ACT Math Summary Sheet
STRATEGIES:
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DEFINITIONS:
Read carefully. Underline the question.
Can you use the answers to plug in?
Quicksolve when answers are variables
Eliminate (cross out!) wrong answers
Skip it if you are not sure. Come back later!
%=
𝑝𝑎𝑟𝑡
𝑤ℎ𝑜𝑙𝑒
○
MA → 𝒙𝒂 ⋅ 𝒙𝒃 = 𝒙𝒂+𝒃
𝒃
○
DS → 𝒙 ÷ 𝒙 = 𝒙
○
(𝒙𝒂 )𝒃
○
PM→
=𝒙
𝑷𝒓𝒐𝒃𝒂𝒃𝒍𝒊𝒕𝒚 =
𝑛𝑒𝑤 # − 𝑜𝑙𝑑 #
𝑜𝑙𝑑 #
Median: Line them up, find the # in the number in the
middle
 Cross off hi-low, hi-low, until one or two left
 If two, take the average
Mode: # appears most often
Range: max – min
Standard Deviation: use range to approximate
MADSPAM
𝒂
Smallest prime # is 2
Distinct = different = unique
Congruent = same size (could be flipped)
Similar means same proportions/angles
PEMDAS
% 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒(𝑀𝑒𝑎𝑛)
𝑡𝑜𝑡𝑎𝑙
=
# 𝑡ℎ𝑖𝑛𝑔𝑠
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𝒂−𝒃
𝒂𝒃
𝒙−𝒎 =
𝟏
○
𝒙𝒎
○
𝟏
𝟐
𝒙 = √𝒙
𝒙𝟎 = 𝟏
𝒘𝒉𝒂𝒕 𝒚𝒐𝒖 𝒘𝒂𝒏𝒕
𝒘𝒉𝒂𝒕 𝒚𝒐𝒖 𝒈𝒐𝒕
Proportions
𝒙𝟏
𝒚𝟏
=
𝒙𝟐
𝒚𝟐
cross multiply -> 𝒙𝟏 𝒚𝟐
= 𝒙𝟐 𝒚 𝟏
*be careful of units! They must match
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵)
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑛!
𝑛𝐶𝑟 = 𝑟!(𝑛−𝑟)! (order doesn’t matter, bowl of ice cream)
𝑛𝑃𝑟 =
𝑛!
(𝑛−𝑟)!
(order matters, ice cream cone)
Lines
slope-intercept formula: y = mx +b
point-slope formula: y-y1 = m (x – x1)
standard form: Ax+ By = C
distance formula is the Pythagorean Theorem in disguise! 𝒅 = √(𝒙𝟏 − 𝒙𝟐 )𝟐 + (𝒚𝟏 − 𝒚𝟐 )𝟐
𝒙 +𝒙
𝒚 +𝒚
midpoint formula ( 𝟏 𝟐 𝟐 , 𝟏 𝟐 𝟐 ) (averages!)
𝑟𝑖𝑠𝑒
m = slope, 𝑟𝑢𝑛 , you can calculate using the formula
𝒚𝟏 −𝒚𝟐
𝒙𝟏 −𝒙𝟐
b = y-intercept, where the line crosses the y-axis (x = 0)
y can also be written as f(x)
parallel lines have the same slope and different y-intercept
perpendicular lines have slopes that are negative reciprocals (e.g. 2 and -1/2)
Quadratics / Parabolas
Circles
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐
−𝒃 ±
√𝒃𝟐 −𝟒𝒂𝒄
𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 =
𝟐𝒂
𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌, 𝒗𝒆𝒓𝒕𝒆𝒙 = (𝒉, 𝒌)
𝒚 = (𝒂𝒙 + 𝒃)(𝒄𝒙 + 𝒅), 𝒓𝒐𝒐𝒕𝒔 =
−𝒃 −𝒅
𝒂
,
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝑺𝒆𝒄𝒕𝒐𝒓 =
𝒄
(𝒂𝟐 − 𝒃𝟐 ) = (𝒂 + 𝒃)(𝒂 − 𝒃)
(𝒂𝟐 + 𝒃𝟐 ) = (𝒂 + 𝒃𝒊)(𝒂 − 𝒃𝒊)
1
Triangle= 2base X height
Circle = 𝜋 𝑟 2
Perimeter
Add up all the side lengths
Circle (circumference)= 2𝜋 𝑟
SOH CAH TOA
𝑠𝑖𝑛2 + 𝑐𝑜𝑠 2 = 1
𝑠𝑖𝑛𝜃
𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃
𝑠𝑖𝑛−1 (𝑥) = 𝜃
𝒄𝒆𝒏𝒕𝒓𝒂𝒍 𝒂𝒏𝒈𝒍𝒆
𝟑𝟔𝟎 (𝒐𝒓 𝟐𝝅)
× 𝝅𝒓𝟐
Ellipses
(𝒙−𝒉)𝟐
𝒂𝟐
Area
Square= 𝑠 2
Rectangle= length x width
Parallelogram= base x height
𝑏 +𝑏
Trapezoid = 1 2 2 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡
𝒄𝒆𝒏𝒕𝒓𝒂𝒍 𝒂𝒏𝒈𝒍𝒆
× 𝟐𝝅𝒓
𝟑𝟔𝟎 (𝒐𝒓 𝟐𝝅)
+
(𝒚−𝒌)𝟐
𝒃𝟐
=𝟏
Triangles
 1800 in a triangle
 Equilateral → all sides are the
same, all angles = 600
 Isosceles → 2 sides are the same,
2 angles are the same
 Scalene → all sides different, all
angles different
 Right Triangles 𝑎2 + 𝑏 2 = 𝑐 2
 Common Triples
3,4,5 * 5,12,13 * 7,24,25
𝑖 = √−1
𝑖 2 =-1
𝑖 3 = −𝑖
𝑖4 = 1
*use your calculator! Above the “.”
𝑙𝑜𝑔𝑏 𝑥 = 𝑛 → 𝑏 𝑛 = 𝑥
math-alpha-math