> _a^ ]`bjbj 4beeL;66\1&1(1(1(1(1(1(1$3c6fL19L11j&1&1-&0 >a.1101.\6<60&06&0L1L1166 V: Algebra I Common Core Curriculum Semester 1
Key: (Aligned to engageny.org modules)
M Module
T Topic
L Lesson
Daily Do Now Problems are encouraged to help students with previous curriculum gaps and to spiral in new curriculum as we move through the course. Initial recommended topics for Do Now problems that are
A focus on fractional work with numeric expressions only (adding/subtracting/multiplying/dividing/exponentiation/easier radical work and converting mixed numbers (both positive and negative) to improper fractions and vice versa). This is a good opportunity to also work on options to use with the graphing calculator when working with fractions.
Rounding (all different place values)
Representing Solution Sets using Set Notation and Interval Notation
Area, Perimeter, and Volume Formulas
Unit Conversion
Radicals (Simplifying, Adding/Subtracting, Multiplying/Dividing, Rationalizing)
After those four which span between 3 4 months, begin spiraling in midterm review questions to help prepare for the midterm. After that, begin spiraling in curriculum from semester one so it stays fresh in student minds to help prepare them for the end of year Regents and/or final.
Unit 1 Sets of Numbers and Properties (12 days)
Overview of Sets of Real Numbers (mention the existence of imaginary numbers, but dont define i)
Incorporate irrational constants, radicals with different indexes (dont simplify radicals here estimate the value of irrational radical expressions), absolute value expressions, and any other alternative method to represent numbers (ie., fractions, powers)
Over view of Properties of Real Numbers
Distributive (and Reverse Distributive), Associative, and Commutative Property Flowcharts must be included - (M1, TB, L6-7)
Zero-Product Property
True and False Equations Determining the truth value of an expression (M1, TC, L10)
Solution Sets for Equations and Inequalities include an infinite number of solutions, a finite number of solutions, and no solutions (M1, TC, L11)
Addition and Multiplication Property of Equality/Inequality (Continue infusing these properties throughout the year with particular emphasis during an equation and inequality solving unit).
Unit 2 Monomials and Polynomials (12 days)
Define terminology Power, Term, Monomial, Binomial, Trinomial, Polynomial, Exponent, Base, Degree (single and multi-variable), etc
Adding and Subtracting Polynomials (M1, TB, L8)
Multiplying Polynomials Monomial times Monomial, Monomial times Polynomial, and Polynomial times Polynomial (M1, TB, L9) (M4, TA, L1)
Dividing Polynomials Monomial divided Monomial, Polynomial divided by Monomial
Exponents
Zero Exponents
Negative Exponents
Rational Numbers Raised to Integer Exponents
Powers Raised to an Exponent
Unit 3 Solving Linear Equations, Inequalities, and Literal Equations (emphasize properties of equalities/inequalities throughout) (14 days)
The state would like to introduce equation solving by having students try to anticipate possible solutions (not all solutions) to equations that look atypical. For example, equations of a higher degree that are in factored form. Find one possible solution to (x4 16)(x5 + 32) = 0. Or, solve for x: 4 (5 + (3 x))) = 12. The idea is to get students to think from the outside in to find possible solutions instead of using standard algebraic methods we are used to. Once students play around with these types of equations, then we introduce them to more systemic ways to solve linear and eventually quadratic equations that yield all possible solutions. In addition to the examples above, include fractional equations, radical equations, absolute value equations, equations of a higher degree not in factored form, exponential equations, etc
Recognizing Linear equations and terminology associated with it (understand that solutions, answers, roots, x-intercepts, zeros are synonymous; knowing where to identify them on a graph, etc.)
Solve Linear Equations (1-step to multi-step, variables on both sides, etc) (M1, TC, L12)
Solve Linear Fractional Equations (cross multiply and common denominator methods) (M1, TC, L18)
Be sure to include potential dangers when solving equations (M1, TC, L13)
Solve Linear Inequalities (M1, TC, L14) (Be sure to know set and interval notation to represent solution sets)
Solve Linear Compound Inequalities joined by and or or (M1, TC, L15)
Graphing Linear Inequalities and Compound Inequalities (M1, TC, L16)
Literal Linear Equations Rearranging Equations/Formulas (M1, TC, L19)
Unit 4 Factoring (M4, TA, L1 4) and Algebraic Fractions (17 days)
Factor by method of Reverse Distribution GCF where the GCF is both a monomial or a polynomial
Factor by method of Reverse Double Distribution
Dividing Monomial by Polynomial
Dividing Polynomial by Polynomial
Multiply/Dividing Algebraic Fractions
Adding/Subtracting Algebraic Fractions
Unit 5 Solving Quadratic Equations with Rational Solutions (Zero-Product Property and Square Root Method only- possible method for equations with rational roots) (11 days)
The state wants students to solve quadratic equations using multiple methods Zero Product Property (rational roots), Completing the Square (rational/irrational roots), and Quadratic Formula (rational/irrational roots). It is recommended that students learn the zero-product property method first and tested only on that one method, before introducing students to the other two methods to avoid confusion.
Recognizing Quadratic equations and terminology associated with it (understand that solutions, answers, roots, x-intercepts, zeros are synonymous; knowing where to identify them on a graph, etc.) (M3, TC, L16)
Solving Quadratic Equations in factored form (M1, TC, L17) (M4, TA, L5)
Solving Factorable quadratic equations in standard form (M4, TA, L6)
Solving Factorable quadratic equations in non-standard form
Solving Fractional quadratic equations be sure to check for extraneous roots
Unit 6 Graphing Linear Functions (16 days)
Introduction to Functions, Function Notation, and Evaluating Functions (M3, TB, L9 10)
Be sure to include function language: ie., domain, range, co-domain, independent, dependent variable, lowercase letter, f of x, input/output values, etc
Show different ways to represent/identify functions; ie., mappings, graphs, tables, charts, etc (M3, TB, L11)
Describe behaviors of a function: ie., increasing/decreasing, positive/negative, constant
Evaluate functions (given domain values, find range)
Solve functions (given range values, find domain)
Graphing Linear Functions and vertical lines
Review concept of slope and slope formula, find slope between two points
Understand connection between slope and rate of change focus on units
Graph linear functions and vertical lines by creating a table of values (M3, TB, L12)
Graph linear functions by using the slope/y-intercept method (M3, TB, L12)
Discuss Parallel and Perpendicular Lines and their solution sets
Determine if a point is a solution to a line
Write the equation of a line in y = mx + b form or point-slope form: y y1 = m(x x1)
Piecewise Linear Functions and their interpretations (M1, TA, L1) (M3, TC, L15)
Interpretations of Linear functions in general (M3, TB, L13)
Algebra I Common Core Curriculum Semester 2
Unit 7 Graphing Quadratic Functions (not in vertex form) (13 days)
Overview of the Parts of a Parabola (ie., turning point/vertex point, line (axis) of symmetry, x and y intercepts, end behavior, etc) (M4, TA, L8)
Finding the parts of a parabola by analyzing its graph
Finding the parts of a parabola without seeing the graph
Graph a parabola in standard form by creating an appropriate table of values (M4, TB, L17)
Graph a parabola in factored form by plotting critical points (M4, TA, L9)
Write the equation of a quadratic in standard form (M4, TA, L7)
Interpreting Quadratic Functions (M1, TA, L2) (M4, TA, L10)
Unit 8 Exponential Functions (10 days)
Graphs of exponential functions (Growth and Decay, transformational shifts included, asymptotes) (M1, TA, L3) (M3, TA, L5)
Exploring different exponential function models and what their variables represent (M3, TA, L6 7)
Understanding final amount, initial amount, growth/decay factor, growth/decay rate, unit of time, etc. and how they relate to word problems.
Students will be able to create exponential functions based on word problems and solve for both the independent and dependent variables. There are moments where students will be able to use guess and check to solve for the independent variable and there will be times when students must use graphing calculator features to solve for the independent variable. Refer to modules for assistance.
Analyzing exponential functions and their relationship compared to other functions. (M3, TB, L14) (M3, TD, L21)
*If time permits or with an honors class, it is recommended to throw in solving exponential equations with common bases and without common bases.
Unit 9 Sequence and Series (6 days)
Understanding what a sequence and series is and the difference between linear and exponential sequences. (M3, TA, L1)
Recursive/Explicit formulas for sequences (M3, TA, L2)
Arithmetic/Geometric sequences (M3, TA, L3)
*It is recommended for this unit to use subscript notation as much as possible as opposed to function notation. Subscript notation (an) is more common for sequences and students need to become familiar with the notation.
Unit 10 Solving Quadratic Equations using Other Methods (possible methods for equations with irrational roots) (11 days)
Vertex Form (Completing the Square) (M4, TB, L11 12)
Identify perfect square trinomials
Rewrite perfect square trinomials as a binomial squared
Rewrite non-perfect square trinomials as a binomial squared plus a constant (vertex form)
Leading coefficient is 1: (x h)2 + k
Leading coefficient is other than 1: a(x h)2 + k
Solving Quadratic Equations by Completing the Square (equations in vertex form) (M4, TB, L13)
Solve quadratic equations that are already in vertex form
Solve quadratic equations that are in standard form by completing the square
Quadratic Formula (M4, TB, L15)
Derive the quadratic formula by completing the square (not recommended) (M4, TB, L14)
Solve quadratic equations in standard form using the quadratic formula
Solve quadratic equations in non-standard form using the quadratic formula
For both the completing the square method and quadratic formula method, roots should be rational and/or irrational. For irrational roots, students should be expected to leave their answers in simplest radical form (which is why it is recommended to spiral radicals in as Do Nows earlier in the year) and to round answers.
Unit 11 Graphing Quadratics and Transformational Shifts (4 days)
Discuss the concept of a Parent Function (shiftless function)
Show all transformational shifts using quadratics and show how they relate to the variables of a quadratic equation in vertex form: y = a(x h)2 + k (M4, TC, L21) (M3, TC, L17 20)
Horizontal Shift (Phase Change)
Vertical Shift (Displacement
Reflection over the x-axis
Dilation where 0 < a < 1, and where a > 1
Discuss the standard rate of change of a parabola on either side of its vertex point (1,3,5) and discuss how that rate of change is affected when a `" 1.
Graph quadratic equations in vertex form using knowledge of shifts (M4, TB, L16)
Writing the equation of quadratics in vertex form.
Unit 12 Graphing Unfamiliar Functions
Students will build an appropriate table of values for each function below. (M4, TC, L18 20) (M3, TD, L24)
Absolute Value Functions
Step Functions
Piecewise Functions
Cube Root Function
Cubic Function
Square Root Function
Students will explore the parent function for each function below and perform indicated transformational shifts on the parent functions. (M4, TC, L18 20) (M3, TD, L24)
Absolute Value Functions
Step Functions
Piecewise Functions
Cube Root Function
Cubic Function
Square Root Function
Students will be able to identify, using interval and/or set notation, the behavior of each function (ie., increasing/decreasing, positive/negative, constant, domain/range, etc...).
Students will be able to see the inverse relationship between quadratic and square root functions in addition to cubic and cube root functions. (M4, TC, L22)
Students will be able to solve equations with inverse relationships graphically and/or algebraically. (M4, TC, L22)
Students will be able to write the equation for each function using knowledge of parent function and transformational shifts.
Unit 13 Systems of Equations (Algebraic and Graphic) (7 days)
Analyze what it means to be a solution to a system of equations with two variables. (M1, TA, L5)
(M1, TC, L20)
Solve systems of equations (M1, TC, L21 23)
Graphically
Algebraically (students must understand how the procedure for solving a system algebraically relates to the addition/multiplication properties of equalities)
Applications of system of equations (M1, TA, L5) (M1, TC, L24)
Unit 14 Statistics (This unit can be consolidated at teacher discretion Recommended scaffolding concepts throughout the year.) All of Module 2 should be read through and adheres to the topics below. Elements of each topic below are mixed throughout the modules. (M2, TA, L1 3) (M2, TB, L4 8) (M2, TC, L9 11) (M2, TD, L12 20) (8 days)
Introduction to Statistics and terminology
Types of Data (Quantitative vs. Categorical)
Sets of Data (Univariate vs. Bivariate)
Graphs of Univariate Sets of Data
Dot/Line Plots
Histograms
Box Plots
Bell Curve
Graphs of Bivariate Sets of Data
Two-Way Frequency Tables
Scatterplots
Measures of Central Tendency
Mean, Median, Mode
Measures of Dispersion
Range, Interquartile Range, Standard Deviation
Types of Distribution
Symmetrical, skewed, U-shaped
Which measure of central tendency best matches the type of distribution
Specifics about each type of graph
Dot/Line Plot
Distribution and measure of central tendency, etc
Histogram
Frequency, Cumulative, Intervals, etc
Box Plot
Five Point Summary, IQR, outliers, etc
Bell Curve
Mean, standard deviation, etc
Two-Way Frequency Tables
Marginal/Joint Frequencies, Relative Frequencies, Conditional Frequencies, etc
Scatterplots
Least Square Regression Line, Correlation Coefficient, Residuals and Residual Plots, etc
*This unit has many components that students need to be able to do on the graphing calculator. It is important to read through all the modules for newer language and expectations for students on the graphing calculator.
Unit 15 Analyzing Graphs in General (if times allows) (M5, TA, L1 3) (M5, TB, L4 9)
Module 5 puts together all the equation and function units from the year. Due to time limits in the 2013-2014 school year, this module was not covered as a separate unit, but rather elements of the modules were embedded throughout other units during the year. It is recommended, if time allows, to review all modules in this uni.mnvwE v w
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