SAT Videos: Twitter Problems of the Day 6-9 and 6-10-10

As we wind down toward the summer my SAT Problems and Videos continue to pick up steam! Below is the latest video from you YouTube channel, MathNotationsVids. I want to thank those who voted in my survey of these videos. I am gratified but I really need more specific suggestions on how to improve these. Your comments on YouTube or here are welcome!

Note: Because I am explaining two problems on one video, I am omitting details and multiple solution paths. Therefore these videos may be useful for your students who want to practice over the summer or revisit in the fall. 


The percent increase problem could be asked in a variety of ways and demonstrated using multiple representations, aka The Rule of Four.  The visualization suggested in the description of the video has students physically demonstrating that doubling the edges of a rectangular solid, a cube in this case, will allow placing not only the original box inside of the bigger box, but SEVEN MORE! There's your percent increase, hands on!

I will be stopping the posted SAT Problem on Twitter on Tue 6-15-10. If I am able to sustain it, I will try to keep this up for the entire 2010-11 school year but who knows...





Finally, as posted on Twitter, I will be offering an individual or small group online course (using Skype) for the SAT or ACT Math this summer on a very limited basis. If you know of any student who might benefit from individualized instruction just email me at dmarain@gmail.com and I will provide details. This must be done ASAP however, as I will be closing this out very quickly.



"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

Another Sample Contest Problem - Counting...

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

BASIC RULES
* Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
* Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
* The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
* Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
* Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
* Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
* Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.


Ok, here's another sample contest problem, this time a "counting" question that is equally appropriate for middle schoolers and high schoolers:

How many 4-digit positive integers have distinct digits and the property that the product of their thousands' and hundreds' digits equals the product of their tens' and units' digits?

Comments
The math background here may be middle school but the reading comprehension level and specific knowledge of math terminology is quite high. This more than counting strategies is often an impediment. If this were an SAT-type question, an example would be given of such a number to give access to students who cannot decipher the problem, thereby testing the math more than the verbal side. On most contests, however, anything is fair game!

Beyond understanding what the question is asking, I believe there are some worthwhile counting strategies and combinatorial thinking involved here. Enjoy it!

Click More to see the result I came up with (although you may find an error and want to correct it!)




My Unofficial Answer: 40
(Please feel free to challenge that in your comments!!_

...Read more

Batteries Required! A Combinatorial Problem MS /HS Students Can Use...

Have you ever inserted batteries in a device only to find that it didn't work? You reverse the batteries and try again, but no luck. You can't find the polarity diagram to guide you and you're dealing with 3 or 4 batteries and all the possible combinations! Well, that just happened to me as I was inserting 3 'C' batteries into a new emergency lantern I just purchased. There was no guide that I could see. I knew there were 8 possibilities but it was late and my patience quickly ran out. I tried it again the following morning, shone my small LED light on it and saw the barely visible diagram.

After seeing the lantern finally operate, I realized I should have used a methodical approach -- practice what I preach!! Then I thought that this might be a natural application of the Multiplication Principle one could use in the classroom. Of course, it would work nicely if you happened to have the identical lantern but you might have some of these in the building or at home which take 2 or more batteries. IMO, there's something very real and exciting about solving a math problem and seeing the solution confirmed by having "the light go on!" I'll avoid commenting on the obvious symbolism of that quoted phrase...

Instructional/Pedagogical Considerations

(1) I would start with a small flashlight requiring only one battery to set up the problem. For this simplest case, students should be encouraged to describe the correct placement in their own words and on paper.

(2) Would you have several flashlights/lanterns available, one for each group of 2-4 students or would you demonstrate the problem with one device and call on students to suggest a placement of the batteries? Needless to say, if you allow students to work with their own flashlights, they will look for the polarity diagram so you will need to cover those somehow. That is problematic!

(3) Do you believe most middle school students (if the polarity diagram is not visible) will randomly dump in the batteries to get the light to go on and be the first to do so? Is it a good idea to let them do it their way before developing a methodical approach? Again, if a student or group solves the problem, it is important to have them write their solution before describing it to the class. If there is more than one battery compartment, students should realize realize the need to label the compartments such as A, B, C , ... Once they reach 3 or more batteries, they should recognize that a more structured methodical approach is needed so that one doesn't repeat the same battery placement or miss one. One would hope!

(4) Is it a drawback that the experiment will probably end (i.e., the light goes on) before exhausting all possible combinations? How would we motivate students to make an organized list or devise a methodical approach if the light goes on after the first or second placement of the batteries?

(5) I usually model these kinds of problems using the so-called "slot" method. Label the compartments A, B, ... for example and make a "slot" for each. For two compartments we have

A B
_ _

Under each slot, I list the possibilities, e.g., (+) end UP or DOWN (depending on the device, other words may be more appropriate). Here I would only concern myself with labeling the (+) end, the one with the small round protruding nub. For this problem I would write the number (2) on each slot since there are only TWO ways for each battery to be placed. Note the use of (..). In general, above each slot I would write the number of possibilities. For two compartments (or two batteries), the students would therefore write (2) (2). They know the answer is 4 but some will think we are adding rather than multiplying. Ask the class which operation they believe will always work. How would you express your questions or explanation to move students toward the multiplication model? The precise language we use is of critical importance and we usually only learn this by experimentation. If one way of expressing it doesn't seem to click with some students, we try another until we refine it or see the need for several ways of phrasing it. This is the true challenge of teaching IMO. We can plan all of this carefully ahead of time, but we don't know what the effect is until we go "live" (or have experienced it many times!).

Perhaps you've already used a similar application in the classroom - please share with us how you implemented it. Circuit diagrams in electronics also lend themselves nicely to this approach. Typically, I've used 2, 3 or more different coins to demonstrate the principle but the batteries seem to be a more natural example, although I see advantages and disadvantages to both. At least with the batteries, students should not question the issue of whether "order counts!"

I could say much more about developing the Multiplication Principle in the classroom, but I would rather hear from my readers.
If you've used other models to demo this key principle, let us know...


REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

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!

Taking Two or More At a Time- What is the Probability You Don't Like These!

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

SIR "OY", E. NOET!

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.

Probability and Counting Challenge -- You'll Need To Sit Down for This One

Five people with first initials A, B, C, D and E were seated randomly in a row in a movie theater with no spaces between them. What is the probability that A, B and C were adjacent to each other in some order? (For example: "DBACE")

A potential SAT problem or is it a level above? A math contest problem or not difficult enough? More importantly, how much experience do most of our students have with these kinds of combinatorial problems? I know that some of our educators who visit here do these with their classes, but is it the norm? My feeling is that students need to see many of these developed over time in more than one course.

What method (s) do you consider the most effective for solving these kinds of problems? For teaching? Are these 2 questions really the same?

After this question is discussed with the class, how does the instructor assess the learning? Give them another one to try immediately or give a worksheet of these (if the text does not provide enough practice)? How could one raise the bar even higher?


Suggested Extension #1: This time we have 10 people seated randomly in a row (no spaces). What is the probability that 4 of them, say A. B, C and D, would be adjacent to each other?

Suggested Extension #2: N people seated randomly in a row (no spaces). What is the probability that a particular subset of M of these people would be adjacent to each other?

Your thoughts...

Geometry WarmUp - A Simpler Integer Triangle Problem

While we're waiting for the 60° integer triangle problem, here's an easier one for both middle schoolers and secondary students. The only fact from geometry that is needed is the all-important triangle inequality:
Any side (in particular, the largest side) of a triangle is less than the sum of the other two sides.
Of course this refers to the lengths of the sides and one can express this in other forms, but I'll leave it at that.
This type of question has become a favorite on the SATs and other standardized tests but, more importantly, it develops clear systematic thinking - the organized list....

How many different triangles have integer side lengths and a perimeter of 5? 10? 15? 20? 25?

COMMENTS/INSTRUCTIONAL HINTS:

• There are really five separate questions here. The instructor can give some or all of these depending on the time allotted. To help the group get started and for clarification, it may be helpful to demonstrate the first question for the group: For a perimeter of 5, there is only one possible triangle, which we can symbolize as {2,2,1}. If these are older students who are comfortable with the triangle inequality, you do not necessarily have to model this one, but that's your call. By modeling the first one, you eliminate some of the ambiguity of ordering the sides.
  
• Since a primary objective here is to make an organized list, you may want to stop after the perimeter of 10 and discuss it at the board. Depending on the ability level of the group, I usually have students work independently, then check each other's work in pairs after they do a couple of these questions. Sort of a think-pair-share approach. Also, don't be afraid to provoke their thinking with questions as they begin to develop their systematic lists (which can get boring for some): "So, do you expect more triangles for a perimeter of 10? Twice as many?"
  
• As each question is reviewed, encourage students to record their results in a table:
  Perimeter..................Number of Triangles
  ......5........................................... 1 ................
  ....10.......................................... 2 ................
  This is critical for middle schoolers in particular, since tables are a basic model for functions! At some point, you can use n or p for the perimeter and symbolize the number of triangles having perimeter n or p as T(n) or T(p).
• Naturally, some students will assume there is a pattern and guess there are 3 possible triangles with a perimeter of 15 - NOT! However, it is natural for all of us to ask: "WHAT'S THE FORMULA?" Well, there is one. It's fairly sophisticated and related to partitions of numbers, but I'll let our readers do their own research for this...
  

An Introduction to the Mathematics of Bingo - Part I: An Investigation for Grades 7-12

While you're celebrating New Year's Eve (meaning you're probably not reading this blog!), or thinking about your favorite math teacher (and probably keeping it to yourself), or considering clicking on the new subscriber chiclets in the sidebar, I thought I would kick off 2008 with something different.

As we were playing family Bingo a few days ago with about three dozen families (my wife was reluctant to go and, of course, she won the first game!), I began thinking about the underlying mathematics of the game and its many variations. I know what my wife would be thinking: "Dave, Why can't you just enjoy the game without analyzing it!"

Here were some thoughts running around in my head, when I should have been concentrating on the two boards I was playing (I won nothing BTW):
(1) Historically: What are the origins of this game? Designed by some brilliant mathematician or was it just some game of chance that evolved?
(2) The number of possible boards must be astronomical. How many different boards are supplied by the companies that manufacture this product?
(3) If I buy a bingo game that comes with, say, 36 boards, are these same 36 boards in every box? If I buy a set from a different manufacturer will the boards overlap or be entirely different?
(4) Are all boards randomly generated by software these days? If a bingo game is to be played in a large hall, with hundreds or even thousands of players, how likely is it that there will be multiple winners in a single game? Do the boards all have different winning lines or are there lots of overlap among boards?

Some probability thoughts:
(5) What is the probability of a winner after the minimum number of balls drawn from the bingo cage, namely four numbers (don't forget the free space!). Since I've never seen this happen, I'm assuming the chances are virtually zero!
(6) More realistically, for about two dozen players, each playing a single board (or one player using 24 boards!), what the expected number of balls drawn before a winner occurs? I was conjecturing less than half of the seventy-five numbers, maybe low thirties.
(7) How did the probabilities change as one increases players and boards? I assumed many mathematicians had already solved all of the intricacies regarding the probability of a winner after 10 numbers, 20 numbers, 30 numbers, etc. Instinctively, I felt that this was a very sophisticated problem, probably beyond my comprehension, but I wanted to know more.

Of course, when we returned home, I did some online research of the game -- fascinating stuff: Origins in Italy, Lotto, Beano, Bingo, Mr. Lowe (the toy manufacturer), Professor Leffler from Columbia University, the fund-raising aspect that started in a church in Wilkes-Barre, PA, and so on...
You can easily find these same sources so I'll leave that for our readers. However, there was a dearth of serious mathematical analysis of the probabilities and the combinatorial aspects. I only found a couple of these and neither went into much explanation of the underlying theory, other than to suggest it is complicated, oh, and data tables generated by some software. Of course, I'm sure I missed some wonderful references that my readers will find.

So I decided to do what I usually do when facing a complicated task (a la Polya): Reduce it to a much simpler problem! Not only to understand it better for myself, but, in the back of my mind, I was thinking of how a middle schooler could begin to understand the complexities of all this.

What could be easier than a 2x2 board - just 4 little numbers on a card and to really oversimplify it, only four numbers will be available: 1,2; 3,4. The semicolon separates the possible values for the first column on the card from the 2nd column. I will use this notation from now on. So here's the first elementary question for the reader and for the student:

STUDENT/READER QUESTION #1:
Assume there is one player with one card using the numbers above. Explain why the probability of winning this simple 2x2 version after two numbers are called is 1, that is, 100%.
You're thinking: Way too obvious a place to start, right! Too boring for the student...

STUDENT/READER QUESTION #2:
Ok, let's dial it up a tad. We'll still keep it a 2x2 card, but, this time, there are three numbers available in each column: 1,2,3; 4,5,6. Remember this notation means that 1,2,3 are the possibilities for the first column and so on.
Again, one player with one card: What is the probability of a win after two numbers are called?
Comment: There are many methods here from listing all of the possibilities to permutations and combinations to multiplication of probabilities (one number at a time without replacement), etc. I believe, pedagogically, it is important for the student to see more than one way!

I could stay with the 2x2 game and add more numbers but the student and our readers are an impatient lot and want to move on to something more interesting, right? So let's move on to a 3x3 board which is much more like the 5x5 board in that it has a free square in the middle. But we have to start slowly here - trust me!

STUDENT/READER QUESTION #3:
Now we have a 3x3 board. The available numbers will be simply 1,2,3; 4,5,6; 7,8,9. Again, one player, one card. Couldn't be easier, right?
(a) What is the probability of a win after TWO numbers are called? That's the minimum number with the free space covered.
(b) A little harder now: What is the probability of a win after THREE numbers are called?

Ok, we'll ask the same two questions with more numbers available:
Suppose the possible numbers are: 1-6; 7-12; 13-18


(c) Now, what is the probability of a win after TWO numbers are called?
(d) What is the probability of a win after THREE numbers are called?

I better stop here! This is enough for Part I. As usual, any results I've stated need to be verified by my readers and don't forget to give proper attribution if using any of this in a classroom setting.

HAPPY 2008!

The 25,000th Positive Odd Integer to Celebrate!

[There's a wonderful discussion in the comments regarding the challenge problem at the bottom of this post. Read tc's and mathmom's astute explanations that generalize to the combinatorial problem of placing k indistinguishable objects into n containers.]


Just a quiet acknowledgment to my readers, an expression of gratitude for helping a math blogger who was unknown before 1-2-07 to reach the 25,000th visit on October 7th. Thank you...

And for our middle schoolers and on up, here's a simple A.P. (that's arithmetic progression, not advanced placement!) problem that is designed to help students see the variety of problem-solving techniques one can employ before they reach for the calculator or plug into a formula.

Could a 6th or 7th grade student or group find a way to determine the 25,000th positive odd integer? How would the instructor guide the process?
Well, let's see...

1st.....2nd.....3rd.....4th.....5th.....
1.......3.......5.......7.......9.......

If students have tackled similar problems and are accustomed to making and analyzing tables, looking for patterns, making conjectures (forming hypotheses) and testing their ideas, perhaps some would arrive at the result. They may even surprise you with their ingenuity!
I'll share my favorite approach but don't expect students to think the same way:

Position...1st...2nd...3rd...4th...5th...25,000th...nth
Even.......2.....4.....6.....8.....10....50,000.....???
Odd........1.....3.....5.....7......9....?????......???

Now, what is the formula for the nth positive even integer? the nth positive odd integer?

Today's problem may not be sophisticated but the issue of pedagogy is never trivial, is it?


Oh, ok, I know my readers want more of a challenge to sink their teeth into. So, I'm adding the following:

The answer to the above problem is 49,999. The sum of the digits of this 5-digit positive integer is 40. Determine the number of 5-digit positive integers with this property. This combinatorial problem should keep you busy for at least a few nanoseconds!

Drawing 3 cards one at a time vs. Drawing 3 cards simultaneously - Those nasty probability questions!

[For a detailed conceptual discussion of several methods, read the excellent comments by Mike and novemberfive.]

It's 'probably' too early in the year for this, but...
This is one of those typical probability questions that all students struggle with until they have enough experience to feel comfortable. When away from these for awhile, most everyone needs to revisit this kind of question and work through the theory again. It's so elusive...

Here's a typical version of this question:

Three cards are drawn at random from an ordinary deck of 52. What is the probability that they will consist of a king, a queen and an ace?

Do you feel totally comfortable when seeing these? Imagine how most students feel. They either have a strong concept here or they get that queasy feeling and express, "I don't like these!"

So here's the issue:
Is there one method you have found for these that makes sense to most students or should multiple methods be demonstrated (and shown to be equivalent)?

To get you started:
Method One: For the denominator, select all 3 cards 'simultaneously' in C(52,3) ways. The numerator is more interesting, so I'll leave that to you.
Method Two: Analyze one card at a time without replacement. Thus, the probability that the first card will be a king is 4/52, etc. Of course, most of you know the 'catch' here, but the method can still work if...
Method Three: Yours!

Fallout from the Sieve of Eratosthenes - Combinatorial Activity for Counting Primes

BACKGROUND and OVERVIEW of ACTIVITY
Middle schoolers are often introduced to the famous sieve mentioned in the title to find which numbers, say from 1 to 100, are prime. This is a common activity in which all the multiples of 2 are first crossed out, then multiples of 3 and so on. The following is a combinatorial (counting) activity that may help them (and more advanced learners as well) appreciate just how beautiful this method is and how it can be generalized to demonstrate the endlessness of primes. In the process, middle schoolers will review the concepts of multiples, common multiples as well as composite vs. prime numbers. The 2nd activity below is for upper-level students although middle schoolers can certainly try it.

Consider the first 3 primes; 2, 3, and 5. Children know what the next one is and that there are many more after that up to 30, but, for this activity, tell them that they will find, by elimination, the remaining primes up to 30. Specifically, using the famous sieve algorithm, they will determine there are SEVEN more primes up to 30 by eliminating all the numbers that are divisible by 2, 3 or 5! Sounds like you've seen this many times? Wait...

Note: DO NOT have students use colored markers or pens to cross out numbers. It tends to obliterate the marks underneath that are needed for analysis.

Middle School Activity (standard sieve approach):
1. List the positive integers up to and including 30.
2. Cross out the multiples of 2 in your list using a slanted /. Explain how you could have determined that 15 numbers were crossed out without using your list.
3. Now cross out all the multiples of 3 from the original list using the \ mark. How many numbers did you cross out this time? How could you have determined this without your list? Count how many numbers have been crossed out twice. How could you have determined this without your list?
Note: In some applications of the Sieve method, students cross out only from the remaining numbers, not from the original list each time. Since our objectives here involve developing the idea of common multiples and also combinatorial methods, students are instructed to cross out some numbers more than once. This is not unusual in many texts or workbooks.
4. Now cross out all the multiples of 5 from the original list. Use the --- mark for this. How many numbers were crossed out? How could you have determined this without your list?
5. Count how many numbers were crossed out exactly once. Describe these numbers.
Note: Students may have difficult expressing this and some discussion is needed. For example, they might at first say "Numbers divisible by only two." This is a fine response but how can the instructor build on this?
6. Count how many numbers were crossed out exactly twice. Describe these numbers.
7. Explain why there was only one number crossed out exactly 3 times.
8. There should now be EIGHT numbers remaining which have not been crossed out. Are these numbers all prime?
9. Ask more questions!

Notes: We know that students (of all ages!) have difficulty with the issue of the number 1 not being regarded as prime. The accepted definition of prime requires that the number have exactly two distinct factors. Seems arbitrary, huh? Besides 1, most students will assume that the remaining seven numbers necessarily have to be prime, however this method does not guarantee that! If we used the above sieve up to 50, then 49 would be left over as well! Subtle...

OVERVIEW OF ADVANCED ACTIVITY
How could older students have attacked this without making a list, using more sophisticated combinatorial methods? The idea behind the above activity was to first identify the numbers that were divisible by 2, 3, OR 5. After eliminating these and the number 1, students were to consider the remaining numbers, which all happen to be prime. The remaining part of this activity deals with combinatorial methods needed to COUNT how many numbers are divisible by 2, 3 or 5 without first making a list. Some will still need to make the list!

MORE ADVANCED ACTIVITY
Consider the list of the positive integers up to and including 30. The following set of questions is designed to get an accurate count of the numbers that are multiples of 2, 3 or 5 and then to consider the remaining numbers.
1. Explain why there are 15 multiples of 2 in this list without actually counting or listing them!
2. Explain why there are 10 multiples of 3 in the original list without...
3. Explain why there are 6 multiples of 5 in the original list without ...
4. Thus far we appear to have accounted for 15+10+6 = 31 numbers in this list which are multiples of 2, 3 or 5. Since there are only 30 numbers to start with, what went wrong! Explain carefully.
Note: The method of 'overcounting' is a critical one for many set-theoretic and counting problems.
5. By now you realize that we need to compensate for the numbers that were counted more than once. Show that 10 numbers were counted more than once.
6. To compensate for these duplications, we can adjust the count: 31 - 10 = 21. Thus it appears that there are 21 numbers in our list that are divisible by 2, 3 or 5. In fact, there are 22! What went wrong! [If you don't believe this, make a list and count!!].
Note: This is subtle. The number 30 still needs to be counted.
7. Ok, we have hopefully established that there are 22 numbers that need to be eliminated from our original list of 30. Thus, there are 30 - 22 = 8 numbers remaining. One of these 8 is not prime. Which one?
8. If you haven't already done so, make a list of the 30 numbers and actually work through each of the steps above to verify your results.
9. What do you think? Is this a good method for counting how many primes there are in a particular list? Would it be practical for counting how many primes there are up to, say, 210 given that 2, 3, 5 and 7 are prime? Where did 210 come from? Why might this method fail to produce only primes?

Summary Comments:
Although this post seemed to be about a sieve for primes, you've probably figured out that it really became an activity to solve the problem of counting how many of the 30 numbers in the list were not divisible by 2, 3 or 5. I'm sure you are thinking that there are many others ways we could have counted the multiples of 2, 3 or 5. Your students may object to the above method and suggest a 'better' one. For example, count all the even numbers up to 30 first. Then count the odd multiples of 3, then the powers of 5 . No duplications, short and sweet, right? However, the set-theoretic method of counting with duplications, then compensating is actually more powerful and can be generalized to longer lists of primes. Embedded in this activity is the subtle notion that there cannot be a finite number of primes. Do you think students would recognize that on their own?

How many even 3-digit positive integers do not contain the digits 2,4, or 6?

More practice for students...
As indicated many many times on this blog, there is no substitute for experience!
The keys to success here are:
(1) Careful reading (do students often miss the key word!)
(2) Knowledge of facts (why is zero the most important number in life!)
(3) Knowledge of strategies, methods (multiplication principle, organized lists, counting by groups, etc)

If these problems are helpful for students, let me know...

Variations on Basic Themes: Digit Problems in the Key of A or G Minor?

Update: Answers, solutions have now been posted in the comments.

I thought of these variations on the well-known combinatorial problems involving 3-digit numbers that pop up frequently as I was teaching arithmetic and geometric series yesterday.
These questions are appropriate for grades 6-12 provided students are given definitions and some practice with arithmetic and geometric sequences, topics that are well within the abilities of middle schoolers. A quick intro to these sequences is all that is really needed OR, as I did below, they can be defined in the problem itself. Thus, these questions provide both practice in arithmetic skills and in combinatorial thinking. Of course, all the experienced or budding programmers out there can write simple code to have their graphing calculators count these, but that should only complement and verify their results, not replace the reasoning needed to solve them, unless these are used for a computer science class (even then, programmers should independently verify their code by solving the problems!).
These are not highly challenging and therefore can be used as Problems of the Day, for extra credit, or enrichment. Our readers will hopefully suggest other extensions and further variations (some are suggested below).

1. The digits of 246 form an arithmetic sequence from left to right because 4-2= 6-4. How many positive 3-digit integers satisfy this condition?


2. The digits of 248 form a geometric sequence from left to right because 4/2 = 8/4. How many positive 3-digit integers satisfy this condition?

Now, how could we make these more challenging? 4-digit numbers or will that make one or both easier, i.e., fewer possibilities? What if the digits were allowed to form these sequences in any order? BTW, I apologize for the music pun in the title. I hope you will respond to that with a positive tone!

Searching continued...Another Challenging Combinatorial Problem

The 'answers' to the problems below are now posted in the comments...

[Update: I modified today's problems for my 9th grade group. An excerpt from the handout is shown at the bottom. How do you think they did? These are youngsters who need very precise instructions and find math more challenging. Guess how they did?]

How many 4-digit numbers have exactly 2 identical digits?

Today's questions were inspired by a Google search from one of the viewers of this blog. No matter how many of these types our students may attempt, there is no substitute for a systematic approach and clear thinking. Professional mathematicians who have done numerous problems of this type and are therefore likely to know an efficient method still can make careless logic errors. Imagine how our students do! They're looking for a quick and dirty approach, one short-cut method or formula that covers all kinds. Not likely!!

There's an interesting semantics problem with this question: Does 3232 count as a possible answer? Your first instinct might be to say, "Of course, it doesn't count!" But how many identical digits does it really have? For this reason, I will reword today's question to:

How many 4-digit positive integers have exactly one pair of identical digits?

[Note: Someone will probably argue about the semantics here as well but I'll let it stand!]
I'll post the result I obtained later but, for now, how many different approaches do you think your students could come up with? I thought of at least 3 not to mention writing a short program to count them!]

I see this kind of problem as appropriate for middle school and high school. For the middle grades, students should begin with the 3-digit version (see worksheet below) and be encouraged to make an organized list. By high school, they should be able to move on to more powerful counting methods but we know some are stuck at the 4th grade level of counting in an unorganized manner. In fact, from my observations, most high schoolers start in the 1000's, count those and multiply by 9. This is actually a fine method but should they know other approaches by the time they complete Algebra 2?

The following is a portion of the worksheet I gave to my group today. It worked out well. Any thoughts? Notice that I modified the 4-digit problem to make it more accessible for them. It provided an easier extension.
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Searching...

Based on a reading of Google searches of visitors to this site, here are a few of the more common topics, rephrased as questions, that I have noticed. I also noted that, aside from pi day, which generated hundreds of visits from those looking for historical information about pi and the names of mathematicians, many are looking for sample MathCounts problems or references to books of these.


1. What is the largest 3-digit prime with 3 prime digits?
Ans: 773 if repeated digits are allowed; 523 if not. This question has appeared in one of my earlier posts. Must be a fairly common one that readers come across from math contests or from class. A quick list of primes up to 1000 can be found at VIAS Encyclopedia.

2. How are the following questions related:
How many different handshakes occur if each of 10 people shakes hands with each of the remaining people in a room? (can be expressed more clearly)
How many different segments can be formed by connecting the vertices of a decagon in all possible ways? (can also be expressed in terms of the number of chords formed by 10 points on a circle)
Ans: Both problems can be expressed as 9+8+7+6+...+2+1 or (9)(10)/2 or 10C2, the number of combinations of 10 objects chosen 2 at a time. In general, for n people or points on a circle: 1+2+3+4+...+(n-1) = (n-1)(n)/2 = nC2. The equivalence comes from the fact that each handshake or each chord is uniquely determined by selecting 2 people or 2 points. To avoid repetition we use combinations rather than permutations.

4. How many different seven-digit phone numbers (ignoring the area code) can be formed?
Ans: If any digit 0-9 were allowed in all positions then there would be 10⁷ possibilities since there are 10 choices for each digit (using the Multiplication Principle here). Subject to restrictions on zeros or other digits or other local considerations there would be fewer.
Here's a related question that I have not yet seen:
How many different IP addresses are possible worldwide if every computer, device etc., must have a unique one?
I believe that IP addresses are always of the form xxx.xxx.xxx where the first digit in each group is allowed to be zero, that is, one could use a 2-digit number in each group. For example, 66.19.35 is acceptable. I didn't research the rules so this may be incorrect.
Ans: If the rule of formation is accurate, there could be 10⁹ or one billion IP addresses. How long will it take for these to be used up? I know someone out there will tell us how this is or will be handled!

Update on IP Addresses: My belief about the form of IP addresses was dead wrong! The protocol should have been 4 groups of integers, each in the range from 0 through 255. There are now newer protocols to allow for the exponential growth of devices needing an address. See the comments for this post to learn more from those far more knowledgeable than myself!

PROBLEMS WITH PRIMES!

Here is a set of middle school problems on primes that require careful reading, knowledge of prime digits, organized listing and other skills. They can also be used to prepare high school students for SATs and other standardized tests which frequently test knowledge of prime numbers. Working without the calculator is strongly recommended. The last couple of questions require a partial list of primes which could be an internet activity. None of these questions is highly challenging but one of the goals is to make children aware of the mysteries of primes, something few appreciate! I guess you could say that learning how to read critically is also a PRIME objective!

1) List the FIVE 2-digit primes whose units' digit is 1.

2) List the FIVE 2-digit primes each of whose digits is NOT prime.

3) List the TWELVE 2-digit nonprimes (composites) each of whose digits is prime.

4) Mentally, determine the largest 3-digit nonprime (composite) each of whose digits is prime.

5) A palindrome is a number like 101 or 222 which reads the same when its digits are reversed.
(a) Explain why there are ninety 3-digit palindromes without listing all of them.
(b) (Internet activity). Search for a list of primes up to at least 1000. Use this list to answer the following: If a 3-digit palindrome were chosen at random, what is the probability that it would be prime?

6) Using your list of primes, determine the largest 3-digit prime having 3 different prime digits. Anything surprise you?

7) (Additional work outside of class) Using the list of primes, devise three problems of your own about 3-digit primes to challenge your classmates. A special prize for the best questions!

SAT-Type Challenge: Exponents and Combinatorial Thinking

These SAT-type questions provide review for the multiplication rule of exponents as well as recognizing the need for using the counting principle vs. careful enumeration in an organized list. Both questions need to be given for the effect. Target: PreAlgebra and beyond...

(a) Consider the list 1,2,3,4,5
If a, b and c are assigned different values from the list above, how many different values will result from the expression a^bc ?

(b) Consider the list 1,2,4,8,16
If a, b and c are assigned different values from the list above, how many different values will result from the expression a^bc ?



Notes:
(i) To encourage use of exponent rules, do not allow calculator. What variations would make this even more powerful?
(ii) Possible extension: For (a), ask students to make a conjecture regarding the largest possible power, i.e., is it obvious which is the greatest among 5¹², 4¹⁵, and 3²⁰ ?

Geometry SAT Problems - Do These Questions Help Students Develop Spatial Sense and Combinatorial Thinking?

Answers and discussion of problems below are now available in Comments.

Some of these questions are reprinted from copyrighted materials from the College Board. In some cases, I've modified the questions for instructional purposes. These questions are linked to more advanced topics involving polyhedra and college geometry but they are appropriate for middle and secondary students as well. When is it valuable for students to actually enumerate the objects asked for? These questions also stress the importance of the phrase 'determined by.'

1. What is the total number of right angles formed by the edges of a cube?

Modified version for classroom use: Show that there are 24 right angles formed by the edges of a cube. You and your partner must find at least TWO different methods. [Note: By giving students the 'answer', the focus is then on process.]

2. How many distinct pairs of parallel edges are there in a cube (or rectangular solid)?

3. How many different planes are determined by the vertices of a cube (or rectangular solid)?

4. How many equilateral triangles are determined by the vertices of a cube?