Weather 101 - Weeblyapickell.weebly.com/.../algebra_i_syllabus_v16-17.docx · Web viewStudents will view Algebra as a tool for analyzing and describing mathematical relationships,

Course Name: Algebra 1(At MS Level: 8th Grade Algebra)

Course Level: 8th/9th gradesCourse Code: MA113/MA123 or MA816

Length of Course: Two Semesters

DescriptionIn this course, students will learn the basics of Algebra. All students will apply the mathematical concepts of Algebra I to enhance their understanding and lifelong use of mathematics. Students will view Algebra as a tool for analyzing and describing mathematical relationships, and for modeling problems that come from the workplace, the sciences, technology, engineering, and mathematics. Mathematics ObjectivesAll students will:

Demonstrate an understanding of, reason about and apply the relationships between different number systems

Represent quantitative relationships using mathematical symbols, and interpret relationships from those representations

Calculate fluently, estimate proficiently, and describe and use algorithms in appropriate situations (e.g., approximating solutions to equations)

Recognize, construct, interpret, and evaluate expressions Transform symbolic expressions into equivalent forms Determine appropriate techniques for solving each type or equation, inequality or system of equa-

tions, apply the techniques correctly to solve, justify the steps in the solutions, and draw conclu-sions from the solutions.

Know and apply common formulas Demonstrate an understanding of functions, their representations, and their attributes Classify functions and know the characteristics of each family. Construct or select a function to model a real-world situation in order to solve applied problems Demonstrate mathematical proficiency through the 8 standards for mathematical practice

1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Grand Rapids Public Schools Algebra 1 2011 Version 8-31.16

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Michigan Educational Technology Standards (METS)9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and independent learning.9.12.CC.2 Use available technologies (e.g., desktop conferencing, e-mail, video conferencing, instant messaging) to communicate with others on a class assignment or project. 9.12.CC.3 Collaborate in content-related projects that integrate a variety of media (e.g., print, audio, video, graphic, simulations, and models)9.12.CC.4 Plan and implement a collaborative project using telecommunications tools (e.g., ePals, discussion boards, online groups, interactive websites, video conferencing)

In order to meet the MET standards in mathematics teachers will need to supplement lessons with online experiences for students to practice with applets, online interactive activities, and communication mediums such as email, forums, and discussion boards. Providing students with experiences online using applets will help prepare them for the Common Core State Standards online assessments that will begin in SY 2014-15. See Algebra I Discovering Algebra Technology Supplement.

TextbookDiscovering Algebra, Key Curriculum Press, 2007

Chapter 0 is a review of fractions and decimals and is an optional chapter. Those lessons can be done at any time during the year. They can also be left as standing lesson

plans for substitute teachers. In the Discovering Algebra book, all of the “Activity Day” lessons are optional because there is

no new learning. The “Take Another Look” lessons at the end of each chapter are not review, but meant to be

extensions, and would be appropriate for designated STEM classes (Science, Technology, Engineering, and Mathematics).

Focus investigations are listed for each unit. o These are the critical investigations to the big concepts of the unit. o Additional investigations are strongly suggested when appropriate and time

permits. Please read the syllabus carefully as there are supplemental lessons that are required to meet the Common Core State Standards.

Technology and Other ResourcesGraphing Calculators, TI-84 or higher with overhead screenDocument cameras with projectorsInteractive white boards (where available)Laptop cartsGeometer’s Sketchpad (District license has been purchased)


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Grading Procedure and Scale 70% - 30%Performance: (Assessment of understanding and knowledge of concepts)

District Common Assessments (DA): 40%Teacher Created Assessments (TCA): 30% (summative quizzes, tests, projects)

Process: 30% Effort and Participation – 20%Classwork and Homework – 10%

Grading ScaleA+ Exemplary Level of Performance

93 –100 A Outstanding level of performance90 – 92 A-

87 – 89 B+High level of performance83 – 86 B

80 – 82 B-77 – 79 C+

Acceptable level of performance

73 – 76 C70 – 72 C-67 – 69 D+

Minimal level of performance63 – 66 D60 – 62 D-0 – 59 E Unacceptable level of performance

Assessments Required for each unit

Common assessment (located on curriculum drive in Algebra 1>Assess-ment folder)

Other teacher created assessments(to be labeled in Grade book with the prefix TCA) Formative assessments

End of Semester Common Exam

Options for Literacy Strategies – See “Literacy Strategies in Mathematics” folder on the curriculum drive for more resources.

Frayer Model Thinking Maps Graphic Organizers SIOP Marzano’s vocabulary strategies SQ3R Reflection strategy: RAFT, Writing to Learn


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Algebra 1 Curriculum Map (See the individual unit details for specific Illuminations lessons)Unit of Study Instructional Materials Big Ideas Pacing

Unit 1Descriptive Statistics Discovering Algebra Chapter 1

1.1 – 1.4, 1.6, 1.7

Analyzing and interpreting data and statistics lead us to informed decisions.Focus Investigation 1.7 & Applet lessons from Illuminations1.1-1.4 should take no longer than 4 days.

2 weeks

Unit 2Relationships Between Quantities and

Reasoning with Equations

Discovering Algebra Chapter 2All sections except 2.6

Algebra allows us to represent real world problems and justify their solutions.Focus Investigations 2.4 and 2.8 & Illuminations LessonAdd in solving equations

5 weeks

Unit 3Linear Relationships

Part 1, Linear Equations

Discovering Algebra Chapter 3All sections except 3.3 and 3.7

This unit builds towards the concept of functions by connecting recursive routines with explicit linear equations.Focus Lessons 3.2 and 3.4 & Illuminations lessonsAdd in Numeracy within warm-ups

4 weeks


Part 2, Fitting a line to data

Discovering Algebra Chapter 4All sections 4.1 – 4.4, 11.1

Optional Sections 4.5 – 4.8

This unit formalizes slope and builds different algebraic representations of linear functions based on the most appropriate form to model a given set of data.Focus Lessons 4.1 and 4.4 & Illuminations Lessons

4 weeks


Part 3, Systems of equations and inequalities


Real world data can be represented by systems of linear equations and inequalities which can be solved for graphically and algebraically.Focus Lessons 5.1 and 5.5 & Illuminations Lessons

4 weeks

Unit 6Exponential Relationships

Discovering Algebra Chapter 6All sections.

Supplement Radical Exponents

Multiplicative recursive routines are explicitly defined as exponential functions. Exponential relationships are functions which allow us to organize data and make informed decisions.Focus Lesson 6.1 & Illuminations LessonsSupplement radical exponents

4 weeks

Unit 7Functions and Transformations

Discovering Algebra Chapter 7All sections

Discovering Algebra Chapter 8Sections 8.1 – 8.4 only

Functions are mathematical relationships where one or more variables are dependent on another variable. Functions can be categorized into families of functions with a parent function which has parameters that can be changed to create new functions.Focus Lessons 7.2 and 8.2

5 weeks

Unit 7.5 Supplemental

Adding, subtracting, and multiplying polynomials result in a new polynomial. The distributive property is used when multiplying polynomials. When we factor polynomials, we are undoing the distributive property. (Converse of the Distributive Property);

1 week

Unit 8Quadratic Functions and Modeling


Quadratic relationships can be represented with multiple representations that can be used to solve real-world problems.

4 weeks

Grand Rapids Public Schools Algebra 1 2011 Version 8-31.16 -4-

Focus Lessons 9.3 and 9.6 & Illuminations Lesson

Grand Rapids Public Schools Algebra 1 2011 Version 8-31.16 -5-

Unit 1: Descriptive Statistics

Focus Question: In analyzing bivariate sets of data, how do scatterplots, correlation coefficients, and linear regression lines help make sense of data and assist in making predictions about data sets?

Big Ideas: Analyzing and interpreting data and statistics lead us to informed decisions

Essential Questions: How can you use data in everyday life? What might be a consequence of misinterpreting data? How can you tell if a linear function is a good model for a given data set?

Learning Outcomes: A data set consists of information that can be grouped into categories and represented visually

perhaps by a pictograph, a dot plot, or a bar graph, which show the number of data points in each category.

Given a data set, some measures of central tendency are more useful than others, especially when there are outliers.

Plots are useful for visually representing measures of central tendency. Box and whisker plots show the quartiles of the ranked data and the outliers. Scatter plots show whether data is correlated if the points are clustered about a line. The correlation coefficient is a measure of how well the linear function models a given set of

data.

Focus Investigation: 1.7 “Guesstimation” Investigation Goal: Students practice estimation skills and understand the meaning of points on the coordinate plane in comparison with the graph of y=x.

Investigation Task: Golf and Divorce -- (Teacher Document)Investigation Goal: Students are given data indicating a correlation between two vari-ables, and are asked to reason out whether or not causation can be inferred.

Instructional Notes: It is recommended that technology be incorporated into unit 1. Most of the Chapter 1 topics are in the middle school standards therefore sections 1.1-1.4 including any assessments should take no longer than 4 days. In addition, students will need to know how to enter a list, create a scatter plot, and perform operations with lists for section 1.7. (See the calculator notes for chapter 1) Section 1.7 is important because it introduces y = x. Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals. A supplemental lesson on correlation and causation will need to be added to this section. Section 1.8 is omitted because Matrices are not part of the college and career readiness standards in the Common Core, however, STEM classes may want to do this section.


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Main Topics Bar graphs and dot plots (DA 1.1) Measures of center (DA 1.2) Five-number summaries and box plots (DA 1.3) Histograms and stem-and-leaf (DA 1.4) Two-variable data: scatter plots (DA 1.6) Focus Investigation 1: Estimating (DA 1.7) Focus Investigation 2: Golf and Divorce

o Does Correlation imply causation???

Key Vocabulary: Bivariate Data Causation Correlation Correlation coefficient Least squares regression line Line of best fit

Measures of Center Measures of Spread Outlier Representations Scatterplot Univariate Data

Common Core State Standards: ASWS.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). (1.1, 1.2,

1.3, 1.4)S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (mean, median)

and spread (interquartile range, standard deviation) of two or more different data sets. (1.2, 1.3, 1.4)

S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (1.2, 1.3 exercises 7 and 10)

S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

b. Informally assess the fit of a function by plotting and analyzing residuals.Assessed: (Lesson 1.7, exercise 7)

S.ID.8 Compute (using technology) and interpret the correlation coefficient.S.ID.9 Distinguish between correlation and causation.N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

(1.7)A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in

the coordinate plane, often forming a curve (which could be a line). (1.7)F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship

it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (1.6)

Mathematical Practice Standards (Look Fors)Make Sense Of Problems And Persevere In Solving ThemStudent Actions Seek and communicate entry points or representations for the problem Communicate observed relationships and constraints Build a solution plan on observed relationships Monitor and evaluate own work, and may report a change of strategy or perspective In examining a proposed solution, ask, 'Does this make sense?'Teacher Actions Monitor students’ thinking and processes to provide scaffolding for students’ conjectures and plans. In summary presentations, require student justifications and reasonableness, and seek alternative solutions


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Construct Viable Arguments And Critique The Reasoning Of OthersStudent Actions While completing an investigation, analyze situations and make or explore conjectures Build a logical progression of statements to justify a conjecture or present a counterexample Listen to or read the arguments of others and ask questions for clarification While completing an investigation, analyze other students’ arguments Reason inductively about data presented in context, making plausible argumentsTeacher Actions Help students make connections between problem setting and symbolic representationsModel With MathematicsStudent Actions Apply prior contextual and mathematical knowledge to solve real-world problems Display relationships among important quantities using tools such as diagrams, graphs, tables, and/or formulas To make sense of Investigations, explore a simpler real-life scenario by making assumptions and using approximations Make sense of an answer according to the context of the problemTeacher Actions During real world Investigations, engage students in recognizing important quantities and exploring ways to represent mathematical rela-

tionships Facilitate discourse around student conjectures about relations and arguments supporting varied modeling representations

METS Standard Explanation & Activities9.12.CT.1 9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and

independent learning. Focus Lesson 1.7 Students use the graphing calculator to create a scatter plot from collected data.

Students then compare to the line y = x and interpret the meaning of their data.(actual vs. estimate) Least Squares Regression (Lesson 7) includes application of correlation. This is a small unit of

lessons. Lesson is found at http://illuminations.nctm.org/LessonDetail.aspx?id=U117Use the graphing applet at http://illuminations.nctm.org/ActivityDetail.aspx?ID=146Java may need to be updated, then right click and choose run this plug in

Barbie Bungie Jump found at: http://illuminations.nctm.org/LessonDetail.aspx?id=L646 Students can use a graphing calculator for creating the scatter plot and linear regression line or they can use the graphing applet at http://illuminations.nctm.org/ActivityDetail.aspx?ID=146 right click and choose run this plug in

AssessmentsAssessment 1: Extended ResponseRepresentations of Univariate Data

Assessment 2: Performance TaskScatterplot Task

Scatterplot Task Rubric

Unit AssessmentAlgebra 1_ Descriptive Statistics Assessment

Assessment Rubric


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Gradebook

http://illuminations.nctm.org/ActivityDetail.aspx?ID=146

http://illuminations.nctm.org/LessonDetail.aspx?id=L646


http://illuminations.nctm.org/LessonDetail.aspx?id=U117

Unit 2: Relationships between Quantities and Reasoning with Equations

Big Ideas: Algebra allows us to represent real world problems and justify their solutions.

Essential Questions: How are equations and inequalities applied in the real world? How can you interpret the solution(s) of equations, inequalities, systems of equations and

inequalities, and quadratics?

Learning Outcomes: A proportion states the equality of two ratios, in which one number may be unknown and

represented by a variable; an unknown variable in a proportion can be solved for by applying the opposite operation.

Proportions can be used to make predictions about a whole population from a sample; a more representative sample will lead to a higher quality prediction.

Conversion factors are ratios that allow the change from one unit to another. If the value change in one variable mirrors the value change in the other, the two directly

proportional variables, x and y, are related by the direct variation equation y = kx. If larger values of one variable correspond to smaller values of the other, they are related by the

inverse variation equation, . An algebraic expression consisting of numbers and/or a variable can be evaluated by and

substituting a number for the variable and following the order of operations to arrive at a single number.

An equation is a sentence that two expressions are equal. A solution is a number that can be substituted for the variable to make the equation a true statement.

Focus Investigation: 2.4 Direct Variation: “Ship Canals”Investigation Goal: Students use data to draw a graph and write an equation that states the relationship between tow variables. Students will see several ways of finding information missing from a table. This investigation uses graphing calculator technology.

Focus Investigation: 2.8 Undoing Operations: “Just Undo It”Investigation Goal: Students discover how using knowledge of the order of operations and patterns in sequences can be used to solve equations.

Instructional Notes: Chapter 2 introduces “undoing” as a method of solving equations, therefore, it is not recommended to “cross multiply and divide” to solve proportions. Section 2.4 introduces y=kx and builds off of 1.7. Students use skip counting to build the concept of slope. Section 2.6 is not recommended because it is quite complicated and not necessary. Students must have a good grasp of order of operations before doing section 2.7. The “Keys to” books can be used for remediation as necessary. In section 2.8, students learn to solve an equation by undoing. This method cannot be used if there are variables on both sides of the equation. Students will learn the balancing method for solving equations in chapter 3. This unit will need supplementation for solving equations.


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Main TopicsProportions (DA 2.1)Sampling and applying ratios and proportions (DA 2.2)Proportions and measurement systems (DA 2.3)Direct variation (DA 2.4)Focus Investigation (Graphing Calculator)Inverse variation (DA 2.5)Evaluating expressions (DA 2.7)Undoing operations (DA 2.8)Focus InvestigationSupplement by adding in solving equationsKey Vocabulary:ProportionVariableDirectly proportionalDirect VariationConstant of Variation

Inverse VariationInversely ProportionalOrder of OperationsEquationSolution

Common Core State Standards: ASWN.Q.2 Define appropriate quantities for the purpose of descriptive modeling. (2.1)A.SSE.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients. (2.4)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. (2.1, 2.2, 2.3, 2.4, 2.5)A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on

coordinate axes with labels and scales. (2.4)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. (2.8)

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (2.8)

A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (2.4)

S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data. (2.2)

S.ID.6 Represent data on two quantitative variables on a scatter ploy, and describe how the variables are related.c. Fit a linear function for scatter plots that suggest a linear association. (2.4)

MET Standard Standard Explanation & Activities9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and independent

learning. Focus Lesson 2.4 Students use the graphing calculator to explore direct variation, create an equation,

and extrapolate values from a given data set. Do I have to Mow the Whole Thing? Lesson on Inverse Variation, found at: (Inverse Variation)

Graphing calculators, graphing applet, or excel is suggested http://illuminations.nctm.org/LessonDetail.aspx?id=L729

Mathematical Practice Standards (Look Fors)Make Sense Of Problems And Persevere In Solving ThemStudent Actions Seek and communicate entry points or representations for the problem Communicate observed relationships and constraints Build a solution plan on observed relationships Monitor and evaluate own work, and may report a change of strategy or perspective In examining a proposed solution, ask, 'Does this make sense?'Teacher Actions Monitor students’ thinking and processes to provide scaffolding for students’ conjectures and plans.


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In summary presentations, require student justifications and reasonableness, and seek alternative solutionsConstruct Viable Arguments And Critique The Reasoning Of OthersStudent Actions While completing an investigation, analyze situations and make or explore conjectures Build a logical progression of statements to justify a conjecture or present a counterexample Listen to or read the arguments of others and ask questions for clarification While completing an investigation, analyze other students’ arguments Reason inductively about data presented in context, making plausible argumentsTeacher Actions Help students make connections between problem setting and symbolic representationsModel With MathematicsStudent Actions Apply prior contextual and mathematical knowledge to solve real-world problems Display relationships among important quantities using tools such as diagrams, graphs, tables, and/or formulas To make sense of Investigations, explore a simpler real-life scenario by making assumptions and using approximations Make sense of an answer according to the context of the problemTeacher Actions During real world Investigations, engage students in recognizing important quantities and exploring ways to represent mathematical rela-


AssessmentsAssessment 1: Performance Task

Do I have to Mow the Whole Thing?Unit AssessmentAlgebra 1_ Relationships between Quantities and Reasoning with Equations

Assessment Rubric


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Gradebook


Unit 3: Linear Relationships (Part 1, Linear Equations)

Big Idea: This unit builds towards the concept of functions by connecting recursive routines with explicit linear equations.

Essential Questions: How can rate of change be used to predict future trends? How can linear functions be used to model real world problems? What are the different ways to represent data with a constant rate of change?

Learning Outcomes: Recursive sequences are generated with a starting point and a designation of how to operate on any

number to get the next number in the sequence. Linear functions form a straight line when graphed, the highest degree of the variable is one, and it

has a constant rate of change. If the slope is negative the line is decreasing, if it is positive the line is increasing, if it is zero the line

will be horizontal and if it is undefined it will be a vertical line. A linear equation in intercept form, y = a + bx, reflects the recursive routine used to generate a

sequence of data values with a constant rate of change. Such a routine begins with a and adds b repeatedly.

You can calculate the rate of change as a difference of output values divided by a difference of corresponding input values. The rate of change determines the steepness of the graph of the linear equation representing the data.

The balancing method is useful for solving an equation if the variable appears more than once in the equation.

Focus Investigation: 3.2 Linear Plots: “On the Road Again”Investigation Goal: Students will discover that quantities generated by a recursive additive sequence (with a constant rate of change) have a linear relationship because their graphs lie in a straight line that is with negative or positive.

Focus Investigation: 3.4 Linear Equations and the Intercept Form: “Working Out with Equations”

Investigation Goal: Students make the transition from using recursive routines to writing linear equations in intercept form. Note that the order is different than in the “slope-intercept form” in that the “starting number” is emphasized and comes first.

Instructional Notes: Section 3.1 looks at recursive sequences which are connected to graphs in section 3.2. Recursion is extremely important as it is listed in the common core state standards. Sections 3.4 and 3.5 introduce y = a + bx as the intercept form. Students learn the balancing method for solving equations in 3.6. To maximize the benefit of 3.6 it would be help to refer to “Moving Straight Ahead”, which is the unit from Connect Math that students completed in Middle School.


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Key Vocabulary:Recursive SequencesRates of ChangeRecursive Routinesy-intercept

Linear RelationshipIntercept FormBalancing

Main TopicsRecursive sequences (DA 3.1)Linear Plots (DA 3.2)Focus LessonLinear equations and the intercept form (DA 3.4)Focus LessonLinear equations and rate of change (DA 3.5)Solving equations using the balancing method (DA 3.6)Common Core State Standards: ASWA.SSE.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients. (3.2, 3.4)A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on

coordinate axes with labels and scales. (3.2, 3.4, 3.5)A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

(3.5, 3.6)A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane,

often forming a curve (which could be a line). (3.4)F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For

example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥1. (3.1)F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified

interval. Estimate the rate of change from a graph. (3.5)F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using

technology for more complicated cases.a. Graph linear and quadratic functions and show intercepts, maxima, and minima. (3.4, 3.5)

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (3.2)

F.BF.1 Write a function that describes a relationship between two quantities.a. Determine an explicit expression, a recursive process, or steps for calculation from a context. (3.1, 3.2, 3.4)

F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (3.2, 3.4, 3.5)

F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals. (3.1)b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. (3.4, 3.5)

F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (3.1, 3.4, 3.5)

S.ID.6 Represent data on two quantitative variables on a scatter ploy, and describe how the variables are related.a. Use a model function fitted to the data to solve problems in the context of the data. Use given model functions or choose a function suggested by the context. Emphasize linear and exponential models. ( 3.4, 3.5)

S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data. (3.4, 3.5)

METS Standard Explanation & Activities9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and independent learning.

Focus Lessons 3.2 and 3.4 Students use scatter plots of recursive sequences to continue to explore the connection between graphs and tables and how they can be used to solve problems.

Exploring Linear Data Lesson can be found at http://illuminations.nctm.org/LessonDetail.aspx?id=L298 Pedal Power, a lesson on interpreting slope as a rate of change, can be found at

http://illuminations.nctm.org/LessonDetail.aspx?id=L586Students compare three graphs and come up with explanations for the slopes of each (no technology


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required) Also can be used in unit 4 as the end asks to calculate slopeMathematical Practice Standards (Look Fors)Make Sense Of Problems And Persevere In Solving ThemStudent Actions Seek and communicate entry points or representations for the problem Communicate observed relationships and constraints Build a solution plan on observed relationships Monitor and evaluate own work, and may report a change of strategy or perspective In examining a proposed solution, ask, 'Does this make sense?'Teacher Actions Monitor students’ thinking and processes to provide scaffolding for students’ conjectures and plans. In summary presentations, require student justifications and reasonableness, and seek alternative solutions



AssessmentsUnit AssessmentAlgebra 1_ Linear Relationships (Part 1, Linear Equations)

Assessment Rubric


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Gradebook

Unit 4: Linear Relationships (Part 2, Fitting a Line to Data)

Big Idea: This unit formalizes slope and builds different algebraic representations of linear functions based on the most appropriate form to model a given set of data.

Essential Questions: What determines whether it is advantageous to use point-slope form or slope-intercept form? How can real world data be modeled with a graph? What is the relationship between correlation and causation? What is the significance of the x and y-intercepts in a linear function? What determines whether two lines are parallel or perpendicular?

Learning Outcomes: The slope of a line is calculated by finding two points on the line and dividing the vertical change

by the corresponding horizontal change. Parallel lines have the same slope and perpendicular lines have slopes that are opposite recipro-

cals. By finding a linear equation (line of best fit) to model the data, we can use the equation to predict

values at any point in time. Linear equations approximate the given data set, but are not an exact match.

Correlation is an indicator of how well the function models data. A strong correlation does not necessarily imply causation.

If you know two points on a line or in a data set, you can find an equation for the line in point-slope form without finding the y-intercept.

Linear equations can be written in the following forms: slope-intercept (y = mx + b), point-slope (y2 - y1= m(x2 - x1), and standard (Ax + By = C). Linear equations are equivalent if their graphs are the same or if symbolic manipulation of one can give the other.

Focus Investigation: 4.1 A Formula for Slope: “Points and Slope”Investigation Goal: Students will explore how to find the slope of a line using two points on the line.

Focus Investigation: 4.4 Equivalent Algebraic Equations: “Equivalent Equations” Investigation Goal: Student will explore how to identify different equations that de-scribe the same line.

Instructional Notes: Chapter 4 formalizes slope and linearity and introduces y = mx + b by comparing a known point and a rate of change. There is a great deal of algebra in section 4.4 and this section will likely take several days. Resources can be found in the “Keys to” books or “Practice your skills.” Sections 4.5 – 4.8 are all activities where students model with linear data. This unit will need supplementation for correlation and causation. Sections 4.5-4.8 are optional.

Key Vocabulary:Slope

Point Slope Form

Distributive Property

Commutative Properties of Addition and Multiplication

Associative Properties of Addition and Multiplication


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Main TopicsA formula for slope (DA 4.1) Focus LessonParallel and perpendicular lines (DA 11.1)Writing a linear equation to fit data (DA 4.2)Teachers will need to supplement for S.ID.8 and 9 in section 4.2 (first introduced in 1.7)Example question can be found at http://illustrativemathematics.org/standards/hsPoint-slope form of a linear equation (DA 4.3)Equivalent algebraic equations (DA 4.4)Focus LessonLinear modeling (DA 4.5 – 4.8, optional)Common Core State Standards: ASWN.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and

interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (4.1)

A.SSE.1 Interpret expressions that represent a quantity in terms of its context.b. Interpret complicated expressions by viewing one or more of their parts as a single entity. (4.3)

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. (4.3, 4.5)

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (4.1)

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. (4.4)

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (4.2, 4.3, 4.4)

F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. (4.1, 4.2, 4.3)

F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima. (4.2)F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another (4.1, 4.2)

F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (4.2)

F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. (4.1, 4.2, 4.3, 4.4, 4.5)S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Use a model function fitted to the data to solve problems in the context of the data. Use given model functions or choose a function suggested by the context. Emphasize linear and exponential models. (4.2, 4.6, 4.7, 4.8)c. Fit a linear function for scatter plots that suggest a linear association. (4.2, 4.3, 4.5, 4.6, 4.7, 4.8)

S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data. (4.2, 4.3, 4.5, 4.6, 4.7, 4.8)

S.ID.8 Compute (using technology) and interpret the correlation coefficient.S.ID.9 Distinguish between correlation and causation.MET Standard Standard Explanation & Activities9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and independent

learning. Focus Lesson 4.1 and 4.4 Students explore the concept of slope via the Dynamic Algebraic

Exploration with Geometer’s Sketch Pad for both lessons. How Tall? A Lesson on the line of best fit, found at http://illuminations.nctm.org/LessonDetail.aspx?


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http://illustrativemathematics.org/standards/hs

id=L776 This uses the graphing applet

Pedal Power, a lesson on interpreting slope as a rate of change, can be found at http://illuminations.nctm.org/LessonDetail.aspx?id=L586Students compare three graphs and come up with explanations for the slopes of each (no technology required)

Mathematical Practice Standards (Look-Fors)Make Sense Of Problems And Persevere In Solving ThemStudent Actions Seek and communicate entry points or representations for the problem Communicate observed relationships and constraints Build a solution plan on observed relationships Monitor and evaluate own work, and may report a change of strategy or perspective In examining a proposed solution, ask, 'Does this make sense?'Teacher Actions Monitor students’ thinking and processes to provide scaffolding for students’ conjectures and plans. In summary presentations, require student justifications and reasonableness, and seek alternative solutions




Pedal PowerUnit AssessmentAlgebra 1_ Linear Relationships (Part 2, Fitting a Line to Data)

Assessment Rubric


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Gradebook




Unit 5: Linear Relationships (Part 3, Systems of Equations and Inequalities)

Big Idea: Real world data can be represented by systems of linear equations and inequalities which can be solved for graphically and algebraically.

Essential Questions: What are the strengths and weaknesses of each method of solving a system of equations? How can you interpret the solution(s) of systems of equations and inequalities? How can systems of linear equations show trends over a of period time in a real world

situation? How can inequalities be used to represent a range of possible solutions in real world data?

Learning Outcomes: Some situations can be modeled with a system of equations. Solutions to systems of two linear

equations can be approximated by seeing where the equations’ graphs intersect or by making tables.

The substitution and elimination methods are two algebraic ways to solve systems. To use the substitution method, one equation must be solved for a variable. To use the elimination method, both equations must be in standard form.

Inequalities help model problems for situations described by phrases such as greater than, less than, no more than, and at least.

To graph an inequality in one variable on a number line, an open circle is used for the greater than or less than inequalities because they do not include the boundary number. A closed circle is when we have the “or equal to” along with the inequalities and the boundary number is included.

Solving inequalities is similar to solving equations except we need to remember that when multi-plying or dividing by a negative, we need to switch the direction of the inequality sign.

To visualize solutions to two variable inequalities, you can graph the line represented by the equation with a solid or dotted line, and then shade in the half-plane on which the solutions to the inequality lie.

You can find the solutions to a system of inequalities by finding the half-planes that give the solutions to the individual inequalities and looking at their intersection.

Focus Investigation: 5.1 Solving Systems: “Where Will They Meet?”Investigation Goal: Students will solve a system of simultaneous equations to find the time and distance at which two walkers meet.

Focus Investigation: 5.5 Inequalities in One Variable: “Toe the Line”Investigation Goal: Students will analyze properties of inequalities and discover some interesting results.

Instructional Notes: Chapter 5 is a traditional build up of systems of equations and inequalities. Section 5.1 uses motion detectors but sample data can be substituted if they are unavailable. Section 5.4 can be skipped, but again, STEM classes may want to do this section which involves using matrices to solve systems. Teachers may want to plan extra days for 5.5 – 5.7. This unit could be split into two sections, systems of equations and inequalities.

Key Vocabulary:


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System of EquationsEliminationSubstitutionInequalities

Half-planeCompound InequalityConstraintsSystems if Inequalities


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Main TopicsSolving systems of equations (DA 5.1)Focus Lesson (could also use 2 storage tanks)Solving systems of equations using substitution (DA 5.2)Solving systems of equations using elimination (DA 5.3)Inequalities in one variable (DA 5.5)Focus LessonGraphing inequalities in two variables (DA 5.6)Systems of inequalities (DA 5.7)Common Core State Standards: ASWA.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that

equation and a multiple of the other produces a system with the same solutions. (5.3)A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear

equations in two variables. (5.1, 5.2, 5.3)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. (5.5)A.CED.3 Represent constraints by equations or inequalities, and/or systems of equations or inequalities and interpret

solutions as viable or nonviable options in a modeling context. (5.1, 5.2, 5.3, 5.6, 5.7)A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by

letters. (5.5, 5.6)A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are

the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (5.1)

A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (5.6, 5.7)


learning. Focus Lesson 5.1 CBR lesson using the motion detector to collect data on distance and

time. Optional Geometer’s Sketch Pad Investigation for 5.6 There has to be a system for this sweet problem, lesson on systems of linear equations which can be found

at http://illuminations.nctm.org/LessonDetail.aspx?id=L766 Talk or Text? Lesson on systems of linear equations which can be found at

http://illuminations.nctm.org/LessonDetail.aspx?id=L780Students will want to use a graphing calculator or the graphing applet

Supply and Demand, a lesson on systems of linear equations which can be found at http://illuminations.nctm.org/LessonDetail.aspx?id=L724

Mathematical Practice Standards (Look-Fors)Make Sense Of Problems And Persevere In Solving ThemStudent Actions Seek and communicate entry points or representations for the problem Communicate observed relationships and constraints Build a solution plan on observed relationships Monitor and evaluate own work, and may report a change of strategy or perspective In examining a proposed solution, ask, 'Does this make sense?'Teacher Actions Monitor students’ thinking and processes to provide scaffolding for students’ conjectures and plans. In summary presentations, require student justifications and reasonableness, and seek alternative solutions

Construct Viable Arguments And Critique The Reasoning Of OthersStudent Actions


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While completing an investigation, analyze situations and make or explore conjectures Build a logical progression of statements to justify a conjecture or present a counterexample Listen to or read the arguments of others and ask questions for clarification While completing an investigation, analyze other students’ arguments Reason inductively about data presented in context, making plausible argumentsTeacher Actions Help students make connections between problem setting and symbolic representationsModel With MathematicsStudent Actions Apply prior contextual and mathematical knowledge to solve real-world problems Display relationships among important quantities using tools such as diagrams, graphs, tables, and/or formulas To make sense of Investigations, explore a simpler real-life scenario by making assumptions and using approximations Make sense of an answer according to the context of the problemTeacher Actions During real world Investigations, engage students in recognizing important quantities and exploring ways to represent mathematical rela-


AssessmentsUnit AssessmentAlgebra 1_ Linear Relationships (Part 3, Systems of Equations and Inequalities)


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Unit 6: Exponents and Exponential FunctionsBig Idea: Multiplicative recursive routines are explicitly defined as exponential functions. Exponential relationships are functions which allow us to organize data and make informed decisions.

Essential Questions: How can data sets and their graphs be interpreted as linear or exponential? How are multiplication and division related to addition and subtraction in the laws of

exponents? How can exponential functions be used to model applications that include growth and

decay in different contexts?

Learning Outcomes: When values change through multiplying by a constant, that constant is called a constant

multiplier. The rate of change increases as the amount increases. When values are increasing through multiplication by a constant that is greater than 1, we say that

we have exponential growth and represent the values by the equation . When multiplying like bases, add the exponents, when you raise a power to a power or a fraction

to a power, multiply the exponents. When a fraction is raised to a power, both the numerator and the denominator are raised to that power.

When dividing powers with like bases we subtract the exponents, that is take the exponent of the numerator and subtract the exponent of the denominator from it.

A negative exponent can be changed to a positive exponent by taking the reciprocal of the base making the exponent positive.

Scientific notation provides a way of naming numbers consistently and is especially useful for numbers that are very close to or far from zero.

Exponential functions are in the general form f(x) = abx. The graph of the exponential parent function (y=bx) decreases at a decreasing rate when b is between 0 and 1 and increases at an in-creasing rate when b is greater than 1.

Focus Investigation: 6.1 Recursive Routines: “Bugs, Bugs, Everywhere Bugs”Investigation Goal: Students will investigate patterns in which a population increases rapidly through the investigation of geometric sequences using recursive routines and constant multipliers.

Focus Investigation: Investigation Goal:

Instructional Notes: This unit begins with recursion, similarly to the linear unit, with multiplying each time instead of adding. Extra days should be built into the schedule for skills practice in sections 6.3 – 6.6. Supplemental lesson(s) on radical exponents will need to be added in to address the common core state standard N.RN.1. The following site http://illustrativemathematics.org/standards/hs gives examples for clarification of the standards. Section 6.4 on scientific notation is not in the Common Core content standards, but it does fit the math practice standards and is used in science classes. Section 6.8 is an Activity Day which are always optional, however, this one is recommended because students experience modeling with exponential decay.

Key Vocabulary:Exponential EquationsBaseExponent

Exponential FormMultiplication, Division and Power Properties of Exponents


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Expanded Form Exponential Growth


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Main Topics Recursive routines (DA 6.1)Focus Lesson Exponential equations (DA 6.2) Multiplication and exponents (DA 6.3) Supplement rational exponents as they relate to the exponential properties and applications

to address N.RN.1 see example at http://illustrativemathematics.org/standards/hs Scientific notation (DA 6.4) Division and exponents (DA 6.5) Zero and negative exponents (DA 6.6) Fitting exponential models to data (DA 6.7) Decreasing exponential models and half-life (DA 6.8)

Common Core State Standards: ASWN.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of

integer exponents to those values, allowing for notation for radicals in terms of ration exponents. For

example, we define 513 to be the cube root of 5 because we want (5

13 )

3to hold, so (5

13 )

3 must equal 5. (6.3

power property)N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. (6.3, 6.5,

6.6)A.SSE.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients. (6.1)b. Interpret complicated expressions by viewing one or more of their parts as a single entity. (6.2, 6.6)

A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

c. Use properties of exponents to transform expressions for exponential functions. (6.3, 6.5)A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the

coordinate plane, often forming a curve (which could be a line). (6.2)A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)

intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions (6.2)

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (6.2, 6.7)

F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥1. (6.1)

F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (6.1, 6.7, 6.8)


e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions showing period, amplitude, and midline. (6.2, 6.7)

F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

b. Use the properties of exponents to interpret expressions for exponential functions. (6.3, 6.5, 6.6, 6.7)

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (6.1)

F.BF.1 Write a function that describes a relationship between two quantities.a. Determine an explicit expression, a recursive process, or steps for calculation from a context. (6.1, 6.2, 6.7)

F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (6.1, 6.2)


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F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals. (6.1)b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (6.2, 6.3, 6.5, 6.6, 6.7)

F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (6.2, 6.3, 6.7)

F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (6.1)

F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. (6.1, 6.2, 6.7)S.ID.6 Represent data on two quantitative variables on a scatter ploy, and describe how the variables are related.

a. Use a model function fitted to the data to solve problems in the context of the data. Use given model functions or choose a function suggested by the context. Emphasize linear and exponential models. (6.7)

MET Standard Standard Explanation & Activities9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and

independent learning. Focus Lesson 6.1 Students use a graphing calculator and recursive routines to explore exponential

growth. Drug Filtering, lesson on exponential decay which can be found at

http://illuminations.nctm.org/LessonDetail.aspx?id=L829Students may want to use an excel spread sheet for their data, or a graphing calculator

National Debt and Wars, curve fitting, exponential growth, and percent change, which can be found at http://illuminations.nctm.org/LessonDetail.aspx?id=L670 Students will need to use an excel spread sheet or a graphing calculator

Predicting your financial future, lesson on compound interest which can be found at http://illuminations.nctm.org/LessonDetail.aspx?id=L761 Students will be introduced to a compound interest simulator appletAt http://illuminations.nctm.org/ActivityDetail.aspx?ID=172


Construct Viable Arguments And Critique The Reasoning Of OthersStudent Actions While completing an investigation, analyze situations and make or explore conjectures Build a logical progression of statements to justify a conjecture or present a counterexample Listen to or read the arguments of others and ask questions for clarification While completing an investigation, analyze other students’ arguments Reason inductively about data presented in context, making plausible argumentsTeacher Actions Help students make connections between problem setting and symbolic representationsModel With MathematicsStudent Actions Apply prior contextual and mathematical knowledge to solve real-world problems Display relationships among important quantities using tools such as diagrams, graphs, tables, and/or formulas To make sense of Investigations, explore a simpler real-life scenario by making assumptions and using approximations


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Make sense of an answer according to the context of the problemTeacher Actions During real world Investigations, engage students in recognizing important quantities and exploring ways to represent mathematical rela-



National Debt and Wars Activity 2: The National Debt and Major U.S. Wars(Completed after students have work on Activity 1 as an investigation.)

Unit AssessmentAlgebra 1_ Exponents and Exponential Functions

Assessment Rubric


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Gradebook

Unit 7: Functions and Transformations

Big Idea: Functions are mathematical relationships where one or more variables are dependent on another variable. Functions can be categorized into families of functions with a parent function which has parameters that can be changed to create new functions.

Essential Questions: How do you determine which type of function to use in modeling a specific real-

world situation? How do we interpret relations and functions through their graphical, algebraic, and

numeric representations? What are the possible transformations that can be performed on a function? How can you write a new function given specific transformation conditions?

Learning Outcomes: A function is a relationship between two sets, called the domain and range of the function, such

that every element of the domain is associated with one and only one element of the range. The domain is a set of input values, and the range is the set of output values.

A graph represents a function if and only if no vertical line through the graph intersects it at no more than one point.

A function increases (rises) if its y-values get larger as its x-values increase. A function decreases (falls) if its y-values get smaller as its x-values increase. A function’s rate of change increases if the slope of the graph becomes larger, and it decreases if the slope becomes smaller.

The notation f(x) is read “f-of-x” and is called function notation. The absolute-value function converts differences between values into distances, which can’t be

negative, by outputting the opposite of negative numbers. The graph of the squaring function is a parabola. Although every positive number has two square

roots, the square root function gives only the positive square root. A translation is a transformation in which the figure is moved in the plane horizontally or

vertically without changing size, shape, or orientation, by adding to the x or y coordinates. When the a, b, h, and k are changed according to the rule g(x)=af(bx-h)+k, the graph changes ac-

cordingly. Transformations are consistent regardless of the family of functions that are being studied.

Focus Investigation: 7.2 Functions and Graphs: “Testing for Functions” Investigation Goal: Students will investigate and use various types of evidence to deter-mine whether relations are functions.

Focus Investigation: 8.2 Translating Graphs: Investigation Goal:

Instructional Notes: Chapter 7 formalizes functions and introduces the vocabulary associated with functions. Some teachers may choose to integrate some of the vocabulary earlier in year. Sections 7.2 and 7.3 should go fairly quickly but section 7.4 will likely take some time. To address the common core state standard F.IF.7b, teachers will need to ensure students are exposed to this. Sections 7.5 and 7.6 introduce functions that are new to students. At the end of chapter 7 a supplemental lesson on inverse functions will need to be added in. Sections 8.1 – 8.4 are on transformations of functions and must be covered, however, the rest of chapter 8 need not be. Since this is a lengthy unit, there are two common assessments associated with the chapter.


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Key Vocabulary:FunctionVertical Line TestIndependent VariableDependent VariableDomainRangeAbsolute Value FunctionSquaring FunctionParabolaSquare Root Function

TransformationImageTranslationsVertexParent FunctionFamily of FunctionsReflection

Main Topics Functions and graphs (DA 7.2) Focus Lesson

o Supplement for vocabulary in 7.2: Domain, Range, Function, Relation, Input, and Output Graphs of real world situations (DA 7.3) Function notation (DA 7.4) Defining absolute value function (DA 7.5) Squares, squaring and parabolas (DA 7.6)

o Supplemental Lesson (p.434) on inverses of functions (equation) Translating points (DA 8.1) Translating graphs (DA 8.2)Focus Lesson (Could use the transformation DL activity) Reflecting points and graphs (DA 8.3) Stretching and shrinking graphs (DA 8.4)

Common Core State Standards: ASWA.SSE.1 Interpret expressions that represent a quantity in terms of its context.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. (8.2, 8.3, 8.4)

F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (7.1, 7.2, 7.4)

F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (7.4, 7.5, 8.1)

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (7.3)

F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (7.2, 7.3)


a. Graph linear and quadratic functions and show intercepts, maxima, and minima.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (7.2,7.3, 8.2)e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (8.2, 8.3)

F.BF.1 Write a function that describes a relationship between two quantities b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (8.1-8.4)

F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and


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algebraic expressions for them. (8.2, 8.3, 8.4)

F.BF.4 Find inverse functionsa. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2x3 or f(x) = (x+1)/(x-1) for x ≠1. Supplemental after chapter 7 sections.

S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (in-terquartile range, standard deviation) of two or more different data sets. (7.5)

METS Standard Explanation & Activities9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and independent

learning. Focus Lesson 8.2 Students use the graphing calculator to explore translations of functions.




AssessmentsUnit Assessment Part 1Algebra 1_FunctionsUnit Assessment Part 2Algebra 1_Transformations Assessment Rubric


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Unit 7.5: Operations with Polynomials (1 Week) The purpose of this unit is to provide supplemental practice to gain prior knowledge and/or additional practice for the unit that follows.

Big Idea: Essential Questions:

How are addition, subtraction, and multiplication with polynomials similar to addition, subtraction and multiplication with numbers?

How does the distributive property support multiplication and factoring of polynomials?

Learning Outcomes: Adding, subtracting, and multiplying polynomials result in a new polynomial. The distributive property is used when multiplying polynomials. When we factor polynomials, we

are undoing the distributive property. (Converse of the Distributive Property);

Main Topics Operations with polynomials Factoring Distributive Property

Common Core State Standards: ASWA-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.MET Standard Standard Explanation & Activities9.12.CT.1 9.12.CT.1 Use digital resources (e.g., educational software, simulations, models) for problem solving and

independent learning.


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Unit 8: Quadratic Functions and Modeling

Big Idea: Quadratic relationships can be represented with multiple representations that can be used to solve real-world problems.

Essential Questions: How do you interpret the solution(s) of quadratic equations? What real-world situations can be modeled with quadratic functions? How can the properties of quadratic functions be used in interpreting data?

Learning Outcomes: If an equation is quadratic (meaning degree of 2), the graph is a parabola which has certain prop-

erties such as a vertex which is a max or min and an axis of symmetry through the vertex.

If an equation has the form , it can be solved by taking the square root of both sides. If a quadratic equation is in factored form, the zero product property can be used to find solutions. Solutions to quadratic equations can be found or verified using graphs. Completing the square creates a perfect square trinomial which is easily factored into the form

f ( x )=a ( x−h )2+k . The vertex form can be used to sketch the graph, by finding the vertex, zeroes, and the axis of

symmetry. To convert a quadratic equation from vertex form to general form, you need to square a binomial.

The quadratic formula, , can be used to solve any quadratic equation that is in standard form.

Focus Investigation: 9.3 From Vertex to General Form: “Sneaky Squares”Investigation Goal: Students practice converting an equation in vertex form to the general form while combining like terms.

Investigation Task: 9.6 Completing the Square: “Searching for Solutions”Investigation Goal: Students investigate using the completing the square method to solve quadratic equations.

Instructional Notes: In section 9.4, factoring is introduced as a skill to identify properties of quadratics and the coefficient on the quadratic term will only be 1. This is consistent with the language in the Common Core. Section 9.5 and Section 9.8 which addresses cubic functions can be skipped. Teachers will need to supplement the materials in the book to meet standard N.RN.3

Key Vocabulary:Quadratic FunctionsRootsX-InterceptsTrinomialPolynomial

Monomials

BinomialTermLine of SymmetryZero-Product PropertyCompleting the SquareDiscriminant


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Main Topics Solving quadratic equations (DA 9.1) Finding the roots and the vertex (DA 9.2) From vertex to general form (DA 9.3)Focus Lesson Factored form (DA 9.4) Completing the square (DA 9.6)Focus Lesson The quadratic formula (DA 9.7) Supplemental Lesson for N.RN.3Common Core State Standards: ASWN.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irra-

tional number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function no-

tation in terms of a context. (9.1, 9.6)F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in

terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity (9.1, 9.2)

F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (9.1, 9.2, 9.3)


a. Graph linear and quadratic functions and show intercepts, maxima, and minima. (9.1, 9.2, 9.3, 9.4, 9.6)F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different

properties of the function.a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (9.1, 9.2, 9.3, 9.4, 9.6)

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (9.1, 9.4, 9.7)

F.BF.1 Write a function that describes a relationship between two quantities.a. Determine an explicit expression, a recursive process, or steps for calculation from a context. (9.1, 9.2, 9.3, 9.4, 9.5, 9.6)

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 ational and exponential functions. (9.1)

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordi-nate axes with labels and scales. (9.2)

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. (9.3, 9.6)

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. (9.1)

A.REI.4 Solve quadratic equations in one variable.a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. (9.6, 9.7)b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. (9.1, 9.2, 9.3, 9.4, 9.5, 9.6)

A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. (9.7)

A.SSE.1 Interpret expressions that represent a quantity in terms of its context.a. Interpret parts of an expression, such as terms, factors, and coefficients. (9.2)

A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). (9.2, 9.3, 9.4, 9.7)


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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. Factor a quadratic expression to reveal the zeros of the function it defines. (9.4)b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (9.6)


learning. Egg Launch, a lesson on quadratic functions explored through different function representations. Lesson

found at http://illuminations.nctm.org/LessonDetail.aspx?ID=L738Mathematical Practice Standards (Look Fors)Make Sense Of Problems And Persevere In Solving ThemStudent Actions Seek and communicate entry points or representations for the problem Communicate observed relationships and constraints Build a solution plan on observed relationships Monitor and evaluate own work, and may report a change of strategy or perspective In examining a proposed solution, ask, 'Does this make sense?'Teacher Actions Monitor students’ thinking and processes to provide scaffolding for students’ conjectures and plans. In summary presentations, require student justifications and reasonableness, and seek alternative solutions

Construct Viable Arguments And Critique The Reasoning Of OthersStudent Actions While completing an investigation, analyze situations and make or explore conjectures Build a logical progression of statements to justify a conjecture or present a counterexample Listen to or read the arguments of others and ask questions for clarification While completing an investigation, analyze other students’ arguments Reason inductively about data presented in context, making plausible argumentsTeacher Actions Help students make connections between problem setting and symbolic representationsModel With MathematicsStudent Actions Apply prior contextual and mathematical knowledge to solve real-world problems Display relationships among important quantities using tools such as diagrams, graphs, tables, and/or formulas To make sense of Investigations, explore a simpler real-life scenario by making assumptions and using approximations Make sense of an answer according to the context of the problemTeacher Actions During real world Investigations, engage students in recognizing important quantities and exploring ways to represent mathematical relation-

ships Facilitate discourse around student conjectures about relations and arguments supporting varied modeling representations

AssessmentsUnit AssessmentAlgebra 1_ Quadratic Functions and Modeling

Assessment Rubric


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Gradebook

http://illuminations.nctm.org/LessonDetail.aspx?ID=L738

Additional Resources for Teachers

Illuminations (NCTM) has the following activities and online manipulatives:

Advanced Data Grapher

Analyzing Data with Box Plots, Bubble Graphs, Scatterplots, Histograms, and Stem-and-Leaf Plots

Algebra TilesManipulating Algebra Tiles to Solve Equations, Substitute in Expressions, and Expand and Factor

Function MatchingFinding the Function Expres-sion That Matches a Generated Function Graph

Pan Balance – Ex-pressions

Investigating the Concept of Equivalence by "Weighing" Numeric and Algebraic Expres-sions

Algebraic Transfor-mations

Exploring Commutativity and Associativity Within a Geomet-ric Situation

Box PlotterCreating a Customized Box Plot with Your Own Data, or Display a Box Plot of an In-cluded Set of Data


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Histogram ToolCreating a Customized His-togram with Your Own Data or an Included Set of Data

Line of Best FitPlotting a Set of Data and De-termining a Line of Best Fit

Linear Regression IInvestigating a Regression Line and Determining the Effects of Adding Points to a Scatterplot

Mean and MedianInvestigating the Mean, Me-dian, and Box-and-whisker Plot for Sets of Data That You Cre-ate

Proof Without Words: Completing the Square

Proving the Algebraic Tech-nique of Completing the Square "Without Words" Using Geom-etry

Square GraphsExploring the Graphs That Re-sult when Two Characteristics of a Square Are Plotted

Websites

Operation Order Algebra Game (order of operations online practice) http://www.funbrain.com/algebra/index.html

You-tube video on domain and range of a function (Khan Academy) http://www.youtube.com/watch?v=NRB6s77nx2g

Samples From PB works (Others available on PB works) http://grpsmath.pbworks.comGrand Rapids Public Schools Algebra 1 2011 Version 8-31.16

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http://grpsmath.pbworks.com/

http://www.youtube.com/watch?v=NRB6s77nx2g

http://www.funbrain.com/algebra/index.html

Expressions and Equations Distributive Property/Box Methodhttp://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/7-7/FOIL.ppt box methodhttp://www.mathatube.com/properties-distributive-box-mothed.htmlYouTube video for the box methodhttp://www.algebrahelp.com/lessons/simplifying/distribution/explanation of distributive property with calculators and other resourceshttp://www.onlinemathlearning.com/distributive-property-math.htmlSummary of math properties (commutative, associative, distributive)http://www.purplemath.com/modules/simparen.htm simplifying with parenthesesExpressions: Writing, Simplifying and Interpretinghttp://www.onlinemathlearning.com/algebra-word-problems.htmlSolving word problem resource.http://www.purplemath.com/modules/translat2.htmSolving word problem resource.

http://www.mathgoodies.com/lessons/vol7/equations.htmlThis site explains how to write equations from words and then has an online practice for students.

http://www.watchknow.org/Video.aspx?VideoID=18964This site has a video for simplifying expressions and a written piece.

http://wps.pearsoned.com.au/md7/0,11310,2695791-content,00.htmlDrag and drop answers to algebraic expressions

http://www.math.com/school/subject2/lessons/S2U2L4DP.html add like terms

http://www.mathsnet.net/algebra/a21.html simplifying expressions interactive

http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/symbolsact.shtml combining terms interactive http://www.regentsprep.org/Regents/math/ALGEBRA/AV2/indexAV2.htm like terms

http://nlvm.usu.edu/en/nav/frames_asid_189_g_1_t_2.htmlalgebra tiles

http://www.onlinemathlearning.com/evaluate-algebraic-expression.htmlThis site has 4 video clips on evaluating expressionsGrand Rapids Public Schools Algebra 1 2011 Version 8-31.16

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http://www.onlinemathlearning.com/evaluate-algebraic-expression.html

http://www.watchknow.org/Video.aspx?VideoID=18964

http://www.mathgoodies.com/lessons/vol7/equations.html

http://www.purplemath.com/modules/translat2.htm

http://www.onlinemathlearning.com/algebra-word-problems.html

http://www.mathatube.com/properties-distributive-box-mothed.html

http://regentsprep.org/Regents/math/ALGEBRA/AOP2/evalPrac.htmEvaluating expressions.

http://www.mr-ideahamster.com/classes/movies/expressions_ti84/ti84expressions.htmlEvaluating expressions on the calculator.

http://mathbits.com/MathBits/TISection/Algebra1/EvaluateExpressions.htmEvaluating expressions on the calculator.

http://www.mr-ideahamster.com/classes/movies/expressions_ti84/ti84expressions.htmlEvaluating expressions on the calculator.

Websites http://www.nwlincs.org/nwlincsweb/EITCdata/PreAlg/PowerPt/3-1-1%20Evaluating%20Expressions.pptPower point presentation about expressions.

http://www.aaamath.com/equ723x2.htmEvaluation practice

http://www.quia.com/mc/319817.htmlMatching words with expressions.

You Tube http://il.youtube.com/watch?v=Sa6bBkztsoo&feature=relatedevaluating expressions - you tube - 9 minutes

http://il.youtube.com/watch?v=H9iUy07QKXQ&feature=relatedevaluating expressions - you tube - 4 minutes - uses four colors

http://il.youtube.com/watch?v=HBaBLax9z-U&feature=relatedEVAL EXPRES - you tube - 4.5 minutes- cute smiley face expressions (engage) http://www.youtube.com/watch?v=XIu5wFT9wS01 minute - one example

http://www.youtube.com/watch?v=Fza_i8hNfsk&NR=138 seconds - verifying your solution to an equation http://www.homeschoolmath.net/teaching/teach-solve-word-problems.phplesson plan on you tube?

http://www.wlcsd.org/webpages/mrjoseph/files/section2.pdfGrand Rapids Public Schools Algebra 1 2011 Version 8-31.16

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http://www.wlcsd.org/webpages/mrjoseph/files/section2.pdf

http://www.quia.com/mc/319817.html

http://www.mr-ideahamster.com/classes/movies/expressions_ti84/ti84expressions.html




This is a printable activity with application problems on evaluating expressions (word problems)


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