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Use the comment section to post a biography fact about John Wilder Tukey. No repetition of facts please. The first to post a fact, gets credit for that fact. Check back to see if your fact has been accepted.
August 31, 2009
Pre-Calculus Assignment List — Chapter 1
Pre-Calculus
Chapter 1
1. Book 1.1 page 9: CV: 1 – 6; E: 2, 25, 29, 39, 43, 49, 53, 59, 63, 65, 66, 67, 68
2. Book 1.2 page 19: CV: 1 – 4; E: 1, 3, 5, 7, 8, 10, 11, 13, 14, 15, 16, 21, 23, 27, 30
3. Book 1.3 page 30: CV: 1 – 6; E: 1, 3, 9, 11 – 25, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 47, 53, 54, 55, 56, 57, 58, 61, 63, 64, 65, 66, 67, 68, 77
4. Book 1.6 page 71: CV: 1, 2, 4, 5; E: 1, 3, 5*, 6*, 9*, 11*, 12*, 13*, 15*, 17*, 19*, 23, 35 – 58, 59*, 65*, 67*, 71*, 73*, 75*, 77*, 79, 80, 89, 91, 93(do not graph), 95(do not graph), read 109 and 110
*graph these*
5. Review 1.1 – 1.3, 1.6 page 78: 53, 54, 55, 57, 60, 63, 67, 68, 69, 72, 73, 75 – 89, 91, 92, 93
TEST
6. Book 1.4.1 page 43: CV: 1, 2, 3; E: 1, 11, 15, 18, 20, 21, 25, 28, 29, 31, 33, 35, 37, 39, 40, 41, 43, 45, 47, 49, 50, 52, 53, 56, 59
7. Book 1.4.2 page 43: E: 63, 65, 67, 69, 71, 73, 76, 77, 79, 81, 82, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 101, 103, 105, 107 (first solve for x, then a, then b), 108 (x, a, 1), 112
8. Handout 1410
9. Book 1.5 page 56: CV: 1; E: 1, 2, 3, 4, 5, 6, 8, 13, 23, 43, 45, 47, 49, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 71, 77, 81, 83, 85, 86, 88, 91, 92
10. Handout 1510
11. Review 1.4 – 1.5 page 77: 1, 3, 5, 7, 9, 11, 12, 13, 14, 16, 19, 22, 25, 26, 31, 39, 42, 47, 49, 94
TEST
August 28, 2009
Lesson Plans August 31 – September 4, 2009
Lesson Plans August 31 – September 4, 2009
Pre-Calculus
Standards: Models for Real World Phenomena; Algebraic Functions; Trigonometric Functions; Sequences and Series
Monday, August 31: 1.1 Use the distance formula and midpoint formula
Tuesday, September 1: 1.2 Graph equations by plotting points; Graph equations using a graphing utility; Find intercepts
Wednesday, September 2: 1.3 Test an equation for symmetry with respect to the x-axis, the y-axis, and the origin; Know how to graph key equations; Write the standard form of the equation of a circle; Graph a circle by hand and by using a graphing utility; Find the center and radius of a circle from an equation in general form
Thursday, September 3: 1.3 Test an equation for symmetry with respect to the x-axis, the y-axis, and the origin; Know how to graph key equations; Write the standard form of the equation of a circle; Graph a circle by hand and by using a graphing utility; Find the center and radius of a circle from an equation in general form
Friday, September 4: 1.6 Calculate and interpret the slope of a line; Graph lines given a point and the slope; Find the equation of vertical lines; Use the point-slope form of a line; Identify horizontal lines; Find the equation of a line given two points; Write the equation of a line in slope-intercept form; Identify the slope and y-intercept of a line from its equation; Write the equation of a line in general form; Define parallel lines; Find equations of parallel lines; Define perpendicular lines; Find equations of perpendicular lines
Geometry
Standards: Number and Operations; Algebra; Geometry; Measurement; Data Analysis and Probability
Monday, August 31: 2.1 Recognize conditional statements; Write converses, inverses, and contrapostives of conditional statements; Write biconditional statements
Tuesday, September 1: 2.2 Write biconditionals; Recognize good definitions
Wednesday, September 2: 2.3 Use the Law of Detachment and Law of Syllogism
Thursday, September 3: TEST Chapter 2 Sections 1, 2, 3, 5
Friday, September 4: 2.4 Connect reasoning in algebra and geometry
Statistics
Standards: Experimental Design; Data Analysis
Monday, August 31: 2.2 How to graph and interpret quantitative data sets using stem-and-leaf plots and dot plots; How to graph and interpret qualitative data sets using pie charts and Pareto charts; How to graph and interpret paired data sets using time series charts
Tuesday, September 1: 2.2 How to graph and interpret quantitative data sets using stem-and-leaf plots and dot plots; How to graph and interpret qualitative data sets using pie charts and Pareto charts; How to graph and interpret paired data sets using time series charts
Wednesday, September 2: 2.3 Find the mean, median, and mode of a population and a sample; Find a weighted mean and the mean of a frequency distribution; Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each
Thursday, September 3: 2.3 Find the mean, median, and mode of a population and a sample; Find a weighted mean and the mean of a frequency distribution; Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each
Friday, September 4: 2.3 Find the mean, median, and mode of a population and a sample; Find a weighted mean and the mean of a frequency distribution; Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each
For all classes.
Activities: Lecture, Board work, Classroom participation
Material: Book problems and Teacher made handouts
Assessment: Daily assignments graded for accuracy. Unit test planned.
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Geometry Notes — Conditionals and Other Statements
A mathematical sentence is one in which a fact or complete idea is expressed. Because a mathematical sentence states a fact, many of them can be judged to be true or false.
Conditional – a statement that clearly states a hypothesis and a conclusion in an if-then form
Conditionals are statements that say if one thing happens, another will follow. Conditionals are symbolically written as p ⟹ q. The ⟹ stands for “implies.” The p stands for the hypothesis, the given or the problem. It is the first part of the conditional. The q stands for the conclusion, the prove, or the answer. It is the “then” part of the if-then statement.
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The definition of a right angle is… an angle that measures exactly 90.
Here is that definition written as a conditional: If an angle is a right angle, then it measures exactly 90.
hypothesis (p): an angle is a right angle
conclusion (q): it measures exactly 90
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Here is the definition of perpendicular lines written as a conditional: If lines are perpendicular, then they form a right angle.
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And here is something silly: If an animal is a whale, then it is a mammal.
If a mathematical sentence is incorrect you must provide a counterexample, which is a single example used to prove a statement false.
An if-then statement or conditional statement is a statement formed when one thing implies another, but not necessarily the other way around.
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Converse – a statement formed by interchanging a conditional’s hypothesis and conclusion. Symbolically this is written as q ⟹ p. q implies p.
Here is the definition of a right angle written as a converse statement: If an angle measures exactly 90, then it is a right angle.
hypothesis: an angle measures exactly 90
conclusion: it is a right angle
The definition of perpendicular lines written as a converse statement: If lines form a right angle, then they are perpendicular.
And then the silly: If an animal is a mammal, then it is a whale.
The two boring math definition are still true statements when written as converse statements. The silly statement is incorrect. To show that, I would present a single counterexample such as a pony — a pretty pony. I could have said geometry student. A pony and a geometry student are both fine examples of creatures that are mammals but are not whales.
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Biconditional – a statement formed by combining a true conditional with a true converse in an if and only if form. Symbolically this is written as p ⟺ q.
The definition of a right angle written as a biconditional: An angle is a right angle if and only if it measures 90.
The definition of perpendicular lines written as a biconditional: Lines are perpendicular iff they form a right angle.
No biconditional can be written about the mammal and whale statement because the statement’s converse was not true.
All geometry definitions can be written as biconditionals.
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A point is a midpoint iff it divides a segment into two congruent segment.
Remember, I like definition that clearly state what the item is followed why that item is important.
A midpoint is first and foremost a point… with a responsibility.
A angle bisector is a ray first… then it does something.
An obtuse angle is first an angle… blah blah blah.
The hypothesis of the conditional and the first part of the biconditional is what the item is first and foremost. The conclusion of the conditional and the second part of the biconditional is the result, the responsibility, the action, the thing that makes the item super special to us all.
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Inverse – a statement formed by negating the original conditional’s hypothesis and conclusion. Symbolically, the inverse is written as ~p ⟹ ~q. The ~ symbol mean “not”. ~p ⟹ ~q is read “not p implies not q.”
Here is the definition of a right angle written as an inverse statement: If an angle is not a right angle, then it does not measure 90.
The definition of perpendicular lines written as an inverse statement: If lines are not perpendicular, then they do not form a right angle.
And again the silly: If an animal is not a whale, then it is not a mammal.
The boring geometry definitions are still true. The silly whale/mammal one is not. A salamander is not a whale and it is not a mammal.
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Contrapositive – a statement formed by interchanging a conditional’s hypothesis and conclusion and by negating both. Symbolically a contrapositive is written as ~q ⟹ ~p. not q implies not p.
Here is the definition of a right angle written as a contrapositive statement: If an angle does not measure 90, then it is not a right angle.
The definition of perpendicular lines written as a contrapositive statement: If lines do not form a right angle, then they are not perpendicular.
And the silly one: If an animal is not a mammal, then it is not a whale.
If you’re thinking about the truth of these statements, you find that they are all true.
The truth of mathematical statement appear in pairs. Conditional, Converse, Inverse, Contrapositive — 4 statements. In pairs, either
- all are true
- 2 are true, 2 are false
- or, all are false
The “verses” go together when it comes to truth. Conditionals and contrapositives also go together when it comes to truth.
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Geometry Notes / Assignment — History — Euclid
The word geometry comes from the Greek word geometrein (geo meaning earth, and metrein meaning to measure). Geometry was originally the science of measuring the land.
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid’s text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history. His method consists of assuming a small set of postulate, and then proving many theorems from those postulate. Although many of Euclid’s results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system.
Euclid was a disciple of the Platonic school, founded by Plato in 387 B.C. in Athens. Neither the year nor place of Euclid birth have been established, nor the circumstances of his death, although he is known to have lived and worked in Alexandria for much of his life. Around 300 B.C. he produced the definitive treatment of geometry in his 13-volume Elements. In compiling his masterpiece Euclid built on the experiences and achievements of his predecessors — Pythagoras, Archytas, Eudoxus, and Theaetetus.
The book synthesized earlier knowledge about geometry, and was used for centuries in western Europe as a geometry textbook. The text began with definitions, postulates, and common opinions, then proceeded to obtain results by rigorous geometric proof. Euclid used the method of proof by exhaustion.
The Elements begins with definitions and five postulates. The first three postulates are postulates of construction. For example the first postulate states that it is possible to draw a straight line between any two points. These postulates also implicitly assume the existence of points, lines and circles and then the existence of other geometric objects are deduced from the fact that these exist. There are other assumptions in the postulates which are not explicit. For example it is assumed that there is a unique line joining any two points. Similarly postulates two and three, on producing straight lines and drawing circles assume the uniqueness of the objects of whose construction is being postulated. The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem obvious but it actually assumes that space in homogeneous — meaning that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid’s decision to make this a postulate led to Euclidean geometry. It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied.
The Elements also included five “common notions”: 1) Things that equal the same thing also equal one another. 2) If equals are added to equals, then the wholes are equal. 3) If equals are subtracted from equals, then the remainders are equal. 4) Things that coincide with one another equal one another. 5) The whole is greater than the part.
The Elements is divided into 13 books.
Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares.
Book three studies properties of the circle.
Book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras.
Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes.
Book six looks at applications of the results of book five to plane geometry.
Books seven to nine deal with number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.
Book eight looks at numbers in geometrical progression.
Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.
Books eleven to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly similar pattern to the two-dimensional analogues previously given in books one and four.
The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using the “method of exhaustion” as invented by Eudoxus.
The Elements ends with book thirteen which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five. This book appears to be based largely on an earlier treatise by Theaetetus.
Little is known about Euclid other than his writings. He is considered the Father of Geometry. Euclid may not have been a first class mathematician, meaning that he did indeed build on the findings of other math greats, but the long lasting nature of The Elements makes him the leading mathematics teacher of ancient times or perhaps of all time. Outstanding.
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Statistics Notes / Assignment — Biography — Vilfredo Pareto
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Use the comment section to post a biography fact about Vilfredo Pareto. No repetition of facts please. The first to post a fact, gets credit for that fact. Check back to see if your fact has been accepted.
August 26, 2009
Geometry Notes — Special Pairs of Angles
Complementary angles – two angles whose measures have a sum of 90°.
Look at the right angle above, ∠1 and ∠2 are complementary. ∠1 and ∠2 are also adjacent. ∠1 and ∠2 are each others’ complement.
∠3 and ∠4, above, are not adjacent, but if m∠3 + m∠4 = 90°, then ∠3 and ∠4 are complementary. They are each others’ complement.
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In the figures above, the two angles ∠PQR and ∠JKL are complementary because they always add to 90°. Often the two angles are adjacent, in which case they form a right angle.
Supplementary angles – two angles whose measures have a sum of 180°.
Look at the figure above, ∠5 and ∠6 are supplementary. ∠5 and ∠6 are also adjacent. ∠5 and ∠6 are each others’ supplement.
∠7 and ∠8, above, are not adjacent, but if m∠7 + m∠8 = 180°, then ∠7 and ∠8 are supplementary. They are each others’ supplement.
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In the figures above, the two angles ∠PQR and ∠JKL are supplementary because they always add to 180°. Often the two angles are adjacent, in which case they form a straight angle.
Linear Pair – two adjacent angles whose non-common sides form a straight line
∠5 and ∠6 above form a linear pair. They are adjacent, they are supplementary, they form a linear pair.
2 important ideas about Linear Pairs
1. Linear Pair Postulate – The two angles that form a linear pair are supplementary.
2. Theorem – The sum of the measures of the two angles that form a linear pair is 180°.
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In the figures above, the two angles ∠JKM and ∠LKM form a linear pair. They are supplementary because they always add to 180° and because they are adjacent, the two non-common legs form a straight line segment JL.
Vertical angles – two non-adjacent angles formed by a pair of intersecting lines
Lines a and e intersect. ∠1 and ∠3 are vertical angles. Another pair is ∠2 and ∠4.
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As can be seen from the figures above, when two lines intersect, four angles are formed. Each opposite pair are called vertical angles and are always congruent. The red angles ∠JQM and ∠LQK are congruent, as are the blue angles ∠JQL and ∠MQK.
Theorem: Vertical angles are congruent.
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