# SAT New Math: Study guide to Score a Perfect 800

### Topic 1. Heart of algebra

• 1.1 Linear Equations
• 1.2 Linear Inequalities
• Create, solve, or interpret linear inequalities in 1 variable.
• Algebraically solve linear  inequalities in 1 variable.
• 1.3 Systems of Linear Equations
• Create, solve, and interpret systems of 2 linear equations in 2 variables.
• Create, solve, and interpret systems of linear inequalities in 2 variables.
• Algebraically solve systems of 2 linear equations in 2 variables.
• Interpret the variables and constants in expressions for linear functions.
• 1.4 Lines in the Coordinate Plane
• Understand connections between algebraic and graphical representations.

### Topic 2. Passport to advanced mathematics

• 2.1 Operations with Polynomials and Rewriting Expressions
• Create an equivalent form of an algebraic expression
• Add, subtract, and multiply polynomial expressions
• Interpret parts of nonlinear expressions in terms of their context
• 2.2 Quadratic Functions and Equations
• Create a quadratic function or equation that models a context.
• Solve a quadratic equation having rational coefficients.
• 2.3 Exponential Functions, Equations, and Expressions and Radicals
• Create a  exponential function or equation that models a context.
• 2.4 Solving Rational Equations
• Create equivalent expressions involving rational exponents and radicals, which includes simplifying or rewriting in other forms.
• Rewrite simple rational expressions
• Solve an equation in 1 variable that contains radicals
• 2.5 Systems of Equations
• Solve a system of 1 linear equation and 1 quadratic equation
• 2.6 Relationships Between Algebraic and Graphical Representations of Functions
• Understand the relationship between zeros and factors of polynomials.
• Use that knowledge to sketch graphs
• Understand a nonlinear relationship between 2 variables by making connections between their algebraic and graphical representations.
• 2.7 Function Notation
• Use function notation, and interpret statements using function notation
• 2.8 Analyzing More Complex Equations
• Use structure to isolate or identify a quantity of interest in an expression or isolate a quantity of interest in an equation

### Topic 3. Problem solving and data analysis

• 3.1 Ratio, Proportion, Units, and Percentage
• Use ratios, rates, proportional relationships, and scale drawings to solve single- and multistep problems.
• Solve single- and multistep problems involving percentages.
• Solve single- and multistep problems involving measurement quantities, units, and unit conversion
• 3.2 Scatterplots, Graphs, Tables, and Equations
• Use scatterplot, linear, quadratic, or exponential models to describe how the variables are related
• 3.3 data presented in a table, bar graph, histogram, dot plot, box plot, line graph
• Use the relationship between 2 variables to investigate key features of the graph.
• 3.4 Table data
• Use 2-way tables to summarize categorical data and relative frequencies
• 3.5 Scatterplots
• 3.6 Key features of graphs
• 3.7 Linear and exponential growth
• Compare linear growth with exponential growth
• 3.8 Data inferences
• Make inferences about population parameters based on sample data
• 3.9 Center, spread, and shape of distributions
• Use statistics to investigate measures of center of data. Analyze shape, center, and spread
• 3.10 Data collection and conclusions
• Evaluate reports to make inferences, justify conclusions, and determine appropriateness of data collection methods. T
• he reports may consist of tables, graphs, or text summaries.
• 3.11 Probability
•  calculate conditional probability.

### Topic 4. Additional topics in math

• 4.1 geometry
• problems using volume formulas
• 4.2 trigonometry
• trigonometric ratios
• Pythagorean theorem
• problems involving right triangles
• Convert between degrees
• Use radians to determine arc lengths.
• Use trigonometric functions of radian measure
• 4.4 triangles
• Use concepts and theorems about congruence and
• similarity to solve problems about lines, angles, and triangles
• Use the relationship between similarity, right triangles, and trigonometric ratios.
• Use the relationship between sine and cosine of complementary angles
• 4.5 circles.
• Apply theorems about circles to find
• arc lengths,
• angle measures,
• chord lengths, and
• areas of sectors.
• Create or use an equation in 2 variables
• to solve a problem about a circle in the coordinate plane

The SAT Math Test also contains six questions in Additional Topics in Math (three in the no-calculator section and three in the calculator section). They may include topics like geometry, trigonometry, radian measure, and complex numbers.

### SAT Math Section Exam Pattern

The Math section of the SAT is used to analyse the mathematical aptitude of the candidates. With the new changes to the SAT exam pattern, candidates will now be able to use calculators for the entire Math section of an SAT.

 Number of Questions 58 Duration 80 minutes Types of Questions Multiple Choice Questions Questions Heart of Algebra Problem Solving Data AnalysisGeometryTrigonometryComplex Numbers

### Solving linear equations and linear inequalities

• Solve linear equations
• Type: $$3x-0.6=1.8$$ What value of $$x$$ satisfies the equation above
• Type: If $$2x-4=5-x$$ what is the value of $$x$$ ?
• Type: What is the solution to the equation $$\frac{1}{2}x+\frac{1}{5}=\frac{1}{7}$$
• Type: $$-2(x-7)=2$$ what is the value of $$x$$
• Type: $$2x+5y=1$$ and $$x=5$$ what is the value of $$2y$$
• Type: If $$3x-1=5$$, what is the value of $$x-40$$ ?
• Type:  solve for $$\left |2x+1 \right |=10$$
• Solve linear inequalities
• Type: What values of $$x$$ satisfy the inequality  $$5x+1>7$$
• Recognize the conditions under which a linear equation has one solution, no solution, and infinitely many solutions
• Type: $$2x−4=a(x−2)$$  If $$a=2$$, is the equation above, what value of $$x$$ satisfies the equation?

### Understanding linear relationships

• basics of linear relationships.
• writing linear equations based on word problems
• important features of linear functions
• Type: Tobias rented a kayak from a sports equipment store. For the rental, the store charged 60 per day plus $$25$$ for delivery. If Tobias was charged a total of 325, for how many days did he rent the kayak?
• Type: The width of a rectangular vegetable garden is $$W$$ feet. The length of the garden is 16 feet longer than its width. Which of the following expresses the perimeter, in feet, of the vegetable garden in terms of $$W$$ ?

### linear inequalities

• Type : Hammer can harvest at least 48 pounds of honey from her bee colony. If he wants to package the honey harvest in $$1.51$$ pound jars, what is the minimum number of jars he can fill?
• Type : Laila wants to buy at least 40 prizes for rewarding her students throughout the semester. The prize pool will be made of small and large prizes, which cost $3 and$5 each respectively. Her budget for the prizes can be no more than \$100. She wants to buy at least 15 small prizes and at least 5 large prizes. Which of the following systems of inequalities represents the conditions described if x is the number of small prizes and y is the number of large prizes?

### Graphing linear equations

• features of linear graphs from their equations
• Write linear equations based on graphical features
• Determine the equations of parallel and perpendicular lines
• Identify solutions to systems of linear inequalities as regions in the $$xy$$ plane

### Solving systems of linear equations

• solve systems of linear equations algebraically: substitution and elimination.
• systems of linear equations have one solution, no solutions, or infinitely many solutions: graphically

### Syllabus Details : Passport to Advanced Math

• Solve quadratic equations in several different ways
• Determine the number of solutions to a quadratic equation without solving

### Non-linear expressions

• Area of a rectangle
• Height versus time
• Population growth and decline
• Compounding interest

### Exponential expressions

• change time units
• Adding and subtracting exponential expressions
• Multiplying and dividing exponential expressions
• exponent operations to rational exponents

• Rational exponents refer to exponents that can be represented as fractions  $$\frac{1}{3},5,\frac{2}{5}$$ etc..
• Radicals are another way to write rational exponents. For example, $$x^{\frac{1}{2}}\; and\; \sqrt{x}$$ are equivalent.
• Exponent operations to rational exponents
• Equivalent rational and radical expressions

### Operations with polynomials

• multiplying two polynomials

### Polynomial factors and graphs

• factors of polynomial functions to the $$x$$-intercepts of polynomial graphs
• polynomial remainder theorem
• The polynomial remainder theorem states that when a polynomial function $$p(x)$$  is divided by $$x-a$$, the remainder of the division is equal to $$p(a)$$, left parenthesis, a, right parenthesis.
• determine the zeros of a polynomial function
• $$x$$-intercepts, $$y$$-intercept and end behavior

• features of a parabola
• function at several different values of $$x$$
• input-output pairs as points in the $$xy$$-plane.
• Sketch a parabola that passes through the points

### Graphing exponential functions

• Graphing exponential growth & decay
• Using points to sketch an exponential graph
• y-intercept ,
• slope of the graph positive or negative,
• value of y as the value of x becomes very large
• shift the horizontal asymptote
• shift the y-intercept

• graphs of linear and quadratic systems
• number of solutions for linear and quadratic systems
• solve linear and quadratic systems algebraically
• solutions to linear and quadratic systems from graphs

• factor by grouping
• Special factoring
• Square of sum
• Square of difference
• Difference of squares

### Isolating quantities

• Using equations and formulas with multiple variables
• Manipulating formulas
• Translate the word problem or given context into an equation.
• Like solving equations, but with more variables

### Function Notation

• Evaluating functions algebraically and using tables
• Determine inputs and outputs using tables
• Evaluating composite functions algebraically and using tables $$f(g(x))$$

### Ratios, rates, and proportions

• Identify and express ratios
• Part-to-part ratio
• Part-to-whole ratio
• Word problems using proportions
• Applying a per unit rate
• $$speed=\frac{distance}{time}$$
• $$price=\frac{total price}{units purchased}$$

### Percentages

•  percentages using part and whole values
• equivalent forms of percentages
• Calculating a percent value
• Finding complementary percentages
• Switching between forms of percentages
• Calculating percent change

### Unit conversion

• Applying unit to unit ratios
• Converting units within rates

### Table data

• calculate proportions and probabilities
• Use proportions and probabilities to find missing values
• find missing values

### Scatterplots

•  line of best fit to describe scatterplots
• Make predictions using the line of best fit
• Fit functions to scatterplots

### Key features of graphs

• Bar graphs, dot plots, and histograms
• draw line graphs based on verbal descriptions
• line graphs

### Linear and exponential growth

•  two variables have a linear or exponential relationship based on their values
• Linear
• Changes (i.e., increases or decreases) at a constant rate
• Changes by $$c$$ per unit of time.
• Exponential
• Changes by ,$$c%$$ percent of the initial value per unit of time
• Changes by a factor of $$c$$ (e.g., halves, doubles) per unit of time
• real-world scenario exhibits linear or exponential growth
• Writing equations based on tables

### Data inferences

• Estimating using sample proportions
• estimate=sample proportion⋅population
• margin of error
• range=estimate ± margin of error

### Center, spread, and shape of distributions

• meanmedian, and  mode.
• range and standard deviation
• The effect of outliers
• Missing value given the mean

### Data collection and conclusion

• good and bad sampling methods
• Sampling methods and their implications
• Correlation and causality
• Identifying study types
• Sample surveys
•  correlation and causation.

### Probability

• $$Probability=\frac{desired \: outcome}{all \: possible \: outcome}$$
• “odds” of any particular event
• get tails or head if you flip a coin
• Either/Or Probability
• Conditional Probability

### Volume word problems

• volumes and dimensions of three-dimensional solids
• dimension changes affect volume
• Volume of a cone
• Impact of increasing the radius

### Right triangle problems

• Pythagorean theorem
• Calculating missing side lengths in right triangles
• Using trigonometric ratios to find side lengths
• Recognizing Pythagorean triples
• sine, cosine, and tangent of similar triangles
•  sine and cosine of complementary angles

### Congruence and similarity

• Finding angles in triangles
• Triangles, vertical angles, and supplementary angles
• Triangles and parallel lines
•  proportional relationships using similarity

### Length, perimeter, Area, Angles, arc lengths, and trig functions

• Convert between radians and degrees
• Special right triangles in circles
• unit circle
• $$x=rcos\theta =cos\theta$$
• $$y=rsin\theta =sin\theta$$
$$\frac{y}{x}=tan\theta$$

### Circle theorems

• central angle, arc length, and sector area
• $$\frac{central \: angle}{360^o} =\frac{arc \: length}{circumference}=\frac{sector \: area}{circle \: area}$$
• Circumference of a circle $$C=2\pi r$$
• Area of a circle $$A=\pi r^2$$
• Number of degrees of arc in a circle $$360$$

### Circle equations

• Standard form equation of a circle
• In the $$xy$$-plane, a circle with center $$(h, k)$$ and radius $$r$$ has the equation:
• $$(x-h)^2 +(y-k)^2 = r^2$$
• Rewriting circle equations in standard form

### Complex numbers

• square roots of negative numbers
• complex number $$z=a+ib$$
• add and subtract complex numbers
• multiply and divide complex numbers