Desmos Common Core Activity Linking Circles, Tangents and Linear-Quadr Systems

Detailed investigation with extensive background notes for instructor and step by step outline for students to follow. Students will be asked to use a slider to approximate the position of a tangent line of slope -1 to a circle centered at (0,0). The tangent line, x+y=k, requires use of a parameter.

Students will begin with a particular radius, 3, then solve a linear-quadratic system to determine the exact equation of one of the tangent lines. They will also be asked to enter an expression for the other tangent line of slope -1 using the same parameter k. After approximating the locations of similar tangent lines for other radii, they will be asked to solve a general system using radius r.

There are different systems offered to the instructor, depending on the sophistication of the student. Finally, a geometric solution is suggested using 45-45-90 triangles.



 Use new contact form at top of right sidebar to contact me directly!


 If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified.

Parametric/Projectile Motion Simulated in Desmos - A Common Core Activity for Algebra/Precalculus

[Updated using folders to reduce amount of visible text. Click on the arrow next to the Folder icon to see the frames below. Thanks to Desmos team for this helpful hint!]

CLICK ON GRAPH TO ACTIVATE DESMOS...

The Desmos activity above is both an investigation of parametric representation and a tutorial for more advanced use of this remarkable WebApp. The The text in the side frames begins with a detailed background of the activity for the instructor and how Desmos can be used to demonstrate projectile motion using both parametric and rectangular coordinates. Some of the uses of slider 'variables' are demonstrated including animation, a powerful feature of Desmos.

In addition to showing how to use parameters in Desmos, the activity itself asks students to compare two different trajectories, representing an object dropped from some initial height, then a 2nd object two seconds later. The horizontal translation of the first graph is juxtaposed against the algebraic representations of these graphs using both system of coordinates.

The student activity starts about halfway down. There is a series of questions and actions the student needs to take in Desmos.

I'm hoping this will prove useful for both the instructor and the student.  Desmos is powerful but, in my opinion, some of the illustrative examples provided by Desmos do not flesh out the ideas behind the various uses of slider 'variables'. I'm hoping this will fill in some of those gaps.  I'm still a novice here so I'm sure more advanced users will be able to improve upon this...

Your comments and reactions are very helpful to me...







If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified.

Three congruent isosceles right triangles walked into a bar...

OVERVIEW
Silly title but you might want to try the following problem with your high school geometry students or with middle schoolers doing a unit on right triangles. Furthermore, elementary school children need many hands-on experiences with pattern blocks, tangrams, pentominos and the like to develop their innate spatial sense. They should also be allowed to experiment with two such triangular pieces to make a square, a parallelogram, a larger isosceles triangle, etc. Then have them work with the 3 triangles to make different polygons including the trapezoid. They don't need to consider the area or the 2nd part of the question.

THE PROBLEM
Three congruent isosceles right triangles are joined to form an isosceles trapezoid having an area of 3 sq units.

(a) Draw a possible diagram.
(b) Determine the perimeter of the trapezoid.

Answer: (b) 6+2√2

REFLECTIONS
•How much time would you allow for a discussion of this problem! 10 min? 15? 20? Guess it depends on whether you see this as just an exercise or as an activity.
• How much difficulty do you think most middle and secondary students would have with drawing an appropriate diagram?
•Do you think most will need to draw several figures before arriving at the isosceles trapezoid? Do you think some will come up with a trapezoid which is not isosceles and think they're finished? Can you anticipate that some will miss one of the key words like isosceles (which occurs TWICE!).
• Do you think the spatial "puzzle pieces" part of the problem is more significant than the numerical part or about equal?
• Do you expect some students to hit a wall and express something like "I forgot the formula for the area of a trapezoid!" We should make this a teachable moment -- "WE DON'T NEED TO RECALL THAT FORMULA! WHY!"
•Do you see benefits from students working in pairs here? Would you have them work independently then come together after a few minutes? My view is the stronger spatial student will "see" the correct figure more rapidly and influence the other who may give up and wait for his/her partner to draw it. So I might ask them to draw a few figures on their own for a couple of minutes.
•Do you think any of the older students need manipulatives?
• What is our role here? Catchphrases like"guide on the side" do not tell us what interventions we should actually use? Part of knowing what to do/say comes from our experience and part from instinct but my rule of thumb was "less is more". Allowing them to struggle for awhile is critical or, to put it another way, "without irritation there would never be a pearl!"
• How would you solve this problem? When planning do you feel it's important to think of alternate solutions or let this flow from the students?
•Finally, I think it's important to identify which of the  Mathematical Practice Standards are brought to play in this investigation. All of them? A couple? Guess that depends on you...

I typically get few if any comments from these detailed investigations. That's ok. Just planting seeds I guess...

Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW

SEASONS GREETINGS!

While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...

The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.

The extension at the bottom goes beyond what might be expected from the beginning of this exploration.

Math educators can adapt this for Algebra 1 through AP Calculus students...

THE PROBLEMS

What are the number(s) described in the following?

1. A number equals its reciprocal.

2. A number equals 25% of its reciprocal.

3.  A number equals twice its reciprocal.

4.  A number equals the opposite of its reciprocal.

5.  A number equals k times it's reciprocal. Restrictions on k? Cases?

Answers:
1. 1,-1
2. 1/2,-1/2
3,  √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined

OVERVIEW and much more...

• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:

(*) New = (Old + k/Old)/2

Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]

Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could  implement this on their graphing calculator or program the algorithm themselves!

The square root of x+1 equals x+1... A Common Core Investigation

OVERVIEW
Fairly straightforward radical equation in the title but there is so much hidden potential here for students in Alg 2/Precalculus.

REFLECTIONS
• The solutions to the equation above are -1 and 0. No big deal, right? The usual algorithm --- just square both sides and solve the resulting quadratic by any one of several methods. Done. Cheerio. But wait...

• We can encourage students to "make it simpler" by substituting 'a' for x+1 obtaining a^(1/2)=a, square both sides yielding a=a^2 which gives 2 easy solutions 0,1 and then x+1=0,1 producing the final result. Not that big a deal though except...

• A graphical interpretation of these equations is illuminating and illustrates multiple representations/The Rule of 4. You could demo this with the graphing calculator displayed on your smart board or have the students graph by hand or on their device. The graphs of y=x^(1/2) and y=x intersect at x=0 and x=1 then, by translation, the graphs of y=(x+1)^(1/2) and y=x+1 will intersect at x=-1 and x=0. Students should be asked for this conclusion BEFORE checking the graphs to verify!

• Is that all there is? Hardly! The current trend on assessments and hopefully in texts is to have students analyze a family of equations using a parameter. But first we can generalize numerically:
Solve
(i) (x+4)^(1/2)=x+2
(ii) (x+9)^(1/2)=x+3

Are there still 2 solutions for each of these? Solving just a couple of these and recognizing extraneous or apparent solutions would traditionally have been the WHOLE lesson! Not any more...

By the way -  why the "4" and "9"? Did I change the pattern from the original equation?

• Now for the parametric form:

(x+k^2)^0.5=x+k

What questions should STUDENTS be asking themselves BEFORE WE ASK THEM?

• Students can certainly be asked to solve the latter equation for x in terms of k. Some will struggle with the procedure/algorithm. Hopefully someone in each group (or the whole class!) will obtain x=0 and x=1-2k. BUT WILL THEY CHECK THE 2nd SOLUTION! The use of a  parameter goes beyond making a better standardized test question. Now the student has to recognize that, in order for there to be 2 solutions, k must be less than or equal to 1 which was suggested by the numerical examples above.

• Of course I'm anticipating most teachers' reactions to an exploration like this. I've provided much more than can reasonably fit in a 40 min lesson. Use it as you see fit or just ignore it. It will go away or will it? 

How (m^2)/(n^2)=(m/n)^2 is Fundamental to Geometry!

OVERVIEW
The Common Core stresses the importance of students developing a deeper understanding of fundamental concepts and to discover/uncover the interrelatedness of mathematics. The discussion below can be used to demonstrate how a basic law of exponents is tied to the geometry of similar figures.

THE PROBLEM/INVESTIGATION
1) If the sides of 2 squares are in the ratio 2:1, show that their areas are in the ratio 4:1
(a) visually
(b) numerically by examining particular cases
(b)  algebraically

2) If the sides are in the ratio 3:1, do you think the areas will be in the ratio 6:1 or 9:1? Now do parts a-c as in 1).

3) If the ratio of the sides is 3:2 show algebraically that the ratio of the areas is 9:4.

4) Show algebraically that if the ratio of the sides of 2 squares is m:n then the ratio of their areas is (m/n)^2.

Note: How does this result connect to the idea that the area of a square varies directly as the square of its side length?

4) If squares are replaced by circles using radii or diameters in place of "sides" show that the results of questions 1-4 are the same.

How does this result connect to the idea that the area of a circle varies directly as the square of its radius or diameter (or circumference)?

REFLECTIONS
• Squares and circles are of course special cases of similar figures. Beyond this investigation lies the BIG IDEA:

The areas of 2-dim similar figures are proportional to the squares of their linear dimensions.

Note: In 3 dim, we can replace 'area' by what?

• Do you see this as one of the fundamental theorems of Euclidean geometry? Is it sufficiently stressed in textbooks and in the standards? Of course you may not feel as I do about all this!

• So what is the geometry connection to
(m/n)^3 = (m^3)/(n^3)...

'.

1,-3,9,-27,... Investigation

No matter where you introduce a lesson on geometric sequences, we can always begin with definitions, rules and formulas. 

The alternative is to build on student intuition and natural curiosity by asking them to write their own observations and questions they would like to have answered.

Imaginary Scenario (or is it?)

Jack: Mom, all the terms are just powers of 3 or their opposites, right?

Mom (Jane): Write your hypothesis, test it and let me know.

If your students or your son is not 15 year old Jack Andraka, here are some suggestions...


1.  What are the next 3 terms?
2.  If the 99th term is x, write an expression for the 100th term? (Recursive thinking)
3.  Which terms are positive? Negative?
4.  Write an expression for the nth term.
5.  How would we graph the sequence?
6.  Are the terms of the sequence increasing? Decreasing? Both? Neither?
7.  Which terms of the sequence are greater than a million? A trillion? Less than -1000000?

Another Imaginary Scenario (or is it?)

Uh, show me where this topic is in the CCSSM.
Uh, where does it say I have to ask all these questions?
Jack who?

Sent from my Verizon Wireless 4GLTE Phone

List the NINE 2-digit PRIMES which...

Here are the last two math challenges I just tweeted for middle schoolers and beyond. You may want to use this as a fifteen minute activity to improve reading, review basic terms and concepts, develop reasoning and writing in math. There was an error on the 2nd question as it originally appeared on Twitter. I then corrected it.

List the nine 2-digit primes which produce prime numbers when their digits are reversed.




List the SIX 3-digit primes which produce primes when their digits are written in ALL possible orders. 137 fails b/c 371 is not prime.

For both questions students should work in teams of 2-4.

For the first question, students should not be allowed to use a calculator!

For the second one, have them experiment with a calculator for a few minutes. If a student thinks they found one, their teammates must verify it! After 3 minutes ask: "Have you noticed that the numbers you're looking for cannot contain certain digits like 2. What digits and why? Discuss it and one member of the team must record the team's findings and provide a written explanation!

After 3-4 more minutes, have them refer to a table of primes online (or print it and hand out a copy to each team). If they don't find it within the 15 min time limit, have them finish it for extra credit for the next day.

Here is one of the numbers: 113. Good luck!


"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught." --from South Pacific

Algebra 2/Precalculus "Extended" Activity Based on an SAT-Type Question

Consider the following problem:

If -5 ≤ x ≤ 4, and f(x) = 2x² - 3, how many integer values are possible for f(x)? 


One can simply view this as a more challenging question to pose to your honors/accelerated students, but, for me, it's an opportunity for all your students to think more deeply about important concepts. I feel strongly that our role here is to ask the key questions which will guide them toward understanding the "big ideas" underlying this problem. In fact, we can turn this question into an extended activity: 15-20 minutes).

Here is one idea for creating the environment currently being recommended. Please keep an open mind before concluding that there is simply not enough time for these explorations...


WITH YOUR LEARNING PARTNER(S):

1.  Sketch the graph of the function on the given domain from recognition of quadratic functions and by making an x-y table with 4-5 points. WRITE YOUR INFERENCES FROM THIS. For example, from the sketch we believe that the greatest y-value on this domain is ___.

WRITE your conjecture for the answer to the problem: ____

2.  Using the TABLE feature of your graphing calculator, with TblStart = -5 and ΔTbl = 1, display the Table.  Now turn TRACE on and analyze the graph on this domain. Does this alter or confirm your conjecture from Step 1?  YES   NO

3.  The following statement is plausible but FALSE.

The domain consists of 10 integer values. Therefore there are also 10 integer values for f(x), so the answer is 10.

Explain why this is wrong. There is more than one error! 

4.  The correct answer is 51. Depending on the class, a few, if not several,  students should be able to come up with the correct answer and provide a thorough explanation.

5.  Group Discussion:

• Ask students how they might have approached this question if it appeared on a standardized test? Plug in x-values? Use the graphing calculator? Guess? Skip it?
• Ask the group what made this questionable formidable for some students? How important was understanding what was asked for?
• Review one successful approach to solving the problem by calling on individual students to give the "next" step.

NOTE: This  problem also presents a highly teachable moment for students to see an application of the Intermediate Value Theorem in Precalculus (or more intuitively in Algebra 2).  Help them make the connection! Is this easy for us to do?

 Your thoughts?



"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

 You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

PI Day, More Videos on Counting, "Odds and Evens"

Since pi day fell on a Sunday this year, we should still be celebrating it today. Besides, March should be declared pi-Month!


It is always fascinating to see how readership (or should I say one-time viewership) always picks up around March 14th every year! I feel obligated to add another pi Day activity or exploration in addition to those I've posted the past three years. By the way, the pi Day Scavenger Hunt is the most popular post by far and I'm not even the one who thought of that idea!


Despite the title of this post, I did not upload a video for this activity. However, there is another video on the MathNotationsVids Channel on YouTube.

Here is an investigation/exploration/activity for middle and secondary:

Part (A)
(i) List all ordered pairs of positive integers (m,n) such that
(1) 1 ≤ m ≤ 10 and 1 ≤ n ≤ 10
(2) m and n are divisible by the same prime p

For example, (m,n) could be (6,9) since 6 and 9 are each divisible by the prime 3.

(ii) Should (9,6) also be counted?

(iii) Another way of expressing Condition (2) is:
The _______________ of m and n is ________  one.
Answer: gcf; not equal to or greater than

(iv) If you listed and counted correctly, you should have found there are 37 ordered pairs which satisfy both conditions. If not, have a partner check your list. Each of you should be checking each other's lists routinely.

Part (B)
(i) Explain, using the multiplication principle, why there are 100 ordered pairs which satisfy Condition (1) above.

(ii) ) What % of all the possible ordered pairs from Condition (1) are relatively prime. If you have immediate access to the internet, research this term before asking your teacher what it means!

(iii) In probability terms, you could say:

If one of the 100 ordered pairs (m,n)  from Part (A) is selected at random, the probability that
m and n are relatively prime is ____%.

Part (C) (more advanced)

If you have access to a graphing calculator, such as the TI-84 or TI-Inspire, enter the following program into memory (call it RELPRIME):

:ClrHome
:Prompt N
:0 → K
:For (X,1,N)
:For (Y,1,N)
:If gcd(X,Y) ≠ 1
:K+1 → K
:End
:End
:Disp K
:Stop

Using this program, complete the following table:

N..........Total # ord. prs..........# of not rel prime prs........% rel prime prs

10.........100.............................37....................................63%

20.........400............................ 145.................................

30

40

50

100

Notes:
K represents the count of ordered pairs which are not relatively prime
N represents the greatest value for the integers
gcd is found by going to MATH, then NUM, then 9:gcd(
The program slows down considerably as N increases. For N = 10, it checks 100 ordered pairs which may take only 2-3 seconds. For N = 100, it checks 100^2 pairs, which could take up to 4-5 minutes. Be patient!!

Conclusion: So what does all of this have to do with π ?
Well, as N increases without bound in the program, the probability that a randomly chosen ordered pair of positive integers (with values up to an including N) will be relatively prime approaches 60.7% rounded.

From out of the blue, compute 6/π²...
Want to know why? Well, that requires some advanced machinery involving infinite products, infinite series, and the Riemann Zeta Function! Perhaps, I'll do an informal development in a video. I love this stuff...


----------------------------------------------------------------------------------
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

HAPPY 2 x 3 x 5 x 67! Let The "Problems" Begin!

May this new year and decade bring happiness and prosperity to each of you now that the 2KO's have come to an end! 

BTW, the italicized symbol in red is my submission for the name we should give to the past 10 years. What do you think of it? Let me know if you came up with one of your own. According to Time Magazine, no one has yet created a name which has caught on (and dozens were listed!).  Also, I will avoid debating those who strongly believe that the first decade of the 21st century ends a year from now!





As MathNotations begins its 4th year, it has become an annual tradition for math ed blogs to challenge their readers to discover interesting facts about the number symbol representing the new year, in this case, 2010, or Twenty-Ten, for those who are as committed to multiple representations as I am!

Those who know me can anticipate that I would recommend making this an exercise for our middle schoolers. Here are a couple of ideas:


"In your group, list as many observations as you can about the number, 2010. Your team's score will be based on both quality and quantity. For example, an observation like "2010 is even" would only earn 1 pt, whereas "2010 must be divisible by 3 because the sum of its digits is divisible by 3" would earn 2 or 3  points since it contains both a fact and an explanation."

Another idea might be to have students write interesting word/number problems involving 2010 for the class to solve. Of course, to obtain credit the student posing the questions must  provide correct answers and solutions!


Your turn...



A final note ---
Some of you may have noticed that I've enabled Comment Moderation due to the number of spam comments which have gotten through. I held out for as long as I could. I do check throughout the day, so, hopefully, this should not prove problematic for my readers.

A Quadratic Trinomial/Factoring Investigation for Algebra I/II

In my Christmas post, I raised the issue of how much time should be spent on factoring quadratic trinomials over the integers in light of the new ADP Standards for Algebra I and II. Hopefully, some of you will provide us with the benefit of your knowledge and experience. I may even make this into a poll or survey to be voted on but, in this post, I will appear to contradict myself and propose an investigation of this topic which requires some effort and time on the part of the student. The target audience would be the regular or accelerated Algebra I/II student.  

We all need to become more creative in the strategic use of time in our classrooms (I still think of myself as being in the classroom!). What are some alternatives to using class time for this? I'll suggest one approach and I'm hoping others will offer their suggestions:


Assign the following as an extra credit or "long-term" project to be due in a week or two. I would not even take classroom time to discuss it. Just hand it out or post it on your website or the department's website if it is to be given to all the Algebra classes. Students can easily download it or print directly if they wish. After they are collected, graded and returned, you may choose to discuss it briefly for about 10 minutes using an overhead transparency, opaque projector or via your computer and a projector. You can also post some  student solutions on the website.


THE INVESTIGATION/PROJECT

[OPTIONAL HINT OR CUE]
The following may require an application of the ac-method learned in class.

(1) Factor the following over the integers and show all steps used in your method of factoring:

(a) 12x² + 27x + 15

(b) 12x² + 28x + 15

(c) 12x² + 29x + 15

(2)
(i) List all positive integers values of b, including the ones from part (a), for which 12x² + bx + 15 is factorable over the integers.
(ii) For each value of b, factor the resulting trinomial.
(ii) How many of these trinomials produce a gcf ≠ 1 for 12, b and 15?

(3) If we knew in advance that 180 has 18 positive integer factors, explain how it follows that there are 9 values for b in part (2).

(4)
(i) If the "12" and "15" were interchanged, explain why this would not change the possible values for b in part (2)?
(ii)For each resulting trinomial such as 15x² + 28x +12, determine its factors and explain how they are related to the factors of the original trinomial (i.e., before interchanging the 12 and 15).


QUESTION FOR OUR DISCUSSION (No, these are not rhetorical! Some are quite knotty)

(1)  What do you see as the benefits of this investigation, if any?

(2) Do the new standards and assessments discourage us from investing time into this type of in-depth problem-solving?

(3) Do you believe this type of assignment should be reserved for the accelerated/honors Algebra I student in 7th or 8th grade or even for the stronger Algebra II student?

(4) With the new ADP Algebra standards, do you believe this type of investigation is reasonable, particularly since it is unlikely that any variation of this would appear on an End of Course Test?

(5) If you were to give this problem, how would you edit the investigation? Parts you would delete or change? Parts you would add?

(6) My goal for this blog has always been to provide you with useful and engaging examples of in-depth problems for your students that require going beyond the mechanical aspects of the course. These problems are developed for this blog -- they do not come from my notes from 30 years ago! Would you be interested in a supplementary resource of such problems for each course you teach? Do you already have one from the publisher or from another source which you really enjoy? Share it!

Demo For Building An Investigation In Geometry For All Levels

Note: Diagram has been modified from original.

 For Figures 1 and 2, the following is given:

AD + AC = BD + BC

Perimeter of triangle = 36

AC = 15

Show that the length of AD = 3.

In other words, demonstrate that the length of AD is independent of sides AB and BC.

Instead of imposing or suggesting my way of using this question to build an investigation, how would you do it?

If you're new to this blog, I have published dozens of examples of investigations which are intended to develop process, conceptual understanding, generalization  and a different view of what mathematics is for our students. An investigation allows students to explore particular cases before attempting to generalize and abstract. Some might call this scaffolding. I see it as creating an experimental environment in the classroom, encouraging our students to become mathematical researchers! I know every argument against this approach but, remember, I'm suggesting that this type of activity only be used perhaps once a month...

The question above can be given as is to some groups of students but may not be appropriate in its present form for many others. The question can be reworded or changed completely.

What would you do?

Please Help Dorothy Go Home - A Probability Fantasy for Middle School and Beyond

Summer vacation is an appropriate time for fantasy. Enjoy the hiatus!



The following investigation is not intended to be a math contest challenge. It reviews fundamental principles of probability and you might want to bookmark it for the fall. We can also simulate the first problem using the programming capabilities of a graphing calculator. I may post a simple program for this later on.


The wizard will let Dorothy go home if she can pass three challenges.

He shows Dorothy 3 playing cards, 2 of which are black and one is red. He shuffles them and turns them face down. "Dorothy, here's your first challenge."

"You will pick a card. If it's red the game ends, you win the game. If it's black, I will remove the card and you will pick a card from the remaining two. If it's red you still win! Ah, but if it's black again you and Toto and your weird friends will remain here for at least one more month."

Well, Dorothy won the game and said, "Now, I want to go home!" But the crafty wizard said, "You weren't listening carefully, Dorothy. I never said you can go home if you won the game. You've only passed the first challenge. You must still pass two more." "That's not fair!" Dorothy protested but the wizard makes his own rules in Oz.

"Alright, Dorothy, you won the game but you knew the odds were in your favor since you had two chances to win. Here's your next challenge:

"What was the probability of your winning and you must give me two correct but different methods?"

Dorothy asked, "These are the remaining challenges, so if I get them right, I can go home, yes??"
"I will not lie to you, Dorothy. This is your 2nd challenge. There will still be one more."

Dorothy was upset but knew she had no choice but to trust him. She thought about the problem for a minute and replied, "The probability of my winning was 2/3. I know I'm right!"
"Very good, Dorothy, but you must explain that answer two different ways." Fortunately, Dorothy was a very responsible middle school student back in Kansas and had learned the methods of compound probabilities and the idea of complementary events (this is a fantasy after all!).

Dorothy was able to provide two correct methods. Can you?

"Very good, Dorothy! You only have one more challenge to conquer and you can go home.
This time there are N cards, one of which is red while the remaining cards are black. N is a positive integer greater than 1. Same rules as before. The cards are shuffled and laid out face down. You pick a card. If it's red the game is over and you win. If it's black, the card is removed and you try again. The game continues until you pick the red card. The only way to lose the game is if you pick all the black cards and the last card remaining is red."

"In terms of N, what is the probability that you will win? Oh, yes, you again have to show two different methods in detail on this magic board over here."

This time, Dorothy needs your help. She can guess the formula but she needs our help to show two ways to derive it. Please help Dorothy go home!

The "POWER" of Circles Part I - An Open-Ended Geometry Challenge and CONTEST Update

Well, registration for MathNotation's 2nd Online Contest has now closed. Not as many participants this time but we do have schools representing several states and one high school from Japan! The questions have been emailed to advisors but results will not be available for a couple of weeks. I also plan on publishing at least one of the contest problems in June. I'm still in the planning stages for running 2-3 of these contests for the 2009-10 school year. Stay tuned...

If you're interested in signing up for upcoming contests, just drop me an email at "dmarain at gmail dot com" and I will put you in my database.

For those of you who haven't been reading about these contests it's the team approach, open-endedness and multi-part nature of some of the problems which separates these contests from most others out there. In other words, these questions reflect the types of investigations I've been publishing since 2007.

Also, after I write 5-10 of these contests, I plan to publish these in a book with detailed solutions and comments. As my wife would say, "I'll believe that when I see it!"
**************************************************************************

Speaking of investigations, here is Part I of an open-ended geometry problem that starts out relatively simply but eventually will lead to deeper results. This investigation will review basic circle concepts involving tangents and secants but will connect to the more advanced ideas of power of a point and the inverse of a point in a circle in later posts.




In the diagram, O is the center of the circle PT is a tangent segment, segment AB is a diameter.







(a) If the radius is 6 and PA = 4, show that the length of tangent segment PT is 8.

Note that (PA)(PB) = (4)(16) = 64 and, from part (a), (PT)² also equals 64. This is a special case of the secant-tangent power theorem you may recall from geometry. Your job in the next part is to demonstrate a particular case of this theorem this using the above diagram.

(b) If the radius of the circle is r and PA = x, show that
(PA)(PB) = (PT)².

Note: Using the secant-tangent power theorem here trivializes this problem. The idea is to demonstrate the result without using that theorem, in effect, proving a special case of this rule!

Click on Read More for further comments and a hint for part (b)...




Comments
(i) Neither of the above parts was intended to be highly challenging. Part (a) is definitely an SAT-type question. A review of geometry never hurts!

(ii) Here's a hint to get started on part (b):
From Pythagorean we know that (PT)² = (PO)² - r². Factor this expression...

(iii) Note that my approach as always is to introduce or develop a theorem or concept such as the secant-tangent power theorem or "power of a point" by looking at particular or special cases (the secant in the diagram contains a diameter) and starting with numerical values rather than variables. This sequence (numerical values, particular case, generalization) forms the basis of the investigations I've been writing for this blog, but, more importantly the underlying foundation for the lessons I planned when I taught. I believed then and now that this approach may be time-consuming (both in planning and implementation) but the payoff is deeper conceptual understanding. Of course I needed to modify the presentation according to the backgrounds and ability levels of my students but I never assumed only my honors and AP classes were up to the challenge.

(iv) In Part II which I will post in a few days, we will discuss the "power of a point" and add a second circle, in which segment PT will be a radius. This will produce a another point, P', the inverse of point P.

...Read more

Analysis of a Series: An Investigation before the AP Calculus BC Exam

The remarkable identity above could be the subject of many math blog posts but we will look at a variation, one that is accessible to precalculus and calculus students. With the AP Calculus BC Exam looming, the following investigation can be used to introduce or to review the topic.

I'm not sure if I have ever made it really clear on this blog that I routinely used these kinds of investigations in the classroom. For those who wonder how I could possibly have completed the required coursework for the AP Calculus BC syllabus or who might question my sanity, a couple of points here:

(1) Of course I didn't do this every day. I might have done an extensive investigation once per unit.
(2) Imagine my surprise when I first saw the Finney, Demana, Waits and Kennedy text, a book that has these kinds of explorations in every chapter! I thought they had found my old lesson plans.
(3) Most of the extensive investigations were assigned for work outside the classroom. In fact, for a while, the first investigation of the year was posted on my web site and emailed to students at the end of August before they arrived in school (I met them in June before they left for the summer or I got their phone numbers from guidance and called each of them to tell them to look for the assignment online, and to download and print it.)
(4) Even if I didn't prepare an exploration every day, most every lesson plan which introduced a new topic included a series of leading questions like these. My intent was always to have them think more deeply about a topic, i.e., to understand

• the historical origins of the topic
  
• how it was connected to their prior learning
• its usefulness and application
• why a method or theorem works (derivation, justification)

Developing these lessons initially was labor-intensive but a work of love. Perhaps no more laborious than what another of my colleagues did for his students: developing a PowerPoint presentation for every lesson for the entire year. As time-consuming as all of that sounds, once you've done a few of these, they start to flow naturally and the following year you only have to revise!

A Series Investigation

Consider the following finite series:



(a) Write the series using summation notation.

(b) Verify the following identity for n > 1:



(c) Use the identity in (b) to show that the value of the series above is

Hint: What was Galileo's most famous invention?

(d) Using a method similar to (c) verify the following for n, even:



Note: If n = 2, the right side would be accurate however the left side would consist of only one term. I could have used summation notation for the left side but I didn't want to give away the answer to part (a).

(e) If n is odd, show that the series on the left of part (d) can be written:



(f) Show that the expression on the right side of the equation in (d) and the expression in (e) are algebraically equivalent.

(g) Use the expressions from (d) and (e) to show that the sum of the following infinite series is 3/4:




(h) There are many ways (p-series, integral test, etc.) to prove that the series

converges. However, for this exploration, we will use the convergence of the series in (g) to do this:
Demonstrate that this series converges using both the Comparison Test and the Limit Comparison Test by using the series in (g).

Notes:

• More commonly, the convergence of the series in (g) is demonstrated by comparing it to the p-series. We're doing the reverse here.
• Another important aspect for precalculus and calculus students is to have them compare the partial sums to the sum of the infinite series. Thus, it's worth taking the time to have them see how close the sum is to 0.75 when adding the first 100 terms, the first 1000 terms etc. Also, indicate that the difference can be thought of as the "error" in the approximation. All of this is needed for further study and it deepens their understanding of infinite series.
  
• As indicated above, this investigation may be too time-consuming for a regular period of 40-45 minutes. I would recommend doing parts (a)-(c) (or (d)) in class and assigning the rest for homework to be collected after 2-3 days.
• Teachers of precalculus can use parts of this investigation when developing the concepts of series. Much of the groundwork for infinite series can be laid before students get to calculus!
  

Math Contest Reminder and a Probability Paradox??

There's still time to register for MathNotation's First Math contest for Grades 7-12 to be held on Tue Feb 3rd. I've decided to extend the registration to Thu Jan 29th. We've had interest expressed from high schools, middle schools, homeschooling teams, even a chapter of an honorary math fraternity! I'd like to see 2-3 more teams compete but I understand that many students and teachers are overextended at this time of year and this was on short notice. Look here for how to register.
----------------------------------------------------------------------------------

So what's the paradox in the title? To someone with a firm grasp of probability there won't be one, but the following series of questions may lead to a surprise for some students.

Overview of Problem
We have two scenarios in this investigation:
A set of five 4-choice multiple-choice questions and a set of five 5-choice multiple-choice questions. Of course the latter is typical of most standardized tests like SATs so this discussion may have relevance to many juniors right now!

Instructional Suggestion
For the following questions, ask students to first make educated guesses before attempting any calculations. The idea is to get them to trust their intuition which often is more accurate than their mathematical procedures!

Background
We know that the probability of correctly guessing, at random, the answer to a 4-choice question is 1/4 which is greater than the chance (1/5) of correctly guessing, at random, the answer to a 5-choice question. That was easy, right? When we ask questions about more than one question the situation becomes more complicated and a deeper understanding of probability concepts is needed: Multiplication of probabilities of independent events, binomial probabilities, etc...

The Investigation

(a) Which of the following is more likely? Randomly guessing all 5 wrong on a 5-choice multiple choice quiz or randomly guessing all 5 wrong on a 4-choice multiple choice quiz?
By intuition (no calculation, respond in 10 sec or less): _________________
Explanation of Intuitive Guess (this may be worthy of class discussion):
Now compute each probability and compare result to your intuitive answer.

(b) Which is more likely? Randomly guessing at least one right out of five on a 5-choice multiple-choice quiz or on a 4-choice multiple-choice quiz?
By intuition: ______________
Explanation of intuitive guess:
By calculating:

(c) How's your intuition doing so far?
Let's try this one:
Which is more likely:
Randomly guessing exactly one right out of 5 on a 5-choice quiz or on a 4-choice quiz?
By intuition:
By calculating:

Any surprises? In case your results don't agree with mine, I will tell roughly you what I got (actual probabilities below). The probability of guessing exactly one right out of five on a 5-choice quiz is slightly more than the probability of guessing exactly one right on a 4-choice quiz! A paradox? An anomaly of the arithmetic involved? Logical? Can you explain it? Try!

(d) Back to normalcy? Compute the probabilities of getting exactly two right out of five on a 5-choice quiz and on a 4-choice quiz. Has the order of the universe been restored!


Selected Answers (not the norm for this blog):
(b) Approx 67.2% on a 5-choice quiz; 76.3% on a 4-choice
(c) Approx 40.96% on a 5-choice quiz; 39.55% on a 4-choice
(d) 20.48% on a 5-choice quiz; approx 26.37% on a 4-choice
Pls check these results for accuracy!!

What are the fundamental concepts in this investigation? What are the learning benefits of this series of questions? Please understand that my intent on this blog is to suggest instructional methods, never to impose. You may find far more effective ways to convey the essential concepts here but, from my experience, there's only sure way to perfect our craft. Keep experimenting and asking questions!!

A Presidential Contest Digit Problem: Fours are Wild!

With MathNotation's First Math Contest less than two weeks away (look here for details), I wanted to provide another sample contest question (multi-part). By the way, we now have several middle schools, high schools and even homeschool teams registered from all overv the country! It takes only a few minutes to register and there's still time!
--------------------------------------------------------------------------------------------------------
For President Obama, the number four has special significance. The most obvious is that he's the 44th president. You can ask your students to think of several other connections between our new president and the number four. But for now, we will focus on 44...

(a) Since 44 = 2^5 + 2^3 + 2^2, 44 equals 101100 in base 2 (binary representation).
Let S be the set of all base-10 numbers (positive integers) whose binary representation consists of six digits, exactly three of which are 1's. Find the sum of these base-10 numbers to reveal part of the mystery behind the title of this post!
Note that the leftmost binary digit must be "1".

Comment: This is a fairly straightforward 'counting' problem accessible to middle schoolers as well as older students. One could simply make a list of the numbers and add them. However, there's a more systematic way to count the 'combinations' and a "different" way of adding here that may help you solve the next problem. Can you find it?

(b) Consider the set of all base-10 numbers (positive integers) whose binary representation consists of ten digits, exactly three of which are 1's. Show that the sum of these base-10 numbers can be written 44(2^9) - 4 - 4. "Fours are wild!"

Note: This seems like a tedious generalization of Part I, but, again, if you find the right way to count and add it won't take long!

BTW, if you're wondering how I came to find all these 4's, well, it might have been serendipity. After all, serendipity has 11 letters and 11 is a factor of 44 and... (Twilight Zone music playing in the background...). Also, if you're wondering what my outside sources for these kinds of problems are, do you really think there's anyone else out there whose mind could be this warped!

MLK, Inauguration, Math Contest and A Radical Investigation!

Don't miss registering for MathNotation's First Math Contest. Registration is as simple as emailing me (dmarain "at" "gmail dot com") to request a form and the Rules. The contest is team-based (up to 6 students), is designed for both middle and high school students and should take 45 minutes or less (extra time is provided for students to enter their answers/solutions on the official answer form in Word). Look here for further info.

I would also like to thank the following blogs and/or webmasters for their graciousness in spreading the word about our first math contest:
Let's Play Math!
MathNexus
Wild About Math
Vlorbik
jd2718
Note: Take a look at jd2718 to see the latest Carnival of Mathematics. Another excellent job by Jonathan!
Homeschool Math Blog

----------------------------------------------------------------------
While we're waiting for the Inauguration on 1-20-09 (12,009 = 3 x 4003 of course), today is Dr. King's birthday, 1-19-09 and 11,909 is prime as it should be! How appropriate it is that we should be honoring today the man who paved the way for our new President...
The title of this post reminds me of an old Johnny Carson routine: Which one doesn't belong with the others! In fact, we can probably make connections among all of these if you're willing to play with words...
------------------------------------------------------------------------------------------------
In case you thought that the Math Contest would lead to a hiatus in publishing investigations and instructional strategy articles, fear not! Today we will once again examine the raison d'etre of this blog:
TEACHING BOTH PROCEDURALLY AND FOR MEANING


Part I
Consider the equation


To reinforce multiple representations (Rule of Four) we can ask students to:

Explain or show why this equation has no real solutions
(a) Graphically
(b) Numerically (TABLE)
(c) Algebraically


At this point I am including some ScreenShots from the TI-84. The bold graph is Y1:











Part II - The Extension!
Consider the equation



(a) For what value(s) of k will the above equation have one real solution? In this case, also determine an expression for that solution in terms of k. Show method clearly.

(b) For what value(s) of k will the above equation have no real solutions. Show method clearly.

(c) Demonstrate your results in (a) and (b) by choosing specific values of k for each case. Use both a graph and a TABLE to support your argument. [Use of the graphing calculator makes sense here.]

-----------------------------------------------------------------------
Pedagogy
Which do you think is more helpful to students -- the graph or the TABLE? From my experience I find that both are important for comprehension and concept. They not only complement each other but each contributes something by itself. The graph not only suggests (not prove!) that the two graphs in part I do not intersect but it leads to a natural questions like: Why is the graph of y = √x + 2 above the the graph of Y1? What do the graphs suggest about the domain of each function? Explain the ERR messages!
Note: I used the word "suggest" because we want our students to understand that graphs do not prove mathematical truth.

When is it appropriate to use this approach? After you've taught the algebraic procedures of solving radical equations? Of course, part (c) of the activity asks for the algebraic explanation, but I've often used the graphical and numerical approach BEFORE teaching the procedure. I believe that it developed meaning for the traditional procedure but, in no way, did it replace the need for carefully explained instruction with a variety of examples! (The "balanced" approach!).

Further, the common reaction I've heard to this kind of instruction is that it is too time-consuming and appropriate only for the honors students. I couldn't disagree more. Developing meaning does take time and is absolutely worth it. It's all part of the "less is more" philosophy and, that, if the foundation is properly put into place, students can develop both the skills of solving radical equations and an understanding of the underlying mathematics. Enough preaching to the choir...


I hope you find this useful when building your next exploration in mathematics! Let me know...

A Preview of the Contest: Probability Investigation with Replacement

Don't forget to email me if you want your students to participate in the first MathNotations online math contest on Tue Feb 3rd. There is still time! Look here for info.


There may not be a probability question on the first contest but the following gives you a flavor of the type of multi-part question I'm talking about -- an investigation in more depth.

You will find many variations of the following problem in texts. From experience we know that the student needs to have numerous experiences with these. How do many students do on this topic when the exam question is slightly different from the ones reviewed in class!

THE PROBLEM STATEMENT
Five cards are numbered 1 through 5 (different number on each card). Typical scenario, right?
George chooses cards randomly one at a time. After he selects a card, he marks a dot on the card, then puts it back (replacement!) in the pile of 5 cards, reshuffles them and draws the next card and so on. The game continues until he selects one of the "marked" cards.

INSTRUCTIONAL STRATEGY
Before a technical analysis of this experiment (sample space, random variable, specific probabilities, expected value), I would typically ask students a broad intuitive question or ask them to suggest questions one might ask about this "game".

Intuitively, I might ask:
In the long run, how many draws would you "expect" it to take for the game to end?

With five cards, what do you think most students would guess? Draw three? Draw four? I think asking this initial question is crucial. In most cases, we want the mathematical result to be reasonable and to roughly agree with our intuition (not always of course, there are paradoxes in math which are counterintuitive!).

THE INVESTIGATION

Part I
What is the probability that George chooses a "marked" card on his second draw for the first time? On the 3rd draw for the first time? 4th draw? 5th draw? 6th draw?
Another way to ask these are: What is the probability that the game "ends" after 2 draws? 3 draws, etc.

Part II
"On average", how many cards would George need to draw to get one of the marked cards for the first time?

Note: In more technical language we are asking for the expected number of draws before the game ends?

Normally, I don't publish answers to these questions but, in this case I will give partial results. Please check for accuracy.

The probability the game ends after 3 draws is 8/25 or 32%.
The expected value for the number of draws for the game to end is approximately 3.51. What does this mean!