# Standardized Test Preparation

## SAT

References: College Board : SAT Math | Khan Academy : SAT |

## Math

Assessment Exam - SAT Math
Heart of Algebra eTAP Lesson
Linear Equations in One Variable
Create and use linear equations in one variable to solve problems in a variety of contexts.
SAT.MATH.1.1.1
Equations as Relations

Basic Operations

Subtracting the Same Number

Dividing the Same Number to Each Side of an Equation, Solving for an Unknown

When to do What

Setting Up and Solving Equations

Completing the Square

Simultaneous Equations With Strange Solutions
Create a linear equation in one variable, and when in context interpret solutions in terms of the context.
SAT.MATH.1.1.2
Equations as Relations

Basic Operations

Subtracting the Same Number

Dividing the Same Number to Each Side of an Equation, Solving for an Unknown

When to do What

Setting Up and Solving Equations
Solve a linear equation in one variable, making strategic use of algebraic structure.
SAT.MATH.1.1.3
Equations as Relations

Basic Operations

Dividing the Same Number to Each Side of an Equation, Solving for an Unknown

When to do What

Setting Up and Solving Equations
For a linear equation in one variable,
1. interpret a constant, variable, factor, or term in a context
2. determine the conditions under which the equation has no solution, a unique solution, or infinitely many solutions.

SAT.MATH.1.1.4
Equations as Relations

Basic Operations

When to do What

Setting Up and Solving Equations
Fluently solve a linear equation in one variable.
SAT.MATH.1.1.5
Equations as Relations

Basic Operations

Dividing the Same Number to Each Side of an Equation, Solving for an Unknown

When to do What

Setting Up and Solving Equations
Linear Functions
In the first case, the variable is the input and the value of the expression is the output. In the second case, one of the variables is designated as the input and determines a unique value of the other variable, which is the output.
Create and use linear functions to solve problems in a variety of contexts.
SAT.MATH.1.2.1
Functions
Create a linear function to model a relationship between two quantities.
SAT.MATH.1.2.2
Functions
For a linear function that represents a context,
1. interpret the meaning of an input/output pair, constant, variable, factor, or term based on the context, including situations where seeing structure provides an advantage;
2. given an input value, find and/or interpret the output value using the given representation

SAT.MATH.1.2.3
Functions

Determine the Equation from a Relation

Solving Equations Using Several Operations
Make connections between verbal, tabular, algebraic, and graphical representations of a linear function by
1. deriving one representation from the other; identifying features of one representation given another representation;
2. determining how a graph is affected by a change to its equation.

SAT.MATH.1.2.4
Functions

Determine the Equation from a Relation

Slope of a Line
Write the rule for a linear function given two input/output pairs or one input/output pair and the rate of change.
SAT.MATH.1.2.5
Functions

Determine the Equation from a Relation

Direct and Inverse Variation

Slope of a Line
Linear Equations in Two Variables
A linear equation in two variables can be used to represent a constraint or condition on two-variable quantities in situations where neither of the variables is regarded as an input or an output. A linear equation can also be used to represent a straight line in the coordinate plane.
Create and use a linear equation in two variables to solve problems in a variety of contexts.
SAT.MATH.1.3.1
Graphing Systems of Inequalities

Solving Equations When Several Terms and Procedures are Involved

Variables and Expressions
Create a linear equation in two variables to model a constraint or condition on two quantities.
SAT.MATH.1.3.2
Solving Equations When Several Terms and Procedures are Involved

Variables and Expressions
For a linear equation in two variables that represents a context,
1. interpret a solution, constant, variable, factor, or term based on the context, including situations where seeing structure provides an advantage;
2. given a value of one quantity in the relationship, find a value of the other, if it exists.

SAT.MATH.1.3.3
Solving Equations When Several Terms and Procedures are Involved

Slope of a Line
Make connections between tabular, algebraic, and graphical representations of a linear equation in two variables by
1. deriving one representation from the other;
2. identifying features of one representation given the other representation;
3. determining how a graph is affected by a change to its equation.

SAT.MATH.1.3.4
Functions as Graphs in the Coordinate System

How to Use Graphs
Write an equation for a line given two points on the line, one point and the slope of the line, or one point and a parallel or perpendicular line.
SAT.MATH.1.3.5
Slope of a Line

Slope Intercept Form of Equations
Systems of Two Linear Equations in Two Variables
Create and use a system of two linear equations in two variables to solve problems in a variety of contexts.
SAT.MATH.1.4.1
Simultaneous Equations
Create a system of linear equations in two variables, and when in context interpret solutions in terms of the context.
SAT.MATH.1.4.2
Simultaneous Equations
Make connections between tabular, algebraic, and graphical representations of the system by deriving one representation from the other.
SAT.MATH.1.4.3
Graphing Systems of Equations
Solve a system of two linear equations in two variables, making strategic use of algebraic structure.
SAT.MATH.1.4.4
The Substitution Method

For a system of linear equations in two variables,
1. interpret a solution, constant, variable, factor, or term based on the context, including situations where seeing structure provides an advantage;
2. determine the conditions under which the system has no solution, a unique solution, or infinitely many solutions.

SAT.MATH.1.4.5
Simultaneous Equations With Strange Solutions
Fluently solve a system of linear equations in two variables
SAT.MATH.1.4.6
Substitution

Elimination
Linear Inequalities in One or Two Variables
Create and use linear inequalities in one or two variables to solve problems in a variety of contexts.
SAT.MATH.1.5.1
Inequalities with Several Operations
Create linear inequalities in one or two variables, and when in context interpret the solutions in terms of the context.
SAT.MATH.1.5.2
Inequalities with Several Operations
For linear inequalities in one or two variables, interpret a constant, variable, factor, or term, including situations where seeing structure provides an advantage.
SAT.MATH.1.5.3
Inequalities with Several Operations
Make connections between tabular, algebraic, and graphical representations of linear inequalities in one or two variables by deriving one from the other.
SAT.MATH.1.5.4
Graphing Systems of Inequalities
Given a linear inequality or system of linear inequalities, interpret a point in the solution set.
SAT.MATH.1.5.5
Graphing Systems of Inequalities
Problem Solving and Data Analysis eTAP Lesson
Ratios, Rates, Proportional Relationships, and Units
Items will require students to solve problems by using a proportional relationship between quantities, calculating or using a ratio or rate, and/or using units, derived units, and unit conversion.
Apply proportional relationships, ratios, rates, and units in a wide variety of contexts. Examples include but are not limited to scale drawings and problems in the natural and social sciences.
SAT.MATH.2.1.1
Ratios & Rates

Direct and Inverse Variation
Solve problems involving
1. derived units, including those that arise from products (e.g., kilowatt-hours) and quotients (e.g., population per square kilometer);
2. unit conversion, including currency exchange and conversion between different measurement systems.

SAT.MATH.2.1.2
Word Problems of Ratios & Rates

Direct and Inverse Variation
Understand and use the fact that when two quantities are in a proportional relationship, if one changes by a scale factor, then the other also changes by the same scale factor.
SAT.MATH.2.1.3
Direct and Inverse Variation
Percentages
Use percentages to solve problems in a variety of contexts. Examples include, but are not limited to, discounts, interest, taxes, tips, and percent increases and decreases for many different quantities.
SAT.MATH.2.2.1
Percent, Decimals & Fractions
Understand and use the relationship between percent change and growth factor (5% and 1.05, for example); include percentages greater than or equal to 100%.
SAT.MATH.2.2.2
Exponential Functions and Data
One-Variable Data: Distributions and Measures of Center and Spread
Choose an appropriate graphical representation for a given data set.
SAT.MATH.2.3.1
Basic Methods of Describing Data
Interpret information from a given representation of data in context.
SAT.MATH.2.3.2
Visually Representing Numerical Data
Analyze and interpret numerical data distributions represented with frequency tables, histograms, dot plots, and boxplots.
SAT.MATH.2.3.3
Showing Orderly Data with Histograms

Seeking Trends with Line Graphs

Boxplot, Interquartile Range, and Midhinge

Methods of Collecting, Representing, and Displaying Data
For quantitative variables, calculate, compare, and interpret mean, median, and range. Interpret (but don’t calculate) standard deviation.
SAT.MATH.2.3.4
Boxplot, Interquartile Range, and Midhinge

Frequency Distribution Table, Summation Notation, and Mean Formula

Standard Deviation of Random Variable
Compare distributions using measures of center and spread, including distributions with different means and the same standard deviations and ones with the same mean and different standard deviations.
SAT.MATH.2.3.5
Standard Deviation of Random Variable
Understand and describe the effect of outliers on mean
SAT.MATH.2.3.6
Paired Data Sets and Scatterplots

Regression Models and Least Square Methods
Given an appropriate data set, calculate the mean.
SAT.MATH.2.3.7
Geometric Mean, Harmonic Mean, and Weighted Arithmetic Mean
Two-Variable Data: Models and Scatterplots
Using a model that fits the data in a scatterplot, compare values predicted by the model to values given in the data set.
SAT.MATH.2.4.1
Paired Data Sets and Scatterplots
Interpret the slope and intercepts of the line of best fit in context.
SAT.MATH.2.4.2
How to Use Graphs
Given a relationship between two quantities, read and interpret graphs and tables modeling the relationship.
SAT.MATH.2.4.3
How to Use Graphs
Analyze and interpret data represented in a scatterplot or line graph; fit linear, quadratic, and exponential models.
SAT.MATH.2.4.4
Correlations

Paired Data Sets and Scatterplots
Select a graph that represents a context, identify a value on a graph, or interpret information on the graph.
SAT.MATH.2.4.5
How to Use Graphs
For a given function type (linear, quadratic, exponential), choose the function of that type that best fits given data.
SAT.MATH.2.4.6
Paired Data Sets and Scatterplots
Compare linear and exponential growth.
SAT.MATH.2.4.7
The number e

Using Logarithms to Model Data
Estimate the line of best fit for a given scatterplot; use the line to make predictions.
SAT.MATH.2.4.8
Correlations
Probability and Conditional Probability
Use one- and two-way tables, tree diagrams, area models, and other representations to find relative frequency, probabilities, and conditional probabilities.
Compute and interpret probability and conditional probability in simple contexts.
SAT.MATH.2.5.1
Conditional Probability

What is Probability?

Conditional Probability
Understand formulas for probability and conditional probability in terms of frequency.
SAT.MATH.2.5.2
Conditional Probability

Discrete Probability Distribution
Inference From Sample Statistics and Margin of Error
Use sample mean and sample proportion to estimate population mean and population proportion. Utilize, but do not calculate, margin of error.
SAT.MATH.2.6.1

Interpret margin of error; understand that a larger sample size generally leads to a smaller margin of error.
SAT.MATH.2.6.2

Evaluating Statistical Claims: Observational Studies and Experiments
With random samples, describe which population the results can be extended to.
SAT.MATH.2.7.1
Multiple Events

Methods of Collecting, Representing, & Displaying Data
Given a description of a study with or without random assignment, determine whether there is evidence for a causal relationship.
SAT.MATH.2.7.2
Planning and Conducting an Experiment
Understand why random assignment provides evidence for a causal relationship.
SAT.MATH.2.7.3
Independent and Dependent Events
Understand why a result can be extended only to the population from which the sample was selected.
SAT.MATH.2.7.4
Independent and Dependent Events
Equivalent Expressions
Make strategic use of algebraic structure and the properties of operations to identify and create equivalent expressions, including
1. rewriting simple rational expressions;
2. rewriting expressions with rational exponents and radicals;
3. factoring polynomials.

SAT.MATH.3.1.1
Rational Expressions with Like and Unlike Denominators

Exponential Functions and Data
Fluently add, subtract, and multiply polynomials.
SAT.MATH.3.1.2

Multiplying and Dividing Polynomials
Nonlinear Equations in One Variable and Systems of Equations in Two Variables
Make strategic use of algebraic structure, the properties of operations, and reasoning about equality to
1. solve quadratic equations in one variable presented in a wide variety of forms; determine the conditions under which a quadratic equation has no real solutions, one real solution, or two real solutions;
2. solve simple rational and radical equations in one variable;
3. identify when the procedures used to solve a simple rational or radical equation in one variable lead to an equation with solutions that do not satisfy the original equation (extraneous solutions);
4. solve polynomial equations in one variable that are written in factored form;
5. solve linear absolute value equations in one variable;
6. solve systems of linear and nonlinear equations in two variables, including relating the solutions to the graphs of the equations in the system.

SAT.MATH.3.2.1
Correlations

Solve Single Variable Equations

Exponential and Logarithmic Equations

Joe's Method

Completing the Square

Absolute Value Inequalities

Simultaneous Equations With Strange Solutions
Given a nonlinear equation in one variable that represents a context, interpret a solution, constant, variable, factor, or term based on the context, including situations where seeing structure provides an advantage.
SAT.MATH.3.2.2
Exponential and Logarithmic Equations
Given an equation or formula in two or more variables that represents a context, view it as an equation in a single variable of interest where the other variables are parameters and solve for the variable of interest.
SAT.MATH.3.2.3
Ratios and Proportions

Direct and Inverse Variation
Fluently solve quadratic equations in one variable, written as a quadratic expression in standard form equal to zero, where using the quadratic formula or completing the square is the most efficient method for solving the equation.
SAT.MATH.3.2.4

Joe's Method

Completing the Square

Nonlinear Functions
Create and use quadratic or exponential functions to solve problems in a variety of contexts
SAT.MATH.3.3.1
Graphing Data
For a quadratic or exponential function,
1. identify or create an appropriate function to model a relationship between quantities;
2. use function notation to represent and interpret input/output pairs in terms of a context and points on the graph;
3. for a function that represents a context, interpret the meaning of an input/output pair, constant, variable, factor, or term based on the context, including situations where seeing structure provides an advantage;
4. determine the most suitable form of the expression representing the output of the function to display key features of the context, including
1. selecting the form of a quadratic that displays the initial value, the zeros, or the extreme value;
2. selecting the form of an exponential that displays the initial value, the end-behavior (for exponential decay), or the doubling or halving time
5. make connections between tabular, algebraic, and graphical representations of the function by
1. given one representation, selecting another representation;
2. identifying features of one representation given another representation, including maximum and minimum values of the function;
3. determining how a graph is affected by a change to its equation, including a vertical shift or scaling of the graph.

SAT.MATH.3.3.2
Function Notation

Exponential Functions and Data

Graphs of Exponential Functions

Using Logarithms to Model Data

Parabolas

For a factorable or factored polynomial or simple rational function,
1. use function notation to represent and interpret input/output pairs in terms of a context and points on the graph;
2. understand and use the fact that for the graph of y = f(x), the solutions to f(x) = 0 correspond to x-intercepts of the graph and f(0) corresponds to the y-intercept of the graph; interpret these key features in terms of a context;
3. identify the graph given an algebraic representation of the function and an algebraic representation given the graph (with or without a context).

SAT.MATH.3.3.3

How to Use Graphs

Function Notation
Additional Topics in Math eTAP Lesson
Area and Volume
Solve real-world and mathematical problems about a geometric figure or an object that can be modeled by a geometric figure using given information such as length, area, surface area, or volume.
1. Apply knowledge that changing by a scale factor of k changes all lengths by a factor of k, changes all areas by a factor of k2, and changes all volumes by a factor of k3.
2. Demonstrate procedural fluency by selecting the correct area or volume formula and correctly calculating a specified value

SAT.MATH.4.1
Triangle Areas

Surface Area

Rectangular Prism Volume

Areas of Rectangles and Parallelograms
Lines, Angles, and Triangles
Use concepts and theorems relating to congruence and similarity of triangles to solve problems.
SAT.MATH.4.2.1
Triangles

SSS and SAS Congruencies
Determine which statements may be required to prove certain relationships or to satisfy a given theorem.
SAT.MATH.4.2.2
Geometric Proofs

Postulates of Geometry

Direct Proofs

Symbols of Logic
Apply knowledge that changing by a scale factor of k changes all lengths by a factor of k, but angle measures remain unchanged
SAT.MATH.4.2.3
Similar Triangles

Properties of Isometrics
Know and directly apply relevant theorems such as
1. the vertical angle theorem;
2. triangle similarity and congruence criteria;
3. triangle angle sum theorem;
4. the relationship of angles formed when a transversal cuts parallel lines.

SAT.MATH.4.2.4
Parallelograms and Parallel Lines Related Proofs

Proving Angle Conjectures

Triangles Sum Conjecture

ASA Congruencies

AAS and HL Congruencies

Similar Triangles
Right Triangles and Trigonometry
Solve problems in a variety of contexts using
1. the Pythagorean theorem;
2. right triangle trigonometry;
3. properties of special right triangles.

SAT.MATH.4.3.1
Special Right Triangles

The Theorem of Pythagoras

Trigonometric Calculations
Use similarity to calculate values of sine, cosine, and tangent.
SAT.MATH.4.3.2
Ratios and Proportions
Understand that when given one side length and one acute angle measure in a right triangle, the remaining values can be determined.
SAT.MATH.4.3.3
The Theorem of Pythagoras
Solve problems using the relationship between sine and cosine of complementary angles.
SAT.MATH.4.3.4
Sine-Cosine Relationships
Fluently apply properties of special right triangles to determine side lengths and calculate trigonometric ratios of 30, 45, and 60 degrees.
SAT.MATH.4.3.5
Special Right Triangles
Circles
Use definitions, properties, and theorems relating to circles and parts of circles, such as radii, diameters, tangents, angles, arcs, arc lengths, and sector areas, to solve problems.
SAT.MATH.4.4.1
Arc Length

Definition - Circles
Solve problems using
2. trigonometric ratios in the unit circle.

SAT.MATH.4.4.2
Tangent Properties

Circumference/Diameter Ratio

Create an equation to represent a circle in the xy-plane.
SAT.MATH.4.4.3
Corresponding Parts
Describe how
1. a change to the equation representing a circle in the xy-plane affects the graph of the circle;
2. a change in the graph of the circle affects the equation of the circle.

SAT.MATH.4.4.4

Understand that the ordered pairs that satisfy an equation of the form (x – h)2 + (y – k)2 = r2 form a circle when plotted in the xy-plane.
SAT.MATH.4.4.5

Convert between angle measures in degrees and radians.
SAT.MATH.4.4.6