ACT math

Formula for inverse variation

y = k/x or xy = k

Dot product (used in multiply matrix by matrix)

(2,5) x (3,1) = 2x3 + 5x1 = 6 + 5 = 11 Notice that a dot "product" is the **addition** of products

Equation of a circle

(h, k) = coordinates of the center of the circle r = radius of the circle (x, y) = a point on the circle

Factors of (x^2 - y^2)

(x + y) (x - y)

Equation of a circle illustrated

(x - h)^2 + (y - k)^2 = r^2 (X^2 means x-squared) (h, k) = coordinates of the center of the circle r = radius of the circle (x, y) = a point on the circle

Whole number

0, 1, 2, 3, .... A whole number is a positive number with no fractional or decimal part. Source: https://www.mathsisfun.com/definitions/whole-number.html

Combination

A collection of things, in which the order does not matter. Example: Your meal comes with 2 side dishes. The menu offers French fries, sautéed spinach, and cole slaw. How many different combinations of 2 sides are there? Answer: 3 French fries and spinach French fries and cole slaw Spinach and cole slaw Source: https://www.mathsisfun.com/definitions/combination.html

Complex number

A combination of a real and an imaginary number in the form a + bi a and b are real numbers, and i is the "unit imaginary number" √(−1) The values a and b can be zero. These are all complex numbers: 1 + i, 2 − 6i, −5.2i, 4 http://www.mathsisfun.com/definitions/complex-number.html

Rational number

A number that can be made by dividing two integers (includes infinitely repeating numbers like 1/3)

Prime number

An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself Source: https://primes.utm.edu/notes/faq/one.html

Area of a rectangle

Base x height

Temperature water freezes, in degrees Celsius & degrees Fahrenheit

Celsius: 0 degrees Fahrenheit: 32 degrees

Factor

Factors are numbers we can multiply together to get another number 5 x 2 = 10 5 and 2 are factors of 10 Source: https://www.mathsisfun.com/definitions/factor.html

Geometric versus arithmetic sequence

Geometric sequence - multiplication - sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3 - Wikipedia Arithmetic progression (AP) or arithmetic sequence - addition - is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with common difference of 2 -Wikipedia

Probability

How likely something is to happen When you toss a coin, what is the probability it will be heads? Answer: 0.5 or 50% Source: https://www.mathsisfun.com/data/probability.html

Combinations & permutations

If the order doesn't matter, it is a combination. (groups) If the order does matter, it is a permutation. (ordered lists, like passwords) With the same initial set of objects, there will be more permutations than combinations. http://www.freelearningchannel.com/l/index.php?s=lesson&c=./Content/Courses/Mathematics/USCommonCore/Math_206_1/Sessions/&f=2.2_Order_and_Probability&a=r&i=1

Probability formula

If you're rolling a die and you need either a a 5 or a 6, what is the probability that you will roll a 6? (6 is the favorable outcome) Answer: 2/6 (2 favorable outcomes out of 6 possible outcomes) Source: http://study.com/academy/lesson/classical-probability-definition-approach-examples.html

Find the least common multiple by listing multiples (usually slower)

Least common multiple of 3 and 5 Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 .... Multiples of 5: 5, 10, 15, 20, 25, 30 ... 3 and 5 both go into 15 3 and 5 both go into 30 3 and 5 both go into 60 15 is the least common multiple

Side lengths of 30-60-90 triangle

Length of sides: a a √3 2a

Matrix calculations

MATRIX ADDITION: add the numbers in each location. Both arrays must have the same dimensions. MATRIX MULTIPLICATION BY A SCALAR: multiply each value by the scalar number. MATRIX MULTIPLICATION: when you multiply two matrices, the result will have as many rows as the first matrix and as many columns as the second matrix. A 2-row by 3-column matrix times a 3-row by 4-column matrix will have 2 rows and 4 columns. AND: the middle numbers (3s in this case) must match or the matrices can't be multiplied. Brian Leaf Khan Academy https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-scalars/a/multiplying-matrices-by-scalars Multiplying matrices by matrices https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-matrices/v/matrix-multiplication-intro

Logarithms

Read "logarithm" as "exponent" "2 raised to what exponent yields 8" (answer: 3)

SOHCAHTOA

Sine = opposite / hypotenuse Cosine = adjacent / hypotenuse Tangent = opposite / adjacent

Find the length of diagonal in parallelogram (74C - #30)

Solutions to #29 & #30 on 74C - ACT - June 2017

What is "the famous log loop"?

Source: http://www.shelovesmath.com/algebra/advanced-algebra/logarithmic-functions/#IntroductiontoLogarithms

Distance formula illustrated

Source: http://keywordsuggest.org/gallery/805320.html

Find the least common multiple by factoring each number (usually faster)

Source: http://math-in-the-middle.com/2015/10/12/5-ways-to-use-factor-trees-in-middle-school-math/

Logarithm laws

Source: http://thecollegepanda.com/books/

Least common multiple

The *smallest* positive number that is a multiple of two (or more) numbers Least common multiple of 2 and 5 is 10 because that is the smallest number both numbers go into evenly Source: https://www.mathsisfun.com/definitions/least-common-multiple.html

How is the formula for direct variation related to the formula of a line?

The direct variation formula is the formula for a line with y-intercept of 0. line: y = mx + b direct variation: y = mx + 0 Source of image: Texas Gateway

What does the equation of a circle mean?

The equation of a circle is a way to express the definition of a circle on the coordinate plane. If the center of the circle is at the origin of the coordinate plane, the equation is r^2 = x^2 + y^2 where r is the radius. http://www.chegg.com/homework-help/definitions/equations-of-circles-63

Greatest common factor

The highest number that divides exactly into two or more numbers. It is the "greatest" thing for simplifying fractions! Source: https://www.mathsisfun.com/greatest-common-factor.html

Least common multiple of 2 numbers (74C - #47)

The smallest number *both* numbers go into OR: The smallest number that can be evenly divided by both numbers OR: The smallest number both numbers can be multiplied to obtain

Natural number

The whole numbers from 1 upwards: 1, 2, 3, and so on

Slope of vertical line

Undefined The slope of a vertical line is undefined because division by zero is undefined Example: x = 2 The line for x = 2 includes (2, 1) and (2,10) Formula: (10 - 1)/(2 - 2) = 9/0

Permutation formula

n = n objects r = taken 5 at a time

Combination formula

n = total number of items r = number of items to be chosen ! = factorial Source: http://statistics.about.com/od/Formulas/ss/Formulas-For-Combinations-And-Permutations.htm#step2

A 2 x 3 matrix times a 3 x 1 matrix will have how many values? (2 rows x 3 columns; 3 rows by 1 column)

two

Formula for direct variation

y = kx NOTE: this is the formula for a line with y-intercept = 0 "When two variables are related directly, the ratio of their values is always the same. So as one goes up, so does the other, and if one goes down, so does the other. Think of linear direct variation as a "y = mx" line, where the ratio of y to x is the slope (m). With direct variation, the y-intercept is always 0 (zero); this is how it's defined." http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/

Formula for Celsius to Fahrenheit & Celsius to Fahrenheit (#29, Test 74H)

C = 5/9(F - 32) F = 5/9C - 32

Fundamental counting principle

Counting by multiplication, not addition (1, 2, 3...) When there are m ways to do one thing, and n ways to do another, then there are m x n ways of doing both. You have 2 pairs of shoes & 3 pairs of socks. That means you have 2 x 3 = 6 combinations White socks, black socks White shoes, black shoes, brown shoes White socks, white shoes White socks, black shoes White socks, brown shoes Black socks, white shoes Black socks, black shoes Black socks, brown shoes (If you tried to count by addition -- 1 + 1 + 1 + 1 + 1, or 1, 2, 3, 4, 5 -- you would get the wrong answer.) Source: https://www.mathsisfun.com/data/basic-counting-principle.html

Simple formula Celsius to Fahrenheit & back NOTE: Use the real formula on the test (but this could be a quick answer check).

Fahrenheit to Celsius (you know degrees Fahrenheit): Subtract 30 from the Fahrenheit temp. and divide by 2 ex: 72 degrees Fahrenheit is equivalent to 21 by this method Celsius to Fahrenheit (you know degrees Celsius) Multiply degrees-Celsius by 2, then add another 30 ex: 22 C is roughly equal to 72 degrees using my method, actual conversion is 74 F = 9/5C + 32 C = 5/9(F - 32) 72 degrees F = 22.2222 degrees Celsius

Is 1 a prime number?

Nope An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. How to remember: "1 is the loneliest number" Sources: https://primes.utm.edu/notes/faq/one.html https://www.youtube.com/watch?v=IQofiPqhJ_s

What is the row-to-column rule for matrix multiplication?

Number of columns in 1st matrix must equal number of rows in second matrix (2 rows x 3 columns) x (3 rows x 3 columns) The 2 "inside" numbers are the same, so the 2 matrices can be multiplied The product of the 2 outside numbers tells you the dimensions of the new matrix (2 x 3 = 6) https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-matrices/v/multiplying-a-matrix-by-a-matrix Matrix multiplication (Math Meeting) https://www.youtube.com/watch?v=kuixY2bCc_0

Consecutive numbers

Numbers which follow each other in order, without gaps, from smallest to largest. 12, 13, 14 and 15 are consecutive numbers. 22, 24, 26 and 28 are consecutive even numbers. Source: https://www.mathsisfun.com/definitions/consecutive-numbers.html

Side-ratio & perimeter problems When you're given: - the length of one side, - the perimeter, and - the ratio of 2 sides

PROBLEM: A triangle with a perimeter of 89 has one side that is 19 inches long. The lengths of the other two sides have a ratio of 3:7. What is the length, in inches, of the longest side of the triangle? (Brian Leaf) Answer: 49 The two unknown sides can be represented as 3x and 7x because 3x/7x = 3/7

Expression for number of permutations of 15 objects taken 5 at a time

Practice Test 2 | Official ACT Prep Guide 2016-2017

Possible future cards - URLs

Ratio problems with triangle perimeters: YouTube lesson: https://www.youtube.com/watch?v=7bK26eXbTgs Purple Math: http://www.purplemath.com/modules/ratio6.htm Rules of exponents & radicals http://slideplayer.com/slide/4752256/ http://www.solitaryroad.com/c629.html http://www.dummies.com/education/math/algebra/how-to-distribute-with-fractional-powers-or-radicals/ Radians https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func Trigonometry https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-sines/v/law-of-sines Law of sines https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-sines/v/law-of-sines Law of cosines https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-example Permutations & combinations https://betterexplained.com/articles/easy-permutations-and-combinations/

Pythagorean theorem

Source: https://mathblog.com/reference/theorems/pythagorean-theorem/

Domain and range

Source: https://www.texasgateway.org/resource/determining-domain-and-range-quadratic-functions-0

i

Square root of negative 1 Source: https://www.codeproject.com/

Start with: 12 women & 18 men. Choose a committee of 5 women & 5 men. - Is this a combination or permutation? - What expression gives the # of different committees that could be selected from these 30 people?

The "C" stands for combination. The order in which the two groups (1 group of women, 1 group of men) are chosen does not matter. The subscript number to the left of the C is the number of men or women you're choosing from. The subscript number to the right of the C is the number of men or women you will choose. The two values--the # of combinations of women and the # of combinations of men--are multiplied together to find the answer.

Real number

The type of number we normally use, such as 1, 15.82, −0.1, 3/4, etc. Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers. They are called "Real Numbers" because they are not Imaginary Numbers. http://www.mathsisfun.com/definitions/real-number.html

Radian (t/k)

The word radian comes from the word radius, and is just another way to measure angles in a "trig circle". One radian means that the length of the radius of that circle is the same as the arc length; and we like to deal with "trig circles" where the radius is 1. http://www.shelovesmath.com/trigonometry/the-unit-circle/ Khan Academy https://www.khanacademy.org/math/algebra2/trig-functions/intro-to-radians-alg2/v/introduction-to-radians Math Is Fun https://www.mathsisfun.com/geometry/radians.html Better Explained https://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/

Area of a triangle

Area of a triangle = 1/2 x b x h (1/2)(base x height) one-half base x height

2 ways to solve a direct variation problem

1. Use a proportion (x1/y1 = x2/y2) 2. Use the direct variation formula (y = kx)

Cake method (GCF & LCM)

1. Write the numbers for which you are trying to find the GCF & LCM next to each other & draw a "layer of cake" around them. 2. Find ANY common factor of the two numbers & write that factor to the left. 3. Divide the numbers by their common factor & write the quotients underneath the numbers. 4. Repeat process with the 2 new numbers. 5. Repeat process as many times as necessary until the 2 numbers at the bottom are 'relatively' prime. 6. To find the greatest common factor, multiply all the numbers to the left of the cake. 7. To find the LCM, draw a big "L" around the cake, and multiply all the numbers in the "L" together. Source: http://math-in-the-middle.com/2014/10/09/finding-the-gcf-and-lcm-using-the-cake-method/

How many degrees in a rectangle

360 (same as a square)

Translate "scale factor" into equation

70G #35 "the scale factor of triangle ABC to triangle DEF is 2/3" ABC/DEF = 2/3

How many ounces in a cup? How many cups in a pint? How many ounces in a pint? Pints in a quart? Quarts in a gallon? Cups in a quart? Cups in a gallon?

8 ounces in a cup 2 cups in 1 pint 16 ounces in a pint 2 pints in 1 quart 4 quarts in 1 gallon 4 cups in a quart 16 cups in a gallon

Multiple

A number that can be divided by another number without a remainder 20 is a multiple of 5 because 20 can be evenly divided by 5 The result of multiplying a number by an integer (not by a fraction) Multiples of 3: 1 x 3 = 3 2 x 3 = 6 3 x 3 = 9 4 x 3 = 12 3, 6, 9, and 12 are multiples of 3 Source: https://www.mathsisfun.com/definitions/multiple.html

Imaginary number

A number that when squared gives a negative result. When we square a Real Number (multiply it by itself) we always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 as well. So how can we square a number and get a negative result? Because we "imagine" that we can. The "unit" imaginary numbers (the same as "1" for Real Numbers) is √(-1) (the square root of minus one), and its symbol is i, or j. http://www.mathsisfun.com/definitions/imaginary-number.html

Integer

A number with no fractional part (includes 0) Includes: • the counting numbers {1, 2, 3, ...}, • zero {0}, • and the negative of the counting numbers {-1, -2, -3, ...} Source: http://www.mathsisfun.com/definitions/integer.html

Permutation

A permutation is an ordered combination Example: passwords ABC is a different password than CBA There are 2 kinds of permutation problems: - repetition is allowed (AAB is an acceptable password) - repetition is not allowed (AAB is not an acceptable password) Source: https://www.mathsisfun.com/combinatorics/combinations-permutations.html

Odds (not tested on ACT or SAT as far as I know)

A ratio of the number of ways something can occur to the number of ways it cannot occur. The odds of a month being August are 1 : 11 (August vs all 11 other months) The odds of a coin landing heads up is 1 : 1 (number of ways it can land heads up/number of ways it cannot land heads up) Source: https://www.mathsisfun.com/definitions/odds.html

Irrational number

A real number that can NOT be made by dividing two integers (an integer has no fractions) A real number's decimal goes on forever without repeating Pi is an irrational number

Fundamental theorem of arithmetic

"Every positive whole number can be written as a unique product of primes" If 1 is *not* a prime number, then 15 is a unique product: 3 x 5 If 1 *is* a prime number, then 15 can be written as: 1 x 3 x 5 1 x 1 x 3 x 5 1 x 1 x 1 x 3 x 5 1 is in a category by itself, neither a prime nor a composite: "1 is the loneliest number" Source: 1 and Prime Numbers - Numberphile https://www.youtube.com/watch?v=IQofiPqhJ_s

Multiply a matrix by matrix

Matrix A 1 7 2 4 Matrix B 3 3 5 2 Matrix A x B 1x3+7x5 1x3+7x2 2x3+4x5 2x3+4x2 Matrix C 38 17 26 14 https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-matrices/a/multiplying-matrices

Midpoint formula

Midpoint formula at Purple Math: http://www.purplemath.com/modules/midpoint.htm

Distance formula

NOTE: the value inside the radical = c in the Pythagorean theorem http://www.purplemath.com/modules/distform.htm

Slope of perpendicular lines

Negative reciprocal When we multiply their slopes, we get -1 https://www.mathsisfun.com/algebra/line-parallel-perpendicular.html