GACE Review
mgarner@kennesaw.edu
Mary Garner, Ph.D.
Professor, Department of Mathematics
and Statistics
Kennesaw State University
Gace Review
• Philosophy/overall plan
• Friends and materials
• Two tests (Number concepts, algebra, precalculus and
calculus on test 1 and geometry, measurement, data
analysis and probability on test 2).
• Test is designed around 16 objectives. Two of the objectives
will be incorporated in the review of all topics:
– Understand how to use a variety of representations to
communicate mathematical ideas and concepts and
connections between them.
– Understand mathematical reasoning, the construction of
mathematical arguments, and problem-solving strategies in
mathematics and other contexts.
GACE Review
Day 1 (Test 1)
• Understand number operations and basic principles of
number theory.
• Understand the real and complex number systems.
• Understand algebraic operations and properties of
functions and relations.
• Understand properties of linear equations, inequalities,
and systems.
• Understand properties of quadratic functions.
• Understand properties of nonlinear functions.
• Understand properties of trigonometric functions and
identities.
GACE Review
Day 2 (Test 2)
• Understand methods of collecting, organizing,
and describing data.
• Understand the theory and applications of
probability.
• Understand the process of analyzing and
interpreting data to make statistical
inferences.
GACE Review
Day 3 (Test 2)
• Understand the principles of measurement.
• Understand principles of Euclidean geometry.
• Understand coordinate and transformational
geometry.
GACE Review
Day 4
You select one or more:
– Understand principles and applications of
calculus.
OR
– Revisit problems concerning complex numbers,
parametric and polar forms, vectors, matrices.
OR
– Go back to a topic from the previous days, or a
topic other than trig or calculus that was treated
lightly or not at all on the previous days.
GACE Review
Day 1 (Test 1)
Number Concepts and Operations
• Understand number operations and basic principles of number theory.
• Understand the real and complex number systems.
Algebra
• Understand algebraic operations and properties of functions and relations.
• Understand properties of linear equations, inequalities, and systems.
• Understand properties of quadratic functions.
Precalculus
• Understand properties of nonlinear functions.
• Understand properties of trigonometric functions and identities.
GACE Review
Day 1 (Test 1)
Number Concepts
• Integers {… -4, -3, -2, -1, 0, 1, 2, …}
• Rational { p/q : p and q are integers and q is not zero}
Note: decimal representation repeating or terminating
• Irrational – not rational, so they cannot be represented in
the form p/q where p and q are integers and q is not zero
Note: decimal representation non-repeating
• Whole numbers {0, 1, 2, 3, …}
• Non-negative integers {0, 1, 2, 3, 4 …}
• Natural numbers (also counting numbers) {1, 2, 3, 4, …}
GACE Review
Day 1 (Test 1)
Number Concepts
• Prime {x : x is not 1 and x is divisible by only x and 1}
• Fundamental Theorem of Arithmetic -- Every positive
integer (except 1) is either prime or can be uniquely
factored as a product of primes
• Real numbers – all rational and irrational numbers
Note: representation on a real number line
• Complex numbers – numbers than can be written in the
form a +bi where a and b are real and i is defined as the
square root of -1
Note: representation in the plane, vector form…
GACE Review
Day 1 (Test 1)
Operations and Algebra
• Properties: (see next slide)
•
•
•
•
•
•
closure
commutative
distributive
inverse
identity
associative
• Complex numbers
•
•
•
•
modulus
conjugate
Trig form, rectangular form
Powers of i
From:
http://www.jamesbrennan.o
rg/algebra/numbers/propert
ies_of_real_numbers1.htm
GACE Review
Day 1 (Test 1)
Algebra
• Simplifying and evaluating algebraic equations and solving
equations – applying properties of real numbers.
• Direct and inverse proportional relationships – specific case
of a linear function.
• Binomial theorem and Pascal’s triangle.
GACE Review
Day 1 (Test 1)
Precalculus=FUNCTIONS
• Linear, quadratic, absolute value, square root, exponential,
logarithmic, polynomial, rational , trigonometric
• Representation in graphs, tables, algebra.
• Domain, range, zeroes, intercepts, intervals of increase or
decrease, maxima and minima, end behavior, symmetry ,
inverse, asymptotes, rate of change
• Composition of functions
• Transformations of functions
• Sequences as functions/recursive form
Linear
F(x) = mx + b
• Largest possible domain: all real numbers
(If domain is {1, 2, 3, …} then function is an arithmetic sequence.)
• Largest possible range: all real numbers as long as m is not zero
(If domain is limited then range will be limited.)
• Zeroes (x intercepts): x = -b/m
• Y intercepts: y = b
• Intervals of increase or decrease: increasing over all real numbers if m is
positive, decreasing over all real numbers if m is negative
• Maxima and minima: none as long as domain is not limited.
• Rate of change: constant
• End behavior
• Symmetry
• Inverse
GACE Review
Day 1 (Test 1)
Transformations
•
•
•
•
Horizontal shift: f(x+c)
Vertical shift: f(x) + c
Vertical stretch or shrink: af(x)
Horizontal stretch or shrink: f(bx)
F(x) = 2x
G(x) = 2(x-3)
F(x) = 2x
G(x) = 2x + 3
F(x) = 2x
G(x) = 3(2x )
F(x) = 2x
G(x) = 2(3x)
F(x) = 22x-4 = 22(x-2)
G(x) = 2(2(x-4))
Start with F(x)
aF(b(x-c)) + d represents what transformations?
GACE Review
Day 1 (Test 1)
Matrices
• Representation
• Operations – addition, multiplication
• Law of large numbers –
– I rolled a die twelve times: 6 6 1 4 3 5 1 3 6 2 5 6
– What is the empirical probability of rolling a 6? Is
it the same as the theoretical probability?
• Central Limit Theorem
– Suppose I repeatedly rolled a die for a really long
time and then took the mean of all the results?
What would be the “expected value”?
– Suppose instead, I roll a die four times, take the
mean, and do this again 40 times.
My results: 2316, 1146, 6221, 1513, 2616, 4265,
2266,1156,4216,2312,6344,1115,5631,3654,4
655,4324,6133, 2563, 3244,5536,4335,…
Mean of 3, 3, 2.75, 2.5, 3.75, 4.25, 4,3.25, 3.25,
2, 4.25, 2, 3.75, 4.5, 5, 3.25, 3.25, 4, 3.25,
4.75, 3.75, …
Statistics
Suppose we want to estimate the average weight of an adult
male in Dekalb County, Georgia. We draw a random sample
of 1,000 men from a population of 1,000,000 men and
weigh them. We find that the average man in our sample
weighs 180 pounds, and the standard deviation of the
sample is 30 pounds. What is the 95% confidence interval.
• (A) 180 + 1.86
(B) 180 + 3.0
(C) 180 + 5.88
(D) 180 + 30
(E) None of the above
GACE Review
Day 3
• Data and Analysis and Probability answers:
• GACE and PRAXIS – note #67 on GACE is also
#16 on Praxis
• Note page 203 GACE birthday problem #72
http://mathforum.org/dr.math/faq/faq.birthd
ayprob.html
Birthday Problem
• “The first person can have any birthday. That gives him 365 possible
birthdays out of 365 days, so the probability of the first person
having the "right" birthday is 365/365, or 100%.
• The second person's birthday has to be different. There are 364
different days to choose from, so the chance that two people have
different birthdays is 364/365. That leaves 363 birthdays out of 365
open for the third person.
• To find the probability that both the second person and the third
person will have different birthdays, we have to multiply:
(365/365) * (364/365) * (363/365) = 132 132/133 225,
which is about 99.18%.
• If we want to know the probability that four people will all have
different birthdays, we multiply again:
(364/365) * (363/365) * (362/365) = 47 831 784/ 48 627 125,
or about 98.36%.”
Suppose a committee of 12 people wants to
form a subcommittee of 3 people. How many
ways can this be done?
Suppose a president, vice-president and
secretary must be selected from a committee
of 12 people?
GACE Review
Day 3
• Data and Analysis and Probability answers:
• GACE and PRAXIS – note #67 on GACE is also
#16 on Praxis
• Note page 203 GACE birthday problem #72
http://mathforum.org/dr.math/faq/faq.birthd
ayprob.html
• Measurement and Geometry
GACE Review
Day 3
Measurement and Geometry
– Parallel lines (2)
– Triangles – congruence & similarity (GACE 57, Ga
constructed response, 15 & 16)
– Circle theorems (and radians vs. degrees) (18) (GACE
56 57)
– 3 D (3)
– Perimeter, area, volume (1, 4, 5)
– Conic sections (24, 25, 27, 22, GACE 63 64)
– Law of sines, law of cosines, Pythagorean theorem,
trig…solving triangles
GACE Review
Day 4
Measurement and Geometry
– Conic sections (24, 25, 27, 22, GACE 63 64)
– Law of sines, law of cosines, Pythagorean theorem,
basic trigonometry…solving triangles
– The unit circle
Calculus
– Limits
– Derivatives
– Integrals
GACE Review
Day 4
http://www.stewartcalculus.com/data/CALCULUS%206E%20Early%2
0Transcendentals/upfiles/ess-reviewofconics.pdf
GACE Review
Day 4
• A circle can be defined as the set of points equidistant from
a “center.”
• A parabola can be defined as all the set of points
equidistant from a “directrix” and a “focus”.
http://mathworld.wolfram.com/Parabola.html
• An ellipse can be defined as the set of all points such that
the SUM of the distances from the point to each “foci” is
constant.
http://mathworld.wolfram.com/Ellipse.html
• A hyperbola can be defined as the set of all points such that
the DIFFERENCE of the distances from the point to each
“foci” is constant.
http://mathworld.wolfram.com/Hyperbola.html
GACE Review
Day 4
Measurement and Geometry
– Conic sections (24, 25, 27, 22, GACE 63 64)
– Law of sines, law of cosines, Pythagorean theorem,
basic trigonometry…solving triangles
– The unit circle
Calculus
– Limits
– Derivatives
– Integrals
GACE Review
Day 4
Given two sides OR a side and an angle of a RIGHT
TRIANGLE, we can find the measures of the other sides
and angles.
If one of the acute angles of a right triangle is congruent
to the acute angle of another right triangle, then the
two triangles are similar. Then the ratios of sides for all
such triangles are exactly the same – so we define sine
and cosines and tangent as ratios of these sides (given
the angle). Sine of an acute angle in a right triangle is
the length of the opposite side over the hypotenuse.
Cosine is the length of the adjacent side over the
hypotenuse. All the other trig functions can be defined
in terms of sine and cosine.
GACE Review
Day 4
Given two sides OR a side and an angle of a
RIGHT TRIANGLE, we can find the measures of
the other sides and angles.
GACE 33, 35
GACE Review
Day 4
Given two sides and the included angle (SAS) OR
three sides (SSS) OR an angle and its opposite
side and one other angle (AAS) OR two angles
and the included side (ASA) of ANY TRIANLGE,
we can find the measures of the other sides
and angles using law of sines and law of
cosines. (Recall ASS is the ambiguous case.)
http://online.redwoods.cc.ca.us/instruct/darnold/HSU/Math115/Spring01/Exa
ms/Exam3/exam3-s.pdf
GACE Review
Day 4
Measurement and Geometry
– Conic sections (24, 25, 27, 22, GACE 63 64)
– Law of sines, law of cosines, Pythagorean theorem,
basic trigonometry…solving triangles
– The unit circle
Calculus
– Limits
– Derivatives
– Integrals
GACE Review
Day 4
The unit circle:
We need to define sine and cosine for angles
other than acute angles! So we change our
definition of sine and cosine to the x and y
coordinates of the unit circle. Note that this
new definition is consistent with the old one,
but extends the possible domain of the sine,
cosine, tangent, …
GACE Review
Day 4
Polar Coordinates:
Convert to rectangular: (r,theta) (6, 112
degrees)
Convert to polar: (x,y) (-3,-4)
Sine and Cosine functions:
FIRST packet p. 38 #28
GACE Review
Day 4
Measurement and Geometry
– Conic sections (24, 25, 27, 22, GACE 63 64)
– Law of sines, law of cosines, Pythagorean theorem,
basic trigonometry…solving triangles
– The unit circle
Calculus
– Limits
– Derivatives
– Integrals
GACE Review
Day 4
Calculus
– Limits – A limit is the number F(x) approaches (the
y value) as x approaches some number or goes to
positive or negative infinity.
Calc packet #2
– Continuity – A function is continuous at a point x –
a if the limit of F(x) as x approaches a is F(a).
Calc packet #3, GACE 36
GACE Review
Day 4
Calculus
– Derivatives – A derivative is the slope of the line tangent to
the curve at a point. It is the instantaneous rate y is
changing per unit change in x. (So you’re already intimate
with derivatives of linear functions …) Recall that the
second derivative is acceleration.
– If the function is increasing, the first derivative is positive;
if the function is decreasing, the first derivative is negative.
If the function is concave up (smile), the second derivative
is positive; if the function is concave down (frown), the
second derivative is negative.
Calc packet 1, 6, 7, 8, 10
GACE 39, 40, 41, 42
GACE Review
Day 4
Calculus
– Integrals – An integral IS the area under a curve.
When we want the integral of a function we need
its “antiderivative”.
Calc packet #9
GACE 43,44, 45, 46, 47, 48, 49
GACE Review
I had fun!!! Thank you for all your hard work.
Please don’t hesitate to call me or email me
with more questions and a report on how you
do on the GACE. Please, I want to know and I
want to help!
mgarner@kennesaw.edu
770-423-6664