CSET Multiple Subjects: Subtest 2 - Math

Changing fractions to decimals

-Divide -Insert decimal points and zeros accordingly Example: 13/20 = 13 ÷ 20 = .65

Adding mixed numbers

Find the LCD and add the whole numbers to get your final answer. Example: 2 1/2 + 3 1/4 = ? 2 1/2 + 3 1/4 = 2 2/4 + 3 1/4 = 5 3/4

Subtracting fractions

Find the LCD and subtract the numerators. Example: 7/8 - 1/4 = ? 7/8 - 1/4 = 7/8 - 2/8 = 5/8

Factoring

Finding two or more quantities whose product equals the original quantity

1 meter (in yards)

1.1 yards

1 gram (in ounces)

1/30 ounce

1 kilometer (in miles)

about 0.6 mile

1 =

1.00 = 100%

Percentage increase or decrease

(change/starting point) x 100 = percentage change Example: What is the percentage decrease of a $500 item on sale for $400? Change: 500-400 = 100 (100/500) x 100 = 1/5 x 100 = 20% decrease

Rational numbers

-All values that can be expressed in the form a/b, where a and b are integers and b ≠ 0 -Or, when expressed in decimal form, the expression either terminates or has a repeating pattern Examples: 4 1/2 = 9/2; therefore, 4 1/2 is a rational number 0.3 is a terminating decimal; therefore, 0.3 is a rational number 0.134343434... is a repeating decimal; therefore, 0.134343434 is a rational number

Some properties/axioms of addition and multiplication

-Communicative property -Associative property -Distributive property

Changing percents to fractions

-Divide the percent by 100 -Eliminate the percent sign -Reduce if necessary Example: 60% = 60/100 = 3/5

Changing percents to decimals

-Eliminate the percent sign -Move the decimal point two places to the left (sometimes adding zeros will be necessary) Example: 5% = .05

Simplifying square roots

-Factor the number into two numbers, one (or more) of which is a perfect square -Take the square root of the perfect square(s) -Leave the others under the √ Example: Simplify √75 √75 = √(25 x 3) = √25 x √3 = 5√3

Other applications of a percent

-For "what," substitute the letter x -For "is," substitute an equal sign -For "of," substitute a multiplication sign -Change percents to decimals or fractions and solve the equation Example: 18 is what percent of 90? 18 = x(90) --> 18/90 = x --> 1/5 = x --> 20% = x

Customary/English system of units

-Length (in., ft., yd, mi.) -Area (sq. in., sq. ft., sq. yd., acre, sq. mi.) -Weight (oz., lb., T) -Capacity (cups, pt., qt., gal., pecks, bushels) -Time (seconds, minutes, hours, days, weeks, months, years, decades, centuries)

Metric/international system of units

-Length (meter) -Volume (liter) -Weight (gram)

Adding and subtracting decimals

-Line up the decimal points and then add or subtract in the same manner you would add or subtract regular numbers -Adding in zeros can make the problem easier to work with -Whole numbers can have decimal points to their right

Changing decimals to percents

-Move the decimal point two places to the right -Insert a percent sign Example: .75 = 75%

Multiplying decimals

-Multiply as usual -Count the total number of digits above the line which are to the right of all decimal points -Place your decimal point in your answer so there is the same number of digits to the right of it as there was above the line Example: 40.012 (3 digits) x 3.1 (1 digit) = 124.0372 (4 digits)

Changing fractions to percents

-Multiply by 100 -Insert a percent sign Example: 2/5 --> 2/5 x 100 --> 200/5 --> 40%

Number line

-Numbers to the right of 0 are positive -Numbers to the left of 0 are negative -Given any two numbers on a number line, the one on the right is always larger, regardless of its sign (positive or negative)

Changing decimals to fractions

-Read it -Write it -Reduce it Example: .8 = eight-tenths = 8/10 = 4/5

1 liter (in quarts)

1.1 quarts

Multiplying or dividing signed numbers

-The product or quotient of two numbers with the same sign will produce a positive answer -The product or quotient of two numbers with the opposite signs will produce a negative answer Example: (-3)(8)(-5)(-1)(-2) = 240

Dividing decimals

-The same as dividing other numbers, except that if the divisor (the number you're dividing by) has a decimal, move it to the right as many places as necessary until it is a whole number -Move the decimal point in the dividend (the number being divided into) into the same number of places -Sometimes you may have to add zeroes to the dividend (the number inside the division sign) Example: 5 ÷ 1.25 = 500 ÷ 125

Parentheses

-Used as grouping symbols -When possible, everything inside parentheses should be evaluated first before doing any other operations -If there are parentheses inside parentheses, start with the most inside parentheses first and work your way out Example: [4 - (11 - 15)] = [4 - (-4)] = 4 + 4 = 8

Adding signed numbers

-When adding two numbers with the same sign (either both positive or both negative), add the pure number portions (absolute values) and keep the sign that is on the numbers Example: (-8) + (-3) = -11 -When adding two numbers with different signs (one positive and the other negative), subtract the absolute values and keep the sign on the number with the larger absolute value. Example: -59 + (72) = 13

Integers

...-3, -2, -1, 0, 1, 2, 3... all the whole numbers together with their opposites

1/100 =

.01 = 1%

1/10 =

.1 = .10 = 10%

1/8 =

.125 = .12 1/2= 12 1/2%

1/6 =

.16 2/3 = 16 2/3%

3/10 =

.3 = .30 = 30%

1/3 =

.33 1/3 = 33 1/3%

3/8 =

.375 = .37 1/2 = 37 1/2%

5/8 =

.625 = .62 1/2 = 62 1/2%

2/3 =

.66 2/3 = .66 2/3 = 66 2/3%

7/10 =

.7 = .70 = 70%

5/6 =

.83 1/3 = 83 1/3%

7/8 =

.875 = .87 1/2 = 87 1/2%

9/10 =

.9 = .90 = 90%

Whole numbers

0, 1, 2, 3, 4,... the natural numbers together with 0

Identity number for addition

0. Any number added to 0 gives that number 0 + a = a + 0 = a Example of using the additive identity: 0 + 3 = 3 + 0 = 3 0 + 3 = 3

Natural numbers

1, 2, 3, 4,... the numbers you would naturally count by 0 is not a natural number.

1 metric ton (in kilograms)

1,000 kilograms

1 mile (in yards, feet)

1,760 yards 5,280 feet

Identity number for multiplication

1. Any number multiplied by 1 gives that number. 1(a) = a(1) = 1 Example of using the multiplicative identity: 1(3) = 3(1) = 3 1(3) = 3

1 foot (in inches)

12 inches

1 square foot (in square inches)

144 square inches

1 pound (in ounces)

16 ounces

1 pint (in cups)

2 cups

1 quart (in pints)

2 pints

1 ton (in pounds)

2,000 pounds

2 =

2.00 = 200%

1/5 =

2/10 = .2 = .20 = 20%

1/4 =

25/100 = .25 = 25%

1 yard (in feet, inches)

3 feet 36 inches

3 1/2 =

3.5 = 3.50 = 350%

1 bushel (in pecks)

4 pecks

1 gallon (in quarts)

4 quarts

2/5 =

4/10 = .4 = .40 = 40%

1/2 =

5/10 = .5 = .50 = 50%

3/5 =

6/10 = .6 = .60 = 60%

3/4 =

75/100 = .75 = 75%

4/5 =

8/10 = .8 = .80 = 80%

1 square yard (in square feet)

9 square feet

Common fraction

A fraction whose numerator is smaller than its denominator; has a value less than 1. Example: 3/5

Improper fraction

A fraction whose numerator is the same or more than the denominator; has a value equal to 1 or more than 1. Examples: 6/6 and 5/4

Composite number

A natural number greater than 1 that is not a prime number (an alternate definition is a natural number that has exactly two different divisors) The first seven composite numbers are 4, 6, 8, 9, 10, 12, and 14

Prime number

A natural number greater than 1 that only has 1 and itself as divisors (an alternate definition is a natural number that has exactly two different divisors) The first seven prime numbers are 2, 3, 5, 7, 11, 13, and 17

Scientific notation

A number in scientific notation is written as a rational number from 1 to 9, and then multiplied by a power of 10 -For original values larger than 1, the exponent on the 10 will be positive -For original values between 0 and 1, the exponent on the 10 will be negative Examples: 2,100,000 = 2.1 x 10^6 Place the decimal point to the right of the first non-zero digit reading from left to right and then count how many places it was moved to get there.

Solving quadratic equations

A quadratic equation is an equation that could be written as Ax² + Bx + C = 0. To solve a quadratic equation: -Put all terms on one side of the equal sign, leaving zero on the other side -Factor -Set each factor equal to zero -Solve each of these equations Example: Solve for x. x² - 6x = 16 x² - 6x - 16 = 0 (x - 8)(x + 2) = 0 x - 8 = 0 --> x = 8 or x + 2 = 0 --> x = -2 Check by inserting your answer in the original equation.

Identity number

A value that, when added to another number or multiplied with another number, does not change the value of that number

Words that signal an operation

Addition: sum, plus, is increased by, more than (example: 3 more than 7 is what?) Subtraction: difference, minus, is decreased by, less than (example: 3 less than 7 is what?) Multiplication: product, times, of, at (examples: 2/3 of 5 is what? 3 at 5 cents cost how much?) Division: quotient, ratio, is a part of, goes into (examples: 3 is what part of 15? 4 goes into 28 how many times?)

Real numbers

All the rational and irrational numbers

Obtuse angle

Any angle whose measure is larger than 90° but less than 180° In the diagram, ∠4 is an obtuse angle.

Acute angle

Any angle whose measure is less than 90° In the diagram, ∠b is acute.

Exponent

Any exponent means to multiply by itself that many times Example: 5³ = 5 x 5 x 5 = 125 x¹ = x and x^0 = 1 when x is any number other than 0

Irrational numbers

Any value that exists but is not rational Examples: π: The decimal name for pi starts out 3.14159265... The decimal name for pi does not terminate nor does it have a repeating pattern √2: The decimal name for the square root of 2 starts out 1.414... The decimal name for the square root of 2 does not terminate nor does it have a repeating pattern

Finding percent of a number

Change the percent to a fraction or decimal and multiply Example: What is 20% of 80? 20/100 x 80 = 1600/100 = 16 or .20 x 80 = 16.00 = 16

Multiplying mixed numbers

Change to an improper fraction and multiply; then change the answer, if in improper form, back to a mixed number and reduce if necessary. Example: 3 1/3 x 2 1/4 = 10/3 x 9/4 = 90/12 = 7 6/12 = 7 1/2`

Fractions

Consist of two numbers separated by a bar which indicates division: the numerator is above the bar and the denominator is below the bar. Denominator: Tells you into how many equal parts something has been divided. Numerator: Tells you how many of those parts are being considered. Example: 3/5 (something has been divided into 5 equal parts, and 3 of those parts are being used)

Changing an improper fraction to a whole or mixed number

Divide the denominator into the numerator Examples: 24/3 --> 24 ÷ 3 = 8 19/5 --> 19 ÷ 5 = 3 4/5

Place value

Example: 210,987,654,321.23456 When the place value name ends with a "ths," it is an indication of being a fraction. Examples: 0.3 is read as "three tenths" and can be expressed in fraction form as 3/10. The value 5.23 is read as "5 and 23 hundredths" and can be expressed as the mixed number 5 23/100.

Reducing when multiplying fractions

Find a number that divides evenly into one numerator and one denominator. Example: 2/3 x 5/12 = 1/3 x 5/6 = 5/18 Reducing early only applies to multiplying fractions, not adding or subtracting.

Adding fractions

First change all the denominators to their least common multiple (LCM), which in fractions is known as the least common denominator (LCD). This value is the least positive value that all the denominators will divide into without a remainder. Example: 5/24 + 7/36 = ? Multiples of 24: 24, 48, 72, 96, 120, 144 Multiples of 36: 36, 72, 108, 144, 180, 216 LCM = 72 5/24 + 7/36 = 15/72 + 14/72 Now that the denominators are the same, add the numerators and keep the denominator. 29/72

Reducing fractions

Fraction must be reduced to its lowest terms. This is done by finding the greatest common factor (GCF) for both the numerator and denominator and then dividing both the numerator and denominator by that value. Example: Reduce 24/36 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 GCF = 12; divide numerator and denominator by 12 and you get 2/3

Simplifying fractions

If either numerator or denominator consists of several numbers, these numbers must be combined into one number, then reduce if necessary Example: (28 + 14) / (26 + 17) = 36/28 = 9/7 = 1 2/7

Dividing fractions

Invert (turn upside down) the second fraction and multiply; reduce if necessary. Example: 1/6 ÷ 1/5 = 1/6 x 5/1 = 5/6

Negative slope

Line falls as it goes to the right

Zero slope

Line is horizontal

Undefined/no slope

Line is vertical

Positive slope

Line rises as it goes to the right

Measures of central tendency

Mean/arithmetic mean, median, mode, range

Order of operations

Mnemonic: Please Excuse My Dear Aunt Sally 1. Parentheses: simplify (if possible) all expressions in parentheses 2. Exponents: apply exponents to their appropriate bases 3. Multiplication or division: do the multiplication or division in the order it appears as you read the problem left to right 4. Addition or subtraction: do the addition or subtraction in the order it appears as you read the problem left to right Example: 10 - 3 x 6 + 10² + (6 + 12) ÷ (4-7) 10 - 3 x 6 + 10² + (18) ÷ (-3) 10 - 3 x 6 + 100 + (18) ÷ (-3) 10 - 18 + 100 + (-6) -8 + 100 + (-6) 92 + (-6) 86

Distributive property

Multiplication outside parentheses distributing over either addition or subtraction inside parentheses does not affect the answer a(b + c) = ab + ac a(b - c) = ab - ac Examples: 7 (3 + 9) = 7(3) + 7 (9) 7 (12) = 21 + 63 84 = 84 5 (12-3) = 5(12) - 5(3) 5(9) = 60 - 15 45 = 45 Note: You cannot use the distributive property with only one operation. 3 (4 x 5) ≠ 3(4) x 3(5) 3 (20) ≠ 12 x 15 60 ≠ 180

Multiplying fractions

Multiply the numerators, multiply the denominators, and reduce to lowest terms if possible Example: 2/3 x 5/12 = 10/36 = 5/18

Changing a mixed number to an improper fraction

Multiply the whole number by the denominator, add the numerator, and then place that value over the denominator Example: 6 2/5 --> (5 x 6) + 2 = 32 --> 32/5

Decimals

Numbers to the left of the decimal point are whole numbers; numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1,000, 10,000, etc. Example: 0.6 = 6/10 = 3/5

Parallelogram

P = 2a + 2b P = 2(a + b) A = bh

Signed numbers

Positive numbers and negative numbers

Slope of perpendicular lines

Slope values will be opposite reciprocals

Approximating square roots

Square roots of nonperfect squares can be approximated. Example: Approximate √83 √81 < √83 < √100 --> 9 < √83 < 10 Since √83 is closer to the √81 than it is to the √100, the √83 will be closer to 9 than it is to 10. The decimal value of the √83 will then be between 9 and 9.5. Therefore, √83 ≈ 9.1 √2 ≈ 1.4 √3 ≈ 1.7

Subtracting signed numbers

Subtract = add the opposite; change the sign of the number being subtracted, and then proceed as an addition problem Example: 12 - (-4) = 12 + 4 = 16

Additive inverse

The additive inverse of a number (also known as the opposite of the number) is a value that, when added to any number, equals 0. Any number added to its additive inverse equals 0. The additive inverse of a = -a For any number a, a + (-a) = 0 Example: 7 + (-7) = 0

Associative property

The grouping, without changing the order, does not affect the answer

Multiplicative inverse

The multiplicative inverse of a number (also known as the reciprocal of the number) is a value that, when multiplied with any non-zero number, equals 1. Any non-zero number multiplied with its multiplicative inverse equals 1. The multiplicative inverse of a (a ≠ 0) is 1/a For any number a (a ≠ 0), a x 1/a = 1 Example: 4/5 x 5/4 = 1

Square root rules: addition and subtraction

The numbers must be combined under the radical before any computation of square roots may be done Example: √(10 + 6) = √16 = 4 √(10 + 6) ≠ √10 + √6

Communicative property

The order in which addition or multiplication is done does not affect the answer

Square numbers

The results of taking integers and raising them to the 2nd power (squaring them) The first seven positive square numbers are 1, 4, 9, 16, 25, 36, and 49 Examples: (-3)² = 9; therefore, 9 is a square number (0)² = 0; therefore, 0 is a square number, but it is not a positive square number

Cube numbers

The results of taking integers and raising them to the 3rd power (cubing them) The first seven positive cube numbers are 1, 8, 27, 64, 125, 216, and 343 Examples: (-3)³ = -27; therefore, -27 is a cube number (0)³ = 0; therefore, 0 is a cube number

Perfect square

The square of a whole number 1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 11² = 121 12² =144

Solid geometry

The study of shapes and figures in three dimensions

Plane geometry

The study of shapes and figures in two dimensions (the plane)

Square root

To find the square root of a number, you want to find some number that when multiplied by itself gives you the original number Perfect (whole number) square roots: √1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 =5 √36 = 6 √49 = 7 √64 = 8 √81 = 9 √100 = 10

Rounding off

To round off any number: -Underline the place value to which you're rounding off. -Look to the immediate right (one place) of your underlined place value. -Identify the number (the one to the right). If it is 5 or higher, round your underlined place value up 1. If the number (the one to the right) is 4 or les, leave your underlined place value as it is and change all the other number to its right to zeros. Example: Round to the nearest thousands 345,678 --> 346,000 928,499 --> 928,000 Example: Round to the nearest hundredth 3.4678 --> 3.47 298,435.083 --> 298,435.08

Square (verb)

To square a number, multiply it by itself Example: 6 squared = 6² = 6 x 6 = 36

Supplementary angles

Two angles whose sum is 180° In the diagram, since ∠ABC is a straight angle, ∠3 + ∠4 = 180°. Therefore, ∠3 and ∠4 are supplementary angles. If ∠3 = 122°, its supplement, ∠4, would be: 180° - 122° = 58°.

Square root rules: multiplication and division

Two numbers multiplied under a radical (square root) sign equal the product of the two square roots, and likewise with division. Example: √(4)(25) = √4 x √25 = 2 x 5 = 10 or √(4)(25) = √100 = 10

Mixed number

When a value is expressed using a whole number together with a common fraction Example: 2 3/4

Subtracting mixed numbers

You may have to borrow from the whole number, just like you sometimes borrow from the next column when subtracting ordinary numbers. Example: 4 1/6 - 2 5/6 = ? 4 1/6 - 2 5/6 = 3 7/6 - 2 5/6 = 1 2/6 = 1 1/3 To subtract a mixed number from a whole number, you have to borrow from the whole number. Example: 6 - 3 1/5 = ? 6 - 3 1/5 = 5 5/5 - 3 1/5 = 2 4/5

Associative property for addition

a + (b + c) = (a + b) + c Example: 8 + (4 + 2) = (8 + 4) + 2 8 + 6 = 12 + 2 14 = 14 Note: Subtraction does not have the associative property. 8 - (4 - 2) ≠ (8 - 4) - 2 8 - 2 ≠ 4 - 2 6 ≠ 2

Communicative property for addition

a + b = b + a Example: 2 + 3 = 3 + 2 5 = 5 Note: Subtraction does not have the communicative property. 3 - 2 ≠ 2 - 3 1 ≠ -1

Associative property for multiplication

a(bc) = (ab)c Example: 8 x (4 x 2) = (8 x 4) x 2 8 x 8 = 32 x 2 64 = 64 Note: Division does not have the associative property. 8 ÷ (4 ÷ 2) ≠ (8 ÷ 4) ÷ 2 8 ÷ 2 ≠ 2 ÷ 2 4 ≠ 1

Communicative property for multiplication

ab = ba Example: (2)(3) = (3)(2) 6 = 6 Note: Division does not have the communicative property. 10 ÷ 2 ≠ 2 ÷ 10 5 ≠ 1/5

1 kilogram (in pounds)

about 2.2 pounds

≈

is approximately equal to

=

is equal to

>

is greater than

Weight (metric system)

kilogram (kg) = 1,000 grams hectogram (hg) = 100 grams decagram (dag) = 10 grams gram (g) = 1 gram decigram (dg) = 0.1 gram centigram (cg) = 0.01 gram milligram (mg) = 0.001 gram

Volume (metric system)

kiloleter (kl or kL) = 1,000 liters hectoliter (hl or hL) = 100 liters decaliter (dal or daL) = 10 liters liter (l or L) = 1 liter deciliter (dl or dL) = 0.1 liter centiliter (cl or cL) = 0.01 liter milliliter (ml or mL) = 0.001 liter

Length (metric system)

kilometer (km) = 1,000 meters hectometer (hm) = 100 meters decameter (dam) = 10 meters meter (m) = 1 meter decimeter (dm) = 0.1 meter centimeter (cm) = 0.01 meter millimeter (mm) = 0.001 meter

Metric system prefixes

milli = 1/1000 centi = 1/100 deci = 1/10 basic unit (meter, liter, gram) = 1 deca = 10 hecto = 100 kilo = 1000

Equation

-A relationship between numbers and/or symbols that says two expressions have the same value -Solving an equation for a variable requires that you find a value or an expression that has the desired variable on one side of the equation and everything else on the other side of the equation -By doing the same arithmetic to each side of the equation, you eventually can isolate the desired variable Example: x-5 = 23. Solve for x. Add 5 to each side of the equation x = 28 Replace the original x with 28 and check to see if the resulting sentence is true. 28 - 5 = 23 23 = 23

Altitude/height

-A segment that goes from a vertex of the triangle and makes a 90° angle with the opposite side, known as the base -Sometimes, the opposite side needs to be extended in order to accomplish this In △ABC, segment AD is the altitude from vertex A. In this case, the base, segment BC, needs to be extended in order for the segment AD to be able to make a 90° angle with the opposite side. In △QRS, segment RT is the altitude from vertex R. Segment RT makes a 90° angle with the base, segment QS.

Inequality

-A statement in which the relationships are not equal -Instead of using an equal sign (=) as in an equation, we use > (greater than) and < (less than), or ≥ (greater than or equal to) and ≤ (less than or equal to). -When working with inequalities, treat them exactly like equations, EXCEPT: If you multiply or divide both sides by a negative number, you must reverse the direction of the sign. Example: Solve for x: -7x > 14 Divide by -7 and reverse the sign x < -2

Proportion

-A statement that says that two expressions written in fraction form are equal to one another. -Proportions are quickly solved using a cross multiplying technique Example: Solve for x. 3/x = 5/7 5x = 21 x = 21/5 or 4 1/5

y-intercept

-The point at which the line passes through the y-axis -The b in the y = mx + b form Example: y = -2x + 6 --> b = 6 Also notice that in the graph, the line passes through the y-axis at y = 6

Origin

-The point at which the two axes intersect -Represented by the coordinates (0,0), often marked simply 0

y-coordinate

-The second number in the ordered pair -Shows how far up or down the point is from 0

Line

-The shortest path connecting two points -Continues forever in opposite directions -Consists of an infinite number of points -Named by any two points on the line -The symbol <—> written on top of two letters is used to denote that line

Monomial

An algebraic expression that consists of only one term Examples: 9x 4a² 3mpxz²

Polynomial

An algebraic expression that consists of two or more terms separated with either addition or subtraction Examples: x + y (a polynomial with two terms) x² + 3x - 4 (a polynomial with three terms)

Quadrants

Four quarters that the coordinate graph is divided into -In quadrant I, x is always positive and y is always positive -In quadrant II, x is always negative and y is always negative -In quadrant III, x is always negative and y is always negative -In quadrant IV, x is always positive and y is always negative

Polygon

-A closed figure formed with line segments which are called the sides -Poly means "many" and gon means "sides"; thus, polygon means "a many-sided figure"

Line segment

-A piece of a line -Has two endpoints and is named by these two endpoints -The symbol — written on top of two letters is used to denote that line segment

Right angle

-Has a measure of 90° -The symbol in the interior of an angle designates the fact that a right angle is formed In the diagram, ∠ABC is a right angle.

Mode

-The value repeated most often -In order to have a mode, some score had to be repeated Example: 2, 2, 3, 5, 2, 6, 2, 6, 7, 9, 11 Mode = 2

Median

A segment that goes from one vertex of a triangle to the midpoint of the opposite side In △ABC, segment BD is a median from vertex B. Point D is the midpoint of segment AC, and so AD = CD.

Adding and subtracting polynomials

Add or subtract the like terms in the polynomials together Example: (3x² - 7x + 12) + (5x² + 9x - 19) = ? (3x² - 7x + 12) + 5x² + 9x - 19 3x² + 5x² - 7x + 9x + 12 - 19 (3 + 5)x² + (-7 + 9)x + (12 - 19) 8x² - 2x - 7

Circle

C = πd C = 2πr A = πr²

Adding and subtracting monomials

Must be like terms (like terms have exactly the same variables with exactly the same exponents on them) Example: 5x and 7x are like terms, but 5x and 7x² are not like terms Example: Simplify the following: 17a + 7b - 12a - 10b Rewrite with like terms near each other 17a - 12a + 7b - 10b 5a - 3b

Rectangle

P = 2b + 2h P = 2(b + h) A = bh

Rhombus

P = 4a A = ah

Point

-The most fundamental idea in geometry -Represented by a dot and named by a capital letter

x-coordinate

-The first number in the ordered pair -Shows how far to the right or left of 0 the point is

Transversal

-A third line that cuts two parallel lines, forming two sets of four angles -Angles in the same relative positions have the same measures -For any two angles you select, if they are not equal to one another, they will be supplementary to one another In the diagram: angle 1 = angle 5 angle 2 = angle 6 angle 3 = angle 7 angle 4 = angle 8 But since vertical angles are equal, angle 1 = angle 3 angle 2 = angle 4 angle 5 = angle 7 angle 6 = angle 8 From this, we can see angle 1 = angle 3 = angle 5 = angle 7 angle 2 = angle 4 = angle 6 = angle 8

Triangle

-A three-sided polygon -It has three angles, or angular rotations, in its interior -The sum of the angles (or angular rotations) is always 180° -The symbol for triangle is △ -Named by naming its vertices or corners The diagram is △ABC.

Linear equation

-An equation whose points, when connected, form a line -Can be written in the form, "y = mx + b" Example: Rewriting 2x + y = 6 in the "y = mx + b" form, you get y = -2x + 6

Coordinates/ordered pairs

-An ordered pair of numbers by which each point on a coordinate graph is located -Coordinates show the points' location on the graph -Shown as (x,y)

Pythagorean triples

-Any three sides of a right triangle -There are many Pythagorean triples with sides that are natural numbers -Common Pythagorean triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17 -Any multiple of one of these will also form a Pythagorean triple (for example, if each side of a 3-4-5 triangle were doubled, it would form a 6-8-10 Pythagorean triple)

Various ways an angle can be named

-By the letter of the vertex; therefore, the angle above could be named ∠A -By the number (or small letter) in its interior; therefore, the angle above could be named ∠1 -By the letters of the three points that formed it; therefore, the angle above could be named ∠BAC, or ∠CAB. The center letter is always the letter of the vertex.

Factoring polynomials that have three terms: Ax² + Bx + C

-Check to see if you can monomial factor (factor out common terms). Then, if A = 1 (the first term is simply x²), use double parentheses and factor the first term. Place these factors in the left sides of the parenthesis. For example, (x )(x ) -Factor the last term, and place the factors in the right side of the parentheses To decide on the signs of the numbers, do the following. If the sign of the last term is negative: -Find two numbers whose product is the last term and whose difference is the coefficient (number in front) of the middle term -Give the larger of the two numbers the sign of the middle term, and give the opposite sign to the other factor If the sign of the last term is positive: -Find two numbers whose product is the last term and whose sum is the coefficient of the middle term -Give both factors the sign of the middle term If A ≠ 1 (if the first term has a coefficient different than 1 — for example, 4x² + 5x + 1), then additional trial and error will be necessary

Factoring out a common factor

-Find the largest common monomial factor of each term -Divide the original polynomial by this factor to obtain the second factor (the second factor will be a polynomial) Example: Factor completely 2y³ - 6y 2y³ - 6y = 2y(y² -3)

Factoring the difference between two squares

-Find the square root of the first term and the square root of the second term -Express your answer as the product of the sum of the quantities from step 1 times the difference of those quantities Example: x² - 144 x² - 144 = (x + 12)(x - 12)

Coordinate graphs

-Formed by two perpendicular number lines (coordinate axes)

Angle

-Formed by two rays that start from the same point -This point is called the vertex; the rays are called the sides o the angle -Measured in degrees -The degrees indicate the size of the angle, from one side to the other

Equiangular triangle

-Has all of its angles of equal measure -Thus, each angle has a measure of 60° -An equiangular triangle is also equilateral

Scalene triangle

-Has all three of its sides of different lengths -The angles in a scalene triangle will all have different measures -The largest angle will be opposite the longest side -The smallest angle will be opposite the shortest side

Equilateral triangle

-Has all three of its sides of equal length, which in turn make each of the three angles equal in measure -Therefore, each angle in an equilateral triangle has a measure of 60° since the sum of the angles in any triangle is 180°

Ray

-Has only one endpoint and continues forever in one direction -Could be thought of as a half-line -Named by the letter of its endpoint and any other point on the ray -The symbol —> written on top of the two letters is used to denote that ray

Isosceles triangle

-Has two of its sides equal in length -The angles opposite the equal sides in an isosceles triangle have equal measure

Vertical angles

-If two straight lines intersect, they do so at a point -Four angles are formed -Those angles opposite each other are called vertical angles -Those angles sharing a common side and a common vertex are adjacent/supplementary angles -Vertical angles are always equal

Pythagorean theorem

-In any right triangle, there is a relationship between the lengths of the three sides -This relationship is referred to as the Pythagorean theorem -If the sides of a right triangle are labeled a, b, and c with c representing the longest of the three sides (the one opposite the right angle), then a² + b² = c²

Probability

-The comparison of the total number of favorable outcomes to the number of possible outcomes -Usually expressed in fraction form: total favorable outcomes / total possible outcomes -Can also be expressed as percents or decimal values

Slope value

-The slope of a line gives a number value that describes its steepness and the direction in which it slants -Positive slope, negative slope, zero slope, undefined/no slope -Slope is calculated by comparing the rise (the difference of the y-values) to the run (the difference of the x-values), when going from one point to another -The m in the y = mx + b Example: Using the points (-2,10) and (-1,8), we can calculate the slope the following way: Slope = rise/run = (10-8)/[-2 - (-1)] = 2/-1 = -2 Regardless of which two points were chosen, the slope value would be the same Line falls as it goes to the right, indicating that the slope is negative

Median

-The value in the middle so that there are an equal number of data values to either side of it -To find the median, arrange the data values from smallest to largest, including any repeats, then find the middle value -If there is an odd number of data values, one of the data values will be in the middle -If there is an even number of data values, take the mean of the two middle values Example: 2, 2, 3, 5, 2, 6, 2, 6, 7, 9, 11 2, 2, 2, 2, 3, 5, 6, 6, 7, 9, 11 Median = 5

The 45-45-90 right triangle

-This right triangle is an isosceles right triangle -If each of the sides that form the right angle has a measure of 1, then using the Pythagorean theorem, you find that the hypotenuse has the value √2 -If an isosceles right triangle had each of the equal sides with a measure of 5, then the hypotenuse would have a measure of 5√2

Perpendicular lines

-Two lines that meet to form right angles (90°) -The symbol ⟂ is used to denote perpendicular lines In the diagram, l ⟂ m.

Intersecting lines

-Two or more lines that cross each other at a point -That point would be on each of those lines

Parallel lines

-Two or more lines that remain the same distance apart at all times -Parallel lines never meet -The symbol || is used to denote parallel lines In the diagram, l || m.

Horizontal axis

-x-axis or abscissa -Numbers to the right of 0 are positive and to the left of 0 are negative

Vertical axis

-y-axis or ordinate -Numbers above 0 are positive and numbers below 0 are negative

Incomplete quadratic

A quadratic with a term missing Example: Solve for x. x² - 16 = 0 Factoring, (x + 4)(x - 4) = 0 x + 4 = 0 --> x = -4 or x - 4 = 0 --> x = 4

Angle bisector

A ray from the vertex of an angle that divides the angle into two equal pieces In the diagram, ray AB is the angle bisector of ∠CAD. Therefore, ∠1 = ∠2.

Angle bisector (triangle)

A segment that goes from one vertex of a triangle and divides the angle at that vertex into two smaller but equal angles In △GHI, segment HJ is an angle bisector from vertex H.

Adjacent angles

Any angles that share a common side and a common vertex In the diagram, ∠1 and ∠2 are adjacent angles.

Graphing equations

Graphs of equations in two variables (usually x and y) can be formed by finding ordered pairs that make the equation true, and then connecting these points Example: Make a graph of the equation 2x + y = 6 One way to do this is to set up a table of values with the x-values first, and then the y-values. You then replace one of the variables with values and find what the other variable would have to be for each replacement. If the x's were replaced with -2, -1, 0, 1, and 2, then find the corresponding y-values If x = -2, then 2(-2) + y = 6 --> -4 + y = 6 --> y = 10 Make a table, plot the points

Straight angle

Has a measure of 180°; also known as a line. In the diagram, ∠BAC is a straight angle.

Acute triangle

Has each of its angles with measures less than 90°

Right triangle

Has one of its angles with a measure equal to 90°

Obtuse triangle

Has one of its angles with a measure greater than 90°

Similar triangles

Have corresponding sides forming proportions

Triangle rules: exterior angle

If one side of a triangle is extended, the exterior angle formed by that extension is equal to the sum of the other two interior angles

Evaluating expressions

Insert the value(s) given for the unknown(s) and do the arithmetic, making sure to follow the rules for the order of operations. Example: Evaluate 2x² - 4y + 11 if x = 3 and y = -5. 2(3)² - 4(-5)+ 11 2(9) - (-20) + 11 18 + 20 + 11 38 + 11 49

Square

P = 4a A = a²

Triangle

P = a + b + c A = (bh)/2

Trapezoid

P = b1 + b2 + x + y A = [h(b1 + b2)]/2

Cylinder

SA = (Base-per)h + 2(Base-area) SA = 2πrh + 2πr² SA = 2πr(h + r) V = (Base-area)h V = πr²h

Prisms in general

SA = (Base-per)h + 2(Base-area) V = (Base-area)h

Rectangular Prism

SA = 2(lw + lh + wh) SA = (Base-per)h + 2(Base-area) V = lwh V = (Base-area)h

Sphere

SA = 4πr² V = (4/3)πr³

Cube

SA = 6a² V = a³

Slope of parallel lines

Same slope values

Range

The difference between the maximum and minimum scores Example: 2, 2, 3, 5, 2, 6, 2, 6, 7, 9, 11 11 - 2 Range = 9

Mean

The sum of the data values divided by the number of data values; average Example: 2, 2, 3, 5, 2, 6, 2, 6, 7, 9, 11 (2 + 2 + 3 + 5 + 2 + 6 + 2 + 6 + 7 + 9 + 11)/11 55/11 Mean = 5

Triangle rules: sides

The sum of the lengths of any two sides of a triangle must be larger than the length of the third side In the diagram of △ABC: AB + BC > AC AB + AC > BC AC + BC > AB

Complementary angles

Two angles whose sum is 90° In the diagram, since ∠ABC is a right angle, ∠1 + ∠2 = 90°. Therefore, ∠1 and ∠2 are complementary angles. If ∠1 = 55°, its complement, ∠2, would be: 90° - 55° = 35°.

Multiplying monomials with polynomials and polynomials with polynomials

Use the distributive property Example: (3x +5)(2x - 7) First distribute the 3x over the (2x - 7), and then distribute the +5 over the (2x - 7) 3x(2x - 7) + 5(2x - 7) 6x² - 21x + 10x - 35 6x² - 11x - 35 This also means that 6x² - 11x - 35 = (3x + 5)(2x - 7)

Multiplying monomials

When an expression has a positive integer exponent, it indicates repeated multiplication (Multiply numbers and add exponents on like term variables) Example: (7x²)(-5x³) (7 · -5)(x² · x³) -35x^5

≥

is greater than or equal to

<

is less than

≤

is less than or equal to

≠

is not equal to

||

is parallel to

⊥

is perpendicular to