A Problem From Afar

My Dear Dear students from last year! It's been a while and I needed to know that you haven't stopped thinking and started memorizing random facts. I miss you all terribly and to show how much I miss you I shall pose a small question to you that should surely look like a "mole hill" to you when compared to your prior achievements. 

Is there a geometric proof of the algebraic statement below?

a² – b² = (a+b)(a-b)

If so, share your suggestion or proof by commenting to this post and remember, you'll have to communicate a visual idea through words because it'll be hard to share images when commenting. Goodluck!

Any Questions?

You can use this post for questions you may have regarding the topics we have covered in both geometry and trigonometry.

Homework due 05/20/09

So, given the radius of a circle:

1. Can you find the area of a sector if you know the central angle that the minor arc makes?

2. Can you find the area of a sector if you know the arc length that the sector intercepts?

(Note: The sector of a circle is like a pizza slice, in case you were wondering what a sector is!)

Homework due 05/15/09

Find the area of the triangle below. (Hint: You will need to find at least one angle, and I would suggest you employ the use of an altitude line.)

Homework due 05/12/09

You have the worksheet!

Homework due 05/11/09

1. On February 10, 1990, high tide in Boston was at midnight. The water level at high tide was 9.9 feet; later, at low tide, it was 0.1 feet. Assuming the next high tide is at exactly 12 noon and that the height of the water is given by a sine or cosine curve, find a formula for the water level in Boston as a function of time, measured in hours since midnight.

2. Of course, there's something wrong with the assumption in the problem above that the next high tide is exactly at noon. If so, the high tide would always be at noon or midnight, instead of progressing slowly through the day, as in fact it does. The interval between successive high tides actually averages about 12 hours 24 minutes. Using this, give a more accurate formula for the height of the water as a function of time.

Simple Harmonic Motion

So you obviously know that you have to plot the tangent functions graph using your unit circles and a table of values. Once you are done with that, you need to establish improvements/refinements to your method of getting precise measurements to craft a function that does a good job of modeling the "simple harmonic motion" of a mass suspended at the end of a spring. Think about:

(a) Number of oscillations in a given time period.

(b) The period (time taken for one oscillation).

(c) The lowest height, the greatest height and the equilibrium position.

(d) How you can utilize all the members of your group in an efficient way.

There may be other considerations and you should certainly discuss them using the blog. On Friday, without a single minute to waste, you must be ready to commence with your function modeling exercise. The end goal is that you come up with a function that fairly accurately models the motion of the suspended mass.

A "Transforming" Experience

So folks, you have some handouts/explorations for recollecting the topic of function transformations. See how much you can recall and grasp. Write down any observations you make. Try taking the test and seeing where you stand. You can use this post to discuss things. Just remember, if you find yourself doing the "if you see this then do this" routine then STOP! Don't do that! I'd rather you not know something then know it that way.

Homework due 05/05/09

You have the worksheet that I gave you in class. Remember, if it takes you longer then that time is worth spending now. Take more time now and you will naturally be able to sort this stuff out with more ease later. Check the conference folder early in the morning or very late at night for the answers to the worksheet.

Homework due 05/01/09

Folks, your assignment for Friday is the handout I gave you in class. If you misplace it or forget it in school you should download it from the AGT conference folder in First Class. If some of you felt overwhelmed by today's class then I suggest looking over the notes and the whole deal with radian measure, degree measure, negative angles and what they mean and what we mean by angles larger than 360 degrees or 2Pi radians. Click here for an image that can help you with the questions about quadrants.

Below are some things to remember for the homework assignment:

In question 1 you should refer to your unit circles and/or your trig tables. Remember that the sign (- or +) for sines, cosines and tangents becomes very important here. If angles are negative, then figure out which positive angles they are "co-terminal" with to figure out the sines, cosines and tangents. If the angles are larger then 360 degrees or much smaller than 360 degrees (such as - 405 degrees) then do the same, i.e. figure out which angle it is co-terminal with and then find the corresponding function values from your unit circle. Below is a link to an image that I think will help you understand what is meant by these new types of angles we are witnessing:
http://media.wiley.com/Lux/65/10665.nfg003.jpg

In question 2 you should note that I have asked for two solutions to the equations. In other words, there are two angles between 0 and 360 degrees that correspond to those particular lengths of lines that each equation depicts. Look at your unit circles and this will make more sense. Make sure to give the angles in both radians and degrees as the question demands. Remember, no calculators here as well.

Question 3 should be a straightforward question that you need no special hints for.

Homework due 04/28/09

Re-read the Jamshed al-Kashi article by Glen Van Brummelen and this time focus on al-Kashi's approximation of Pi. Re-try his method in your homework journals, grappling with all the arguments and seeing how he geometrically derives his formula. It will require your prior knowledge of geometry. Be ready to discuss your exploration in class.

Homework due 04/27/09

Your homework is the worksheet handed out to you in class and in case you misplace it or leave it behind in school, you can download it from the class conference folder, AGT - Kerai. Please do the work in your homework journals and show all your work.

Quiz on Friday, April 24th

You will be quizzed on whatever we have covered in trigonometry and also on our work with circles. You can use this post to compare answers for the questions you did in class involving right triangles.

Assignment due 04/20/09

This assignment is to be done on seperate sheets of paper as it will be collected on Monday.

1. Using only the formulas we’ve derived so far and the computations of sines of angles we have done, determine the value of sin(3°). (Use your algebraic skills at first and then use your calculators to get a numerical answer.)

2. Read the article on Jamshed al-Kashi.

3. Go back to where the formula involving sin(3 theta) occurs in the article and then attempt problem 4 of this assignment.

4. Using the sine and cosine addition and subtraction identities, prove the formula .

5. Use fixed-point iteration, as outlined in the article, to get a value for sin(1°) to about 6 decimal places of accuracy.

Sine Half-Angle Formula Discussion Post

Hint #1:
Identify which line segments on the diagram are equal to sin(alpha/2) and cos(alpha/2), and note how sin(alpha) and cos(alpha) show up in the diagram.

Hint #2:
What is the value of angle ECB? Are there similar triangle possibilities here?

Hint #3:
CE=CD-AB or AB=CD-CE

Hint #4:
Get an expression for cos(alpha) from triangle ECB. Every term in your expression should be convertible to sines or cosines of alpha and (alpha/2).

Hint #5:
From here it's just algebra. Solve for sin(alpha/2), remembering that you can convert a cosine to a sine using the pythagorean theorem.

Sine Addition Law Discussion Post

Hint #1:
Identify which line segments on the diagram are equal to sin(beta), cos(beta), and the quantity we want, sin(alpha+beta).

Hint #2:
Can you think of a ratio that equals sin(alpha) and one that equals cos(alpha)?

Hint #3:
What is angle CDF equal to ?... are there any similar triangles in the diagram? Which ones?

Hint#4:
See if you can determine values for the lengths FC, FD, EC, and OE in terms of the sines and cosines of alpha and beta.

Homework due 04/14/09

So, now that you have your wonderful trig-table (zij) you can write in the sines, cosines, and tangent values of the special angles +36 degrees, their supplements, and then tick mark all the other sine, cosine and tangent values that you could possibly find using the formulas that you have been given. Remember, that if you can find the sine of a new angle then you can surely use that to find one of yet another new angle. What you CAN find, you can use.

When doing this exhaustive checking off of angles, think of what rational angles sines, cosines and tangents can be found for and what would be necessary in order to be able to find values for all angles.

Note: If you heard what Jack uttered today, then he has already clued you in to the second part of what I am asking you to consider.

Homework due 04/14/09

Consider our table of values that we have started to generate for the sines and cosines of certain angles. So far we have 0, 30, 36, 45, 60, 90 and we can surely find more based on these. However, we then have all the other angles (infinitely many to be precise) for which we do not know sines and cosines and are concerned about how we could generate these values using geometric approaches.

You have been given a formula sheet that consists of relationships that the Muslim astronomers and their Indian and Greek predecessors were well aware of. Although we may not use these without the necessary derivations that we will undertake soon, we can get an idea of the possibilities of angles for which sines and cosines could be determined based on these formulas.

So, list as many angles between 0 and 360 degrees that we can find sines and cosines for based on the formulas you have been given. Also, ponder the first two formulas and determine whether they are just stating what you had already observed before or are they completely new formulas for you. We will discuss this further in class.

Homework due 04/13/09

Geometric Solution to the Sine of 36 Degrees

Okay folks, you have a neat challenge that awaits you on the geometric solution to the sine of 36 degrees. Please use this blog to post comments if you have questions. Remember, this will require you to use all your knowledge of geometry and don't be surprised if a quadratic equation pops up somewhere. You can use your calculators for the computations (to avoid spending hours doing arithmetic as the 10th century astronomers did) but only round off quantities at the very end. In fact, don't round off anything until you get your final answer. Work in your homework journals and show all your work.

Then, read the article on Islamic Astronomy by Owen Gingerich that I handed out at the end of class. This reading will provide you with all the necessary context for the work we are conducting.

(Now please do the reading or I'll be forced to give you another "reading quiz" and then let the dangling question of whether it'll be counted or not be much cause for emotional instability!)

Recommended Practice Questions (continued from class)

Folks, these are the exercises from class. Please make sure to do them if you wish to understand how the theorems pertaining to circles are used. Remember, you will be quizzed on this next week (and it will not be open book).

Page 365-366; #3, 8, 9, 10 and 12
Page 368-369; #4, 9, 13 and 23
Page 371; #8 and 9.

Take-Home Test Extension Granted!

You may turn in the take-home test Monday, April 6th, but since this is an extension you can expect additional homework over the weekend.

(Note: This extension has not been granted due to Jack's request. His is a case of time-management that does not necessarily warrant an extension. The extension has been granted on account of extenuating circumstances that some others have communicated and due to the fact that the Spring Play is up for three nights during the week.)

Take Home Test #5, Question 1 Discussion Post

You can use this post to discuss Question 1 on Test #5 (Take-Home).

Take Home Test #5, Question 2 Discussion Post

You can use this post to discuss Question 2 on Test #5 (Take-Home).

Take Home Test #5, Question 3 Discussion Post

You can use this post to discuss Question 3 on Test #5 (Take-Home).

Take Home Test #5, Question 4 Discussion Post

You can use this post to discuss Question 4 on Test #5 (Take-Home).

Homework due 03/27/09

Do problems 1, 2, 4 and 7 on page 355. Remember, you can refer to your notes from the class that Ms. Huber taught or you can also refer to the textbook from pages 351 onwards if you neglected to take notes that day.

Note:Please do make sure to finish the previous homework first. I have not assigned too much on this one so that you can finish that one yourself. If you have other questions then post them on the blog.

Homework due 03/24/09

Do problems 1, 2, 4, 7, 16 and 18 on pages 273-274 and then problems 10, 13, 15 and 17 on pages 278-279.

Homework due 03/20/09

Do problems 2, 4, 5, 7 & 9 on page 278.

Homework due 03/18/09

Do problems 1, 3, 7, 10 and 13 on page 171.

Class on 2/27/09

Read the example problem shown on page 244 and then do problems 4-8 on page 245 of your textbook.

(Note: This work will obviously be done in your notebooks.)

Class on 2/25/09

[Please make sure to read the entire post!]

Read pages 239 to 240 of the textbook. We proved the theorem and its three corollaries that follow in class today. So, once you have proved the theorem and read about mean proportionals you should try to prove the corollaries by using the proportions of the similar triangles you prove in the theorem. Make sure to prove each case and understand why it works and then read the words of the corollaries to make sense of the what you have proved. The words of the corollaries actually help remember the relationships.

Next, do problems 5, 7, 9, 15 and 16 on page 241 using the theorem and its corollaries that you just read (the questions are pretty easy and should not take you time). Question 16 is both interesting and crucial at the same time. If you cannot prove the statement in that question, then don't worry about it because if you turn to pages 242-243, then this famous theorem and its converse have been proven for you and you can read the proof and understand how it is done. (If you are Connor Miksch and have proved this theorem another way, you are still required to prove it this way and understand it the books way!)

You must complete the work above today so that you can do the work for tomorrow which includes a reading packet that the sophomores already got today to go home and read. You will have to pick up your reading packet (titled The Pythagorean Theorem) from the front desk before, during or after school tomorrow. Make sure to pick it up and read it as you will be assessed on this reading on Monday (reading quiz). The reading will provide you with important background information and context regarding the theorem and will also paint some of the landscape that will be necessary for trigonometry later. Do not save it for the weekend as you will have homework over the weekend.

Note: You are not required to take notes on this reading.

Class on 2/24/09

Do problems 1, 4, 5, 12 & 15 on page 235 of your textbook. You may use this post to discuss any of the questions.

Solution to #12 on Page 234

For this problem you must remember the theorem that states that the altitudes of similar triangles are proportional. To keep he post short I have summarized the solution. The diagram to te right is what I will be referring to for this solution. Some points were in the description in the question and others are introduced as a result of the auxiliary lines.

You know that QDC is similar to QAC since their angles are equal. Thus, AQ/DQ = BQ/CQ = AB/DC. The ratio of AB/DC is 3/2 from the given. Moreover, the altitudes of similar triangles are also proportional and so JQ/KQ = AQ/DQ. Therefore, JQ/KQ = 3/2. We are first interested in finding the length of JQ, which consists of JK+KQ. JK is 1.5 inches since the distance between DC and AB is 1.5 inches. Thus, 1.5 +KQ = JQ. Then, (1.5 + KQ)/KQ = 3/2. You can now solve this equation for KQ. When you have KQ then you can know JQ, which is the distance of Q from AB.

Now consider my auxiliary line through P that is perpendicular to both AB and DC and serves as the distance between AB and DC just as JK does. Now due to alternate interior angles formed by the dagonals and equal opposite angles you know that ABP and CDP are similar triangles. Also, you will find that PY and PX are altitudes for those triangles, respectively. If the triangles are similar then the altitudes are similar. Thus, AB/DC = PY/PX. AB/DC = 3/2 and so PY/PX = 3/2. We know that PY+PX = 1.5 inches from the given. So, you have two equations and two variables. Now all you have to do is find PY.

Homework due 2/17/09

Start by doing exercises 4-10 on page 224 and then from Exercises [A] on pages 225-226 do problems 1-4, 7 and 9. Next, read pages 226-228 and make sure to understand the proofs for the theorems regarding similar triangles discussed on those pages. You will be required to take notes on this reading. In your notes please try and re-write, in paragraph form, the proofs for the theorems.

[Note: For those students that are absent on Friday, Feb 13th, you can download the handout we will have done in class from the class conference folder and you will be expected to complete the handout and then do the homework that is assigned in this post.]

Quiz on Friday, February 20th

You will have a quiz on Ratio and Proportions, and whatever we will have done with Similar Figures till then.

Homework due 2/13/09

Read Case B of the theorem about angle bisectors on page 217 and understand the proof for it, making sure to read why Case B does not work for when the sides of the triangle are equal. Then do problems 3, 5 and 10 on pages 218-219.

Homework due 2/11/09

Read pages 212-214 and then read the blog post from last semester that talked about the issue of "number vs. magnitude". You can search it in the top left hand corner of the blog where there is a search field.

Homework due 2/10/09

Do exercises 3, 5, 6 & 9-12 on pages 208-209.

Tips for Complex Proofs

So, we may have reached a stage where just crafting a proof in our mind and then putting it down on paper is not the best way for more involved and complicated proofs. In essence, we have come to what geometry is all about; exploration! To make the process of discovery more effective, try some of the suggestions below.

1. Draw a reasonably large diagram so that relationships between lines and angles are easy to spot and label and that auxiliary lines find adequate space to be drawn.

2. Label all that you can on the diagram and if you're afraid of clutter then consider using some alternatives for marking congruent angles and lines. (Jack's suggestion of using roman numerals is effective!)

3. Start forming a list of relationships you do see as being true. If possible, jot down some reasons so that the task of writing a formal proof is not too daunting. The list is important because often the diagram cannot house all the relationships and the list could help you establish relationships, especially algebraic ones.

Homework due 02/02/09

Do problems 12, 13 & 15 on page 191 and then do problems 1-10 on page 203.

Homework due 01/30/09

Do problems 3, 4, & 6 on page 187 and problems 1, 3, 5 & 9 on pages 190-191.

Homework due 01/28/09

Folks, below is the diagram that you must use to prove the concurrence of the altitudes of a triangle. Remember that the dotted lines are generated by making auxiliary lines from the vertices of the triangle such that each line is parallel to the side opposite each vertex. These auxiliary lines form a triangle of their own. Now, remember that your proof cannot

(a) assume the concurrence of the altitudes when proving precisely that and

(b) will employ the use of the theorem that the perpendicular bisectors of a triangle are concurrent at a point that is equidistant from the vertices.


Note: Please do not look at the book for this proof; you'll take all the fun out of proving it yourself!

Homework due 01/26/09

Do exercises 2, 5 & 8 on page 181 and exercises 2, 3, 4 & 7 on page 183.

Group #3 Discussion Post

The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices.

Prove this theorem!

Group #2 Discussion Post

The median of a trapezoid is parallel to its bases and is equal to one half of their sum.

Prove this theorem using the diagram provided in class.

Group #1 Discussion Post

In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices.

Prove this theorem and its converse.

Homework due 01/20/09

The theorems you proved in the assignment in class today are commonly referred to as the “midpoint theorems.” Use the results of these theorems as well as other theorems you have learned before and do exercises 1, 3, 4, 5, 15 and 22 on pages 177-179 of your textbook. Then, read pages 179-180 and fill in the missing reasons (stated as “Why?”) in the proof for Theorem 30. Be sure to read and understand the definition and corollary that follow from it.

Quiz on Friday, January 16th

You will have a quiz on parallel lines and planes and some of our work with geometric solids.

Homework due 01/13/09

Read pages 168-169 and do problems 1, 2, 9, & 16-18 on pages 170-171.

Homework due 01/12/09

Do exercises 1-31 on pages 166-167. Be sure to read the postulates and theorem covered in the text just before the exercises.