Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables

Save this PDF as:

Size: px
Start display at page:

Transcription

1 of 26 8/20/2014 2:00 PM Answers Teacher Copy Activity 3 Lesson 3-1 Systems of Linear Equations Monetary Systems Overload Solving Systems of Two Equations in Two Variables 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 should recall that an absolute value of a number is its distance from zero on a number line. Have students evaluate the following: [6] 6 [6] Then have students solve the following equation x = 6 College [x = 6 or Board. x = 6] All rights reserved.

2 of 26 8/20/2014 2:00 PM Example A Marking the Text, Interactive Word Wall Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check their results. Developing Math Language An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two numbers that have a specific distance from zero on a number line. 1 Identify a Subtask, Quickwrite When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary. Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to write each equation and then discuss how the solution set is represented by the graph. Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler example such as: The average cost of a pound of coffee is \$8. However, the cost sometimes varies by \$1. This means that the coffee could cost as little as \$7 per pound or as much as \$9 per pound. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line. Have groups present their findings to the class. ELL Support For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think of it. Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount (greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).

3 of 26 8/20/2014 2:00 PM Developing Math Language An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <, >,,, or. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related. Example C Simplify the Problem, Debriefing Before addressing Example C, discuss the following: Inequalities with A > b, where b is a positive number, are known as disjunctions and are written as A < b or A > b. For example, x > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x < 5 or x > 5. See graph A. This also holds true for A b. Inequalities with A < b, where b is a positive number, are known as conjunctions and are written as b < A < b, or as b < A and A < b. For example: x < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution is 5 < x < 5. See graph B. This also holds true for A b. Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the whole class. Teacher to Teacher Another method for solving inequalities relies on the geometric definition of absolute value x a as the distance from x to a. Here s how you can solve the inequality in the example: Thus, the solution set is all values of x whose distance from is greater than. The solution can be represented on a number line and written as x < 4 or x > 1.

4 of 26 8/20/2014 2:00 PM 2 Quickwrite, Self Revision/Peer Revision, Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving inequalities. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of students present their solutions to Item 4. Assess Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities. Activity Standards Focus In Activity 3, students write and graph systems of equations. They solve the systems of equations using graphing, substitution, and elimination. They also use technology and matrices to solve systems of equations. Throughout this activity, emphasize that there is more than one way to solve a system of equations and that some methods are more efficient in certain situations. Plan Pacing: 2 class periods Chunking the Lesson #1 2 #3 Check Your Understanding

5 of 26 8/20/2014 2:00 PM #7 Example A #11 Example B Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Have students list five solutions to the equation 2x + y = 14. Then pose and discuss the following questions: Will all students have the same five solutions? How many solutions exist for the equation? How can you visually show all of the existing solutions for the equation? Developing Math Language Be sure students understand that a solution to a system of equations is any ordered pair that, when substituted into each equation in the system, results in a true statement for every one of the equations in the system. If an ordered pair makes one equation true, but not all of the equations in the system, it is not a solution. 1 2 Shared Reading, Close Reading, Interactive Word Wall, Create Representations These first few items introduce solving systems of linear equations by graphing. Item 1 also demonstrates the limitations of graphing as a solution method. It asks students to approximate the solution by identifying a point of intersection that is not a lattice point in the coordinate plane. Review with students that a lattice point is a corner or intersection of two grid lines on the Cartesian plane. Common Core State Standards for Activity 3 HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3 Create Representations Remind students that to graph an equation, they should either write the equation in slope-intercept form or find the x- and

6 of 26 8/20/2014 2:00 PM y-intercepts. Technology Tip Students can use graphing calculators to graph each system and determine its solution. On TI calculators, the intersect option is found as option 5 under the [2nd][CALC] menu. On a TI-Nspire, this is done under the analyze option in the Graphs&Geometry tool. For additional technology resources, visit SpringBoard Digital. Developing Math Language Make sure that students understand that although there are four terms used when describing the solution set for a system of equations, there are only three classifications for a solution set: (1) inconsistent, (2) consistent and independent, (3) consistent and dependent. Mini-Lesson: Solving Systems Using a Graphing Calculator If students need additional help solving systems of equations using a graphing calculator, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to classifying a system of equations by the number of its solutions. To reinforce Item 5, have students make a sketch of the situation. 7 Predict and Confirm, Discussion Groups, Look for a Pattern Prior to using analytic geometry to solve this item, focus student attention on the starting amounts for both plans as well as the rate of change for both accounts. Students may note that they begin \$3,600 apart and that the gap will narrow by \$100 each month. Therefore, it will take 36 months for the accounts to be equal. Connect the initial amounts to the y-intercept and the rates of change to the slopes when solving using analytic geometry. Differentiating Instruction Tables of values can be used to answer Item 7. Creating and populating the tables of values often helps students who struggle with algebraic modeling to write equations correctly.

8 of 26 8/20/2014 2:00 PM Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Teacher to Teacher In Item 18, let one variable represent the number of engineers who will stay at level I, and let the other variable represent the number of engineers who will be promoted to level II. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of equations and solving systems of equations by graphing, substitution, and elimination. If students are having difficulty writing equations that model a situation, review the steps of identifying what you know and what you want to know, assigning variable names and writing equations based on what you know. Learning Targets p. 29 Use graphing, substitution, and elimination to solve systems of linear equations in two variables. Formulate systems of linear equations in two variables to model real-world situations. Shared Reading (Learning Strategy) Definition Reading the text aloud (usually by the teacher) as students follow along silently, or reading a text aloud by the teacher and students Purpose Helps auditory learners do decode, interpret, and analyze challenging text Close Reading (Learning Strategy) Definition

9 of 26 8/20/2014 2:00 PM Reading text word for word, sentence by sentence, and line by line to make a detailed analysis of meaning Purpose Assists in developing a comprehensive understanding of the text Create Representations (Learning Strategy) Definition Creating pictures, tables, graphs, lists, equations, models, and /or verbal expressions to interpret text or data Purpose Helps organize information using multiple ways to present data and to answer a question or show a problem solution Discussion Groups (Learning Strategy) Definition Working within groups to discuss content, to create problem solutions, and to explain and justify a solution Purpose Aids understanding through the sharing of ideas, interpretation of concepts, and analysis of problem scenarios Role Play (Learning Strategy) Definition Assuming the role of a character in a scenario Purpose Helps interpret and visualize information in a problem Think-Pair-Share (Learning Strategy) Definition

10 0 of 26 8/20/2014 2:00 PM Thinking through a problem alone, pairing with a partner to share ideas, and concluding by sharing results with the class Purpose Enables the development of initial ideas that are then tested with a partner in preparation for revising ideas and sharing them with a larger group Quickwrite (Learning Strategy) Definition Writing for a short, specific amount of time about a designated topic Purpose Helps generate ideas in a short time Note Taking (Learning Strategy) Definition Creating a record of information while reading a text or listening to a speaker Purpose Helps in organizing ideas and processing information Look for a Pattern (Learning Strategy) Definition Observing information or creating visual representations to find a trend Purpose Helps to identify patterns that may be used to make predictions Suggested Learning Strategies

11 1 of 26 8/20/2014 2:00 PM Shared Reading, Close Reading, Create Representations, Discussion Groups, Role Play, Think-Pair-Share, Quickwrite, Note Taking, Look for a Pattern Have you ever noticed that when an item is popular and many people want to buy it, the price goes up, but items that no one wants are marked down to a lower price? p. 30 Connect to Economics The role of the desire for and availability of a good in determining price was described by Muslim scholars as early as the fourteenth century. The phrase supply and demand was first used by eighteenth-century Scottish economists. The change in an item s price and the quantity available to buy are the basis of the concept of supply and demand in economics. Demand refers to the quantity that people are willing to buy at a particular price. Supply refers to the quantity that the manufacturer is willing to produce at a particular price. The final price that the customer sees is a result of both supply and demand. Suppose that during a six-month time period, the supply and demand for gasoline has been tracked and approximated by these functions, where Q represents millions of barrels of gasoline and P represents price per gallon in dollars. Demand function: P = 0.7Q Supply function: P = 1.5Q 10.4 Math Terms A point, or set of points, is a solution of a system of equations in two variables when the coordinates of the points make both equations true.

12 2 of 26 8/20/2014 2:00 PM To find the best balance between market price and quantity of gasoline supplied, find a solution of a system of two linear equations. The demand and supply functions for gasoline are graphed below. 1. Make use of structure. Find an approximation of the coordinates of the intersection of the supply and demand functions. Explain what the point represents. Sample answer: (9.15, 3.3); At a price of \$3.30, people will demand 9.15 million gallons of gas, and companies will be willing to supply it. 2. What problem(s) can arise when solving a system of equations by graphing? Sample answer: Graphing is not very accurate if the intersection is not on a lattice point, or the scaling of the graph is not accurate enough. Technology Tip You can use a graphing calculator and its Calculate function to solve systems of equations in two variables. Math Terms Systems of linear equations are classified by the number of solutions. Systems with one or many solutions are consistent. Systems with no solution are inconsistent. A system with exactly one solution is independent.

13 3 of 26 8/20/2014 2:00 PM A system with infinite solutions is dependent. 3. Model with mathematics. For parts a c, graph each system. Determine the number of solutions. a. one solution b. no solutions c. infinitely many solutions d. Graphing two linear equations illustrates the relationships of the lines. Classify the systems in parts a c as consistent and independent, consistent and dependent, or inconsistent. a. b. c. consistent and independent inconsistent consistent and dependent Check Your Understanding p Describe how you can tell whether a system of two equations is independent and consistent by looking at its graph. If the system is independent and consistent, the graph will show a pair of lines that intersect at a point.

14 4 of 26 8/20/2014 2:00 PM 5. The graph of a system of two equations is a pair of parallel lines. Classify this system. Explain your reasoning. The system is inconsistent. A pair of parallel lines never intersect, which means that the graphs of the equations have no points in common and the system has no solutions. 6. Make sense of problems. A system of two linear equations is dependent and consistent. Describe the graph of the system and explain its meaning. The graph of the system is a single line; there are an infinite number of solutions. p. 32 Connect to Personal Finance A down payment is an initial payment that a customer makes when buying an expensive item, such as a house or car. The rest of the cost is usually paid in monthly installments. Discussion Group Tips As you work with your group, review the problem scenario carefully and explore together the information provided and how to use it to create a potential solution. Discuss your understanding of the problem and ask peers or your teacher to clarify any areas that are not clear. 7. Marlon is buying a used car. The dealership offers him two payment plans, as shown in the table. Payment Plans Plan Down Payment (\$) Monthly Payment (\$)

15 5 of 26 8/20/2014 2:00 PM Marlon wants to answer this question: How many months will it take for him to have paid the same amount using either plan? Work with your group on parts a through f and determine the answer to Marlon s question. a. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 1. y = 300x b. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 2. y = x c. Write the equations as a system of equations. Math Tip When graphing a system of linear equations that represents a real-world situation, it is a good practice to label each line with what it represents. In this case, you can label the lines Plan 1 and Plan 2. d. Graph the system of equations on the coordinate grid.

16 6 of 26 8/20/2014 2:00 PM e. Reason quantitatively. What is the solution of the system of equations? What does the solution represent in this situation? (36, 10,800); In 36 months, the total cost of both plans will be \$10,800. f. In how many months will the total costs of the two plans be equal? 36 months Check Your Understanding 8. How could you check that you solved the system of equations in Item 7 correctly? Sample answer: Check that the ordered pair (36, 10,800) satisfies both of the equations in the system. 9. If Marlon plans to keep the used car less than 3 years, which of the payment plans should he choose? Justify your answer. Plan 1; The graph shows that when the time is less than 36 months (or 3 years), the total amount paid for Plan 1 is less than the total amount paid for Plan Construct viable arguments. Explain how to write a system of two equations that models a real-world situation. Sample answer: Identify the two quantities in the situation that can vary. Assign variables to these quantities. Write an equation in terms of the two variables that models part of the situation. Then write a second equation in terms of the two variables that models another part of the situation. Finally, write the two equations as a system.

17 7 of 26 8/20/2014 2:00 PM p. 33 Math Terms In the substitution method, you solve one equation for one variable in terms of another. Then substitute that expression into the other equation to form a new equation with only one variable. Solve that equation. Substitute the solution into one of the two original equations to find the value of the other variable. Investors try to control the level of risk in their portfolios by diversifying their investments. You can solve some investment problems by writing and solving systems of equations. One algebraic method for solving a system of linear equations is called substitution. Example A During one year, Sara invested \$5000 into two separate funds, one earning 2 percent and another earning 5 percent annual interest. The interest Sara earned was \$205. How much money did she invest in each fund? Step 1: Let x = money in the first fund and y = money in the second fund. Write one equation to represent the amount of money invested. Write another equation to represent the interest earned. Step 2: Use substitution to solve this system.

18 8 of 26 8/20/2014 2:00 PM Step 3: Substitute the value of x into one of the original equations to find y. Math Tip Check your answer by substituting the solution (1500, 3500) into the second original equation, 0.02x y = 205 Solution: Sara invested \$1500 in the first fund and \$3500 in the second fund. Try These A Write your answers on notebook paper. Show your work. Solve each system of equations, using substitution. Check students work.

19 9 of 26 8/20/2014 2:00 PM a. ( 14, 13) b. (12, 1) c. (3, 7) d. Model with mathematics. Eli invested a total of \$2000 in two stocks. One stock cost \$18.50 per share, and the other cost \$10.40 per share. Eli bought a total of 130 shares. Write and solve a system of equations to find how many shares of each stock Eli bought. p. 34

20 0 of 26 8/20/2014 2:00 PM 11. When using substitution, how do you decide which variable to isolate and which equation to solve? Explain. Sample answer: Choose a variable that is easy to isolate by finding the equation with a variable that has a coefficient of 1 or 1. Another algebraic method for solving systems of linear equations is the elimination method. Example B p. 35 Math Terms In the elimination method, you eliminate one variable. Multiply each equation by a number so that the terms for one variable combine to 0 when the equations are added. Then use substitution with that value of the variable to find the value of the other variable. The ordered pair is the solution of the system. The elimination method is also called the addition-elimination method or the linear combination method for solving a system of linear equations. A stack of 20 coins contains only nickels and quarters and has a total value of \$4. How many of each coin are in the stack? Step 1: Let n = number of nickels and q = number of quarters. Write one equation to represent the number of coins in the stack. Write another equation to represent the total value.

21 1 of 26 8/20/2014 2:00 PM Step 2: To solve this system of equations, first eliminate the n variable. Step 3: Find the value of the eliminated variable n by using the original first equation. Step 4: Check your answers by substituting into the original second equation. Solution: There are 5 nickels and 15 quarters in the stack of coins. Try These B Solve each system of equations using elimination. Show your work. Check students work. a. (5, 5)

22 2 of 26 8/20/2014 2:00 PM b. (2, 4) c. ( 3, 4) d. A karate school offers a package of 12 group lessons and 2 private lessons for \$110. It also offers a package of 10 group lessons and 3 private lessons for \$125. Write and solve a system of equations to find the cost of a single group lesson and a single private lesson. Check Your Understanding 12. Compare and contrast solving systems of equations by using substitution and by using elimination. Sample answer: In both methods, you start by solving for the value of one of the variables and then use that value to solve for the value of the other variable. In the substitution method, you use substitution to get rid of one of the variables. In the elimination method, you add equations to get rid of one of the variables.

23 3 of 26 8/20/2014 2:00 PM 13. Reason abstractly. Ty is solving the system using substitution. He will start by solving one of the equations for x. Which equation should he choose? Explain your reasoning. Sample answer: The first equation; to solve the first equation for x, Ty only needs to add 2y to both sides, but to solve the second equation for x, Ty would need to do two steps: first, subtract 6y from both sides, and then divide both sides by Explain how you would eliminate one of the variables in this system: Sample answer: Multiply the second equation by 2 to get. Then add the equations to eliminate the variable y and get 8x = 33. Lesson 3-1 Practice 15.Solve the system by graphing. ( 2, 5) 16.Solve the system using substitution.

24 4 of 26 8/20/2014 2:00 PM ( 5, 6) 17.Solve the system using elimination. (3, 4) 18. Make sense of problems and persevere in solving them. At one company, a level I engineer receives a salary of \$56,000, and a level II engineer receives a salary of \$68,000. The company has 8 level I engineers. Next year, it can afford to pay \$472,000 for their salaries. Write and solve a system of equations to find how many of the engineers the company can afford to promote to level II., where x is the number of engineers who will stay at level I and y is the number of engineers who will be promoted to level II; solution: (6, 2); The company can afford to promote 2 engineers to level II. 19. Which method did you use to solve the system of equations in Item 18? Explain why you chose this method. Answers will vary.

25 5 of 26 8/20/2014 2:00 PM

26 6 of 26 8/20/2014 2:00 PM

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

Tennessee Department of Education. Task: Sally s Car Loan

Tennessee Department of Education Task: Sally s Car Loan Sally bought a new car. Her total cost including all fees and taxes was \$15,. She made a down payment of \$43. She financed the remaining amount

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

Isolate the absolute value expression. Add 5 to both sides and then divide by 2.

11 of 21 8/14/2014 2:35 PM of a variable expression. You can use the definition of absolute value to solve absolute value equations algebraically. Since then the equation ax + b = c is equivalent to (ax

Unit 1 Equations, Inequalities, Functions

Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious

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

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 7 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

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

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

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,

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

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:

Teacher: Maple So School: Herron High School. Comparing the Usage Cost of Electric Vehicles Versus Internal Combustion Vehicles

Teacher: Maple So School: Herron High School Name of Lesson: Comparing the Usage Cost of Electric Vehicles Versus Internal Combustion Vehicles Subject/ Course: Mathematics, Algebra I Grade Level: 9 th

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

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.

Mathematics Curriculum Guide Precalculus 2015-16. Page 1 of 12

Mathematics Curriculum Guide Precalculus 2015-16 Page 1 of 12 Paramount Unified School District High School Math Curriculum Guides 2015 16 In 2015 16, PUSD will continue to implement the Standards by providing

A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

Lesson 22: Solution Sets to Simultaneous Equations

Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

Absolute Value of Reasoning

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

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

PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

Letter to the Student... 5 Letter to the Family... 6 Correlation of Mississippi Competencies and Objectives to Coach Lessons... 7 Pretest...

Table of Contents Letter to the Student........................................... 5 Letter to the Family............................................. 6 Correlation of Mississippi Competencies and Objectives

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

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

Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan

Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve

Overview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices

Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with

DRAFT. Algebra 1 EOC Item Specifications

DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

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:

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,

Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

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

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

2-8 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Solve. 1. y + 7 < 11 2. 4m 12 3. 5 2x 17 y < 18 m 3 x 6 Use interval notation to indicate the graphed numbers. 4. (-2, 3] 5. (-, 1] Objectives Solve

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

3.1. Angle Pairs. What s Your Angle? Angle Pairs. ACTIVITY 3.1 Investigative. Activity Focus Measuring angles Angle pairs

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Use Manipulatives Two rays with a common endpoint form an angle. The common endpoint is called the vertex. You can use a protractor to draw and measure

Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

Content Emphases for Grade 7 Major Cluster 70% of time

Critical Area: Geometry Content Emphases for Grade 7 Major Cluster 70% of time Supporting Cluster 20% of Time Additional Cluster 10% of Time 7.RP.A.1,2,3 7.SP.A.1,2 7.G.A.1,2,3 7.NS.A.1,2,3 7.SP.C.5,6,7,8

IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

Algebra 1-2. A. Identify and translate variables and expressions.

St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities

Chapter 6 Notes Name Section 6.1 Solving One-Step Linear Inequalities Graph of a linear Inequality- the set of all points on a number line that represent all solutions of the inequality > or < or circle

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

Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

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.

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

I Can Do This! Systems of Equations Grade Eight

Ohio Standards Connection: Patterns, Functions and Algebra Benchmark H Solve systems of linear equations involving two variables graphically and symbolically. Indicator 10 Solve 2 by 2 systems of linear

Let s explore the content and skills assessed by Heart of Algebra questions.

Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

Solving Systems of Linear Equations Elimination (Addition)

Solving Systems of Linear Equations Elimination (Addition) Outcome (lesson objective) Students will accurately solve systems of equations using elimination/addition method. Student/Class Goal Students

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 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

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

Acquisition Lesson Plan for the Concept, Topic or Skill---Not for the Day

Acquisition Lesson Plan Concept: Linear Systems Author Name(s): High-School Delaware Math Cadre Committee Grade: Ninth Grade Time Frame: Two 45 minute periods Pre-requisite(s): Write algebraic expressions

Return on Investment (ROI)

ROI 1 Return on Investment (ROI) Prepared by Sarah Major What is ROI? Return on investment (ROI) is a measure that investigates the amount of additional profits produced due to a certain investment. Businesses

The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

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

Task: Will and Latisha s Tile Problem Algebra I

Tennessee Department of Education Task: Will and Latisha s Tile Problem Algebra I In math class, Will and Latisha were challenged to create their own pattern with tiles. Latisha built the first three arrangements

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

Make sure you look at the reminders or examples before each set of problems to jog your memory! Solve

Name Date Make sure you look at the reminders or examples before each set of problems to jog your memory! I. Solving Linear Equations 1. Eliminate parentheses. Combine like terms 3. Eliminate terms by

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

Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

My Notes CONNECT TO SCIENCE

Mood Rings, Part SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/ Retell, Vocabulary Organizer Tyrell and four of his friends from West Middle School went to a craft fair and they all decided to buy

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

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

Systems of Linear Equations and Inequalities

Systems of Linear Equations and Inequalities Recall that every linear equation in two variables can be identified with a line. When we group two such equations together, we know from geometry what can

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

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

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

Slope Investigation. Grade 8, Algebra 1, or Math 1 in a unit with rate of change and slope

NATIONAL MATH + SCIENCE INITIATIVE Mathematics Slope Investigation LEVEL Grade 8, Algebra 1, or Math 1 in a unit with rate of change and slope MODULE/CONNECTION TO AP* Rate of Change *Advanced Placement

Algebra Bridge Project Cell Phone Plans

Algebra Bridge Project Cell Phone Plans Name Teacher Part I: Two Cell Phone Plans You are in the market for a new cell phone, and you have narrowed your search to two different cell phone companies --

Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY

RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY I. INTRODUCTION According to the Common Core Standards (2010), Decisions or predictions are often based on data numbers

3.1 Solving Systems Using Tables and Graphs

Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,

Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

10.1 Systems of Linear Equations: Substitution and Elimination

726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

Word Problems Involving Systems of Linear Equations

Word Problems Involving Systems of Linear Equations Many word problems will give rise to systems of equations that is, a pair of equations like this: 2x+3y = 10 x 6y = 5 You can solve a system of equations

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

Algebra 1 Topic 8: Solving linear equations and inequalities Student Activity Sheet 1; use with Overview

Algebra 1 Topic 8: Student Activity Sheet 1; use with Overview 1. A car rental company charges \$29.95 plus 16 cents per mile for each mile driven. The cost in dollars of renting a car, r, is a function

Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.

Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable

High School Functions Interpreting Functions Understand the concept of a function and use function notation.

Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.

Systems of Linear Equations

DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

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

Describing and Solving for Area and Perimeter

Grade 3 Mathematics, Quarter 2,Unit 2.2 Describing and Solving for Area and Perimeter Overview Number of instruction days: 8-10 (1 day = 90 minutes) Content to Be Learned Distinguish between linear and

Unit 1: Place value and operations with whole numbers and decimals

Unit 1: Place value and operations with whole numbers and decimals Content Area: Mathematics Course(s): Generic Course Time Period: 1st Marking Period Length: 10 Weeks Status: Published Unit Overview Students

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

The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students