_ Approximate Length of Unit _6 lessons_ Unit Organizer How do we concretely and symbolically represent quantities in math? What are the basic rules of math? Standards Content Standards Standards Addressed A-SSE.1. Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 8.EE.7. Solve linear equations in one variable. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 6.EE.2. Write, read, and evaluate expressions in which letters stand for numbers. Write expressions that record operations with numbers and with letters standing for numbers.for example, express the calculation Subtract y from 5 as 5 y. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s 3 and A = 6 s 2 to find the volume and surface area of a cube with sides of length s = 1/2. 6.EE.3. Apply the properties of operations to generate equivalent expressions. For example, apply the 1

3 distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. 6.EE.4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities. 6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form x +p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. MP2. Reason abstractly and quantitatively. MP4. Model with mathematics. MP5. Use appropriate tools strategically. Targeted Standards- content and skills/processes to be taught and assessed A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 8.EE.7. Solve linear equations in one variable. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. CUNY Elementary Algebra Proficiency Standards 2) a. Translate a quantitative verbal phrase into an algebraic expression. 3) a. Translate verbal sentences into mathematical equations. 3) b. Solve all types of linear equations in one variable. Supporting Standards - content that is relevant to the unit but may not be assessed; may include connections to other content areas A-SSE.1. Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients 2

4 represented by letters. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 6.EE.2. Write, read, and evaluate expressions in which letters stand for numbers. Write expressions that record operations with numbers and with letters standing for numbers.for example, express the calculation Subtract y from 5 as 5 y. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s 3 and A = 6 s 2 to find the volume and surface area of a cube with sides of length s = 1/2. 6.EE.3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. 6.EE.4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities. 6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form x +p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. What Do Students Have To Know and Be Able To Do in Order to Meet the Targeted Standards? Students will know and understand: The basic algebraic properties The significance of equality Students will be able to: Represent algebraic quantities symbolically and with algebra tiles Apply the basic algebraic properties Simplify and manipulate algebraic expressions and equations Solve linear equations with whole number coefficients using algebra tiles and symbolically Solve linear equations with integer coefficients using algebra tiles and symbolically Essential/Guiding Questions 3

5 What makes two things equal? How do we find something that is missing or we don t know? How do we use symbols to represent something concrete? Summative/End of Unit Assessment Students must complete common CUNY-wide standardized assessment at the end of the term. Students will complete a battery of questions similar in form on the relevant content. In the unit exam, as opposed to the final exam, students will be given both multiple choice and free response questions. Students will be given a chance to revise their exam responses in online homework. Students will solve matched items from sample CUNY Elementary Algebra Final Exams and Review materials. In support of the multiple choice format questions, students will show their work for additional grading. Question: How can we integrate deeper, more meaningful assessments given time and standardization restraints? Does the assessment: assess all targeted standards? align to Depth of Knowledge level? demonstrate critical thinking skills? demonstrate learning in different ways? allow for diverse needs of students? Scoring Criteria Develop a scoring criteria tool that will evaluate your summative/end of unit assessment. Students must average a correct response rate of 74% on all unit assessments to be considered proficient. Questions for Consideration How well do we want them to know it and be able to do it? What do we want students to know and be able to do? How will we know when they know it or do it well? Entry-level Assessment Once learning targets are determined and your summative assessment has been designed, students are pre-assessed to determine their strengths, weaknesses, understandings and misconceptions in order to inform instruction. See Appendix E As this is the first unit of the semester, students will just have been given a sample exam based on the final CUNY-wide exam. Data will be pulled on relevant questions. 4

6 Students must have mastery of whole number operations with single digit numbers. How do I find out what my students already know and are able to do? find out what additional support students need to meet a given learning target? form flexible groups for instruction based on what students know and are able to do? Type of Assessments In addition to your summative/end of unit assessment, what other assessments will you use throughout the unit (e.g., formative, summative assessments, diagnostic assessments, pre-assessment aligned with learning targets, classroom assessments, learning checks?) See Appendix E Assessment Anecdotal records Class discussions Conferences and interviews End of unit tests (including MC and OR) Journals, learning logs Performance events Performance tasks Projects Running records Selected and/or constructed responses Self-assessment/reflection Student revision of assessment answers Student work folder Learning target aligned to assessment x x x x x Write F for Formative an S for Summative (may be both) F F S F F S F _F S How Often? daily _once 3x/unit _once_ 3x/unit regularly Other: Learning Experiences- When designing learning experiences, consider varied and rigorous instructional strategies to teach content. As you design instruction, plan learning experiences that reinforce and enrich the unit while connecting with the standards and assessments. Specific details can be recorded in lesson plants. See Appendix F 5

7 Indicate your unit learning experiences here. Through inductive work with physical models, students will make the transition to common mathematics symbolism. Activities include: 1) algebra tile guessing game 2) pan balance activities 3) symbolic practice 4) algebra tile integers task 5) review 6) summative assessment 7) revision of summative assessment How do the learning experiences address individual student needs? consider the perspective of the learner? include varied and rigorous experiences? incorporate appropriate literacy strategies/skills? incorporate appropriate content literacy strategies/skills? connect to other content areas as appropriate? integrate technology as appropriate? Unit Sequencing- Order/sequence your lessons after determining your assessments and learning experiences. This sequencing should build upon students previous knowledge, allowing them to make connections to their learning. See Appendix F 1. Solving linear equations with whole number coefficients using algebra tiles 2. Solving linear equations with whole number coefficients symbolically 3. Solving linear equations with integer coefficients using algebra tiles 4. Solving linear equations with integer coefficients symbolically 5. Synthesis and review 6. Unit assessment Resources/Tools List resources/materials that are needed to support student learning. Algebra tiles Algebra tile scale Copies of relevant handouts/activities 6

8 Reflection After teaching the unit, reflect on the strengths and weaknesses of the lessons, activities and assessments. How can I make the unit more effective? * What worked well and how do I know this? Students were engaged and naturally formed groups to work together. Some (not all) appreciated the graphical way to think about equations with algebra tiles, and it helped them understand the algebra *What lessons/activities do I need to revise? Why? How? Signed numbers activity needs to be revised, because students continue to make standard errors throughout semester. How? Not sure - maybe a variety of different methods prevented, until students identify one that works for them. Instructional time is an issue *How did my assessments (formative/summative) guide/alter my instruction? Students continued to make standard errors, necessitating mini-session reinforcement throughout semester in context of later topics (ex. graphing and polynomials) * Should/Could I involve other teachers in this unit (cross-content connections) Yes, algebra tiles certainly have potential - need more input, and possibly ways to integrate into later topics for evidence (pro and con) * Are there any additional resources I need to include? No? *What might I do differently next time? Integrate other learning ideas, revise examples to address most common errors. Questions for Reflection: What worked well and how do I know this? What lessons/activities do I need to revise? Why? How? How did my assessments (formative/summative) guide/alter my instruction? Should/Could I involve other teachers in this unit (cross-content connections)? Are there any additional resources I need to include? What might I do differently next time?

9

10

11

12

13

14

15

16

17

18

19

20

21 Linear Equations Lesson 2 (1) Notes for Instructor: Distribute worksheets to students. Go through the worked example on the board while students follow from sheet (without writing.) Then have students work through problems in pairs/groups while instructor circulates around class. If students are making conceptual errors, have them repeat problems using algebra tiles method. Make sure students write out the checking of their answer. (2) Worked Example: Solve and check: 3x + 14 = 8x + 4 3x + 14 = 8x + 4 3x = 8x Subtract 4 on both sides. 3x + 10 = 8x 3x x = 8x 3x Subtract 3x on both sides. 10 = 5x 10 5 = 5x Divide by 5 on both sides. 5 2 = x Check: 3x + 14 = 8x + 4 3(2) + 14 =? 8(2) + 4 Substitute the value of 2 for x =? = 20 1

22 2 (3) Solve for the variable using algebra, and Check your answer: (a) 5n = 45 (b) 3x + 5 = 14 (c) 23 = 6y + 11 (d) 8n + 6 = 3n + 21 (e) 7t + 1 = t + 19 (f) 3(x + 7) = 8(x + 2) (g) 7 + 4(x + 2) = 3(2x + 1) + 2

23 (4) The product of a number and seven is equal to the sum of the number and twelve. What is the number? 3 (5) When the sum of a number and five is multiplied by three, the answer is the same is when the sum of the number and three is multiplied by four. What is the number?

24 4 (6) Lesson 2: Formative Assessment (a) Solve for n using the algebra tiles method: 3(4 + n) = 4n (b) Solve for n using the symbolic algebra method: 3(4 + n) = 4n + 10 (c) Check your answers by substitution. (d) Multiple Choice. Solve for x: 2(5 + x) = 6 + 3x A) x = 4 B) x = 2 C) x = 1 D) x = 2

25 Lesson 3: Representing Integers and Integer Coefficients with Algebra Tiles Part I: Integers and their Opposites Fill in each blank to make each statement true. - 4 = = = 0 0 = = = = 0 0 = + Part II: Reconsidering Subtraction Complete the following sentence: Subtracting a number is like its. Explain how this is demonstrated in Part I. Part III: Representing negative numbers and subtraction with algebra tiles When we are using algebra tiles, we use the red side to represent negative values. So, for example, we could show as: We know that = 0, so for each positive unit we can pair with a negative unit, they will equal 0, and we can remove them. We call this a zero pair. Using the example from above, we can find three zero pairs (each one is circled below). Draw the leftover tiles below: + = What is left after you remove all the circled zero pairs above? Therefore, =. 1

26 This also works when we use the x tiles: + = 3x + -5x = -2x We can represent the opposite of a number by flipping over the algebra tile. If we want to show the opposite of a positive number, we flip the tile to its red (negative) side. If we want to show the opposite of a negative number, we flip the tile to its yellow (positive) side. We found above that the subtraction is the same as adding the opposite of a number. Therefore, we could show 3x 4 = 2 in two ways: - = OR 3x - 4 = 2 + = 3x + -4 = 2 2

27 This helps us more than just taking away, since we don t have a great way of representing subtraction with algebra tiles (but we do have a way to represent negatives). Additionally, sometimes we don t have any of the quantity we want to take away. Practice: Fill in each missing equation or representation with algebra tiles. Use adding the opposite to represent subtraction. (If you don t have different colored pens, shade in the negative tiles and leave the positive tiles as an outline.) Equation 5x 2 = 13 Algebra Tiles + = 5 = 2x - 3 = + + = Part IV: Solving Equations with Integer Coefficients In the previous lesson, you learned how to solve equations with whole number (positive) coefficients. Here, we will skip the basics of solving equations using algebra tiles and encourage you to review your work from the last section if you need your memory refreshed. Say you are confronted with the equation 3x + 5 = -1. You would represent and then solve the equation as follows. Remember that the goal is to isolate all units on one side 3

28 of the equation and all variable terms on the other, and that you can transfer quantities from one side of the equation to the other by adding the opposite. Algebra Tiles Algebraic symbols and verbal description -3x + 5 = -1 + = -3x is represented by 3 red bars, 5 is represented by 5 yellow squares, and 1 is represented by 1 red square -3x = = + In order to transfer the 5 to the other side of the equation, we add the opposite of 5 (which is 5) to both sides. We can write this as subtraction. -3x = -6 = We remove all zero pairs and have separated the variable terms from the units. = -x = -2 = = = We divide the terms so that each bar is connected to an equal number of units. BUT WAIT! Those bars don t represent x, but x, or the opposite of x. x = 2 To find the opposite of x, we must turn over that tile. But whatever we do to one side, we must do to the other, so we will also turn over the units. Now, we have solved for x. 4

29 Practice: Complete each cell in each table below. Use actual algebra tiles if they help you with the drawing. Algebra Tiles Algebraic symbols and verbal description 6 = -4x + 10 = + 6 is represented by 6 yellow squares, -4x is represented by 4 red bars, and 10 is represented by 10 yellow squares 5

30 Algebra Tiles Algebraic symbols and verbal description -7 3x = 2-7 is represented by 7 red squares, subtracting 3x is represented by adding 3 red bars, and 2 is represented by 2 yellow squares When you have a negative coefficient, what is the last step when solving using algebra tiles? Why? 6

31 Linear Equations Lesson 4 (1) Notes for Instructor: Distribute worksheets to students. Go through the worked example on the board while students follow from sheet (without writing.) Then have students work through problems in pairs/groups while instructor circulates around class. If students are making conceptual errors, have them repeat problems using algebra tiles method. Make sure students write out the checking of their answer. (2) Worked Example: Solve and check: 3x + 4 = 2x x + 4 = 2x x + 3x + 4 = 2x + 3x + 19 Add 3x on both sides. 4 = 5x = 5x Subtract 19 on both sides. 15 = 5x 15 5 = 5x Divide by 5 on both sides. 5 3 = x x = 3 Check: 3x + 4 = 2x ( 3) + 4 =? 2( 3) + 19 Substitute the value of 3 for x =? = 13 1

32 2 (3) Solve for the variable using algebra, and Check your answer: (a) 3n = 21 (b) 2x + 5 = 4x 7 (c) 7 = 35 4w (d) 3q 5 = q (e) p + 5 = 37 9p (f) 4(1 x) = 2(3x 2) + 2 (g) 3n 7(2n 1) = 45 3(4n + 1)

33 (4) The product of a number and -5 is equal to twice the number subtracted from 9. What is the number? 3 (5) Eight less than 5 times a number is -33. What is the number? (6) Multiple Choice. Solve for x: 3(2x + 5) = 5 + 4x A) x = 1 B) x = 1 C) x = 2 D) x = 2

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results

Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At

### Pearson Algebra 1 Common Core 2015

A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).

### Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in

### Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### Problem of the Month: Perfect Pair

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Mathematics. Mathematical Practices

Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### Tennessee Department of Education

Tennessee Department of Education Task: Pool Patio Problem Algebra I A hotel is remodeling their grounds and plans to improve the area around a 20 foot by 40 foot rectangular pool. The owner wants to use

### GRADE 6 MATH: GROCERY SHOPPING AND THE QUILT OF A MATH TEACHER

GRADE 6 MATH: GROCERY SHOPPING AND THE QUILT OF A MATH TEACHER UNIT OVERVIEW This unit contains a curriculum- embedded Common Core aligned task and instructional supports. The task is embedded in a 2 3

### CHICAGO PUBLIC SCHOOLS (ILLINOIS) MATH & SCIENCE INITIATIVE COURSE FRAMEWORK FOR ALGEBRA Course Framework: Algebra

Chicago Public Schools (Illinois) Math & Science Initiative Course Framework for Algebra Course Framework: Algebra Central Concepts and Habits of Mind The following concepts represent the big ideas or

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University

VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS

### Accentuate the Negative: Homework Examples from ACE

Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),

### GRADE 8 MATH: TALK AND TEXT PLANS

GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculum-embedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems

### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

### Learning Objectives 8.2. Media Run Times 8.3. Instructor Overview 8.8 Tutor Simulation: Roman Numerals and Polynomials

Unit 8 Table of Contents Unit 8: Polynomials Video Overview Learning Objectives 8.2 Media Run Times 8.3 Instructor Notes 8.4 The Mathematics of Monomials and Polynomials Teaching Tips: Conceptual Challenges

### Learning Objectives 9.2. Media Run Times 9.3

Unit 9 Table of Contents Unit 9: Factoring Video Overview Learning Objectives 9.2 Media Run Times 9.3 Instructor Notes 9.4 The Mathematics of Factoring Polynomials Teaching Tips: Conceptual Challenges

### High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.

Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations

### Math at a Glance for April

Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common

### Prentice Hall. California Edition of Algebra 1 - Classics Edition (Smith/Charles) 2008. Grade 8

Prentice Hall Grade 8 California Edition of Algebra 1 - Classics Edition (Smith/Charles) 2008 C O R R E L A T E D T O California s Map for a Basic Grade Level Program Grade 8 PROGRAM DESCRIPTION Prentice

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### Current California Math Standards Balanced Equations

Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### MAT 096, ELEMENTARY ALGEBRA 6 PERIODS, 5 LECTURES, 1 LAB, 0 CREDITS

1 LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK MATHEMATICS, ENGINEERING and COMPUTER SCIENCE DEPARTMENT FALL 2015 MAT 096, ELEMENTARY ALGEBRA 6 PERIODS, 5 LECTURES, 1 LAB, 0 CREDITS Catalog

### Algebra I Teacher Notes Expressions, Equations, and Formulas Review

Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

### Polynomials and Polynomial Functions

Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions Overview Number of instruction days: 13-15 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Prove

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Mathematics Grade-Level Instructional Materials Evaluation Tool

Mathematics Grade-Level Instructional Materials Evaluation Tool Quality Review GRADE 8 Textbooks and their digital counterparts are vital classroom tools but also a major expense, and it is worth taking

### Problem of the Month The Wheel Shop

Problem of the Month The Wheel Shop The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

### Algebra 1. Curriculum Map

Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Using Algebra Tiles for Adding/Subtracting Integers and to Solve 2-step Equations Grade 7 By Rich Butera

Using Algebra Tiles for Adding/Subtracting Integers and to Solve 2-step Equations Grade 7 By Rich Butera 1 Overall Unit Objective I am currently student teaching Seventh grade at Springville Griffith Middle

### Successful completion of Math 7 or Algebra Readiness along with teacher recommendation.

MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 8-11 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION

### Opposites are all around us. If you move forward two spaces in a board game

Two-Color Counters Adding Integers, Part II Learning Goals In this lesson, you will: Key Term additive inverses Model the addition of integers using two-color counters. Develop a rule for adding integers.

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### ALGEBRA I (Created 2014) Amherst County Public Schools

ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### Math Journal HMH Mega Math. itools Number

Lesson 1.1 Algebra Number Patterns CC.3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. Identify and

Academic Standards for Grades Pre K High School Pennsylvania Department of Education INTRODUCTION The Pennsylvania Core Standards in in grades PreK 5 lay a solid foundation in whole numbers, addition,

### DENTAL IMPRESSIONS TARGET COMMON CORE STATE STANDARD(S) IN MATHEMATICS: N-Q.1:

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career

### CORE Assessment Module Module Overview

CORE Assessment Module Module Overview Content Area Mathematics Title Speedy Texting Grade Level Grade 7 Problem Type Performance Task Learning Goal Students will solve real-life and mathematical problems

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### Problem of the Month: Digging Dinosaurs

: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of

### Problem of the Month: Double Down

Problem of the Month: Double Down The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

### A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles

A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Open-Ended Problem-Solving Projections

MATHEMATICS Open-Ended Problem-Solving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEM-SOLVING OVERVIEW The Projection Masters for Problem-Solving

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### Curriculum Alignment Project

Curriculum Alignment Project Math Unit Date: Unit Details Title: Introduction to Functions Level: College Introductory (Pre-Calculus) Team Members: Stanley Ocken Mathematics, City College of New York,

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA. Semester Hours Credit: 3

CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA Semester Hours Credit: 3 (This course is equivalent to DSMA 0301. The difference being that this course is offered only on those campuses

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

### Indiana Academic Standards Mathematics: Algebra I

Indiana Academic Standards Mathematics: Algebra I 1 I. Introduction The college and career ready Indiana Academic Standards for Mathematics: Algebra I are the result of a process designed to identify,

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### Algebra Unit Plans. Grade 7. April 2012. Created By: Danielle Brown; Rosanna Gaudio; Lori Marano; Melissa Pino; Beth Orlando & Sherri Viotto

Algebra Unit Plans Grade 7 April 2012 Created By: Danielle Brown; Rosanna Gaudio; Lori Marano; Melissa Pino; Beth Orlando & Sherri Viotto Unit Planning Sheet for Algebra Big Ideas for Algebra (Dr. Small)

### Analyzing and Solving Pairs of Simultaneous Linear Equations

Analyzing and Solving Pairs of Simultaneous Linear Equations Mathematics Grade 8 In this unit, students manipulate and interpret equations in one variable, then progress to simultaneous equations. They

### High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.

Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### https://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...

of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students

### How To Understand And Solve Algebraic Equations

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### EE6-5 Solving Equations with Balances Pages 77 78

EE6-5 Solving Equations with Balances Pages 77 78 STANDARDS 6.EE.B.5, 6.EE.B.6 Goals Students will use pictures to model and solve equations. Vocabulary balance equation expression sides (of an equation)

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### Lesson 18: Introduction to Algebra: Expressions and Variables

LESSON 18: Algebra Expressions and Variables Weekly Focus: expressions Weekly Skill: write and evaluate Lesson Summary: For the Warm Up, students will solve a problem about movie tickets sold. In Activity

### INDIANA ACADEMIC STANDARDS. Mathematics: Grade 6 Draft for release: May 1, 2014

INDIANA ACADEMIC STANDARDS Mathematics: Grade 6 Draft for release: May 1, 2014 I. Introduction The Indiana Academic Standards for Mathematics are the result of a process designed to identify, evaluate,

### Solving Systems of Linear Equations Substitutions

Solving Systems of Linear Equations Substitutions Outcome (lesson objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking

### Grade 7 Mathematics. Unit 2. Integers. Estimated Time: 15 Hours

Grade 7 Mathematics Integers Estimated Time: 15 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization Grade

### DELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))

Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 9-10) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### WritePlacer Sample Topic. WritePlacer. Arithmetic

. Knowledge of another language fosters greater awareness of cultural diversity among the peoples of the world. Individuals who have foreign language skills can appreciate more readily other peoples values

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### How To Factor Quadratic Trinomials

Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

### Algebra II Unit Number 4

Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees.

### Big Ideas in Mathematics

Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### PHILOSOPHY OF THE MATHEMATICS DEPARTMENT

PHILOSOPHY OF THE MATHEMATICS DEPARTMENT The Lemont High School Mathematics Department believes that students should develop the following characteristics: Understanding of concepts and procedures Building

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Unit 12: Introduction to Factoring. Learning Objectives 12.2

Unit 1 Table of Contents Unit 1: Introduction to Factoring Learning Objectives 1. Instructor Notes The Mathematics of Factoring Teaching Tips: Challenges and Approaches Additional Resources Instructor

### This lesson introduces students to decimals.

NATIONAL MATH + SCIENCE INITIATIVE Elementary Math Introduction to Decimals LEVEL Grade Five OBJECTIVES Students will compare fractions to decimals. explore and build decimal models. MATERIALS AND RESOURCES

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests