Integrated Algebra II

Course Description

This course is designed specifically for the student who needs to enhance his/her past learned algebraic skills, and the course prepares students for the rigor of Regular Algebra II. Emphasis will be placed on relating concepts of Algebra 1 to the real world, previewing basic statistical concepts, and developing problem-solving and thinking skills to be applied to all math situations and problems.

Students will explore the connections between various functions and the real world, and investigate the similarities between the different types of functions and equations discussed and learned in this course.

Course Big Ideas

All functions can be investigated for the same basic features.
Understanding inverse operations can help solve any equation.
Understanding basic math rules and facts can help build knowledge for the bigger picture and help create a stronger foundation for learning new math concepts.

Course Essential Questions

What are the inverse operations and methods used to solve different types of equations?
What are the common features and characteristics you can identify for all functions?
How do you know what operations you can use when working with various forms of functions, expressions, and terms?

Course Competencies

Graph functions, identify or calculate slope, end behavior, intercepts, domain, range, horizontal and vertical shifts, reflections, and the vertex and axis of symmetry of a function.
Students will be able to solve radical equations, exponential equations, linear equations, and quadratic equations.
Understand function notation and how functions relate to real-world scenarios.

Course Assessments

Summative unit tests and quizzes.
Check-for-understandings.
iXL.
Writing prompts.
Independent practice worksheets/assignments.
Midterm exam (units 1 - 4).
Final Exam (units 5 - 9).

How do you multiply and divide fractions?
How do you add and subtract fractions?
How do you turn values in unlike forms to like forms to compare them?
How do you determine if terms are "like"?
What is the correct order of operations when simplifying an algebraic expression?
How do you determine the order of values when comparing them from least to greatest?

CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable.
A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation-solving process (linear equations only).
A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation (linear equations only).
CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method.
A1.1.3.1.1 Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities).

Solve multi-step equations.
Solve equations with one fraction.
Write a multi-step equation given a real-life situation.
Solve a single inequality.
Solve a compound inequality.
Solve absolute value equations.
Solve absolute value inequalities.
Represent a solution to an inequality using interval notation.
Represent a solution to an inequality using a number line, inequality, and interval notation.

Unit Essential Questions

How do we know if a solution to an equation or inequality is correct or makes sense in context to the problem?
What is the difference between solving an equation vs. an inequality?
What does absolute value represent, and how does it affect the solutions to equations?
What does it mean for one quantity to be greater or less than another, and how is that represented mathematically?

CC.2.1.HS.F.7 Apply concepts and complex numbers in polynomial identities and quadratic equations to solve problems.
A2.2.1.1.3 Determine the domain, range or inverse of a relation.
CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships.
A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
A2.1.3.2.2 Use algebraic processes to solve a formula for a given variable (e.g., solve d = rt for r).
CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions.
A1.2.1.2.1 Create, interpret and/or use the equation, graph or table of a linear function
CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities.
A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.
A1.2.1.1.2 Determine if a relation is a function given a set of points or a graph.
CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables.
A2.2.3.1.1 Draw, identify, find and/or write an equation for a regression model (lines and curves of best fit) for a scatter plot.

Know

Function notation.
Domain intervals.
Range intervals.
Function evaluations.
Function operations.
Function compositions.
Different types of slope.
Slope calculation.
Slope-intercept equation of a line.
Point-slope equation of a line.
Intercepts.
Scatter-plot comprehension.
Linear inequalities with two variables.
Absolute value graph characteristics.
Inverse functions.

Understanding/Key Learning

Do

Identify domain and range of a relation.
Evaluate functions using an equation and a graph.
Perform function operations.
Perform function compositions.
Identify slope given an equation or a graph.
Identify x- and y-intercepts given an equation or a graph.
Graph a line in slope-intercept form.
Graph a single inequality with two variables.
Write an equation of a line given two points.
Write an equation of a line given a scatter plot.
Graph absolute value functions given an equation.
Identify the vertex and axis of symmetry of an absolute value graph.
Find the inverse of a function given a graph.
Find the inverse of a linear function given an equation.

Unit Essential Questions

How can we use functions and their operations to model, analyze, and solve real-world and mathematical problems involving relationships between quantities?
What do the characteristics of a graph tell us about the equation or inequality it represents?
In what ways are linear functions and linear inequalities similar, and how are they different when represented graphically?
How are inverse operations and inverse functions similar?

Lesson Essential Questions

How do we perform operations on functions to create new functions?
What are the different ways to evaluate a function?
What is the input and output of a function?
Why do linear inequalities have shaded regions, and what do they represent?
What pieces of information do you use to graph a line?
What pieces of information do you use to graph a linear inequality with two variables?
How do you graph the inverse of a function?
How do you find an inverse equation?

CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.
- A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
- A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation (linear equations only).
- A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method.
CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- A1.1.3.2.1 Write and/or solve a system of linear inequalities using graphing (limit systems to 2 linear inequalities).

How can systems of equations and inequalities be used to model and solve real-world problems involving multiple conditions or constraints?
What does the solution to a system of equations represent, and how can we find it using different methods?
How are systems of equations and systems of inequalities similar and different in terms of their solutions and representations?

Lesson Essential Questions

How do you determine a solution to a system of inequalities?
How do you determine a solution to a system of equations by graphing?
What is the process to set up and solve a system of equations using elimination?
What is the process to set up and solve a system of equations using substitution?
How do you check or verify a solution to a system of equations?
What is the difference between no solution and infinite solutions?

CC.2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations.
A2.1.3.1.4 Write, solve and/or apply linear or exponential growth or decay (including problem situations).
A2.2.1.1.4 Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increasing/decreasing, intercepts, zeros, and asymptotes).
A2.1.3.1.3 Write and/or solve a simple exponential or logarithmic equation (including common and natural logarithms).
CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems.
CC.2.2.HS.D.3 Extend the knowledge of arithmetic operations and apply to polynomials.

Identify bases and exponents of an expression.
Solve equations with variables within the exponent.
Simplify expressions using exponent rules.
Add, subtract, and multiply polynomials of various degrees.
Calculate balances and interest in accounts.
Calculate the general exponential growth or decay of a given amount.
Graph exponential functions, and identify domain, range, and asymptotes.

Unit Essential Questions

Lesson Essential Questions

How do we identify and interpret the key features of an exponential graph?
What pieces of information do you need to calculate compound interest?
What pieces of information are needed to calculate the exponential growth and decay of a quantity?
What are the base and exponent of an exponential expression?
How do we determine if a function is growth or decay?
What are the rules for multiplying and dividing exponential terms/expressions?
What are the rules for adding and subtracting exponential terms/expressions?
How do you solve for an equation with a variable as an exponent?

CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems.
CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable.
CC.2.2.HS.C.4 Interpret the effects transformations have on functions and find the inverses of functions.
CC.2.2.HS.C.6 Interpret functions in terms of the situation they model.

CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems.
CC.2.2.HS.D.4 Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs.
CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions.
CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems.
A2.1.3.1.1 Write and/or solve quadratic equations (including factoring and using the Quadratic Formula).

Identify the vertex of parabola.
Identify maximum/minimum vertex of parabola.
Identify the axis of symmetry.
Graph quadratic functions.
Factor quadratics with lead coefficient of 1.
Solve quadratic equations using the zero product property.
Solve quadratic equations using the quadratic formula.
Solve quadratic equations by taking the square root.
Simplify non-perfect square root expressions.

Unit Essential Questions

Lesson Essential Questions

How can you determine the direction a parabola opens?
How do you set up and solve a quadratic equation using the quadratic formula?
How do you set up a problem to solve using square roots?
How do you set up a quadratic expression to factor it into two binomials?
What are the steps to use the zero product property to solve an equation?
How do you identify the vertex of a parabola from an equation in vertex form?
What is a vertex of a parabola?
What is the difference between a maximum and a minimum vertex?
How do you write/put an equation in standard form?

CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays.
CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables.
CC.2.4.HS.B.6 Use the concepts of independence and conditional probability to interpret data.
A2.2.3.2.1 Use combinations, permutations, and the fundamental counting principle to solve problems.
A2.2.3.2.2 Use odds to find probability and/or use probability to find odds.
A2.2.3.2.3 Use probability for independent, dependent or compound events to predict outcomes.
CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities and compound events in a uniform probability model.
A1.2.3.3.1 Find probabilities for compound events (e.g., find probability of red and blue, find probability of red or blue) and represent as a fraction, decimal or percent).

Know

Measures of central tendency.
Categorical vs. quantitative data.
Creating and interpreting box-and-whisker plots.
Different data representations, like frequency tables, histograms, and stem-and-leaf plots.
Theoretical/experimental probability.
Compound probability.
Probability of mutually exclusive vs. overlapping events.
Probability of independent and dependent events.
The process for calculating permutations/arrangements.
The process for calculating combinations/selections.
The process for calculating odds.

Understanding/Key Learning

Do

Calculate the different measures of central tendency.
Create box-and-whisker plots.
Interpret/understand box-and-whisker plots.
Construct/interpret stem-and-leaf plots.
Construct/interpret histograms.
Construct/interpret a frequency table.
Calculate simple experimental and theoretical probability.
Calculate the theoretical probability of overlapping events.
Calculate the probability of independent and dependent events.
Calculate arrangements/permutations.
Calculate the number of combinations of an event.
Calculate the odds of an event.

Unit Essential Questions

How does understanding probability help us make informed decisions under uncertainty?
How can we communicate data effectively using appropriate representations?
How can we organize and represent data to provide insight into the relationship between quantities?
How can we use theoretical and experimental probability to predict the likelihood of real-world events?

Lesson Essential Questions

What do you need to incorporate when calculating the probability of overlapping events?
What is different about calculating odds vs. probability?
What is the definition of quantitative and categorical data?
How do you create a stem-and-leaf plot?
How do you create a frequency table?
How do you create a histogram?
When do you use a permutation?
When do you use a combination?
How do you calculate the probability of more than one event?
How do you calculate the odds of an event?