Middle School Math NT203
{1,2,4,...} How much total when each square on a chess board (64 squares) when each square doubles each time the amount of rice on each?
(Has 64 squares, so numbers do not go on infinitely) a = 1 (the first term) r = 2 (doubles each time) n = 64 (64 squares on a chess board) n-1 over (sigma sign) over (k=0) times (ar^k)=a(1-r^n/1-r) plug in numbers: 64-1 over(sigma sign) over k=0 (1x2^0) = 1 (1-2^64/1-2) = 1-2^64/-1 = Answer: 2^64 -1 or 18,446,744,073,709,551,615
294 base 10 convert to base 2
(LSB) 294/2= 147 R.0 147/2=73 R. 1 73/2=36 R.1 36/2=18 R. 0 18/2=9 R.0 9/2=4 R. 1 4/2=2 R. 0 2/2= 1 R.0 1/2=0 R. 1 (MSB-Most Significant Bit) Answer: 100100110 base 2
A squared radical puts the square in the radical; same with index.
(√a^n)^m=√a^nm m√n^√a=mn^√a
1. Giga- 2. Mega- 3. Kilo- 4. Hecta- 5. Deca- 6. Unit/Meter/Liter 7 .Deci- 8. Centi- 9. Milli- 10. Micro- 11. Nano-
1. One billion= 1,000,000,000 -10^9 2. One million =1,000,000=10^6 3. One thousand= 1,000=10^3 4. One hundred =100=10^2 5. Ten=10=10^1 6. One=1=10^0 7. One tenth =0.1=10^-1 8. One hundredth = 0.01=10^-2 9. One thousandth =0.001=10^-3 10. One millionth 0.000001=10^-6 11. One billionth= 0.000000001=10^-9
Division with radicals involves "rationalising the denominator" since we remove all irrational numbers in the denominator of the fraction by using the denominator's conjugate: (a + b)(a − b) = a^2 − b^2 Solve: 1/ √3 - √2
1/ √3 - √2= 1/√3 - √2 ( √3 + √2)/(√3 + √2)= √3 + √2/ (√3)^2 - (√2)^2= √3 + √2/ 3-2= √3 + √2
Solve: Sin A?/a? = Sin B 25/b 15 = Sin C 107/c?
180-107-25=48 = Sin A 48/a? Sin A 48/a? = B Sin 25/b 15 = 48(15)=25a a=28.8 so... Sin A 48/28.8 = Sin B 25/b 15 = Sin C 107/c Find c: Sin B 25/b 15 = Sin C 107/c= 15(107)=25c= 64.2 Answer: Sin A 48/28.8 = Sin B 25/b 15 = Sin C 107/c 64.2
Radicand same, solve outside normal (same if have index): 2√7 − 5√7 + √7
2√7 − 5√7 + √7= (2 − 5 + 1)√7=−2√7
2√x / √x - √y
2√x / √x - √y times (√x + √y)/(√x + √y)= Answer: 2x + 2√xy / x-y
Simplify 3√125
3√125= 3√25 x 5= 3(5√5)= Answer: 15√5
Different radicand and so we cannot do anything with it: √5 + 2√3 - 5√5
5 + 2√3 - 5√5= 2√3 - 4√5
Asymptote
A line that a graph approaches but does not reach. It may be a vertical, horizontal, or slanted line. A function which continuously approaches a line or axis without meeting it at any finite distance.
Transversal line
A line that intersects 2 lines to form 8 angles. A line that intersects two or more lines
Slope
A measure of the steepness of a line. Given two points with coordinates (X1,Y1) and (x2,y2) on a line the slope, m, of the line is given m = rise/run= y2-y1/x2-x1
Rational Numbers
A number that can be written as a fraction with a numerator and a denominator that are integers. The decimal representation of a rational number either ends or repeats.
Central Tendency
A number that describes something about the "average" score of a distribution.
Midpoint
A point that divides a segment into two congruent segments
Composite Number
A positive integer that has at least one positive divisor other than one or the number itself. In other words, a composite number is any integer greater than one that is not a prime number.
Sum of an Arithmetic Series
A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series. S n= n/2 [2a + (n-1)d] = n(a + a n/2) a= 1st term d= common difference between terms n= how many terms are being added up in the sequence S= the whole sum
Divisibility Rules
A set of rules that guide you to find out if a number is divisible by another. Example: last digit even then divisible by 2, last digit 0 or 5 divisible by 5, last digit 0 divisible by 10, sum of digits divisible by 3 or 9 then divisible by 3 or 9
Prime Numbers
A whole number greater than 1 with only two factors, 1 and itself (2 is the only even prime number)
A car is traveling at a speed of 100 kilometers per hour. What is its approximate speed in meters per second? A. 28 meters per second B. 36 meters per second C. 280 meters per second D. 360 meters per second
A. This question requires the examinee to analyze the use of various units and unit conversions within the customary and metric systems. Both units in the given quantity must be converted: kilometers to meters and hours to seconds. (100 kilom/1 hr) (1000 m/1 kilom) (1hr/3600 seconds) = 27.8 meters/second (wavy)= 28 meters/second or (100)(1000)(3600)=27.8 meters per second
A car that is advertised for $28,900 is sold after an 8% discount. If the commission rate is 0.75% of the sale price, how much money will the salesperson earn? A. $199.41 B. $216.75 C. $19,941.00 D. $21,675.00
A. This question requires the examinee to solve a variety of problems involving ratios, proportions, and percents. The final sale price of the car is 92% of the asking price (100% - 8% discount) and the salesperson earns 0.75% of the sale price. Commission = ($28,900)(0.92)(0.0075) = $199.41.
Lateral surface area of a right circular cone
A= (pie)r√r^2 + h^2 A is Area R is Radius H is Height
Surface area of a sphere
A=4(pie)r^2
Natural Numbers
Also called the counting numbers, this set includes all of the whole numbers except zero (1, 2, 3, ....)
Acute Angle
An angle less than 90 degrees
Obtuse Angle
An angle that measures more than 90 degrees but less than 180 degrees
Nonlinear equation
An equation or inequality that, when represented graphically, results in a graph that is not a straight line
Linear inequalities
An inequality containing one or more terms in which the variable(s) is/are raised to the power of one but no higher Example: x ≤ 1 and 2n - 3 > 9 Like solving linear equations, except when you multiply or divide an inequality by a negative number, direction of inequality must change. Inequality sign flips when divide or multiple by a negative.
Find x if the line through the points (6, x) and (1, -5) has a slope of 2.
Answer: x = 5 m = rise/run= y2-y1/x2-x1 so do 2= -5-(x)/1-6 to 2=-5x/-5 to 2*(-5)=-5-x/-5 * (-5) to -10+5=-5-x+5 to -5*-1=-x*-1 to 5=x or (6,5) and (1,-5). Check answer by solving to get 2 as the slope.
Real Numbers
Any number that is positive, negative, or zero, and is used to measure continuous quantities. All rational and irrational numbers.
The Slope of Parallel Lines
Are identical
Mean
Average
Five lines intersect as shown. If x and y are angle measures in degrees, which of the following equations relates y to x in the diagram above? A. y=1/3 x B. y=x-90 C. y=1/2 x D y=180-x
B. This question requires the examinee to analyze properties of points, lines, planes, and angles. If two lines are both perpendicular to the same transversal, the two lines are parallel. Thus there are two pairs of parallel lines in the figure. By the alternate interior angle theorem, x=y+90 and y=x-90 (90 degrees that is).
Bit, Byte, Word
Bit=Single binary digit Byte=8 bits (binary digits) Word=16 bits (binary digits)
A way to display a distribution of data values but using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.
Box and whisker plot
Prime Factorization
Breaking down a composite number until all of the factors are prime
A hiker climbs uphill at a steady pace, rests at a scenic spot for a while, then continues at a slower pace to the top of the hill. The hiker stops for lunch at the top, then decides to run down to the base of the hill. Which of the following graphs best expresses the hiker's speed as a function of time? [Draw. Think of speed as Y (vertical) and time as X (horizontal)]
C. This question requires the examinee to analyze the properties of relations and functions in multiple representations (e.g., tables, graphs, equations, words). The "steady pace" held by the hiker reflects a constant speed over time; thus the first section of the graph is a horizontal line (slope equals zero) at some constant value of speed. The hiker slows to a resting position with a speed of zero, though time passes so the value for time continues to increase. The approach to the top of the hill shows a positive slope, with increasing speed and time as the hiker resumes walking, but since the pace is slower, the graph does not reach the previous level of speed. This pattern repeats when the hiker stops for lunch, but the run to the base reflects a greater speed than was achieved on the initial climb.
Binary Numbers
Can add and subtract. Each column of a binary number represents double the column to its right. Is Base of 2. Each place can not go higher than one. If reaches 2, that place goes to zero and a 1 is added to the left (2=10 (said one zero not ten)). If three, then zero becomes 1 (3=11(said one one not eleven)). For four, can't go higher then 1, so add 1 on left and others return to zero (4=100 (said one zero zero not one hundred)). Etc.,
Proportions
Concerned with the size relationships of one part to another.
If the limit of function h(x) equals a real number k as x goes to negative infinity, which of the following is the equation of an asymptote of the graph of h(x)? A. x = -k B. y = -k C. x = k D. y = k
D. This question requires the examinee to analyze function behavior in terms of limits, continuity, and rates of change. The situation described can be represented as . This means that as x approaches negative infinity, lim (with x, arrow pointing right, negative sign, infinity sign all underneath) h(x) approaches but never reaches k and that y = k is a horizontal asymptote.
Which of the following is an equation of a line perpendicular to the line 4x - 9y = 12? A. y=9/4 x + 6 B. y = 4/9 x +3 C. y= -4/9 x -4 D. y= -9/4 x -2
D. This question requires the examinee to analyze the relationship between a linear equation or inequality and its representations. The slope, m^1 , of the given line is found by isolating y in the given equation. 4x-9y=12 to 9y=4x-12 to y= 4/9 x -12/9 and m^1= 4/9. The slope, M^1, of a line perpendicular to the given line is the negative reciprocal of the slope of the given line M^1. Thus m^1= -1/m^1 so m^1= -9/4 and y=- 9/4 x -2 must be the equation of the perpendicular line.
In the base-2 number system, the sum of 101 and 1011 is: A. 1000 B. 1112 C. 2000 D. 10000
D. This question requires the examinee to analyze the role of place value in any number system. Align addends 101 and 1011 such that place values are in the appropriate columns. Add the two digits in the right column, 1 + 1 = 2. The 2 is regrouped as equaling 1 × 2^1 + 0 × 2^0 with the 1 carried to the next column (Table 1). Continue the process until complete (Table 2), regrouping as needed, yielding the sum 10000. 1o11 +101 ____ 10000 (make 2, or one zero, requiring the one to be raised. Occurs with each as each then make 2, or one zero)
The end points of one diagonal of a parallelogram are (1, 3) and (1, -3). The end points of its other diagonal are (3, 1) and (-1, -1). What is the perimeter of the parallelogram? A. 2 √ 2 + 2√ 5 B. 10 C. 20 D. 4 √ 2 + 4 √ 5
D. This question requires the examinee to connect algebra and geometry by applying concepts of distance, midpoint, and slope to classify figures and solve problems in the coordinate plane. Use points (1, 3) and (3, 1), then points (3, 1) and (1, -3), and the distance formula to find the length of two different sides of the parallelogram: √ (1-3)^2 + (3-1)^2 which = 2 √2 and √ (3-1)^2+[1-(-3)]^2 which = 2 √ 5 . There are two of each of these sides, thus the perimeter of the parallelogram is (2) (2 √2 then +2) (2 √5) which = 4 √2 + 2 √ 5.
If a person can buy up to 3 times as many desktop computers as laptop computers with the same amount of money, then which of the following inequalities relates the price of a desktop computer D to the price of a laptop computer L? [The < and > have lines underneath them] A. D<3L B. D>3L C. 3D<L D. 3D>L
D. This question requires the examinee to connect appropriate algebraic notation to phrases and sentences. The condition of "up to 3 times as many" is the same as 1 laptop computer having the value of at most 3 desktop computers. Thus for the price of a desktop computer, D, and the price of a laptop computer, L, 3D>L . [line under > sign]
Geometric Sequence- Solve: x n = ar^(n-1) {10, 30, 90, ?, 810, 2430, ...}
Each term is found by multiplying the previous term by a constant. {a, ar, ar2, ar3, ... } a=the first term r=the factor between the terms (called the "common ratio")When r=0, we get the sequence {a,0,0,...} which is not geometric. So r cannot be 0. a = 10 (the first term) r = 3 (the "common ratio") n= term x n = 10 × 3^(n-1) x4 = 10×3^(4-1) = 10×27 = 270 is answer for 4th term
Quadratic Formula
Formula used to find the solution to a quadratic equation when it is set equal to zero. x = -b ± √(b² - 4ac)/2a Tells the point of x where line crosses the x line. The Discriminant is (b^2-4ac): positive- there are 2 real solutions, zero- there is one real solution, negative- there are 2 complex solutions (negative as the answer of the discriminant is a negative number).
Horizontal
Going straight across from side to side
< with line underneath or > with line underneath means...
Greater/less than or equal to. Just < or> mean greater/less than.
Base Ten
Have columns or "places" for 10^0 = 1, 10^1 = 10, 10^2 = 100, 10^3 = 1000, and so forth. A system in which each number place is worth 10 times the place to its right. Also called the decimal system.
Congruent
Having the same size and shape
Pythagorean Thereom Converse
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. c^2=a^2 + b^2
Lateral surface area
In a solid is the sum of the surface areas of all its faces excluding the base of the solid.
Vertical
In an up and down direction
Perpendicular
Intersecting at or forming right angles
Irrational Numbers
Is any real number that cannot be expressed as a ratio of integers or as a fraction. Cannot be represented as terminating or repeating decimals as they are fractions. Therefore, a number that has a remainder, for example. Can be a non-whole number that can't be a fraction. Example: imperfect square root.
Law of Cosines
Know 2 sides and 1 included angle or know all three sides. Is useful for finding: the third side of a triangle when we know two sides and the angle between them.
Law of Sines
Know 2 sides and one opposite angle and you want to find the measure of an angle opposite a known side. Or when you know 2 angles and 1 opposite side and want to get the side opposite a known angle. In both cases, you must already know a side and an angle that are opposite of each other. Provides a formula that relates the sides with the angles of a triangle. a/sin A = b/sin B =c/sin C a, b, c are the sides and A, B, C are the angles. Side a faces angle A, side b faces angle B and side c faces angle C. When we divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C.
Meter
Length
y=mx+b
Linear equation. m=slope and b=y intercept y=f(x) when identity function
Gram
Mass/Weight
Sigma sign
Means "sum up". Could signify as "S".
Median
Middle # when put in high to low order, or the average of the 2 middle #'s
Mode
Most frequently occurring score
Absolute Value
Never negative as only asks "how far away from zero?" not "what direction?" Example: I3I=3 (3 away from zero) I-3I=3 (no negative with absolute value).
Index
Number exponent outside radical. Example: m^√a
Radicand
Number in radical. Example: the a in √a can multiply, divide, add, subtract radicands of different numbers as long as index and exponents are same. Just be sure to simplify answer. Example: √33 x √3= √99 = √9 x 11 =3√11
Integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals) ...cannot be a fraction or a decimal
Direct Proportional Relationships
Relationship between two variables whose ratio is a constant value
Sum of Geometric Series
S n = a(1-r^n)/1-r
Parabolas
Some Nonlinear Graphs That Are U-Shaped Curves; Y= X Squared
Quartiles
The 25th, 50th, and 75th percentiles, referred to as the first quartile, the second quartile (median), and third quartile, respectively. The quartiles can be used to divide a datat set into four parts, with each part containing approximately 25% of the data The values that divide the data into four equal parts
The Midpoint Formula
The midpoint of a segment with endpoints (x1 , y1) and (x2, y2) has coordinates. Works for all line segments: vertical, horizontal or diagonal. Formula: (x1+x2/2 , y1+y2/2)
Radical Sign
The sign √, which indicates the square root of the number following (or a higher root indicated by a preceding superscript numeral). To denote that the root is to be extracted or that the root marked by an index (as in ∛ ̅ or ∛ for the cube root) is to be extracted. Square root a radical to get rid of it: √ multiplied by 1^2 gets rid of it, however it changes positive to negative and vice versa. When plugging number back into equation ensure to revert back to original positive. Square root always positive.
Slope of a Perpendicular line the other one given
The slope of one line will be the reciprocal to the other. -1 would become 1 and vice versa.
180 degrees is the amount of degrees in what shape?
Triangle when all angles are added together.
6^√√2 - 12^√2^13
Use m√n^√a=mn^√a and (√a^n)^m = √a^nm 6^√√2= 6 x 2^√2 = 12^√2 and 12^√2^13= Make index and exponent same √2^12 x √2^2 so is 12^√2^12 x 12^√2= 2 12^√2= put together 12^√2 - 2 12^√2= Answer: -12^√2
Volume of a right cone and a pyramid
V= 1/3 BH
Volume of a sphere
V=4/3 (pie) r^3
Liter
Volume
Infinite Geometric Series
When n goes on for infinity. Well ... when r is less than 1, then r^n goes to zero and we get: (infinity sign) over (sigma sign) over (k=0) times (ar^k)=a(1/1-r) a = (the first term) r = (the "common ratio") n= term r must be between (but not including) -1 and 1 and r should not be 0 because we get the sequence {a,0,0,...} which isn't geometric
Whole Numbers
Zero and the positive integers are the whole numbers.
Solve 5x² + 2x + 1 = 0
a = 5, b = 2, c = 1 Note that The Discriminant is negative: b^2 - 4ac = 22 - 4×5×1 = -16 x = [ -2 ± √(-16) ] / 10 The square root of -16 is 4i (i is √-1, read Imaginary Numbers to find out more) x = ( -2 ± 4i )/10 Answer: x = -0.2 ± 0.4i The graph does not cross the x-axis. That is why we ended up with complex numbers.
Solve this equation for x: 5x² + 6x + 1 = 0
a = 5, b = 6, c = 1 x = [ −b ± √(b^2 − 4ac) ] / 2a x = [ −6 ± √(6^2 − 4×5×1) ] / (2×5) x = [ −6 ± √(36−20) ]/10 x = [ −6 ± √(16) ]/10 x = ( −6 ± 4 )/10 x = −0.2 or −1
Linear Equations
an equation containing one or more terms in which the variable(s) is/are raised to the power of one but no higher algebraic equation in which the variables are raised only to the first power, (y=4x+3), and the graph of it is a straight line.
Quadratic Formula in Standard Form
ax^2 + bx + c = 0 where a, b, and c are real numbers with a ≠ 0.
Pythagorean theorem
a²+b²=c² A formula for finding the length of a side of a right triangle when the lengths of two sides are given.
Distance Formula
d =√ (x2-x1)^2 + (y2-y1)^2 Find the distance between two points (x1,y1) and (x2,y2), all that you need to do is use the coordinates of these ordered pairs and apply the formula.
Euclidean geometry
geometry based on Euclid's axioms: e.g., only one line can be drawn through a point parallel to another line; system based on a set of points in space called a plane. Sometimes called "plane geometry".
Find Slope Solve: What is the slope of line (-1,-2) and (1,2)?
m = rise/run= y2-y1/x2-x1 Answer: -2-(2)/-1-(1) = 2+2/1+1 = 4/2= 2 Answer: m=2
Algebraic Notation
mathematical shorthand used to express numerical properties, patterns, and relationships
If n^√a and n^√b are natural numbers, and index n is a natural number, then...
n^√a times n^√b = n^√ab Example: 4^√4a times 4^√7a^2 b = 4^√28a^3b
Arithmetic Sequence
the difference between one term and the next is a constant.In other words, we just add the same value each time ... infinitely. 1, 4, 7, 10, 13, 16, 19, 22, 25,... Can write as {a, a+d, a+2d, a+3d, ... } with a is the first term, and d is the difference between the terms (called the "common difference") With numbers above would be... a = 1 (the first term) d = 3 (the "common difference" between terms) so result is {a, a+d, a+2d, a+3d, ... } to {1, 1+3, 1+2×3, 1+3×3, ... } to {1, 4, 7, 10, ... }
Inverse Proportional Relationships
when one variable increase proportionally as the other variable decreases
Arithmetic Sequence Rule for finding specific number in sequence. Solve: 3, 8, 13, ?, 23, 28, 33, 38, ... What is the 4th term?
x n = a + d(n-1) (We use "n-1" because d is not used in the 1st term). a= 3 (1st term) d=5 [the common difference between term (which 1st term doesn't have)] n= 4 (as in 4th term/number in sequence) = 3 + 5(n-1) = 3 + 5n - 5 = 5n - 2 Answer: x 4 = 5×4 - 2 = 18
Simplify a radical by solving for factors-1st step when solving problem
√18= √9 x 2= 3√2 as 2 cannot be squared
√2/3a - 2√3/2a
√2/3a - 2√3/2a=multiply top and bottom of each fraction with their respective denominators. This gives us a perfect square in the denominator in each case, and we can remove the radical. √2(3a)/3a(3a) - 2√3(2a)/2a(2a)= √6a/9a^2 - 2√6a/4a^2= Simplify: √1/9a^2 (not 6a as cannot be √) and √1/4a^2= 1/3a √6a - 1/a √6a= Common Denominaor is 3a so... √6a/3a - 3√6a/3a (a goes into 3a 3 times is why)= So... √6a - 3√6= -2√6/3a so answer is... Answer: -2/3a √6a