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