Setting the Tone in Precalculus - Another Coordinate Investigation

Note: Read the first comment I posted which suggests a purely Euclidean geometry approach to this problem...

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:

PRINCE? NAH! E-ROI!

Tangent problems are usually the domain of calculus but we can keep them within the reach of geometry and algebra if we restrict our attention to circles. The calculus student spends a considerable amount of time solving a wide variety of "tangent to the curve" exercises. As any calculus instructor will tell you, many of the harder problems ask students to determine the equations of the tangent to a curve from a point not on the curve. The issue there is not the calculus. It's all about an understanding of the interface between the algebra and geometry, the essence of coordinate methods. I developed this investigation specifically to address this issue before students enter calculus. Might be another "fun" problem to start the year off with. If nothing else, it will establish the rigor of your precalculus course early on!

Part I of Investigation
Determine the coordinates of the points of tangency for the tangent lines to the unit circle from the point (0,2).
Note: Unit circle refers to the circle of radius 1, center (0,0).

The remaining parts will all refer to this same circle.

Part II of Investigation
Repeat part I for (0,3) and (0,4).
Write your observations, conjectures.

Part III of Investigation
Show that the y-coordinate of the points of tangency for the tangent lines to the unit circle from the point (0,k) is 1/k, where k ≥ 1.

Notes, Comments...
(1) The result of Part III suggests that as k increases, the y-coordinate of the point of tangency decreases (inverse ratio). Ask students what happens as k approaches 1.
Students should make sense of this visually by sketching tangent lines from various points on the y-axis above the circle.
(2) There are several effective methods for solving the above parts, however, one needs to know the fundamental relationship between a tangent line and the radius drawn to the point of tangency. From that point on, one can represent the slope in two ways or represent the y-coordinate of the point of tangent in two ways. This requires strong understanding of coordinates, graphs and algebraic relationships. You may find other methods -- share them! BTW, one could also use trig methods.
(3) I chose the unit circle and a point on the y-axis for simplicity so that the student could focus on essential ideas. However, one could generalize the result to any circle and any point outside. Have fun with that!
(4) Anyone mildly surprised by the reciprocal relationship between the y-coordinate of the point on the y-axis and the y-coordinate of the point of tangency? Can anyone make sense of that?

There are twice as many girls as boys: 2G = B or G = 2B?

The English language has many confusing phrases but "as many as" IMO has blighted the youth of many an algebra student. Perhaps you think I'm exaggerating this? At the beginning of the school year, write the phrase in the title of this post on the board and have your PreAlgebra/Algebra I (or higher) students write one of the two equations on their paper. Give them only a few seconds, then compile the results. Let us know if the vast majority choose the correct equation. Of course, the outcome depends on the group and many other factors but if we have enough data it might prove interesting. I'm basing this on many years of questioning students. Perhaps I am the only one who has experienced this phenomenon!

The abstraction of algebra is difficult enough for some youngsters. Students who are new to our language have particular difficulty with idiomatic phrases but those born here also seem to struggle with the verbal parts of word problems - that's completely obvious to any algebra teacher of course. If only we could remove the words from a word problem!

Certainly teaching vocabulary and math terminology is an essential part of what we do as instructors. We should also hold students accountable for this vocabulary by assessing it directly.

In this post, I'm inviting readers to share some of the coping mechanisms and pedagogical strategies they use in the classroom to help students survive phrases like "as many as." What phrases seem to cause the most confusion among your students? How about "x is four less than y?"

Here is my initial offering. Let me know if you do something similar or if you feel this might be helpful (or if you vehemently disagree!).

KEY STEP: First decide from the wording of the problem if there are more girls or more boys. In fact, this should have been my original question -- not the equations! It is critical for students to be able to translate the verbal expression into a comparative relationship: Which is the larger quantity? Number of boys or number of girls? Hopefully, most youngsters would interpret the original problem to imply that there are more girls than boys. Hopefully! Ask this question first (metacognitively, students need to learn to ask themselves questions like this when they are reading).

NEXT STEP: Now the issue is where to place the "2" in the equation. Based on the key step above, we know that the number of girls is the larger quantity. Ask them why 2G = B would be incorrect.

Better alternative for some:
We all know that those who have difficulty handling abstraction benefit from concretization, i.e., using numerical values:

Have them write both possibilities:
B = 2G and G = 2B
Now have them substitute values for G and B that make sense for the original problem, say
G = 12, B = 6. Some struggle with this!
By substituting (students like the phrase "plug in") these values into both equations, they should see that 6 = 2⋅12 does not make sense. The correct equation should become apparent. Should...
Of course, most youngsters need to practice many of these before they reach comfort level.

Your thoughts, suggestions, anecdotal evidence???

Starting the Year off in Precalculus/Algebra 2 - A Little Review?

Don't forget our August-September MathAnagram. No responses yet but this mathematician deserves our recognition. 'Relatively' speaking, this mathematician was truly unique and, perhaps, the last of a dying breed.

PRINCE? NAH! E-ROI!

Sometimes I found that a coordinate problem was a good way of reviewing the geometry and algebra needed to refresh memories after a 2-month layoff. Here's a fairly straightforward one that admits many different approaches and might be used to set the tone. Encourage students, working in groups, to find at least THREE different methods. This will extend the thinking of those who "solve" it rapidly and sit there complacently. This problem is a bit more appropriate for the students who completed Algebra 2 although Geometry and Algebra 1 methods are possible.
To reiterate: The problem itself is not particularly challenging. The purpose here is to provide review of several ideas, methods, theorems and strategies.

Given points A(0,0) and B(12,0). Determine the coordinates of all points C(x,y) such that ∠ACB is a right angle and ΔACB has area 18.

2008 has 2 digits that are the same -- A Probability Investigation For Middle Schoolers And Beyond

As the school year is beginning...

Which would you conjecture is more likely:
No digits the same in a 3-digit number or no digits the same in a 2-digit number?

You have 30 seconds to choose one of these - - - - - - - - - -
NOW WRITE YOUR GUESS ON YOUR PAPER and compare with your partner. Take one minute to discuss your thoughts...

Alright, I know some of you take exception to wasting these 30 seconds. What could be gained from such 'blind' guessing without the time to really think it through and work it out. I often used device this to encourage youngsters to react instinctively and to learn to trust their intuition. How many times have all of us had the experience of not trusting ourselves, only to find later that we were right. If it turns out that this gut reaction is not supported by the data, then the mathematical researcher (or the experimental mathematician in this case) revises the hypothesis. Ultimately, one attempts to validate one's conjectures via logic (deduction, induction, etc.). If you're still not convinced this is worthwhile, it's only a suggestion...

Now we're past the prelims. Our goal is to have our students begin with solving a particular case of the problem above and then to develop a general relationship for:
The probability that an N-digit positive integer will have N different digits. Of course, N is restricted to be in the range 1..10. We would hope our students from middle school on would recognize that the probability for N = 1 is 100%, whereas the probability for N = 11,12,13,... would be zero! Yes, we would hope!

(1) Show that the probability a 2-digit positive integer has different digits is 90%.
Comments: This is a well-known and fairly basic problem, but this is just the jumping-off point for this investigation. Various methods are likely here, depending on the background of the student. The middle school student (and many secondary as well) would likely list or count the number of 2-digit numbers with different digits. Some would realize that it might be easier to count those with identical digits and subtract from the total. More advanced students may use more sophisticated approaches for this and the other parts below. One could use this activity to develop the multiplication principle, permutations, use of factorials, etc. However, there is much to be gained from 'first principles.' Careful counting and making an organized list never go out of style!

(2) Show that the probability a 3-digit positive integer has 3 different digits is 72%.

(3) Complete the following table up to N = 10:
Note: P(N...) denotes the probability of the indicated outcome.

Number of Digits N.....P(N different digits)

...........1......................100% or 1
...........2..................... 90% or 0.9
...........3......................72% or 0.72
...........4......................50.4% or 0.504......
.
.
..........10.....................................................


(4) Time to revisit your original conjecture.... Explain why the probabilities decrease as the number of digits increase.
Note: One could give a purely descriptive explanation here.

(5) For more advanced groups:
Develop a formula for P(N).

(6) For more advanced groups:
Enter your expression from (4) into Y₁ of the Y= menu in your graphing calculator. Set up a TABLE with Start value of 1, increment (Δ) = 1 and Auto for Indpnt and Depend. Display your table and check the values you found from your own table.

(7) [Optional]
Closure: Write 3 ideas, methods, strategies, mathematical principles, etc., you have learned from this activity.

Back-to-School Geometry - Rectangles, Squares and Deeper Challenges

The connections between geometry and other rich areas of mathematics are boundless. Here is a fairly straightforward set of problems that can be explored as far as your eye and mind can see. On the surface, we have three rectangles each of which has a half-diagonal of length 6. Students can be asked to find the area of each without using any trigonometry. On a deeper level, one can ask students to explain or prove why the square has the greatest area for a given
diagonal length. This is straightforward using the well-known trig formula for area of a triangle, K = (1/2)absin C, however the challenge here is to use non-trig methods (although the student can use special right triangles) to compute the areas and demonstrate the maximum. The maximum piece is more sophisticated and the idea of bringing this in before precalculus and calculus has many benefits.

The instructor might begin by asking students to draw any rectangle with a diagonal of 12. How many such rectangles could there be? Which one would appear to have the greatest area? Ok, now let's explore a few special cases.

This problem allows the creative student to devise a visual way of explaining the maximum. It also allows the instructor to bring in the Arithmetic Mean-Geometric Mean Inequality for enrichment. So many methods and approaches are possible...

PRINCE? NAH! 'E-ROI'! August MathAnagram Finally!

Better late than never...
This is definitely not one of my finest anagrams but it's the best I could do. Hopefully, the descendants of this eminent mathematician will forgive me.

As always, ignore the punctuation and I trust many of you will solve this 'rapidement'.

Please follow the usual procedures when submitting...

DO NOT SUBMIT YOUR ANSWER AS A REPLY TO THIS POST

• Remember to email me at dmarain at geeeemail dot com
• Pls use Mystery Mathematician August 2008 in the subject line!
  

(PLEASE ENUMERATE!)

(1) The name of the mathematician
(2) Some interesting info/anecdotes re said person
(3) Explain the code embedded in the anagram! (Not too exciting here this month)
(4) Your sources (links, etc.)
(5) Your full name and the name you want me to use when acknowledging your accomplishment
(6) If you're new, how you found MathNotations
(7) If new, your connection to mathematics

A Triangle Inequality and Algebra Application

Students learn the triangle inequality in geometry and might even recall it correctly 5 minutes after the final exam ("Let's see -- I think it's something like 'two sides of a triangle are more or is it less than the other side'"). Of course, if they can visualize it, they might retain it longer, but, in the end, they should know it as well as they know their basic arithmetic facts. (Uh oh -- bad analogy!).

Here's a fairly straightforward application although many students will answer it incorrectly even if they correctly recall the key geometric fact. Can you see where the traps lie?

A triangle has sides of lengths 12-x, 12 and 12+x. How many integer values of x are possible?


Now for the generalization:

A triangle has sides of lengths a-x, a and a+x, where a is a positive even integer. In terms of a, how many integer values of x are possible?


Comments:
(1) Do you believe we should more heavily emphasize the triangle inequality? The SAT certainly does (not that that is any justification, right?).
(2) What teaching strategies have worked best for you in the classroom when introducing this theorem? How much time is generally devoted to this topic?

SIR "OY", E. NOET! --- OYSTEIN ORE REVEALED!

Clues revealed....
(1) When a student at the University of Gottingen, Ore learned Emmy Noether's new approach to Abstract Algebra, which influenced his own work in the field. Later, he and Emmy collaborated.
(2) For his dedication to his native country of Norway, Ore was knighted (the 'SIR' reference).


Note: I highly doubt that I will be able to produce future anagrams with 2 or more embedded clues!

I chose Professor Ore for several reasons, not the least of which was his influence on me. Somewhere in my library I still have his classic Graphs and Their Uses, a wonderful monograph for young learners published back in the 60's by the New Mathematical Library (MAA). I would also highly recommend his Invitation to Number Theory from the same series.

Our winners for the July contest were

Eric Jablow

I vaguely remembered a mathematician with 'ore' and 'oy'. Then, I started guessing. 1. Øystein Ore 2. The only things I know about him are from his Wikipedia article. You don't need me for that. 2.5. He was decorated as a Knight of St. Olav. He worked with Emmy Noether. 3. http://en.wikipedia.org/wiki/Øystein_Ore

Kevin Cheslack-Postava

Dear Dave: Oystein Ore. Imbedded clue is E. Noet" Connection between the two: From the MacTutor history on Ore: "...he spent time at Göttingen University where he was influenced by Emmy Noether finding her new approach to algebra particularly exciting... In 1930 the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether. " I remember Ore's high school level book on graph theory from my youth - very clear exposition.

Paul Miller

(1) The name of the mathematician: Oystein Ore
(2) Some interesting info/anecdotes re said person: Collaborated w/ Emmy Noether, was knighted

Paul discovered the two clues in the anagram -- congratulations! Ironically, he apologized for not discovering additional "code"!

I'm putting the finishing touches on our August MathAnagram. Should be up in a couple of days or sooner...