Download rational number- a number that can be written as a fraction

Glossary for CTE Instructors
Academic Standards in Mathematics
Standard
DEFINITIONS
6.1.1.1
rational number: a number that can be written as a fraction
6.1.1.1
plot pairs of numbers on a coordinate grid - place a dot on a
coordinate graph to represent the location of an ordered pair.
Ordered pairs are written in the format (x, y) where x is the
horizontal distance and y is the vertical distance. The x and y
are coordinates of the point.
6.1.1.2
EXAMPLES
1/2; 7 ( 7/1); 9.6 (96/10);
0.333…(1/3)
(5,2)
inequality symbols:
< is less than
> is greater than
≥ is greater than or equal to
≤ is less than or equal to
≠ is not equal to
3<7
7>3
4≥2; 2≥2
2 ≤4; 2≤2
2≠4
6.1.1.3
ratio: a comparison of two numbers
can be written as a fraction, such as
4/3; with a colon, such as 4:3; or
with the word “ to”, such as 4 to 3
6.1.1.5
whole numbers:
0, 1, 2, 3, 4, …
6.1.1.5
prime factors:
factors: numbers that divide into another number with
no remainder
the factors of 6 are 1, 2, 3, and 6
prime numbers: numbers whose only factors are 1
and itself
17
6.1.1.5
exponent: a number that tells how many time a number is
multiplied by itself
34 = 3∙3∙3∙3
6.1.1.6
greatest common factor (GCF): the largest number that is the
factor of two (or more) numbers, used in reducing fractions
the GCF of 12 and 16 is 4 because
4 is the largest number that will “go
into” both 12 and 16
6.1.1.6
least common multiple (LCM): the smallest number that two
(or more) numbers will “go into”, used in finding common
denominators
the LCM of 12 and 16 is 48
6.1.2.3
variable: a symbol (usually a letter) used to represent a
number
6.2.2.1
associative property of addition: (a+b)+c = a+(b+c)
grouping symbols can be rearranged when numbers are added
(1+2) + 3 = 1+ (2 + 3)
6.2.2.1
associative property of multiplication: (ab)c=a(bc)
Note: (ab means a times b; if a=3 and b=4 then ab = 12, not 34)
(1∙2)3 = 1(2∙3)
X , Y, M……
1
6.2.2.1
commutative property for addition or multiplication: a+b =
b+a and ab=ba, order can be changed when adding or
multiplying
2+3=3+2
2∙3=3∙2
6.2.2.1
distributive property: a(b+c) =ab+ac
3(x+2y) = 3x + 6y
6.2.3.1
equations / inequalities: equations use an =, inequalities use
one of the inequality symbols
Equation; y = 3x
Inequality; y ≤ 3x
6.3.1.1
prism: a three dimensional figure, where the two bases are
congruent and the sides are rectangles The most common
prism is a box, but others are possible.
polygons: closed figures made up of non-overlapping
segments
;
6.3.1.2
quadrilateral: a four sided polygon
6.3.1.2
rhombus: a quadrilateral with four sides
of equal length
6.3.1.2
parallelogram: a quadrilateral where the opposite sides are
parallel
6.3.1.2
trapezoid: a quadrilateral with exactly one pair of parallel sides
6.3.1.2
kite: a quadrilateral with two pairs of adjacent sides that are
the same length
6.4.1.1
sample space: all possible outcomes
coin sample space is heads or tails
6.4.1.2
probability: the number of favorable outcomes divided by the
total number of possible outcomes
When rolling a die, the probability of
rolling the number 4 is P(4)= 1/6
6.4.1.4
experimental probability: the probability you get when you
actually do an experiment
toss a coin 10 times and get heads
four times, the experimental
probability of getting heads is 4/10
theoretical probability - see defn. of probability, no actual
experiment is done The theoretical probability is always
between 0 and 1
the theoretical probability of getting
heads is 1/2
2
7.1.1.1
pi: the number you get if you divide the circumference
(distance around the outside) of a circle by its diameter (the
distance across a circle, going through the center) written in
decimal form the value of pi is approximately 3.14 (the actual
decimal goes on forever)
7.1.2.2
inverse relationships: operations that “undo” each other
multiplication & division,
addition & subtraction,
squaring and taking the square root
7.1.2.5
proportion: setting two ratios equal to each other
4/8 = 1/2
7.1.2.6
absolute value of a number: the distance a number is from
zero; the absolute value is always a positive number
I -4l = 4 , l3l = 3
7.2.2.1
unit rate, constant of proportionality, slope: the rate at
which something changes
25 miles per hour
7.2.3.1
algebraic and numerical expression: does not contain an =
or an inequality symbol
3x+7 or 17 – y
7.2.3.1
like terms: terms that have exactly the same variables
in the expression 3a + 2b – 7 – 2a ,
the like terms are 3a & – 2a
7.2.3.3
order of operations: the order used in simplifying an
expression.
Given radius = 4
C= 2 π r
C = 2 (3.14) 4
= 25.12
Frequently remembered by using the first letter of each word in
“Please Excuse My Dear Aunt Sally.”
1.
1. Parenthesis (grouping symbols)
(2+3)2 – 12 / 6 = 23
2.
2. Exponents
52 – 12 / 6 = 23
3.
3. Multiply or Divide from left to right
25 – 12 / 6 = 23
4.
4. Add or Subtract from left to right
25 – 2
= 23
7.2.2.3
grouping symbols: parenthesis or brackets- used to indicate
order of operations
7.3.2.1
similar figures: figures that are the same shape (sides are
proportional and corresponding angles are the same)
7.3.2.2
scale factor: a ratio comparing a length on a drawing to the
actual length of an object
(2+3)
On a map 1 inch = 15 miles
3
7.3.2.4
transformation: change of position or size of a figure
7.3.2.4
translation: shifting an object horizontally, vertically or both
7.3.2.4
reflection:
rotation: when a figure is turned (rotated) around a given point
See examples below
This star
degree
rotation.
shows 72
(360 / 5)
dialation: produce similar figures (see above)
A figure can be enlarged or reduced.
7.4.1.1
mean: average – add the numbers together and then divide by
the number of numbers
3, 4, 1
Mean = 8/3
7.4.1.1
median: the center number when the numbers are listed in
order (I use median in a road to help my students remember)
3, 4, 1
Median = 3
7.4.1.1
mode: the value that occurs most often
It does not need to be a number.
3, 4, 1, 1
7.4.2.1
histogram: a bar graph without spaces between the bars; the
heights of the bars give the frequency of the data
Mode = 1
47
46
1st Qtr
45
2nd Qtr
44
3rd Qtr
43
4th Qtr
42
North
8.1.1.1
Rational numbers: Any number that can be represented as a
fraction
8.1.1.1
irrational number: a number that cannot be represented by a
fraction
8.1.1.2
square root: if b2 = a then b is the square root of a
symbol:√
8.1.1.1
integer: whole numbers and their opposites
, , , , -3, -2, -1, 0 1, 2, 3, . . .
8.1.1.2
real number – all the rational and irrational numbers
See above
8.1.1.5
scientific notation: a number written in the format so that
there is one number to the left of the decimal point x some
power of 10. This format is usually used with very large or very
2.978 x 105 = 297800
2.978x10-5 = 0.00002978
8/4, 5, 9.7
π,
9 , 3.1978354…
9=3
4
small numbers.
8.2.1.1
independent and dependent variable: The value of the
independent variable determines the value of the dependent
variable.
When graphing; the independent variable is along the horizontal
axis and the dependent variable along the vertical.
(Hint: f(x) is the same as “y”)
In equations, y = 3x+9, the
independent variable is the x and
the dependent variable the y,
.
If you are graphing foot length and
shoe size, shoe size is the
dependent variable and foot length
the independent variable because
shoe size depends on foot length
8.2.1.1
function notation: f(x)
8.2.1.2
constant: a number with no variable
8.2.1.3
linear function: its graph is a line; frequent formats include
y = mx + b or ax + by = c ( there are no exponents on the
variables, the variable is not an exponent, the variables do not
appear in the denominator of a fraction and there are no
absolute value symbols around the variable)
8.2.1.4
arithmetic sequence: a sequence of numbers, where the
difference between consecutive terms is a constant number
1, 6, 11, 16, 21, 26 . . . (the
common difference is 5)
8.2.1.5
geometric sequence: a sequence of numbers, where the ratio
between consecutive terms is a constant number
1, 3, 9, 27, 81. . . (the common ratio
is 3)
8.2.2.2
Slope (m) rate of change = change in the y values =
change in the x values
given the points (-1,5) and (5,7)
m=7-5 = 2
= 1
5- 1
6
3
8.2.2.3
slope intercept form: Equations of the format y = mx + b, m is
the slope & b is the y- intercept
8.2.2.2
intercepts: x-intercept: where a graph crossed the x-axis
y-intercept: where a graph crossed the y-axis
if f(x) = x+3 , then f(7) = 7+3
3x+7y+2; 2 is the constant
y2 -y 1
x2 -x 1
Y =1x-1
y = -2x +4;
-2 is the slope
4 is the y-intercept
X-int =
1
Y-int =
-1
8.2.2.2
rate of change: another name for slope, how fast something
is changing
25 miles per hour
8.2.2.3
coefficient: a number in front of a variable
example: 4xy, 4 is the coefficient
8.2.4.3
point slope: given a point (a,b) on a line, and its slope m
the equation of the line is y – b = m ( x – a)
Find the equation of a line going
through point (2,3) with a slope of 4
y – 3 = 4(x-2)
8.2.4.3
standard form of a line: ax + by = c
3x + 4y = 12
8.2.4.7
system of equations: two or more equations that use the
Find a solution for the equations
5
same variables to describe a situation
8.3.1.1
Pythagorean theorem: used with right triagles,
a & b are the lengths of the legs
c is the length of the hypotenuse
a2 + b2 = c2
y = 4x
&
y = 3x +2
c
a
b
If a = 3 and b = 4 then
32 + 42 = c2
9 + 16 = c2
25 = c2
5=c
8.3.2.1
parallel lines:
lines in the same plane that do not intersect
Parallel planes: planes that do not intersect
8.3.2.1
perpendicular: meet at a right angle
8.4.1.1
scatter plot: a graph that displays related data as ordered
pairs
8.4.1.2
line of best fit: a line that most closely fits a set of data
grades 9-11
9.2.1.1
function: a relationship where there is a unique output for
each input
f(x) = 3x + 2
9.2.1.1
domain: all possible values for the independent variable
Possible shoe sizes include 0-30
9.2.1.5
vertex: the point where a graph shifts from going up to going
down or visa versa; usually used with a parabola when looking
for the maximum or minimum value
6
9.2.1.5
line of symmetry: a line that divides a figure with reflectional
symmetry into two congruent (same size, same shape) halves
9.2.1.6
zeros of a function: the x intercepts (where y = 0)
9.2.1.5
quadratic function: the equation is of the form
ax2 + bx + c = 0
the graph is a parabola (kind of like a U)
Zero of
function
=1
3x2 + 2x + 4 = 0
9.2.1.7
asymptote: a line which the graph of a function approaches
but never intersects
9.2.2.3
exponential function: the equation is of the form y = a(bx)
a is the initial amount; b is the growth rate if b>1 or the decay
rate if 0<b<1.
y = 4(2x)
9.2.2.4
recursive formula: a formula that states the initial amount and
then a rule to get from one term to the next
P0 = 10 Population at time 0 is 10
Pn = P n-1 +5 population at time 1
is 10 + 5 = 15
population at time 2 is 15 + 5 = 20
…
Your hair is 3 inches long.
Every month it grows .5 inches
9.2.2.4
explicit formula: a formula in the format you usually see it
example C = π d
9.2.3.1
polynomial: algebraic or numeric terms that are added or
subtracted
example: 3s + 2t – 3
9.2.3.1
rational expressions: fractional expressions where either the
numerator, denominator or both contains a variable
3x +2
9.2.3.5
complex number: a number in the form a + bi where a and b
are real numbers and √ -1 = i
It is very unlikely you would use a complex number in your
classes.
7
9.2.4.1
quadratic formula: a formula that allows you to solve a
quadratic equation.
for the equation ax
ax2 + bx +c = 0
x=
 b  b 2  4ac
2a
9.3.1.2
decompose two and three dimensional figures: break down
complex figures into workable shapes i.e. triangles, squares,
circles, cylinders
9.3.2.1
axiom: a basic assumption
Through any two points there is
exactly one line
9.3.2.1
undefined terms: Terms that are not formally defined.
Point, line, plane
9.3.2.1
theorem: a statement that can be proved
Pythagorean Theorem
9.3.2.2
inverse of an if – then statement: Statement: If A then B
Inverse: If not A then not B
If I get a job then I will pay you back
If I don’t get a job, then I won’t pay
you back
9.3.2.2
9.3.2.2
9.3.2.3
converse of an if-then statement: Statement: If A then B
Converse: If B then A
If a statement is true, the inverse and converse may or
may not be true.
If I get a job then I will pay you back
contrapositive: Statement: If A then B
contrapositive : If not B then not A
If a statement is true, the contrapositive is true.
If I get a job then I will pay you back
proof by contradiction: An assumption is made that what you
are trying to prove is false. In the course of the proof, you
should come up with a contradiction, Since there is a
contradiction, the original statement must be true.
(This form of proof is rarely used in high school.)
Prove Jim is a man.
If I pay you back then I have a job
If I don’t pay you back then I don’t
have a job
Assume Jim is a woman. Jim is not
a woman, therefore he is a man.
8
9.3.3.1
transversal - a line that intersects two (or more) lines (most often
used when the two lines are parallel
two parallel lines cut by a transversal
2
1
4
3
6
5
7
8
9.3.3.2
corresponding angles: angles in the same relative position. If
two parallel lines are cut by a transversal, corresponding angles
are congruent (of equal measure).
angles 1& 5
9.3.3.2
interior angles: angles on the inside
angles 3, 4, 5, & 6
9.3.3.2
alternate interior angles: interior angles on opposite sides of the
transversal. If two parallel lines are cut by a transversal, alternate
interior angles are congruent (of equal measure).
angles 3 & 6
4&5
9.3.3.2
Same side interior angles- intertior angles on the same side of
the transversal. If two parallel lines are cut by a transversal, same
side interior angles are supplementary (add up to 180o)
angles 3 & 5
4&6
9.3.3.2
vertical angles - Given two intersecting lines, the angles that are
across from each other
angles 1 & 2 are vertical angles
2&6 3&7 4&8
1
2
9.3.3.2
complementary angles: angles that add to 90o
9.3.3.2
supplementary angles: angles that add to 180o
9.3.3.3
equilateral triangle: triangle with three sides of equal length. The
three angles of an equilateral triangle are also congruent.
9.3.3.3
isosceles triangle: a triangle with at least two sides of equal
length
9.3.3.3
scalene triangle: a triangle with no sides of equal length
9.3.3.5
45-45-90 triangle: a triangle with angles measuring 45o, 45o, &
90o
The two legs of the triangle are the same length and the
hypotenuse is the length of the leg times √2
A
B
A
B
9
9.3.3.5
30-60-90 triangle: a triangle with angles measuring 30o, 60o, &
90o
The longer leg of the triangle is the length of the short leg times √3
and the hypotenuse is the length of the short leg times 2.
9.3.3.6
congruent: same size and same shape
9.3.3.7
regular polygon – a polygon where all the sides and all the angles
are the same size
a square
9.3.4.1
acute angle: an angle whose measure is less than 90o
<
9.3.4.1
right angle: an angle whose measure is 90o
L
obtuse angle: an angle whose measure is greater than 90o but
less than 180o
9.3.4.2
sine (sin): in a right triangle it is the length of the leg opposite a
given angle divided by the length of the hypotenuse
See tangent example
9.3.4.2
cosine (cos): in a right triangle it is the length of the leg adjacent
to a given angle divided by the length of the hypotenuse
See tangent example
9.3.4.2
tangent (tan):: in a right triangle it is the length of the leg opposite
a given angle divided by the length of the leg adjacent to the angle
3
4
9.4.1.1
box and whisker plot: a graph that shows, the minimum data
value, the first quartile (25th percentile), the median, the 3rd quartile
(75th percentile) and the maximum data value
(The first quartile can also be called the lower quartile, and the thrid
quartile can also be called the upper quartile.)
sin(A )= 3/5
cos (A) = 4/5
tan (A) = 3/5
5
A
min. Q1
Q3 max.
med
9.4.1.1
measures of center: (mean, median, mode, quartile, percentile)
See earlier definitions
9.4.1.1
measures of spread: Measurements used to describe the spread
of the data
standard deviation, range (maximum
value – minimum value), interquartile
range (upper quartile – lower quartile)
10
9.4.1.3
regression line: another name for line of best fit
See earlier definitions
9.4.1.3
correlation coefficient: a number between -1 and 1 that
indicates how closely the data fits the regression line .
Negative one and positive one would
indicate a perfect fit. 0 indicates no
relationship.
9.4.1.4
normal distribution: data that fits a bell shaped curve
IQ scores
9.4.3.1
counting procedures: combinations and permutations
combinations: how many different groups one can get
from the data
permutations: similar to combinations, except a different
order counts as a different group
Using the numbers 1, 2, and 3 - the
number of combinations of two
numbers is three (1&2, 1&3, 2&3)
the number of permutations is 6 (1&2,
2&1, 1&3, 3&1, 2&3, 3&2)
9.4.3.4
Law of Large Numbers: If an experiment is repeated MANY
times, the experimental probability of an event approaches the
theoretical probability.
Probability of heads = ½
9.4.3.5
9.4.3.6
intersection: symbol ∩ used with the word AND the elements in a set that they have in common
set A= {a,b,c,d} and B = {c,d, e}
A ∩ B = {c,d
9.4.3.5
9.4.3.6
union: symbol U used with the word OR- all the elements included
in one, the other, or both sets
set A= {a,b,c,d} and B = {c,d, e}
A U B = {a,b,c,d, e}
9.4.3.7
complement of an event: the probability of an event NOT
happening
The formula is 1 – probability the event will happen.
Prob of rain = .7
Prob of NOT rain = .3
9.4.3.9
conditional probability: The probability that something will
happen given the fact that something else has happened.
Probability of buying a snack given
you have stopped for gas
11