Class Expectations

Geometry/Trigonometry/Algebra 3


According to Professor Lee of Washington University, "Mathematics is the single field of human endeavor in which we are the most certain of the correctness of our knowledge.  We use algebra everyday.

The main goal of this class is to help you acquire a deep understanding of and appreciation for  algebra. You will learn to think about it in a more confident and focused manner.

A secondary goal of this class is to help you become adept at mathematical communication. Opportunities to improve your communication skills on several levels will arise throughout the class (see below for details): speaking precisely about mathematical ideas in class; writing informally about mathematical ideas on the A2A Blog; and writing careful mathematical conclusions on homework assignments.

Grading Setup
o   

Test approx 40% up to 100 points each
o    Benchmarks approx. 10% (min.  20) up to 5 points each
o    Google Docs Total approx.  20%(HOMEWORK approx. 12.5%, NOTES approx. 7.5%)
o    Blog Total approx. 7%
o    Internet approx. Total 4%
o    Final Exam approx 20%
•    =Total approx 101%


Homework

All homework will be submitted through Google docs for grading. You should include your name, lesson number, and the assigned problems.

You must submit a minimum of 25 assignments for homework on docs.google.com for credit.  Any extra assignments submitted will contribute towards up to 5 extra points per assignment.  You must check each problem (option + v) has check mark. Chapter Reviews will be worth 6 points each if completed!

Homework will be graded as follows out of 5 points:

1 did something

2 did half

3 did most

4 did all

4.5 did all and checked

5 did all and formatted correctly

Notes

For notes, you should include your name, the pages, and the lesson title.

You must submit a minimum of 15 notes on docs.google.com for credit. You will get your notes from information given in class as well as the assigned reading from the lesson.

Notes will be graded as follows out of 5 points:

1 did something

2 half information is covered

3 most of material covered

4 All material Covered

5 Neat, Images included, organized

Blog

Significant Contributions approx 2% (min. 2 per unit)

Tag all contributions with your name for credit.  Each Posting is worth 2 points.

Comments approx 1%  (min. 2 per unit)

Tag all comments with your name for credit. Each quality comment is worth 1 point.

Posting for Points approx. 4%

Each comment is worth 5 points.

Internet -Details to follow

Youtube Review Video approx. 2%

Podcast Review of a Lesson approx 2%

Attendance
You must come to class if you are at school. If you are not here when I take attendance you will be marked absent. If you arrive late I will change the mark to tardy. If you have a note your tardiness will be excused. (Note: excessive absences can prohibit you from passing the class) For every unexcused absence you will lose a whole point from your final average. For every three unexcused tardies you will lose a whole percentage point to your final average.

If you miss class you are still responsible for the homework. The best way to prepare for a trip is to get your assignment in advance.

BEHAVIOR
Disruptive or offensive behavior will not be tolerated. Offensive language is not used in my room.

NO CHEATING! Do not roam your eyes around the room. Please keep your eyes on your own paper to avoid even the appearance of looking at the answers of classmates it may be mistaken as cheating.

CELL PHONES
Cell phones must be turned off and put away at all times during class. A violation of this policy will result in the removal of the phone and it will be given to Paul Hayward.

EXTRA HELP
We all need a little extra help every now and then. While every effort will be made to answers questions during class, there may be times when you need further clarification. I will be regularly available to help students with math during:

Wednesday @ Lunch

Please contact me as soon as you become uncomfortable and we will find a time even outside of these times to get you help. I’m more than willing to work with you, but I also expect you to make the effort and show me you are invested in learning. If you are having trouble with any part of my class, let me know and I will help you. I am a big helper, that’s my job.

The Path to Success!!

You can succeed in math and in this course if you:

1. Respect everyone in the class at all times. We all work at different paces and use different methods. Please respect those differences.
2. Always be on time and prepared for class.
3. Have read the lesson and done the homework.
4. Show all your work
5. Check your work carefully so that you know what’s hard for you and mark the items on your homework that you want to discuss in class.
6. Review your class notes every night before going to bed.
7. Always get extra help from the teacher when you feel you are falling behind.
8. Find one or two people to be your study partners and form a study group.
9. Make the class work for you by making sure you get your questions answered, and listening to what other students have to say. Your classmates have great ideas, sometimes better than my ideas, so, be attentive. They may also ask questions you want to ask. Listen to your classmates.
   
10. Participate regularly on the class blog
    
     

How to get an A……

If your percentage of accomplishment is higher than 92% you get an A.

Or, if your total points for the term reach 1510 Points you will receive an A.

Extra Points

2 Paideia Math Tweets (min 4, .5 points)

50 Flickr Diary (up to 50 points)

36 Test Aftermath 9 points (required, unless you score higher than a 91 on the test)

Scribing 5 points (no min.)

Tutorial 5 Points  (no min.)

20 BOB  5 points (max. number of tests times 5 points)

Challenges (varying point totals)

Accurately Completed Scoring Spreadsheet 25 points

Web Page Answer top rank  (up to 20 points)

1 point Extra Comments

1 point Extra Significant Contributions

Amp & SHift

http://www.youtube.com/watch?v=bow4hGXbqZ4

showing how to move the the equation w/ the amp. & shift

final!!!!!!!

The law of sines

sin A
a = sin B
b = sin C
c
This can also be interpreted as three equations:

sin B
b = sin C
c , sin A
a = sin C
c , and sin A
a = sin B
b

The law of cosines

c2 = a2 + b2 – 2ab cos C
There are two other versions of the law of cosines,

a2 = b2 + c2 – 2bc cos A and b2 = a2 + c2 – 2ac cos B.

n a circle whose radius is 10 cm, a central angle θ intercepts an arc of 8 cm.

a) What is the radian measure of that angle?
Answer. According to the definition:
θ = s
r = 8
10 = .8
b) At that same central angle θ, what is the arc length if the radius is
b) 5 cm?

Answer. For a given central angle, the ratio of arc to radius is the same. 5 is half of 10. Therefore the arc length will be half of 8: 4cm.
Converting radians and degrees
Convert 200° into radian measure:
200° (Π/180°) = 200/180Π radians or 3.49 radians
Convert 1.4 radians into degrees: 1.4 (180°/Π) = 80.2 °

http://www.math.uakron.edu/~tprice/Trig/Graphs.pdf Great powerpoint on sine and cosines waves.

Scribe Post review 5.1-5.3 Nigel.

So in class tuesday we looked over some problems that could possible be on the test for a while then we had time to work on our own once we asked all of our questions.

The ambiguous case
The so-called ambiguous case arises from the fact that an acute angle and an obtuse angle have the same sine. If we had to solve
sin x = ½,
for example, we would have
x = 45° or x = 135°.
(Topic 6, Example 1.)
In the following example, we will see how this ambiguity could arise.

In triangle ABC, angle A = 30°, side a = 1.5 cm, and side b = 2 cm. Let us use the law of sines to find angle B.
sin B
sin 30° = 2
1.5

Since sin 30° = ½ (Topic 7, Example 2),
sin B = ½· 20
15
= 10
15
= 2
3
.666
On inspecting the Table for the angle whose sine is closest to .666, we find
B42°.
But the sine of an angle is equal to the sine of its supplement. That is, .666 is also the sine of 180° − 42° = 138°.
This problem has two solutions. Not only is angle CBA a solution,

but so is angle CB'A, which is the supplement of angle CBA. (We can see that it is the supplement by looking at the isosceles triangle CB'B; angle CB'A is the supplement of angle CB'B, which is equal to angle CBA.)
Given two sides of a triangle a, b, then, and the acute angle opposite one of them, say angle A, under what conditions will the triangle have two solutions, only one solution, or no solution?

Let us first consider the case a < b. Upon applying the law of sines, we arrive at this equation:
1) sin B = sin A· b
a .
Now, since h
b = sin A , where h is the height of the triangle (Fig. 1),
then
b sin A = h.
On replacing this in the right-hand side of equation 1), it becomes
sin B = h
a .
There are now three possibilities:

h
a < 1, which implies h < a (Fig. 1),
h
a = 1, which implies h = a (Fig. 2),
h
a > 1, which implies h > a (Fig. 3).
In the first of these -- h or b sin A < a -- there will be two triangles.
In the second -- h or b sin A = a -- there will be one right-angled triangle.
And in the third -- h or b sin A > a -- there will be no solution.

The future!

Study video

I hope this video helps to explain drawing cosine and sine functions on a graph.

http://www.youtube.com/watch?v=Lsea3tMN99k

How to know if an equation is even, odd, or neither

http://www.youtube.com/watch?v=7cWoQJydTgw

. simple. but helpful . :)

What you need for your FINAL!

I will provide each student with a sheet of formulas including the trigonometric identities and the unit circle. Students are responsible for all other formulas and equations.

Each student will be allowed one sheet from twitter at the normal font size.

You will need a calculator for the final along with 2 pencils.

I will see you all in class....Good Luck!

Review

Law of sines

Law of cosines

Area of a triangle

Area of a triangle using Heron's formula

(we did not learn this one in class but i thought it might be useful)

You can use herons formula if you are given all sides and no angles

where  

Review

Remember SOH CAH TOA to solve right triangles.

Also remember special right triangles.

30º-60º-90º Triangles

45º-45º-90º Triangles

Review of the Ambiguous Case

SAS - Side, Angle, Side ASA - Angle, Side, Angle AAS - Angle, Angle, Side SSS - Side, Side, Side HL - Hypotenuse Leg for Right Triangles.

We also discovered that SSA did not work to prove triangles congruent. We politely called it the Donkey Theorem ; - )

By definition, the word ambiguous means open to two or more interpretations.
Such is the case for certain solutions when working with the Law of Sines.

	• If you are given two angles and one side (ASA or AAS), the Law of Sines will nicely provide you with ONE solution for a missing side.
	• Unfortunately, the Law of Sines has a problem dealing with SSA. If you are given two sides and one angle (where you must find an angle), the Law of Sines could possibly provide you with one or more solutions, or even no solution.

Before we investigate this situation, there are a few facts we need to remember.

Facts we need to remember: 1. In a triangle, the sum of the interior angles is 180º. 2. No triangles can have two obtuse angles. 3. The sine function has a range of . 4. If the = positive decimal can lie in the first quadrant (acute <) or in the second quadrant (obtuse <).

Let's look at some cases. In each example, decide whether the given information points to the existence of one triangle, two triangles or no triangles.

Example 1: In , a = 20, c = 16, and m = 30º. How many distinct triangles can be drawn given these measurements?

Use the Law of Sines:

C = sin^-1 (0.4) = 24º (to the nearest degree) - in Quadrant I.
Sine is also positive in Quadrant II. If we use the reference angle 24º in Quadrant II,
the angle C is 156º.
But, with m = 30º and m = 156º the sum of the angles would exceed 180º.
Not possible!!!!

Therefore, m = 24º, m = 30º, and m= 126º and only ONE triangle is possible.

Example 2: In , a = 7, c = 16, and m = 30º. How many distinct triangles can be drawn given these measurements?

Use the Law of Sines:

Since sin C must be 1, no angle exists for angle C.
NO triangle exists for these measurements.

Example 3: In , a = 10, b = 16, and m = 30º. How many distinct triangles can be drawn given these measurements?

Use the Law of Sines:

B = sin^-1(.8) = 53.13010 = 53º.
Angles could be 30º, 53º, and 97º : sum 180º
The angle from Quadrant II could create angles 30º, 127º, and 23º : sum 180º

TWO triangles possible.

This example is the Ambiguous Case. The information given is the postulate SSA (or ASS, the Donkey Theorem), but the two triangles that were created are clearly not congruent. We have two triangles with two sides and the non-included angle congruent, but the triangles are not congruent to each other.

Review

How to convert radians to degrees and degrees to radians.

Degrees into radians

360° = 2π. 180° = π.

Radians into degrees

COOOOOOOL STUFFFFFF

Thing you can do with trig. It is HOT!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Astronomy and geography

Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn't yet considered to be a planet since it was the center of the universe, and the outer planets weren't discovered then.) The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry. Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy.

As the earth is also a sphere, trigonometry is used in geography and in navigation. Ptolemy (100-178) used trigonometry in his Geography and used trigonometric tables in his works. Columbus carried a copy of Regiomontanus' Ephemerides Astronomicae on his trips to the New World and used it to his advantage.

Engineering and physics

Although trigonometry was first applied to spheres, it has had greater application to planes. Surveyors have used trigonometry for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long.

Physics lays heavy demands on trigonometry. Optics and statics are two early fields of physics that use trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding space. Related fields such as physical chemistry naturally use trig.

Review

Basic Trigonometric Values

http://staff.jccc.net/swilson/trig/basicvalues.htm

Review

Standard position

We say that an angle is in standard position when its vertex A is at the origin of the coordinate system, and its Initial side AB lies along the positive x-axis. We say that AB has "swept out" the angle BAC, and that AC is its Terminal side.

Coterminal angles

Angles are coterminal if, when in the standard position, they have the same terminal side.

Acute angle

An angle whose measure is less than 90^o.

Obtuse angle

An angle whose measure is grater than 90^o but less than 180^o.

Right angle

An angle whose measure is 90^o.

Reflex angle

An angle whose measure is greater than 180^o but less than 360^o.

A straight angle

An angle whose measure is 180^o.

Central Angle

A central angle is an angle with endpoints and located on a circle's circumference and vertex located at the circle's center . A central angle in a circle determines an arc . For an inscribed angle and central angle with the same endpoints,