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Multiple Pathways To Success Quarter 1 Learning Module Algebra 1 Equations, Inequalities, and Functions Copyright July 31, 2014 – Drafted October 29, 2015 Prince George’s County Public Schools Board of Education of Prince George’s County, Maryland
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Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

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Page 1: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

Multiple Pathways To Success

Quarter 1 Learning Module

Algebra 1

Equations, Inequalities, and Functions

Copyright July 31, 2014 – Drafted October 29, 2015

Prince George’s County Public Schools

Board of Education of Prince George’s County, Maryland

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Dear Scholars, As you move through the Algebra I curriculum, the level of academic rigor will increase. This could potentially lead to gaps in your understanding. Therefore, this learning module has been designed to assist you in acquiring and strengthening the essential skills needed for successful completion of Algebra I Common Core. Your experiences with this module will also help to remediate misconceptions, confusion, and rebuild areas of weakness. Sincerely, Writers of the Multiple Pathways to Success Modules

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Part I: Seeing Structure in Expressions

Student Learning Outcomes ● Write equivalent numerical and polynomial expressions in one variable, using addition,

subtraction, multiplication, and factoring, including multi-step problems in mathematical and contextual situations.

● Evaluate expressions, including for accuracy within context, and justify the results. ● Identify the vocabulary for the parts that make up the whole expression and interpret

their meaning in terms of a context. ● Understand that linear expressions have real world applications, which lead to the

solution of problems. Mathematical Practices

● MP1: Make sense of a problem and persevere in solving them ● MP2: Reason abstractly and quantitatively ● MP3: Construct a viable argument and critique the reasoning of others ● MP4: Model with mathematics ● MP6: Attend to precision ● MP7: Look for and make use of structure

Resources/Websites

● The following page contains the graphic organizer Common Vocabulary Associated With Expressions. This organizer provides the definitions of basic vocabulary terms associated with solving equations and offers visual representations of how the terms look are actually applied in mathematics.

● The Powerpoint will demonstrate Common Vocabulary and Examples Associated with Expressions. (https://drive.google.com/a/pgcps.org/file/d/0BwV-rRDNmsGlLU1LeTBfUGNMb3M/view?usp=sharing)

● The Video “Writing Algebraic Expressions from Words” will demonstrate how to translate English to algebraic expression. (http://mathbitsnotebook.com/Algebra1/AlgebraicExpressions/AEtranslations.html)

● The Video “What are variables, expressions, and equations?” will demonstrate how different parts of an expression effects the expression. (https://www.khanacademy.org/math/algebra-basics/core-algebra-expressions)

● The Cosmeo Video “Evaluating Algebraic Expressions with Rational Numbers” will demonstrate how to evaluate algebraic expression. (http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-154s.html)

● The Cosmeo Video “Simplifying Algebraic Expressions with Exponents Using Distributive Property” will demonstrate how to use distributive property when simplifying algebraic expressions.

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(http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-157s.html)

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1) a) Write an algebraic expression to model the following phrase: the price of a meal plus a 15% tip for the meal.

b) What could the expression represent, including units? c) What if the tip is 20% instead of 15% and 3 people are sharing the cost evenly. How can you represent the amount each person pays with a simplified algebraic expression? Identify the units for the expression.

2) Why is the expression not the correct factored form for ? 3)

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Given the expression: a) What is the largest coefficient? b) How many terms are in the polynomial? 4) Simplify and justify steps: 4(a + 2b) = 3(2a – b) + 6a – 7b Given 4a +8b – 6a + 3b + 6a – 7b _________________________________ 4a – 6a + 6a + 8b + 3b – 7b _________________________________ a(4 – 6 + 6) + b(8 + 3 – 7) _________________________________ a(4) + b(4) _________________________________ 4a + 4b _________________________________

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Baseball Math

5) A pitcher’s Winning Percentage is determined by dividing the number of games the pitcher won by the sum of the number of games won and lost. If Randy Johnson has won 18 games and lost 6 games, what is his Winning Percentage? (Write the answer to the nearest thousandth.) 6) A player’s Fielding Average is determined by adding the number of putouts and assists the player gets, divided by the total number of putouts, assists, and errors the player has. If Nomar Garciaparra has 185 putouts, 11 assists, and 4 errors, what is his Fielding Average? (Write the answer to the nearest thousandth.)

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Part II: Reasoning With Equations and Inequalities

Student Learning Outcomes: ● Solve linear equations and inequalities is one variable. ● Solve multi-variable formulas or literal equations for a specific variable in a linear

equation. ● Create and solve linear equations and inequalities in one variable and use them in

contextual situations to solve problems. ● Create and solve equations in two or more variables to represent relationships

between quantities. ● Interpret the solutions of equations and inequalities to determine their meanings

within the context of a real world situation. ● Select appropriate units for a specific formula and interpret the meaning of the unit in

that context. Mathematical Practices: ● MP1: Make Sense of a Problem and Persevere ● MP2: Reason Abstractly and Quantitatively ● MP3: Construct a viable argument and critique the reasoning of others ● MP4: Model with Mathematics ● MP6: Attend to Precision ● MP7: Make Use of Structure

Resources/Websites: ● The next page contains the graphic organizer Common Vocabulary Associated With

Solving Equations. This organizer provides the definitions of basic vocabulary terms associated with solving equations and offers visual representations of how the terms look are actually applied in mathematics.

● The YouTube Video “Solving Equations With Variables on Both Sides” will demonstrate how to solve equations through the use of the Distributive Property, the combination of like terms, and inverse operations. (https://www.youtube.com/watch?v=ZC4zUoLkq28)

● The PowerPoint Presentation, “Solving Literal Equations” will revisit how to use the Distributive Property, the combination of like terms, cross multiplication, and inverse operations to solve equations containing more than one variable. (https://drive.google.com/a/pgcps.org/file/d/0Byu2srspQwCmQ2c5bTQybnkyV00/view?usp=sharing)

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Common Vocabulary Associated With Solving Equations

Word Definition Example

Coefficient

A number multiplied with a

variable

3x + 5 = -73

Inverse Operation

An operation that reverses the

effect of another operation

3x + 5 = -73 – 5 – 5

____________

Distributive Property

The way in which multiplication is

applied to the addition or subtraction of two or more numbers

inside a set of parentheses a(b + c) = ab + ac

3(x + 6) = 12 3x + 18 = 12

Like Terms

Terms that have the same variable

and power, and can thus be combined

3x + 5y + 2x = 7

Solution

Any value for a variable that, when substituted, makes an equation true

3x – 7 = 17

3x = 24 x = 8

Isolation

To use inverse operations so that a variable is by itself on one side of

the equation

x + 7 = 8 – 7 – 7

X = 8 – 7

Opposites

Terms with the opposite sign but

the same absolute value

6n + -6n = 0

1. Acceleration is the measure of how fast a velocity is changing. The formula for

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acceleration is a = Vf − Vi, where a represents the acceleration rate, Vf is the final t

velocity, Vi is the initial velocity, and t represents the time in seconds.

a. Solve the formula for Vf. b. What is the final velocity of a runner who is accelerating at 2 feet per

second squared for 3 seconds with an initial velocity of 4 feet per second? 2. Deandre's doctor must decide how much medicine he needs for each dosage.

The dosage (d), in milligrams, depends on Deandre's body mass (m), in kilograms. The formula below is used to calculate the dosage of his medicine.

d = 0.1m2 + 5m

a. Explain what the variables “d” and “m” represent within the context of the problem. Underline the textual evidence that supports your response.

b. Explain what 80 kilograms means within the context of the problem. Circle the textual evidence that supports your response.

c. Which variable are you required to solve for? Box the textual evidence that supports your response.

d. Determine the dosage needed, in milligrams, if Deandre's body mass is 80 kilograms.

3. Solve for “x”: - (x – 3) = 2 (x – 5) + 4x. 5 4

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4. The following is a student solution to the inequality

There are two mathematical errors in the student’s work. Identify at what step each mathematical error occurred and explain why they are mathematically incorrect.

a) The first mathematical error occurred going from line ____ to line ____. The

student should have ______________________________________________

_______________________________________________________________

_______________________________________________________________.

b) The second mathematical error occurred going from line ____ to line ____. The student should have __________________________________________

_______________________________________________________________ _______________________________________________________________.

5. A high school is having a talent contest and will give different prizes for the best five acts in the show. First place wins the most money, and each place after that wins $50 less than the previous place.

Part A: Create an equation that can be used to determine the total amount of prize money based on the value of the first place prize. Enter the model in the space provided below.

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Part B: The talent contest has a total of $1000 in prize money. Solve the equation from Part A to determine the amount of money that exists for each of the five prizes. Enter your answer(s) and calculations in the space provided below.

Part C: Reason abstractly and quantitatively by explaining what your solution(s) means within the context of the problem.

Part D: Use validation methods to justify that your solutions are correct. Use may use words, symbols, or both in your response

Part III: Interpreting Function In high school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression. For example, the time in hours, T, it takes for a car to drive 100 miles is a function of the car’s speed in miles per hour, v; the rule T(v) = 100 v expresses this relationship algebraically and defines a function whose name is T. The set of inputs to a function is called its domain. We often infer the domain to be all inputs for which the expression defining a function has a value, or for which the function makes sense in a given context. A function can be described in various ways, such as by a graph, by a verbal rule, tables or charts, mapping and by an algebraic expression like f(x) = a + bx; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of the function models,

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and manipulating a mathematical expression for a function can throw light on the function’s properties.

Student Learning Objectives: ● Identify the domain as the x values of an equation (graph is left to right movement) ● Identify the range as the y values of an equation (graph is down to up movement) ● Define a function as mapping one element in the domain to exactly one element in

the range (the x value doesn’t repeat in the domain if the graph passes the vertical line test)

● Write equations in proper function notation ● Graph a function in the form y = f(x)

Mathematical Practices: ● MP1: Make Sense of a Problem and Persevere ● MP2: Reason Abstractly and Quantitatively ● MP3: Construct a viable argument and critique the reasoning of others ● MP4: Model with Mathematics ● MP6: Attend to Precision ● MP7: Make Use of Structure

Important Vocabulary Define each term or concept.

Term Definition Example

Relation A relation is simply a set of ordered pairs

{(3,2), (1,6), (-2,0)}

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Function A function is a special relation that assigns (maps) every element in

the domain to 1 and only 1 element of the range.

Domain The set of all x values of a relation or function

Given the relation {(3,2), (1,6), (-2,0)} Domain { -2, 1, 3 }

Range The set of all y values of a relations or functions.

Given the relation {(3,2), (1,6), (-2,0)} Range { 0, 1, 2 }

Independent variable

A variable that represents a quantity that is being manipulated

in an

The number of hours (h) you study and your test grade (g):

The hours you study is the independent variable. The test grade is the dependent variable because your test grade will depend on how long

your studied for the test.

Dependent variable

A variable that depends on one or more other variables

Topic Video Resource Website Links

Coordinate Plane:

Math is fun: https://www.mathsisfun.com/data/cartesian-coordinates.html

1 2 3 4

1 0 -1 3

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PowerPoint: https://drive.google.com/open?id=1r64VjztSoSMI5NwiK0F8n3Zlkr2yInM8OKgiR2dEEQc

Relations: Video- PH Determining a Reasonable Domain and Range for a Situation

Regents Prep: http://www.regentsprep.org/regents/math/algtrig/atp5/lfunction.htm PowerPoint: https://drive.google.com/open?id=1YifPWDnglT1lknKGYSjO258e4YpjKR2zXKvVvlSWkhI

Functions: Video: Introductions to functions https://youtu.be/VhokQhjl5t0 Video: PH Identifying Functions using a Mapping Video: PH Identifying Functions using a Vertical Line Test Video: PH Making a Table from a Function Rule Video: PH Finding the Range of a Function Given the Domain

Math is fun: https://www.mathsisfun.com/sets/function.html Math bits notebook: http://mathbitsnotebook.com/Algebra1/Functions/FNDomainRange.html

1)

The Customer A certain business keeps a database of information about its customers. a) Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning.

Customer Name Home Phone Number

Page 16: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

Heather Baker 3105100091

Mike London 3105200256

Sue Green 3234132598

Bruce Swift 3234132598

Michelle Metz 2138061124

b) Let P be the rule which assigns to each phone number in the table above, the customer

name(s) associated with it. Is P a function? Explain your reasoning.

c) Explain why a business would want to use a person's social security number as a way to identify a particular customer instead of their phone number.

2) Suppose f is a function. a.) If 10=f(−4), give the coordinates of a point on the graph of f. b.) If 6 is a solution of the equation f(w)=1, give a point on the graph of f. 3) John makes DVDs of his friend’s shows. He has realized that, because of his fixed costs, his average cost per DVD depends on the number of DVDs he produces. The cost of producing x DVDs is given by C(x)=2500+1.25x. a) John wants to figure out how much to charge his friend for the DVDs. He’s not trying to make

any money on the venture, but he wants to cover his costs. Suppose John made 100 DVDs. What is the cost of producing this many DVD’s? How much is the cost per DVD?

b) John is hoping to make many more than 100 DVDs for his friends. Complete the table showing his costs at different levels of production.

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Number of DVDs

0 10 100 1000 10000 100000 1000000

Total Cost

Cost per DVD

c) Explain why the average cost per DVD levels off. d) Find an equation for the average cost per DVD of producing x DVDs. 4)

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a)

b)

Page 19: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

Statements Answers

The car is not moving. Graph

The car is traveling at a steady speed. Graph ________ and Graph ________

5)

a)

Graph Equation Table Rule

A

B

C

D

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Page 21: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

b) Graph A c) Graph B d) Graph C e) Graph D

6)

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a)

b)

c)

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d)

Maryland College and Career Readiness Standards

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. (major) A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (major) A.CED.4 Rearrange formulas to highlight a

N.Q.1 Use units as a way to understand problems and guide the solution of multi-

step problems.�

(supporting) N.Q.2

Define appropriate units for the purpose of descriptive modeling. �

(supporting)(cross-cutting) N.Q.3 Choose a level of accuracy appropriate to limitations on measurement

when reporting quantities.�

(supporting) A.CED.3 Represent constraints by linear equations and interpret solutions as viable or non-viable options in a modeling context. (major)

A.SSE.1a Use the structure of an expression to analyze linear equations in order to

develop a plan for solving the problem.�

(major)

A.SSE.1b

Interpret complicated expressions by viewing one or more of their parts as

single entity. �

(major)(fluency)

Page 24: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

quantity of interest, using the same reasoning as in solving equations. (major)

A.REI.3 Solve linear equations in one variable including equations with coefficients represented by letters. (major)

A.REI.1 Explain the steps in solving linear equations in one variable. (major) (cross-cutting) A.SSE.1a Use the structure of an expression to analyze linear equations in order to

develop plan for solving the problem.�

(major) A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as

a single entity�

. (major) (fluency)

Page 25: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

Scoring Rubric / Success Criteria

Conceptual Understanding

42 Total Points

Part I: Seeing Structure in Expressions (1a, 1b, 1c, 2, 3a, 3b, 4, 5, 6)

9

One point for each part of each problem

Part II: Reasoning With Equations and Inequalities

(1a, 1b, 2a, 2b, 2c, 2d, 3, 4a, 4b, 5a, 5b, 5c, 5d)

13

One point for each part of each problem

Part III: Interpreting Functions (1a, 1b, 1c, 2a, 2b, 3a, 3b, 3c, 3d, 4a, 4b, 5a,

5b, 5c, 5d, 5e, 6a, 6b, 6c, 6d)

20 One point for each part of each problem

Execution of Mathematical Practices

12 Total Points

MP1: Make sense of a problem and persevere in solving them

● Analyze and explain the meaning of the

problem ● Actively engage in problem solving

(Develop, carry out, and refine a plan)

2

one point per bullet

MP2: Reason abstractly and quantitatively ● Represent a problem with symbols ● Explain their thinking ● Examine the reasonableness of their

answers/calculations

3

one point per bullet

MP3: Construct a viable argument and 1

Page 26: Algebra 1 QLM Quarter 1 - PGCPS · Quarter 1 Learning Module Algebra 1 Equations, ... translate English to algebraic expression. ... The test grade is

critique the reasoning of others ● Justify solutions and approaches

MP4: Model with mathematics ● Use representations to solve real life

problems ● Apply formulas and equations where

appropriate

2

one point per bullet

MP6: Attend to precision ● Calculate accurately and efficiently ● Explain their thinking using mathematics

vocabulary ● Use appropriate symbols and specify units

of measure

3

one point per bullet

MP7: Look for and make use of structure ● Use knowledge of properties to efficiently

solve problems

1

Final Score /54