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GACE Review [email protected]

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GACE [email protected]

Mary Garner, Ph.D.

Professor, Department of Mathematics and Statistics

Kennesaw State University

- 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.

- 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.

- 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.

- Understand the principles of measurement.
- Understand principles of Euclidean geometry.
- Understand coordinate and transformational geometry.

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.

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.

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, …}

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…

Operations and Algebra

- Properties: (see next slide)
- closure
- commutative
- distributive
- inverse
- identity
- associative

- modulus
- conjugate
- Trig form, rectangular form
- Powers of i

From: http://www.jamesbrennan.org/algebra/numbers/properties_of_real_numbers1.htm

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.

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

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

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?

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, 2266,1156,4216,2312,6344,1115,5631,3654,4655,4324,6133, 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, …

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

- 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.birthdayprob.html

- “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?

- 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.birthdayprob.html

- Measurement and Geometry

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

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

http://www.stewartcalculus.com/data/CALCULUS%206E%20Early%20Transcendentals/upfiles/ess-reviewofconics.pdf

- 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

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

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.

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

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/Exams/Exam3/exam3-s.pdf

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

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, …

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

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

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

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

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

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!

770-423-6664