Math

manipulatives

physical objects that can be used by students to illustrate and explore math concepts; counting cubes, pattern blocks, popsicle sticks...

models

objects, pictures, or drawings that represent a concept pictorially

proportional materials

consist of objects that are proportional to each other with respect to shape and size

non-proportional materials

materials like color counters where color is the only thing that distinguishes one from another

formal measurement

uses traditional measurement tools: rules, tape measures, meter sticks, protractors...

informal measurement

readily available objects at hand: footsteps, arm lengths, book lengths...; precedes formal measurement

formal formative assessment

quiz, paper, or tangible student work that would be graded

informal formative assessement

observation, questions, checking for understanding

informed instruction

ability to analyze where students are in their understanding of learning, misconceptions, adjusting accordingly

summative assessments

final assessment for a unit of study

pre-assessment

process of determining where students are with respect to their understanding of the prerequisites necessary to move into the new unit

levels of geometricreasoning

not age dependent
level 0- visualization (what shape looks like)
level 1- analysis (attributes)
level 2- informal deduction (relationships)
level 3- deduction (axiomatic system)
level 4- rigor (compare contrast in axiomatic system)

Polya's 4 steps in problem solving

1. understand the problem
2. make a plan
3. carry out the plan
4. analyze and reflect

deductive

reasoning in which a conclusion is reached by stating a general principle and then applying that principle to a specific case (The sun rises every morning; therefore, the sun will rise on Tuesday morning.)

indirect proof

assuming the opposite of what we are asked to prove and reasoning logically until a contradiction to a given is encountered proving our assumption wront

formal reasoning

reasoning that is deductive and uses known info

informal reasoning

reasoning that is inductive and may be true or false

Hindu-Arabic system

our number system; base 10

magnitude

the size of numbers

natural numbers

the counting numbers 1, 2, 3, 4, 5...

whole numbers

the natural numbers and 0

integers

positive and negative whole numbers and 0

rational numbers

real numbers that can be expressed as a ratio of 2 integers; integers, fractions, mixed numbers, 0, decimals... can be graphed on a number line

irrational numbers

real numbers that can't be represented as a ratio of 2 integers; infinite, non-repeating decimals

complex numbers

numbers are written in the form a +bi (a and b are real numbers and i is the imaginary unit/square of -1)

reflexive property

a = a

transitive property

if a = b and b = c, then a = c

symmetric property

if a = b, then b = a

substitution

if a + b = c and a = x, then x + b = c

closure

if the operation on 2 elements in the set always yields a member of the set; addition and multiplication (you can get a non whole number with subtraction/division)

closure for addition

if a and b are real numbers and a + b = c, then c is also real

addition property of equality

if a = b, then a + c = b + c

subtraction property of equality

if a = b, then a - c = b - c

multiplication property of equality

if a = b, then ac = bc

division property of equality

if a = b and c doesn't = 0, a/c = b/c

closure for multiplication

if a and b are real and ab = c, then c is also real

commutative property

order property; changing the order doesn't change the end results
a x b = b x a

commutative property for addition

a + b = b + a

commuutative property for multiplication

ab = ba

associative property

grouping property; when there are 2 or more occurrences of an expression that can be done in any order as long as the sequence of operations doesn't change
(a + b) + c = a + (b + c)

associative property for addition

a + (b + c) = (a + b) + c

associative property for multiplication

a(bc) = (ab)c

distributive properities

an expression can be expanded in a specific way to form a different yet equivalent expression
a(b + c) = ab + ac

additive identity

a set of numbers which contains the operation of addition; a number that when added to any element x in a set always yields x
5 + 0 = 5

identity property for addition

a + 0 = 0 + a = a

multiplicative identity

a set of numbers that involves multiplication; when you multiply the number to another number the other number is always the result
5(1) = 5

identity property for multiplication

a x 1 = 1 x a = a

additive inverse

opposite; when added to a number it yields 0
5 + -5 = 0

multiplicative inverse

reciprocal; when multiplied by x yields 1; 1/x
a/b --> b/a because a(b) / b(a) = 1

property of denseness

between any 2 rational numbers there is always at least one additional number; whole numbers aren't dense because there isn't a number between 2 whole numbers (ex. 7 and 8)

algorithm

a step-by-step procedure for doing something

implicit rules

rules where implied form elements shownarrangement where order is considere

explicit rules

determine that rule and define it algebraically

arithmetic sequence

terms have a constant difference

geometric sequence

a common scale factor (ratio) is multiplied by a term to get the next term

PEMDAS

parenthetical expressions, exponential expressions, multiplication or division (in order left to right), addition or subtraction (in order left to right)

slope-intercept form

y = mx + b

standard form

Ax + By = C

point-slope form

y - y?= m(x - x?)

parallel lines

lines that have the same slope; never intersect

perpendicular lines

lines that have slopes that are negative reciprocals of each other

direct variation

y = kx; k is the constant of variation; graphs are always linear and pass through the origin

distance

a function of rate and time; d = rt

independent linear system

system that has one, unique solution

dependent linear system

system that has no unique solutions; results when 2 equations are equivalent to each other

inconsistent linear system

system that has no solution

quadratic formula

-b�[?b�-4ac]/2a

discriminant

the ?b�-4ac part of the quadratic formula; if positive the quadratic has 2 real roots, if negative has 2 complex roots, if 0 has one solution/a double root

exponential function

y=ab^x

limit

upper boundary of the sequence

asymptote

type of limit; the line that a graph approaches but never reaches

sin

opposite side / hypotenuse

cos

adjacent side / hypotenuse

tan

opposite side / adjacent side

secant

line that intersects a curve in 2 places

tangent

line that intersects a curve at exactly one point

tessellation

figures that can be replicated to cover a serface without gaps

regular polygons

polygons with equal sides and equal angles; tessellate without dissection; equilateral triangle, a square and a regular hexagon

4 forms of symmetry in a plane

rotation, reflection, translation, and glide translation

translation

movement right or left, up or down

relfection

produces a mirror image of the original

glide reflection

involves both a translation and a reflection

rotation

done about a point, usually the origin

line of symmetry

fold line"; left and right side will mirror

dilations

cause a figure to change in size by a scale factor

transformation

change in form, shape, or location of a function

measurements of length

english: inch, foot, yard, mile
metric: millimeter, centimeter, meter, kilometer

measurements of weight/mass

english: ounce, pound, ton
metric: milligram, gram, kilogram, metric ton
(weight is affected by gravity vs mass is constant/space you occupy)

measurements of capacity

english: ounce, cup, pint, quart, gallon
metric: milliliters, liter

dimensional analysis

unit analysis; the process of converting within or between systems or solving problems with several different units by multiplying by factors of 1
ex. 12in/1 ft, 2.45cm/1 in

Pythagorean Theorem

relationships among sides of a right triangle
a�+b�=c�
c=hypotenuse=longest side of triangle/opposite right angle

point

a location in space; undefined term

plane

no depth but has infinite "surface"; undefined term

line

infinite set of points extending straight in opposite directions; undefined term

line segment

a line with specific beginning point and ending point; can be measured

ray

has an endpoint and extends infinitely in one direction

angle

formed by the intersection of 2 rays, lines, or line segments

acute angle

an angle that has a measure less than 90 degrees

right angle

an angle has a measure of exactly 90

obtuse angle

an angle that has a measure between 90 and 180

straight angle

an angle that has a measure of exactly 180

reflexive angle

an angle that has a measure between 180 and 360

collinear

points that lie on the same line

coplanar

points on the same plane

congruent angles

equal angles

complimentary angles

angles that add to 90

supplementary angles

angles that add to 180

transversal

a line that intersects both parallel lines

congruent triangles

triangles whose corresponding sides and corresponding angles have exactly the same measurements

SSS

theorem to prove triangles are congruent; each pair of corresponding sides are congruent

SAS

theorem to prove triangles are congruent; 2 pairs of corresponding sides are congruent and angles included among them are congruent

ASA

theorem to prove triangles are congruent; 2 pairs of corresponding angles are shown to be congruent and the sides included between he angles are congruent

AAS

theorem to prove triangles are congruent; 2 pairs of corresponding angles and any additional pair of corresponding sides are congruent

isosceles triangle

a triangle with two equal sides

equilateral triangle

a triangle with 3 equal sides

scalene triangle

a triangle with no equal sides

axiom

a truth that is accepted without formal proof

Euclidean Geometry

2-dimensional geometry; points, lines, angles, planes, areas...; first axiomatic system that was formally written down

postulates

truths that are accepted without proof

theorems

must be proven before they can be used

perimeter

distance around the outside of a figure
rectangle: 2 (w + l)
square: 4s
circle: c=2?r or ?d
sum of all sides

area

the amount of surface inside the figure
rectangle: lw
square: s^2
parallelogram: bh
triangle: 1/2bh
circle: ?r^2
trapezoid: [(b? +b?)/2] x h

surface area

sphere: 4?r�
cone: ?ra + ?r�
cylinder: 2B + 2?rh
rectangular prism: 2(wh + lw + lh)
triangular prism: 2B + (s?h + s?h + bh)
rectangular pyramid: B + 2(1/2wa + 1/2la)
triangular pyramid: B + 1/2 (ba + s?a + s?a)
*a = slant height

lateral surface area

sum of the areas of all faces excluding the base

volume

sphere: 4/3?r�
cone: 1/3 ?r�h
cylinder: Bh
rectangular prism: Bh
triangular prism: Bl
rectangular pyramid: 1/3 (B) h
triangular pyramid: 1/3 (B) h

slant height

distance from the apex (top) to an edge of the base

net

a 2D representation or pattern for a 3D object

unit circle

a circle with a radius of 1; hypotenuse is equal to 1

mean

average

median

middle number of a set

mode

most frequent number in a set

range

difference between highest and lowest value in a set

outlier

piece of data that lies outside the norm and will skew measures like mean but little effect on median and mode

theoretical probability

what is likely to happen in theory

experimental probability

what actually occurs and record the results

interquartile range

difference between Q3 and Q1

variance

a measure of how much individual scores differ from the mean

standard deviation

positive square root of the variance

skewness

amount of asymmetry in a distribution

positive skew

a tail that pulls to the right towards larger numbers; cause man to be greater than the median

negative skew

a tail that pulls to the left toward the smaller numbers; bunching of data at upper end of distribution

probability

# of successful outcomes/# of outcomes possible

odds

# of favorable outcomes/# of unfavorable outcomes

compound event

event occurs when 2 or more simple events are performed together; ex. pair of dice
P (A and B) = P(A) x P(B)

compound and inclusive event

events that can happen at the same time
P (A or B) = P(A) + P(B) - P (A and B)

permutation

probability where order matters
nPr=n!/(n-r)!

combination

probability where order doesn't matter
nCr=n!/r!(n-r)!

geometric probability

probability that involves comparing a part of an area to the entire area; ex. probability you'll hit the dart board
P(circle) = area of the circle/area of entire rectanble

binomial distribution

used only when exactly 2 outcomes are possible

geometric distribution

two mutually exclusive outcomes, independent trials with equal probability of success. count trials until first success

normal distribution

distribution associated with most real-word data and is commonly referred to as a bell curve

confidence interval

margin of error; tells us about the accuracy of the data from the sample