**Algebra 1**

UNIT 1: Foundations: Relationships Between Quantities

Every unit will focus on a model that demonstrates the Standards of Mathematical Practice.

This proposed unit map is a synthesis

of the Chicago Public Schools Planning

Guide 1.0, the CCSS Traditional Pathway for Algebra 1, and CME.

CME Objectives

• Identify, describe, and justify patterns in addition and multiplication tables.

• Subtract using integers, which is the same as to add the integer’s opposite.

• Apply the basic rules of arithmetic to integers.

• Represent fractions and real numbers as points ona number line.

• Represent a rational number in many different ways.

• Choose a representation for a number that is appropriate for the exercise at hand.

• Extend the basic rules of addition and multiplication from the integers to the real numbers.

• Perform arithmetic with integers and fractions.

CPS Objectives

Evaluate algebraic expressions, including those with exponents and those requiring knowledge of order of operations.

Apply the properties of operations to generate equivalent expressions.

Find the absolute value of a rational number.

Add rational numbers.

Explain why the sum of two opposite numbers is 0.

Apply the skill of adding rational numbers to the real world.

Subtract rational numbers.

Apply the skill of subtracting rational numbers to the real world.

Use properties of operations when adding and subtracting rational numbers.

Multiply rational numbers.

Interpret products of rational numbers in real world contexts.

CCSS Objectives

Interpret the structure of expressions.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Model: How did the Ancient Mayans create their own mathematical system that in turn allowed them to create a calendar that has been accurate for thousands of years?

Create your own mathematical system that incorporates negatives, fractions, order of operations, and the basic rules of arithmetic.

http://www.radicalmath.org/docs/MayanMath.pdf

Benchmarks

1. I can perform basic operations on real numbers.

2. I can apply the order of operations to simplify expressions.

3. I can apply rules of exponents.

4. I can perform common conversions.

UNIT 2: Foundations: Reasoning with Equations

CCSS Objectives

Reason quantitatively and use units to solve problems.

CME Objectives

• Express word problems using variables andmathematical notation.

• Write formulas using two or more variables.

• Understand the relationship between an equation and its solutions.

• Understand the basics of equations, including when equations are always true and always false.

• Solve equations using the basic moves.

• Understand that equations can have multiple solutions or no solutions.

• Solve an equation involving many variations of the Distributive Property.

Benchmarks

5. I can apply the distributive property.

6. I can solve equations with one variable involving multiple steps.

7. I can solve inequalities with one variable.

8. I can solve literal equations.

Model: Organizing for the Future

In this activity, students are working for a union that is organizing for a large rally. Students are given a set of financial and time constraints, and have to determine which organizing strategies to use and how much time to spend on them.

Topics: Feasible Region, Inequalities, Solving Equations

Unit 3: Graphs

CME Objectives

• Plot points and read coordinates on a graph.

• Understand how an ordered pair representsa point.

• Connect graphs to sets of data.

• Test a point to determine whether it is on thegraph of an equation.

• Graph an equation by plotting points.

• Write the equation of a vertical or horizontal linegiven its graph or a point on its graph.

• Read a graph to identify points that are solutionsto an equation.

• Identify characteristics of a graph given itsequation.

• Find the intersection point of two graphs andunderstand its meaning.

• Sketch the six basic graphs

• Recognize the distinguishing features of the basicgraphs, such as their general shape, and pointsand quadrants they pass through.

Benchmarks

9. I can use a graph to display the connection between variables on a coordinate plane.

10. I can explain that the coordinate pairs of a graph are the solutions to an equation.

11. I can identify the six basic graphs given a graph, table, or equation.

Model: Incarceration & Crime Rates

This model explores the connection between Crime Rates and Incarceration Rates in the U.S.

Math involved includes line graphs, introduction to slope, writing functions, and rates.

CPS Objectives

• Show understanding of the concepts of speed and rate.

• Create motion graphs (distance vs. time).

• Describe how changes in motion affect the graph.

• Use common or first differences to determine if a function is linear.

• Identify relationships as linear or nonlinear using a table, graph, orequation.

• Find rates for data in tables, graphs and problem situations.

• Create equations or inequalities to describe situations in real-world or mathematical problems.

• Write equations in two variables to describe real world situations.

• Graph a relationship between two variables in the coordinate plane.

• Analyze the relationship between the two variables using both tables and graphs.

• Use graphing calculator to analyze relationship.

• Define graphs in terms of domain and range, input and output, notation.

• Understand how to decide upon the domain of a graph as it relates to a real-world context.

**Unit 4: Linear Equations**

CPS Objectives

• Solve a linear system of equations with two variables.

• Graph solutions to linear inequalities in one variable on a numberline.

• Graph solutions to linear inequalities in two variables on a coordinateplane.

• Graph solutions to systems of linear inequalities in two variables on a coordinate plane.

CPS Objectives

• Analyze situations involving linear functions and formulate linear equations to solve problems.

• Use different methods for solving linear equations: using concrete models, graphs, and the properties of equality.

• Choose an appropriate method, and solve the equations.

• Apply techniques for solving equations in one variable to solve literal equations.

• Compare and contrast to determine the advantages and limitations ofusing a particular representation to answer a question.

• Analyze and create equivalent algebraic expressions and rules.

• Write inequalities in one and two variables to represent problemsituations.

• Solve linear inequalities in one variable using tables, graphs, andalgebraic operations.

CPS Objectives

• Identify and compare the constant of proportionality in a proportional relationship between two variables by examining:

• Tables , Graphs, Equations , Diagrams , Verbal description

• Graph a proportional relationship and understand that the unit rate in the proportion is the slope of the graph of the relationship.

• Write the equation of a line in different forms (slope-intercept, standard, and point-slope forms).

• Identify slope and y-intercept from graphs, tables and problem situations

.• Identify equations as linear or non-linear

• Understand the effects of changing m or b on the graph of y = mx + b.

• Transform the parent function y = x to create other linear functions.

• Interpret the meaning of m and b from tables and graphs.

CME Objective

• Describe the steepness of a line.

• Calculate the slope between two points.

• Understand slope as a rate of change.

• Calculate the average speed between two points on a distance-time graph.

• Find other points on a line when given a slope and a point.

• Write linear equations.

• Sketch graphs of linear equations.

• Determine the slope of a line from its equation.

• Determine whether a runner will overtake another runner.

• Solve inequalities algebraically and by using graphs.

• Graph the solution set of an inequality.

CCSS Objectives

• Interpret linear models.

CCSS Objectives

• Represent and solve equations and inequalities graphically

• Build a function that models a relationship between two quantities

Benchmarks

12. I can find the slope (rate of change) between two points

13. I can graph linear equations.

14. I can generate an equation of a line.

15. I can write an equation that shows the connection between the variables; then use a graph to display the connection on a coordinate plane.

**Unit 5: Systems**

Model: Sandkofa

To be used as cumulative final project.

http://www.radicalmath.org/docs/Sandkofa.pdf

4 weeks

5 weeks

3 weeks

**7 weeks**

CME Objectives

• Solve systems of linear equations with twovariables by using substitution and elimination.

• Determine whether two lines are parallel orintersecting using the slope of each line.

• Write and solve word problems for systems of equations.

• Sketch the solution of inequalities of two variablesand systems of inequalities of two variables

CCSS Objective

• Solve systems of equations.

Benchmarks

16. I can estimate the intersection of a system of linear equations by graphing.

17. I can find the intersection of a system of linear equations by using algebraic methods.

18. I can find the solution region of a system of linear inequalities by graphing.

**4 weeks**

Buy A Car Webquest

Americans are dependent on their cars to get them to school, work, and everywhere else they need to go in the course of their daily lives. Your family is no exception. Recently, your family car stopped running and you are in the market for a new vehicle.

Your challenge will be to find a car or truck that will be cost effective to purchase and cost effective to operate. Using the cost of the vehicle, the annual cost of gas, and the annual cost of insurance you will analyze which vehicle is the best choice for your family.

http://carquest.marckerschner.com/

**Unit 6: Functions**

**6 weeks**

**Unit 7: Exponents and Polynomials**

**5 weeks**

**Unit 8: Quadratics**

**6 weeks**

CCSS Objectives

• Interpret functions that arise in applications in

terms of a context.

• Analyze functions using different representations

• Build a function that models a relationship

between two quantities.

• Build new functions from existing functions.

• Construct and compare linear, quadratic, and exponential models and solve problems.

CPS Objectives

• Represent functions using words, tables, graphs, and symbols.

• Identify independent variables in functional relationships.

• Use function notation.

• Keep track of steps or processes to reflect and observe patterns thatwill aid in the formulation of equations that arise from functions.

• Understand the concept of a functional relationship as well as thebasic aspects of linear relationships.

• Demonstrate an understanding of “reasonable inputs” and discreteand continuous data.

• Solve real-world problems and model real-world situations using patternsand mathematical relationships.

• Make connections among representations of mathematical relationships,using words, pictures, tables, graphs, and algebraic rules.

• Determine if a relationship represented by a table, rule, graph, or statement can be represented by an exponential function.

• Determine if a table, graph, rule or statement can be represented by a linear or exponential function.

• Use functions of the form y = ab^x to represent exponential relationships.

• Explain how changes in the parameters a and b for y = ab^x affect the graph of an exponential function.

CME Objectives

• Find a rule to describe input and output values.

• Build a function from a word problem.

• Determine whether a relationship is a function

based on its description or its graph.

• Make input-output tables.

• Provide applications of mathematical functions

and nonfunctions.

• Find the domain of a function.

• Graph a function.

• Work with functions expressed in various form (e.g.,

x A notation, f(x) notation, tables, and graphs).

• Evaluate compositions of functions.

• Estimate values from the graph of a function.

• Determine whether a table represents alinear function.

• Fit a linear function to a table where possible.

• Translate a word problem into an equation.

• Build and understand a function-machine network.

• Match tables with constant ratios to exponential functions.

Benchmarks

19. I can determine and explain why a given relationship is a function.

20. I can use function notation to evaluate functions.

21. I can write and graph linear functions.

22. I can write and graph exponential functions.

23. I can use functions to model situations (both mathematical and contextual).

Functions and Everyday Situations This lesson unit is intended to help you assess how well students are able to:• Articulate verbally the relationships between variables arising in everyday contexts.• Translate between everyday situations and sketch graphs of relationships between variables.• Interpret algebraic functions in terms of the contexts in which they arise.• Reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.

http://map.mathshell.org/materials/download.php?fileid=1259

CCSS Objectives

• Extend the properties of exponents to rational exponents.

• Perform arithmetic operations on polynomials

CPS Objectives

• Add and subtract polynomials, simplifying with a variety of methods, including (but not limited to) using concrete models and algebraically

combining like terms.

• Classify polynomials by type and degree.

• Model a situation with a polynomial expression.

• Multiply monomials, binomials, and trinomials with a variety of methods, including (but not limited to) using concrete models and directly applying the distributive property.

• Develop and understand the laws of exponents.

• Simplify numerical and variable expressions involving exponent.

CPS Objectives

• Solve quadratic equations by factoring.

• Identify and make connections among factors, solutions, x-intercepts, and zeros.

• Solve quadratics by graphing.

• Explain the meaning of solutions for given situations.

• Solve quadratic equations using the quadratic formula.

• Use the discriminant to determine the number of solutions for a quadratic equation.

• Explain the meaning of solutions for given situation.

• Factor quadratic expressions.

• Determine if a relationship represented by a table, rule, graph, or statement can be represented by a quadratic function.

• Use functions of the form y = ax^2 + c to represent some quadratic relationships.

• Explain how changes in the parameters a and c for y = ax^2 + c affect the graph of the parent quadratic function y = x^2.

• Identify and make connections between solutions and x-intercepts.

• Simplify square roots algebraically and connect the simplified form to the geometric models for square roots.

• Use the discriminant to determine the number of solutions for a quadratic equation.

CME Objectives

• Make calculations involving integral exponents.

• Simplify expressions involving integral exponents.

• Explain and apply the basic rules of exponents.

• Calculate with the zero exponent and negative exponents.

• Factor expressions by identifying a common factor.

• Apply the Zero-Product Property to

factored expressions.

• Recognize and provide examples of polynomials.

• Understand the definition and importance of the

terms coefficient and degree.

• Expand polynomials and express them in

normal form.

• Determine whether polynomials in different forms

are equivalent.

• Add, subtract, and multiply polynomials.

Benchmarks

24. I can apply the basic rules of exponents.

25. I can perform aritmetic operations on polynomials.

26. I can identify different ways to write/rewrite equvalent forms of a polynomial by factoring.

Benchmarks

27. I can identify a quadratic function from a graph, table or equation.

28. I can identify key features of the graph a quadratic function.

29. I can factor quadratic functions.

30.

CME Objectives

• Use the quadratic formula to solve equations

or determine whether an equation has no real

solutions.

• Construct a quadratic equation given the equation’s two roots.

• Factor nonmonic quadratics.

• Use your knowledge of quadratics to optimizesome quadratic functions.

• Graph quadratic functions and examine the graph to find the vertex.

• Explore word problems involving quadratic functions.

• Use difference tables to analyze quadratics and other polynomials.

How Do Poor People Bank? (internal link)

Abstract: A 2 - 4 day lesson on the banking structures available for poor people. The lesson starts by explaining how traditional banks provide people with interest on savings accounts and then explores how this is flipped on poor people forced to use check cashers, pawn shops, rapid refund loans, etc. Math involved includes percents, interest, compound interest, APR.

Category: Interest, Compound Interest, Annual Percent Rate, APR

http://www.radicalmath.org/browse_category.php?cat=Compound%20Interest

CCSS Objectives

Construct and compare linear, quadratic, and exponential models and solve problems