## Algebra IA Intensive

## Course Syllabus

**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

Ø 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.

Ø Rounding (all different place values)

Ø Representing Solution Sets using Set Notation and Interval Notation

Ø Area, Perimeter, and Volume Formulas

Ø Unit Conversion

Ø Radicals (Simplifying, Adding/Subtracting, Multiplying/Dividing, Rationalizing)

Ø 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 don’t define i)

o Incorporate irrational constants, radicals with different indexes (don’t 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

o Distributive (and Reverse Distributive), Associative, and Commutative Property – Flowcharts must be included - (M1, TB, L6-7)

o Zero-Product Property

o True and False Equations – Determining the truth value of an expression (M1, TC, L10)

o Solution Sets for Equations and Inequalities – include an infinite number of solutions, a finite number of solutions, and no solutions (M1, TC, L11)

o 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 –

o Zero Exponents

o Negative Exponents

o Rational Numbers Raised to Integer Exponents

o 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)

o 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)

o Solve Linear Compound Inequalities joined by “and” or “or” – (M1, TC, L15)

o 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

## Assignments

## Class Resources

## Algebra IB Intensive

## Course Syllabus

Key: (Aligned to engageny.org modules)

M – Module

T – Topic

L – Lesson

**Unit 6 – Graphing Linear Functions **

- Introduction to Functions, Function Notation, and Evaluating Functions (M3, TB, L9 – 10)

o Be sure to include function language: ie., domain, range, co-domain, independent, dependent variable, lowercase letter, f of x, input/output values, etc…

o Show different ways to represent/identify functions; ie., mappings, graphs, tables, charts, etc… (M3, TB, L11)

o Describe behaviors of a function: ie., increasing/decreasing, positive/negative, constant

o Evaluate functions (given domain values, find range)

o Solve functions (given range values, find domain)

- Graphing Linear Functions and vertical lines

o Review concept of slope and slope formula, find slope between two points

o Understand connection between slope and rate of change – focus on units

o Graph linear functions and vertical lines by creating a table of values (M3, TB, L12)

o Graph linear functions by using the slope/y-intercept method (M3, TB, L12)

o Discuss Parallel and Perpendicular Lines and their solution sets

o Determine if a point is a solution to a line

o Write the equation of a line in y = mx + b form or point-slope form: y – y1 = m(x – x1)

o Piecewise Linear Functions and their interpretations (M1, TA, L1) (M3, TC, L15)

o Interpretations of Linear functions in general (M3, TB, L13)

#### Unit 7 – Graphing Linear Functions

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 – y
_{1}= m(x – x_{1}) - Piecewise Linear Functions and their interpretations (M1, TA, L1) (M3, TC, L15)
- Interpretations of Linear functions in general (M3, TB, L13)

#### Unit 8 – Graphing Quadratic Functions (not in vertex form)

- 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 9 – Graphing Quadratics and Transformational Shifts

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 10 – Graphing Unfamiliar Functions and Transformational Shifts

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 write the equation for each function using knowledge of parent function and transformational shifts.

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)

#### Unit 11 – Exponential Functions

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 12 – Systems of Equations (Algebraic and Graphic)

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 13 – Statistics (This unit can be consolidated at teacher discretion.) 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)

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
- Frequence, 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.