Teaching Probabilities and Strategies Via Games!

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!

Instructional Strategy Series for Middle School and Beyond: Developing Direct & Inverse Ratio Concepts

Three beagles can dig 4 holes in five days. How many days will it take 6 beagles to dig 8 holes?

Standard Assumptions

Note: It may be highly instructive to ask the students what natural assumptions (stated below) are being made here before you tell them!
(1) All beagles work at the same rate. (If you understand beagle behavior intimately, you might question this). Seriously, it's the underlying assumption of constant "rate of work" that is so fundamental here.
(2) All holes are the same size.

Instructional Commentary
Well, at least, I didn't ask the classic: "How many eggs can 1.5 hens lay in 1.5 days (my all-time favorite word problem)!

The focus of this post will be on the first two stages of concept development using a concrete numerical example. You may take strong exception to the approach below of combining both direct and inverse variation in the same lesson, but, remember, the goal here is concept development, not proficiency with an algorithm! The algebraic stage will be deferred or left to the reader. The algebraic relationships are extremely important and worthy of extended discussion but that needs to be a separate discussion.

Stage I: Building on Intuition
Before developing a strict mathematical procedure involving direct, inverse or joint variation I feel it is critical for students to trust their "math sense." Encourage this with comments like:
"Forget calculations here, boys and girls, just think about this problem, use commonsense, and you might be able to arrive at the answer in less than 10 seconds!"
Don't think they can? No harm in trying...

I believe that when we tell them to trust their intuition, some will arrive at the correct answer of 5 days. Encourage those who "see" it to share their reasoning: WHY will the number days not change! This will vary according to the ability level and confidence of individuals in the group but, even more importantly, according to the environment you create in the classroom (accepting non-judgmental climate leads to greater risk-taking).

When review of homework, content coverage and time for guided practice (before the assignment is given) are the highest priorities of our lessons, then it is natural to question the wisdom of the above strategy.
This is obvious from typical comments like:

"Very nice, Dave, but who has the time for that, it's not going to be tested on the State Test and, moreover, I'm not teaching gifted kids like you must have had."

I won't react to my own Devil's Advocate arguments. Those you who know my philosophy of education know what my response would be!

Stage II: Beyond Intuition - Developing Proportionality Concepts via a Systematic Approach
"Well, boys and girls, now that we believe the answer is still FIVE days, let's try to approach this more mathematically, that is, more logically and systematically, in case the answer cannot be 'guessed' so easily."

I have found over the years that the following TABLE or matrix approach is a powerful model for devleoping proportionality concepts before the student sees a single algebraic relationship:

EVERY DOG HAS HIS DAY!
Beagles	Holes	Days
3	4	5
3	8	??
6	8	???

Note how this approach avoid changing both the number of holes and the number of dogs in the same step! By keeping one quantity fixed, the student may better be able to focus on the relationship between the other two. Thus, in the second row I kept the number of dogs constant, changing only the number of holes:
"Boys and girls, if the number of dogs stays the same and we double the number of holes, then what will happen to the number of days ?"(they will double).
[Note that I asked for the effect on the the number of days before I asked for the actual number of days, namely 10 days.]

This approach develops the idea of direct variation before we express the relationship algebraically: As one quantity increases, so does a second quantity proportionately.

Now that we have filled in the second row (replace the ?? with 10 days), we can move on to another relationship:
"Boys and girls, look at the 3rd row. What quantity (variable) did we not change (keep constant)? What quantity did change? If we double the number of dogs, what should happen to the number of days needed to dig the same number of holes?"
(Yes, some will think 'double', since direct variation is often the initial reaction of many students).

Thus we are literally constructing direct and inverse variation via numerical computation before we develop any general relationships. Yes, this is time-consuming, but hopefully you will see the payoff in comprehension.

Stage III: Expressing Relationships Algebraically
Not in this post!
Important Note:
Normally, we would be very reluctant to mix both types of variation in one lesson, choosing to develop mastery of just direct variation first, then inverse much later on. Yes? Therefore you might feel that combining these will lead to confusion on the part of most students in most classes. Remember, though, the intent here was to develop a strong intuitive base for different types of variations before attempting to formalize any of this! You may not agree, but I'm proposing it anyway. I have done this with good results. Once the concept foundation is laid, students can take off with all the formulas!

A Different "Approach" to 0.99999999...??

An investigation for middle schoolers? Precalculus students? Calculus students? Anyone who is fascinated by patterns and an understanding of the infinite and infinite processes?? Enjoy this at any level or depth you wish...

Take out your calculators folks....
Determine the first dozen decimal places, then the exact decimal for each of the following:

1 - 1/9

1 - 1/99

1 - 1/999

1 - 1/9999

1 - 1/99999

Continue this pattern until the denominator has a string of 9 nines.

Questions:
(1) Describe any patterns you observe. What if the denominator had a string of 100 nines? A string of N nines?
(2) What does all of this suggest (not prove) about the meaning of 0.999999... (repeating)?
(3) Oh, and by the way, you may also want to examine the decimal expansions of 1/9, 1/99, 1/999, 1/9999, ... How would you describe the exact decimal representation of 1/9999...9, where the denominator has 100 nines? N nines?

Is there anything new under the sun here? OR just another view of well-known facts about infinite repeating decimals, sums of infinite geometric series, limits and real numbers???
Your thoughts...

Odds and Evens - FEATURED BLOGS and Update for Dec 2008

Some thoughts and loose ends...

It's long overdue but better late than never - I've decided to feature two math websites/blogs for this month. I will try to do this on a monthly basis. It's important to me to acknowledge the outstanding work of other math bloggers particularly those who have been kind enough to support my efforts over the past two years. There are so many excellent sites to choose from and I don't want anyone to think I'm ranking these by this selection. I will acknowledge several others but I had to start somewhere.

FEATURED MATH BLOG/WEBSITES FOR DEC.


Maria Miller's HomeSchool Math blog.

In Maria's own words:
I love teaching and I love math. This blog is my way of reaching out and helping you to teach it too.

Maria has hundreds of posts covering many topics in math and math education as can be seen from the extensive Labels section in the right sidebar. Maria particularly enjoys demonstrating the bar model method from Singapore Math. Well worth reading...

Maria also has a comprehensive math resource site with links to free materials as well as other excellent products. In particular, I'd like to bring the following page to your attention: Problem Solving, Word problems and Math Projects - Free resources on line for K-8 students and beyond. Maria is very thorough and writes a short review of each. Make no mistake -- these resources are for public school parents, students and educators as well. Maria's enthusiasm, dedication, and knowledge are incomparable.

Finally, Maria has developed a series of highly regarded workbooks, her Math Mammoth series, for Grades 1-5. Affordable, easy to follow, a complete curriculum... Enjoy!


MathNEXUS
MathNotations was the featured website of the week on this excellent site. I would strongly recommend this site for all math educators. Jerry Johnson who maintains it somehow manages to update frequently with useful links to just about everything going on in math education. His weekly features include Problem of the Week, Quote of the Week, Statistic of the Week, Humor of the Week, Website of the Week (MathNotations made it!), Calendar Events and Famous Birthdays to name a few! This site receives thousands of visits each week. You will not be disappointed.

Talk about weird coincidences. Jerry found my blog and somehow remembered me from a presentation I gave back in the 90's at the Annual NCTM Convention in San Diego. Turns out that he was the program chair!

UPDATE
1. Anyone remember a MathNotations post from awhile ago regarding the classic boring of a hole through a sphere problem. This well-known Geometry/Calculus conundrum continues to tantalize and intrigue readers. Here's the idea of the puzzle if you forgot it:

Imagine you have a bowling ball with a diameter of 8.5 inches and you drill a hole exactly through the middle so that the remaining part of the ball is 6 inches high. Now imagine boring a large hole though a giant bowling with a diameter of 8.5 feet so that the part of the ball left behind is still only 6 inches high. Incredibly, the volumes remaining of these two drilled-out bowling balls are equal! Can you picture this? The scientific approach here might be to physically drill through two such objects of equal density and compare the volumes remaining by weighing them, but mathematicians usually try to avoid such 'hands-on' experiments, choosing to prove the result using methods from geometry or calculus. From the number of views this puzzle has received, I'm working on a video demonstrating the result using elementary calculus (volumes by cylindrical shells or disks). Because the writing on the board I'm using tends to be small, it will be difficult to read all the details but I'll do my best. I recognize that these videos are not of the quality you can find elsewhere on YouTube but I still think that working through such a fascinating problem makes it worthwhile and somewhat unusual. Considering that it reviews fundamental integral methods it might also be helpful for students who will soon be getting to this topic (or reviewing for exams). If you watched any of my other videos you know I explain the details thoroughly, trying to avoid skipping any key steps. That guarantees that the hole in the sphere video will be quite long (I'll probably break it up) and it will take some time to bring it to fruition. Despite the technical issues, if you want to see more of these, let me know. I'm also considering a few dedicated to the lower grades, e.g., how to develop algebra sense in middle schoolers (from numerical patterns to algebraic expressions). Any interest?

2. The problem from the other day regarding the angles of an isosceles triangle received some nice responses but the question would not have made a good contest or test problem. There's nothing worse than an apparently challenging question that students can answer correctly using incorrect reasoning! Here again is the problem:

Exactly two of the sides of a triangle are congruent and one of the equal angles is known to have degree measure greater than 50. How many integer values are possible for the measure of the remaining angle (the one that is different from the base angles)?

I had indicated there were a couple of sticky points one had to be wary of (my readers caught on to the fact that the equal angles do not have to have integer measure and that the triangle is not equilateral), but, if the student overlooks both of these, the correct answer can still be obtained! Thus, if the student starts with a base angle of 51°, one obtains a vertex angle of 78°. Reasoning quickly, one arrives at 78, the correct answer, as the number of possible values (all integers from 1 to 78). Two errors which nullify each other and produce a correct result! That's why it's always preferable to have assessments in which students are required to explain their methods but it's simply impractical to have standardized tests with 45-50 such questions. I should have caught that but it does demonstrate the challenges of writing high-quality short-answer (objective-type) questions!

3. By now you've probably all forgotten about the MathAnagram for Oct-Nov-Dec.
Check the link if you're interested. Here it is again:

CHARMED ERA

FRACTIONS, FRACTIONS, FRACTIONS EVERYWHERE - A MIDDLE SCHOOL ACTIVITY

What middle schooler doesn't get that warm and fuzzy feeling when we tell them we're going to play with fractions! Here's a small investigation to capture that mood of euphoria...

RULES OF THE GAME:

• No Calculators - No decimals!
  
• All fractions must be expressed in lowest terms
  

On a number line mark off 0 and 1 and approximately locate points A and B whose coordinates are 1/4 and 1/3 respectively.

1. Write the fraction that divides the segment between A and B into two equal parts. How can we verify that this point satisfies the desired condition.
Complete: This fraction is the _______ of 1/4 and 1/3 and is ____-way between 1/4 and 1/3. The corresponding point is the _________ of segment AB.

2. Write the two fractions that divide the segment between A and B into three equal parts.

3. Do the same for four, five and six equal parts.

4. Reach/Extension: Describe a general procedure for dividing the segment AB into any desired number of equal parts (mathematically speaking, we would say n equal parts).


Comments:
(1) What prerequisite skills do students need to have in order to attempt this investigation? When planning a lesson like this, I found I had to consider this question first and review those needed skills. This avoided many issues that would otherwise slow down the lesson. I always tried to avoid the "You don't remember this?" comment. Sometimes this took superhuman effort on my part!

(2) Students should be organized into pairs or teams of 4. They can "divide" up the labor.

(3) There are several approaches to these problems. Many confident students with strong foundation skills (ok, this narrows it down to one student in the back of the room), recognize that a common denominator approach makes the most sense. You might see some very clever resourceful methods coming from your youngsters.
Note that 1/4 = 3/12, 1/3 = 4/12. In order to place a fraction in the 'middle', rewrite 1/4 = 6/24, 1/3 = 8/24. Don't be surprised to see some students invent similar methods for the other divisions.

Geometry Problem Requiring Critical Reading & Thinking

Exactly two of the sides of a triangle are congruent and one of the equal angles is known to have degree measure greater than 50. How many integer values are possible for the measure of the remaining angle (the one that is different from the base angles)?

Comments:
(1) This is not intended to be a significant challenge. Rather it is meant as a warmup or for a slightly more extended discussion.
(2) Since middle school students know the sum of the angles of a triangle and can be told the basic fact about the base angles of an isosceles triangle, this problem is appropriate for them too.
(3) How would you expect most students to approach this? Do you think the majority would start by plugging in 51, 52, 53, etc.?
(4) Does this type of question promote important problem-solving skills and strategies? Do students recognize the significance of the 'boundary values' 50 and 90, values that are not in the domain of the base angles yet can be critical for the analysis?
(5) There are at least two 'traps' set in this problem that are intended to help students become more critical thinkers and not jump too quickly to conclusions. After all, what is a trap? If one is circumspect, details are not so easily overlooked.

The Product of a 40-Digit Integer and a 60-Digit Integer has ___ or ___ Digits. A Problem for the Calculating Middle School Mind!

Too ambitious as a WarmUp for the 6th or 7th grader? Would they immediately employ the "Make it simpler and look for a pattern" strategy? Is the calculator appropriate for this activity? Is this really an activity/investigation?

Since I'm already regarded as an anachronism, I guess it wouldn't hurt to play word games here:

Is this problem un'characteristic' of MathNotations!?!

Hey, there's a whole generation (or more) who may have no idea what that means! If you do know, you can always say you heard about it from your great-grandfather who carried around his slide rule! Hey, anyone have their Keuffel & Esser handy?

Just Another SAT-Type Combinatorial Problem

How many 3-digit positive even integers have at least one digit equal to 2?

Comments:
(1) Talk about intense information overloading! Students need considerable exposure to these short but concentrated exercises. There are at least 6 pieces of information packed into these 14 words! What % of students do you think would miss or misinterpret one or more of these 'clues'?
(2) Do they really put questions like this on the SATs? Ask any student who recently took the PSAT!
(3) There are many strategies possible here. Many students (if they fully comprehend the question) will start listing 102,112,120,... but how successful will they be using this approach? There is a powerful approach for the "at least one" types of counting problems. I strongly advocate this starting in middle school.
(4) I have published many other similar problems (look under combinatorial math in the index). Do these get easier with practice over time? I think so but there always needs to be clarity of thinking and a careful organized approach. The quick clever student often falters when detail is required. This may help that student to mature!
(5) Are you thinking I'm making too big a deal over SAT-types of questions? What if your students won't even take these tests, choosing ACTs instead? Hopefully, you will come to believe that my purpose is to use these kinds of problems simply as a vehicle for taking students to a higher level of thought. What would be the harm of using these for the occasional class opener (aka Warm-Up, Problem of the Day, Do Now, etc.). In fact I would encourage this at least once a week!

Addendum
Another compelling reason to discuss more than one method of solution for combinatorial problems: One is rarely 100% certain of the accuracy of one's answer without doing the problem by an alternate method and getting the same result!