Math Rules for Digital SAT
1. Avoid algebraic operations whenever possible. Instead, plug in values for the
variables and solve.
2. Use the Desmos calculator whenever possible. To solve for an unknown variable, put
all the terms on one side of the equation and graph that equation: the solution(s) will be
the x-intercepts of the graph. For systems of equations, graph both equations, and the
solution(s) will be at the intersection of the lines.
3. For Geometry problems, always draw a picture of the figures described in the
question! You have to conceptualize the question the right way first before you can
solve it! Also consider using visual estimation whenever possible.
4. Give yourself some Freebies™ (high-level difficulty questions that you can skip with a
smile!). For most students, start with 2 Freebies™ on Module 1 and 6 Freebies™ on
Module 2. You can always come back to these questions if you have time, but you NEED
to focus on the easy/intermediate-level questions you can get for sure. If you have no
idea how to solve a difficult problem (and you can’t eliminate any incorrect answer
choices), just pick “A”.
Basic Concepts Cheat Sheet:
A.
Functions
● Understand 𝑥 as the input,𝑓(𝑥) as the output. Also understand 𝑓(𝑥) as “fancy 𝑦”
(they are interchangeable when graphing on an x-y axis).
● For 𝑝(3) = 0, (x - 3) must be a factor of 𝑝(𝑥).
For p(-7) = 0, (x+7) must be a factor of p(x).
● For f(g(x)), work inside-out. Solve for g(x) first, then plug in that value for f(x)
B. Exponents
● Remember that x² will always be positive.
● If dividing (with a like base), subtract bottom exponent:
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More properties:
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For fractional exponents, turn the fraction into a flower: flower=power/root=root
(numerator=power/denominator=root)
C. Graphing
● If you can graph it, graph it.
● For 𝑦 = 𝑚𝑥 + 𝑏 , 𝑚 is slope (rise/run), 𝑏 is the y-intercept.
● ALWAYS PAY ATTENTION TO THE UNITS ON THE X-Y AXIS!!! They are not
always intuitive (One square does not necessarily represent one unit). When
calculating slope, it’s not the number of squares, it’s the number of UNITS.
● Parallel lines have the same slope.
● A Perpendicular line has the negative inverse slope.
● For the line 𝑦 = 2𝑥 + 5 shifted down 4 units, the new line is 𝑦 = 2𝑥 + 1. If the
line is shifted up 4 units, the new line is 𝑦 = 2𝑥 + 9
● The y value at the x-intercept is always 0. The x value at the y-intercept is 0.
● For parabolas, the x value of the vertex is always the halfway point between the
two x-intercepts.
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For parabolas in 𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 form, if the “a” value is positive, the parabola
opens up (smiley face). But if the “a” value is negative, the parabola goes down
(frowny face).
Quadrants:
D. Word Problems
● If you can draw a picture, draw a picture.
● Use basic logic whenever possible, especially to eliminate incorrect answer
choices.
E. Systems of equations
● If they say “ the system has infinite solutions” then the two equations are equal.
● If a linear system has “no solutions” the lines are parallel - they have the same
slope!
● Solve by plugging values in place of the variables if possible.
● Solve systems of equations word problems by testing values.
● You can also use the Desmos Graphing Calculator to solve systems of
equations.
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F. Matching Coefficients
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For 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 15𝑥 + 41𝑥 + 14 → 𝑎 = 15, 𝑏 = 41, 𝑐 = 14
G. Trigonometry
● Use SOHCAHTOA for Trig. 𝑠𝑖𝑛(𝑥) = 𝑜𝑝𝑝/ℎ𝑦𝑝, 𝑐𝑜𝑠(𝑥) = 𝑎𝑑𝑗/ℎ𝑦𝑝,
𝑡𝑎𝑛(𝑥) = 𝑜𝑝𝑝/𝑎𝑑𝑗
● (90°- x°) is code for “the remaining angle in a right triangle” — the one that is not
90° or x°
● Look out for 3,4,5 right triangles! (or multiples of 3,4,5 like 6,8,10 or 9,12,15) and
5, 12, 13 right triangles!
H. Statistics
● Negative association = when one variable increases, the other decreases
Positive association = when one variable increases, the other increases
● Mean = classic average, Median = middle number, Mode = most frequently
occuring number, Range = difference between highest and lowest number
● Low Standard Deviation - More values in a set are close to the average
High Standard Deviation - More values in a set are away from the average
● Avoid answer choices with absolute statements (this MUST be the case)
I. Absolute Value
● Calculate what’s inside the bars. If it’s positive, keep it positive. If it's negative,
MAKE it positive. |3 − 5| = 2
J. Plug in points
● Whenever they say “This point (0,0) or (a,b) is on this line or is a solution to this
system of equations,” plug in those values for x and y!
K. Inequalities
● follow the same rules as equations, except when multiplying or dividing by a
negative - REVERSE THE SIGN!
● When graphing a system of inequalities, the solution to the system is where both
sets OVERLAP!
L. Probability
● Probability = winners/(total # of possible outcomes)
M. Ratios
● 1:2 = ½ = 0.5
N. Equations You Need to Know - Equation of a circle, Vertex form of a parabola,
Quadratic Formula, Difference of Squares
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O. Percent Increase and Decrease
● 25% increase of 𝑥 = 𝑥(1. 25) 25% decrease of 𝑥 = 𝑥(. 75)
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$100 invested at a 7% increase compounded annually is 100(1. 07)
If A is 20% more than B, then A = B(1.2)
𝑡
P. Radians and Degrees
𝛑
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𝛑 radians = 180°, 2𝛑 radians = 360°, 2 radians = 90°
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Go from radians to degrees Go from degrees to radians -
Q. Quadratic Equations
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Always simplify quadratic equations, and put in 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 form BEFORE
using the quadratic formula
For y = (x - 2)(x - 4) the “roots”, aka “zeros”, aka “x-intercepts” are 2 and 4.
For parabolas, the x value of the vertex is always the halfway point between the
two x-intercepts.
−𝑏
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The x value of the vertex can be calculated with the expression 2𝑎 taken from
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the form 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0.
To find the number of solutions to a quadratic equation (or a system), use the
2
2
2
discriminant 𝑏 − 4𝑎𝑐 for the form 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0.
2
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If 𝑏 − 4𝑎𝑐 is positive, there are 2 real solutions.
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If 𝑏 − 4𝑎𝑐 is zero, there is one real solution.
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If 𝑏 − 4𝑎𝑐 is negative, there are no real solutions.
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R. Conversion Units
1.5 hours into minutes: 1. 5 ℎ𝑟 ×
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only minutes!
Put the unit you want to go into in the numerator. The unit you want to go away
from in the denominator!
S. F.O.I.L. Method - First, Outer, Inner, Last
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60 𝑚𝑖𝑛
1 ℎ𝑟
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Hours cancel out, and you are left with
T. Pythagorean Theorem
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2
2
● 𝑎 +𝑏 =𝑐
U. Alternate Interior Angles
P.E.M.D.A.S. Order of Operations! - When simplifying expressions, simplify in this order:
● Parenthesis - calculate what is inside the parentheses FIRST, before doing
anything else
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Exponents - calculate anything raised to a power (for example, ((− 9) = 81)
Multiplication
Division
Addition
Subtraction
Glossary of Important Terms:
Equivalent: equal
Origin: (0,0) on the x-y axis
Undefined: means you get a zero in the denominator
Negative association: when one variable increases, the other decreases
Positive association: when one variable increases, the other increases
Mean: classic average
Median: middle number
Mode: most frequently occuring number
Range: difference between highest and lowest number
Constant: known or unknown value that does not change - e.g. the b value in y = mx+b
Variable: value that changes - e.g. x or y values in y = mx + b
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Solution Set: All possible solutions to an equation. For 𝑥 = 16 , the solution set is {4, -4}
Roots: values for x that give you an output of 0 for y or f(x)
Zeros: values for x that give you an output of 0 for y or f(x) same as a root.
X-intercepts: where a line crosses the x-axis. Also values for x that give you an output of 0 for y
or f(x)
Slope: rise/run or “rate of change”
𝑥
Exponential Growth: 𝑦 = 2 . Will look like this:
Linear Growth: 𝑦 = 2𝑥. Will look like this:
Equilateral Triangles: All sides are equal and all angles are 60 degrees.
Margin of Error: An expression of confidence in the accuracy of survey results. If the margin of
error is 2, the actual values are likely to be +2 or -2 from the survey results.
Similar Triangles: All angles are the same (though lengths of sides may be different)
Congruent: Same length or size
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