MathNotations

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 expectednumber 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!

Posted by Dave Marain at 6:31 AM 0 comments

Labels: discrete math, investigations, math contest problems, probability

NCTM Teaching Standards:

Develop and evaluate inferences and predictions that are based on data

Understand and apply basic concepts of probability

Target Audience: Grades 7-12

Tools Needed: Graphing calculator with a random integer generator or an online random number generator (look here for example)

Classroom Organization
(After demo mode): Students working in groups of 3 (two opponents, the 3rd calls out the numbers and keeps score; roles are rotated)

Sample Classroom Scenario
Who thinks they can beat me at a game of chance? I will demo the game, then I will play against an opponent. If you beat me two out of three, you are the new champ and you can pick your opponent. After 10 minutes, you will be playing in small groups and recording the results.

The Play
Using a random integer generator we will generate random digits, one at a time, from 1 through 9, inclusive (no zeros). The object is to build a 5-digit integer which is greater than your opponent's by placing each 'called' digit into one of the five place-value positions. Once you place a digit you cannot change it!

Let's try it... Ok, Marissa, turn on the random integer generator, press Enter and call out the first integer. FOUR!
Ok, I'll place it here: ___ ___ 4 ___ ____
Call out the next integer: SIX!
I'll place it here: ___ 6 4 ___ ___
Next: TWO!
___ 6 4 ___ 2
Next: FOUR!
___ 6 4 4 2
Last digit! FIVE!
5 6 4 4 2
How did I do? Could I have used a better strategy? Do you think you could have beaten me?
Who wants to play! To win, you have to beat me two out of three. Ok, Dimitri, I will work on my paper and you work on yours. Remember, you cannot change a digit's position once you place it...

Brief Discussion of Strategy Based on Probability Arguments:
Suppose the first two digits called are 3 and 6 in that order. Would it be better to place the 6 in the thousands' place or the ten thousands' (leftmost) position? If you place the digits here:
___ 6 ___ 3 ___, what is the probability that at least one of the next three digits chosen will be 6,7,8, or 9. (Otherwise, your strategy would have backfired). To compute this, we look at the complementary condition, i.e., we determine the probability that the next 3 digits chosen will all be in the range 1 through 5. The probability of this is (5/9)(5/9)(5/9) or approximately 17%, so the probability that our strategy works is about 83%, odds that seem worth playing! Experienced game players often compute these probabilities mentally or have seen these situations so many times they know these probabilities by heart!

Notes
(1) Students may not know there is a Random Integer generator built into many graphing calculators. For example on the TI-84, press MATH, then PRB, then 5:randInt(.
From the home screen, Enter randInt(1,9), ENTER. Each time you press ENTER another "random" digit will be displayed. The person calling these out must be instructed to announce only ONE digit at a time!
(2) Why 5-digit numbers? This seems to make the game fairly interesting and moving at a good pace. Expect ties of course!

Perhaps this is a good activity before the holidays. Have fun and let me know how it goes!

Posted by Dave Marain at 9:32 AM 11 comments

Labels: discrete math, games, probability, winning strategies

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.

Posted by Dave Marain at 4:33 PM 3 comments

Labels: counting problems, digit problems, discrete math, investigations, probability, writing in math

Only a few days left for your July MathAnagram. Either it's harder than I thought or no one is paying attention!

Look here for details.

Probability questions will forever addle the minds of students and adults alike. If all problems could involve selecting one object randomly, life would be good. Unfortunately, selecting 'more than one' is becoming common on standardized tests these days. Taking two or more objects immediately ratchets up the difficulty:
Does order count?
Multiple solution paths
Making a list
Combinations? Permutations? Multiplication Principle? Using "rules" of probability?

In what course do students receive sufficient instruction in this important area? Algebra 1? Algebra 2? Precalculus? A probability/statistics/discrete math class? AP Stat? IMO, the lack of standardization in secondary curriculum can lead to some topics getting short shrift.

I've come to the conclusion that middle schoolers should devote more time to some of this, since 4th graders are generally expected (in most states' standards) to solve the "select one object" type. What do you think? Since most readers enjoy the math challenges and not this kind of curriculum discussion, here are our offerings for today...

A bag contains eight coins: two each of pennies, nickels, dimes and quarters.

Question 1: If two coins are randomly selected, show that the probability that the two coins will total at least 20 cents in value is 1/2.

Question 2: If 4 coins are randomly selected, show that the probability of getting exactly one of each kind of coin is 8/35. (At least two methods please!)
Note: This result implies that the chances of getting at least one matching pair of coins among the 4 coins is greater than 75%!!

Question 3: Invent your own!

Comments:
There are many problems of this type on this blog (see probability, combinatorial math in the index). Further, there are many other excellent blogs and web sites that address these topics and provide wonderful challenges and explanations. Two of the best are Jonathan's over at jd2718 and Isabel over at God Plays Dice.

Posted by Dave Marain at 6:19 AM 11 comments

Labels: combinatorial math, discrete math, probability

Here is the link to the Carnival of Math Edition X.

The following is the first in a series of investigations in recursive sequences and functions for middle school and secondary students. This apparently advanced topic is accessible to prealgebra students at an introductory level. The first few parts of the investigation below are appropriate for the younger students. The remaining parts require more algebraic facility and reasoning. The problem in the title of this post doesn't begin until more than halfway down the page (after some background is developed). Do not skip the background below since it's referred to frequently in the activity. My personal experience is that this topic is highly engaging to students. Considering the connection between recursion and fractals, this topic is certainly part of most standards-based curricula. The terminology of recursion (recursively-defined sequences, recursive description, recursive function, recurrence relations, etc.) is quite confusing at first. Many confuse these ideas with iteration, a general term for describing repetitive algorithms.
Finally, from a pedagogical point of view, please note how the Rule of Four is implemented in the activity below: We start with a verbal description of the rule of formation of a sequence (in natural language), followed by a concrete numerical representation of the terms, followed by symbolic representation. One could also depict the terms graphically on a number line or in the coordinate plane if the function model is used for the sequence.
----------------------------------------------------------
I should probably save this for the new school year but it's hard for me to suppress ideas when they begin to crystallize. I've been thinking for some time about how we can introduce recursive functions in prealgebra through advanced algebra and beyond. I enjoy taking sophisticated ideas and reducing them to basic principles, then developing lessons that explore the topic in some depth. Moreover, this particular topic reveals the interconnectedness of mathematics in a particularly elegant and beautiful way.

Background (Needed for the Investigation Below!)
Consider the sequence 1,2,4,8,...
Elementary students can generally guess the most likely value for the next term, 16. They also are expected to identify the 'rule' of forming the 'next' term, namely doubling or multiplying by 2. This is an important stage in their development of algebraic reasoning - abstraction or generalization. In addition, they should begin to recognize that the terms of the sequence can be described generally as powers of 2, even though a formal introduction to exponents normally begins in 7th grade.
Middle school students should progress to the function table format of a sequence:
n.....a_n
1.....1
2....2
3....4
4....8
5....16
...

Elementary and middle school students should be able to verbalize in natural language that 'you double the terms'. As educators, we need to lead them to a more formal relationship by a line of Socratic questioning like: "Double what? To get what?" Students should then be able to express the idea that each term is twice the previous term. We can ask, "Which term doesn't follow that rule?"
To symbolically describe this sequence, we can write:
a₁ = 1
a_n+1 = 2 ⋅ a_n, n = 1,2,3,...
This is known as a recursive description of the sequence. Try it - replace n by 1,2, and 3 and see if it produces the terms above.
[Note: Later on, in more advanced algebra, students should be able to express this as a recursive function: f(1) = 1; f(n+1) = 2f(n), n = 1,2,3,...]

The closed or general form requires a knowledge of exponents but is accessible to 7th graders
a_n = 2^n-1, n = 1,2,3,... Try it!
(If you're questioning my sanity (you wouldn't be the first!) about introducing such sophisticated mathematics to general 7th graders, well, I do have a legitimate basis for this curricular decision - more later...).

Powers lend themselves naturally to a recursive description and this is why I begin with the above example. Recursive thinking develops when we ask questions like:
If we know what 2⁵ is, how would we obtain 2⁶?
To deepen this understanding further:
If we know what 2⁹⁸ is, how would we obtain 2¹⁰⁰?
Does the exponent key on a calculator help students see these relationships? Not really! The calculator is useful to demonstrate powers and exponents but not for this discussion. Later on, the graphing calculator can be used to enter recursively-defined functions (after they've learned the ideas!).

If you're very familiar with recursively defined sequences and functions, you've probably left this page already! However, the idea of an operation or function being defined in terms of itself is a beautiful and very important notion in mathematics. This type of thinking was necessary for Mandelbrot to develop the notion of fractals, which defines a process in which each stage is defined in terms of the preceding stage or stages - that is recursive thinking!

Ok, by now you're wondering what happened to the title of this blog!
THE PROBLEM:
TAKE ANY NUMBER, ADD THREE, DIVIDE (OR MULTIPLY) THE RESULT BY -1. NOW REPEAT THIS SEQUENCE OF OPERATIONS ON THE RESULT YOU OBTAINED.

Student Activity:
1. Start with the number 6 and follow the instructions above. Repeat this 2 more times. List the first 4 terms of the sequence obtained. Write a brief description of what you observe about this sequence.
2. This time start with a different integer. Again, list the first 4 terms of the sequence obtained and your observations.
3. By now you've concluded that the sequence obtained will alternate in the form a,b,a,b,...
Which one of the original operations (add 3, multiply result by -1, etc.) do you believe is causing the sequence to repeat like this?

The remaining parts require algebra background.

4. If the first term is x, verify algebraically that the sequence will alternate.
5. You've now determined that the sequence appears to be repeating but not constant like a,a,a,a,... For what value of x, the first term, will the sequence be constant, i.e., all terms will have the same value?
6. Write a recursive definition (refer to how we did this for powers) for our sequence whose first term is x. We'll start you off:
a₁ = ____
a_n = __________, n = _______
7. BACKGROUND
The algebraic formulation of the recursive description in #6 was fairly straightforward, since it is just a symbolic representation of the verbal "Take any number, add three, then divide the result by -1." As useful as this may be, we often a need a general formula for the nth term as a function of n, rather than in terms of preceding terms of the sequence. That last sentence was fairly complex, so here's an illustration:
Let's assume the first term is 6. Then the nth term can be described as:
a_n = 6 if n = 1,3,5,7,... (i.e., n is odd)
a_n = -9 if n = 2,4,6,... (i.e., n is even).
This would allow us to find any particular term, say the 100th term, without knowing the values of preceding terms. Such a description is known as a general description or the closed form of the sequence. Such a formula is often very hard to determine, whereas the recursive form is easier to formulate. In our problem, the general formula for the nth term had to be given in two cases or piecewise as mathematicians term it.
It is possible to give a single formula for all of the terms of our sequence as a function of n:
a_n = -7.5(-1)ⁿ - 1.5, n = 1,2,3,...
Verify this formula for our sequence above: 6,-9,6,-9,6,-9,...
8. Change the original problem to:
Take any number, add 2 to it and multiply the result by -1. Repeat.
(a) Starting with an original value of 6 (as the first term), list the first 5 terms of the sequence.
(b) Write a recursive description for this sequence.
(c) Write a piecewise formula for the nth term as in the background example above.
(d) Write a single formula (closed form) for the nth term in terms of n for this sequence.

9. (More Challenging)
A sequence is defined verbally by:
Take any number, add k to it and multiply the result by -1.
(a) If the first term is x, write a recursive description for this sequence.
(b) Write a piecewise formula for the nth term.
(c) (Super Challenge) Write a single formula (closed form) for the nth term in terms of n for this sequence.

Posted by Dave Marain at 4:00 PM 7 comments

Labels: advanced algebra, discrete math, middle school, recursive functions, SAT-type problems

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