MathNotations

Posted by Dave Marain at 7:57 AM 2 comments

Labels: primes, SAT-type problems

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

Posted by Dave Marain at 10:22 AM 2 comments

Labels: explorations, investigations, math challenge, middle school, primes

PLS NOTE THE EDIT TO THE PROBLEM BELOW. THE ORIGINAL WORDING WAS INACCURATE.

The following is not a classic function question even though it uses function notation. This is an original problem I wrote but it is the kind of question that might appear. The level of difficulty would be medium. The math content is middle school level but the wording and notation are the challenge for most students. Beyond preparing students for a test like the SATs, my strong belief is that such questions should be included in textbooks from middle school on (even with that function notation!). This question reviews basic math concepts (primes, factors, gcf) and can also be used as a springboard for discussion of the concept of "relatively prime", Euler's phi function, π(x) and other number-theoretic topics.
Note: The "For example" hint may or may not be included in the question. It certainly makes the notational issue less formidable.

If n is a positive integer greater than 1, then the sets F(n) and P(n) consist only of positive integers and are defined as follows:

A positive integer, k, belongs to the set F(n) if k ≤ n and the greatest common factor of k and n equals 1.
A positive integer, k, belongs to the set P(n) if k ≤ n and k is prime.
For example, F(6) contains the numbers 1 and 5 and therefore has two elements. P(6) contains the numbers 2, 3 and 5 and therefore has three elements.

What is the ratio of the number of elements in F(20) to the number of elements in P(20)?

Click Read more below to see answer (suggested solution will be posted later).

Answer: 1
Explanation: Not yet...

Posted by Dave Marain at 9:19 AM 6 comments

Labels: divisibility, functional notation, functions, gcf, more, primes, SAT-type problems

Let the amusement begin with all of the cutesy questions and math contest problems involving our new calendar year, 2009.

Shall we begin, looking for curiosities. Perhaps our students in grades 4-8 can discover their own. Why not post their best ideas or perhaps I may create a contest right here at MathNotations! Hmmm...

Ok, let's get started:

1) The difference of the units' digit and thousands' digit is 7, the smallest prime factor of 2009.

NOTE: An important benefit of these kinds of observations is that it helps students learn how to formulate and express their ideas using correct mathematical language. This is as hard for many high schoolers as it is for middle schoolers!

2) Who actually knows a divisibility rule for 7? (Proving it is another matter).
How about 200 - 18 = 182, then 18 - 4 = 14 which is divisible by 7, so 2009 is also!
No idea what I just did? You'll just have to research it, boys and girls! Ok, an excellent resource is the Math Forum of course. Look here.

3) When 2009 is divided by its units' digit, 9, the remainder is 2, the thousands' digit. Not surprising if you know about remainders when dividing by 9.

4) The product of the distinct prime factors of 2009 is 41x7 = 287, my favorite highway in NJ. This is probably not the curiosity I would be looking for from my students!!

Ok, enuf' of this silliness. I'll leave it to my astute readers to bring in the New Year in their own unique fashion. BTW, a useful site for a list of primes is here. Keep it handy and enjoy!

Posted by Dave Marain at 7:58 AM 6 comments

Labels: fun problems, investigations, middle school, primes

[Have you voted yet in the survey in the sidebar? Time is running out...]


Just an isolated middle school mini-challenge to get the day started? Perhaps...

Those of you who are familiar with this blog know that MathNotations is dedicated to providing activities/investigations for middle and high school teachers to use or modify (provided proper attribution is given of course). In this post, I will demonstrate how one can build an extended or richer activity from a math contest or standardized test problem.

It is important to remind our readers here that these kinds of activities and problems do not constitute a curriculum. Students need to first develop proficiency with skills and procedures. These explorations are only intended to extend and enrich student learning. They can be used in part or in whole, as a long-term project outside of class, a team activity in the classroom or a myriad of different ways. All of this is at the discretion of the educator.

First of all, the problem in the title, in its present format, would not be an SAT or a standardized test question, unless the standardized test included free-response or open-ended questions.

In SAT format, the question might be changed to:

Which of the following can be expressed as the sum of three distinct primes?
(A) 6 (B) 9 (C) 12 (D) 15 (E) 17

Not a particularly challenging problem, but some students would struggle with comprehending the wording or paying attention to details ('distinct') or because of lack of knowledge about primes. This type of question is fairly common.

Let's return to the original question:

Find all combinations of 3 distinct primes whose average is 13.

I've administered this type of question to students and observed their methods. Sadly, some do not immediately recognize that the problem is equivalent to:

Find all combinations of 3 distinct primes whose sum is 39.

Most students do see this at once, but there are a few in middle and high school who have not developed sufficient conceptual understanding of averages or have simply not been exposed to enough problems.

As far as methods and approaches go, I'm always surprised that many middle and high school students use fairly random listing methods rather than a systematic approach. After all the years now of instruction in problem-solving techniques, one should expect that students would make an organized list as follows:

2,2,35 Discard this for two reasons! Would most students recognize the logic behind concluding that 2 cannot be one of the three primes?

3,5,31
3,7,29
3,11,25 (discard)
3,13,23
(I'll let the reader finish the list!)

If I were to assess the value of this single question, I might give it a 7 on a scale of 10. I'm sure some would rate it as 1 or 2 since some perceive these kinds of questions as useless. However, my feeling is that the question does develop mathematical thinking and there's something to be said for attention to detail and a systematic approach.

But this is not the end. Suppose the educator finds this problem in a book or math contest or online. How can one extend it to a richer experience for all students, not just the accelerated, honors or gifted child? Although it may appear at first that the primary intent of the question is to encourage a systematic approach (making an organized list) or reviewing ideas about averages or primes, the content of the question is essentially about writing a number as a sum of 3 primes, distinct, in fact. Is this an important question that has occupied the minds of our greatest mathematicians for years? Uh, actually, yes! Look here!

Students need to be encouraged to ask more questions after the problem is solved. The instructor guides this exploration by modeling some of the questions students need to ask: Is there anything special about 13? Can every prime be written as a sum of three distinct primes? Every odd? Three primes, not necessarily distinct? Does the original number 13 have to be prime or even odd for that matter? Why are we using three primes in the sum? Why not two? Your turn, boys and girls!

You get the idea. This isolated problem becomes a springboard for deeper mathematical research. Here is one possible assignment:

Write your own challenge problem of this type? Make sure you can solve it and be prepared to present it to the class!

What would you expect your students to come up with? You can't be sure until you try it of course, but can you anticipate some of the responses?

By the way, I have already heard most of the arguments for why this type of research is impractical in a math classroom:

"My students don't even know their basic facts and you want them to become mathematicians!" "This is for the math team geniuses."
"I don't have time for this - I have a real curriculum to cover and if this not going to be tested..."
"Teach children the basics, not this 'fuzzy' math!"

Oh well, enjoy it anyway!

Posted by Dave Marain at 9:30 PM 8 comments

Labels: investigations, middle school, primes, research

There are countless problems involving the factors of a positive integer we're seeing in middle school classrooms and on standardized tests these days. They are often used as challenges or warm-ups and questions similar to the one(s) below have appeared frequently on this blog. Students become more proficient with this type of question by doing many variations repeatedly over time. As they mature, they will come to appreciate a more general approach to finding the number of factors of any positive integer. Number theory is now included in most states' standards so there needs to be some time devoted to this topic on a regular basis.

STUDENT PROBLEM/READER CHALLENGE

This problem/activity is often best implemented in small groups. Each member of the group should make their own list and then compare, however, they might want to divide the labor by having some students do the numbers up to 50 and others do the rest.
Suggested Time for Activity: 15-20 minutes (the problem can be explored further for homework or a challenge, then revisited the following day for 5 minutes).

The number 12 has 6 positive integer factors: 1,12;2,6;3,4.
(a) List all positive integers up to and including 100 that have exactly four factors.
(b) Higher-order: These numbers fall into 2 categories. Describe these categories.

Alternate Problem (shorter time needed): What is the largest 2-digit positive integer that has exactly 4 factors?

Posted by Dave Marain at 7:52 AM 5 comments

Labels: factors, number theory, primes, warmup

Many math educators use warmup problems to review, challenge or set the tone as students walk in the room. Routines like this are effective in having students 'hit the ground running'. These mini-problems can be on the board, while an overhead transparency of selected answers are displayed. The instructor then has time to circulate, check homework, engage students, and get a feel for the difficulties they had with the assignment. By the way, do most of your students take these warmups seriously?

Here is a warmup that requires more active participation on the student's part. I may suggest a different one in a later post, but I'd really like readers to share some of their favorites!

Math Bee
All students stand at their seats. They are told they will be have to give the next number in sequence, according to some rule that will be explained. They will have 3 seconds to respond (can be adjusted but no more than 5). If incorrect or time runs out, they will be instructed to sit down and the next person will have to give the correct answer and so on. You may want to start them off by giving them the first number (judgment call here). I suggest you allow a maximum of 5 minutes for this activity.

Here's the problem I gave to a group of high schoolers but it is highly suitable for middle schoolers as well:

Using positive integers, think of primes ending in 1 or 3 (you may want to use the technically precise phrase 'whose units digit is 1 or 3'). For example, 21 'ends in' 1, but it is not prime. You must go in order and you are not allowed to ask the number the previous person gave! You will have 3 seconds to respond. If incorrect or time runs out, I will ask you to sit down and the next person will need to give the correct number. We will continue until there is only one star shining or time runs out and we have co-champions!

Learning Objectives:
(a) Reviews primes (How many of your students do you think would be eliminated early by starting with 1? You can always start them off with 3 if you feel this will help. Some students will begin with 11, assuming that you meant 2-digit numbers. By the way, 51 and 91, in particular, typically knock out many, if they get that far! Finally, are you thinking this question is not particularly relevant for high schoolers? Count how many questions relate to primes on the SATs!)
(b) Improves listening skills and concentration (How many of your students do you think will forget the last number given either because their minds are wandering or from trying to think ahead to their turn?)
(c) Learning how to think under pressure. (Although we know some students will 'freeze up' or be embarrassed if they are eliminated, they will not be alone! Typically, about half of the students in an above-average class will be sitting down on the first pass through the class! With a high-achieving class of very strong students, you may need to make several passes to reach a winner. If there are a couple left after 3-4 minutes, proclaim co-champions.

Let me know how this goes if you try it after the Thanksgiving break. What variations would you use to make this more effective for your students? What other kinds of problems are suitable for this 'Bee?' Were my predicted statistics way off? Did you predict that students would get past 100?
Again, please share some of your favorite warmups!

Posted by Dave Marain at 6:50 AM 2 comments

Labels: math challenge, primes, warmup

[Now that the 'Carnival is Over' (is that another song title?), it's time to return to the essence of this blog.]

There is no end to the number of articles one can find on the internet and in the literature regarding prime numbers, from famous theorems to unsolved problems that seduce budding young mathematicians.

The following investigation is intended for middle school students, working in research teams, but can be extended to secondary students who want to explore mathematics further in their classroom or in their Math Club.

Student Investigation

Notice that
4 = 2+2
6 = 3+3
8= 3+5
10 = 3+7 = 5+5

The purpose of this investigation is to explore part of the world of prime numbers and become a mathematical researcher. Mathematicians, like scientists, observe phenomena, look for patterns, make conjectures and generalizations and try to prove them. Mathematicians seek to understand the secrets (general truths) underlying patterns and relationships in numbers and shapes.

1. Based on the first few examples above, do you think your mathematical research team has enough information to make a conjecture, or educated guess, about even positive integers? By the way, why didn't we begin the pattern from the first even positive integer, 2?

2. Let's continue the exploration. Begin by making a list of all primes up to 100. Why does it make sense to have this list available?

3. A table is very useful to organize your data and form hypotheses. A suggested table is provided below. One of the column headings needs to be completed. Then complete the table for even integers up to 30. Your team leader should assign a few of these to each member of the team.

Even Positive Integer........Number of Ways to ___............List of ways
4..................................................1........................................2+2
6
8
10...............................................2........................................3+7;5+5
12...............................................1.........................................5+7
.
.
.
30

4. Here's an example of a conjecture:
Based on
22 = 3+19; 5+17; 11+11
24 = 5+19; 7+17; 11+13
26 = 3+23; 7+19; 13+13
28 = 5+23; 11+17
one might conjecture that even numbers can be written as a sum of two primes in at most three ways.
Do you think it's easier to prove this 'educated guess' or disprove it? Try it! You may need to extend your table!

5. Extend your table to even positive integers up to and including 60.

6. Based on this table, your research team now has to make at least three conjectures, then attempt to disprove them or provide an explanation for why they may be true. You may need to go beyond your table.

7. Jeremy determined that
60 = 7+53; 13+47; 17+43; 19+41; 23+37; 29+31 and
100 = 3+97; 11+89; 17+83; 29+71; 41+59; 47+53
He conjectured that six is the greatest possible number of ways that an even number up to 100 can be written as a sum of two primes. Disprove it! Again, you might need to extend your table.

8. Is your research complete? Do you think a mathematician would make other conjectures about even numbers or think of other problems related to sums of primes? Perhaps, numbers that can be written as a sum of three primes? Sums of consecutive primes like 3+5+7. Perhaps you'd like to continue....

9. [When the activity is complete] Research Goldbach's Conjecture on the web and write a brief description of its history. Has it been proved?

As usual, make suggestions for improving this; revise, edit, enjoy...
If you use this in the classroom, please share the experience. The feedback is invaluable to me.

Posted by Dave Marain at 7:19 AM 11 comments

Labels: conjecture, Goldbach Conjecture, investigations, middle school math, 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?

Posted by Dave Marain at 6:21 AM 1 comments

Labels: combinatorial math, multiples, primes, sieve

[Update - The answers to the questions below appear in the comments section.]

I am continually amazed by some of the search phrases that lead to this blog. Although many are math topics about which readers are looking for more information, some are actual math problems that intrigue me. Here is one for today that led me to probe more deeply. On the surface it doesn't seems to require that much analysis, just an understanding of the rule for finding the number of factors of a positive integer from its prime factorization (see my earlier post on Fun with Factoring) and a good list of primes, but you may see something deeper here. At the least, it looks like an interesting investigation for middle schoolers and beyond with a webquest built in. I am indebted to the searcher whoever she or he may be!

Here's the actual search phrase I found:

1. What is the largest 3-digit integer with exactly four factors?

Before revealing the answer, I decided to expand this a bit:

2. (Easier but still worth doing) What are the largest and smallest 2-digit integers with exactly four factors?

3. Ok, so naturally, we would also ask: What is the smallest 3-digit integer with exactly four factors?

5. Keep going... What are the largest and smallest 4-digit integers with exactly four factors?

Of course, a simple factoring program written on a graphing calculator or in C++, etc., would suffice, but see how long it takes you to search and how logical reasoning and analysis can save some time. Of course it always helps to have a list of primes handy so don't forget the ultimate primes list from the U. of Utah.

Before you decide this is just a way to keep kids busy, try it. If you see a pattern or wish to expand this, go for it!

Posted by Dave Marain at 5:49 AM 4 comments

Labels: factors, investigations, number theory, primes

[You may want to read the comments for this post. Some useful devices to help students recall important rules/facts from trig & calculus.]


Regardless of whether one approves of giving students mnemonics to help them recall various math facts or terms, students do use some of these and, in fact, don't we all! I know many math teachers detest PEMDAS because it can mislead students but the 'positives may outweigh the negatives'!
Here are a few of my favorites, some of which I've devised and some I've learned from creative teachers and students. I know some of you have your own pet phrases - pls share!!
With the May SATs only a few days away, perhaps one of these will stick in a student's head and help...

1. Zero is a WEIRDO (last 2 letters need to have a strikethrough)
Ok, here's how this works: Each letter helps students recall an important fact about the number ZERO which many students seem to forget almost daily! I'll start you off - try to guess the rest:
W: Whole (i.e., Zero belongs to the set of whole numbers)

2. Spell the word 'WHOLE'. The middle letter reminds us that ZERO is WHOLE and (E)VEN.

3. INTEGER (underline the N, E, and G) - to help students recall that integers can be NEGative.

4. PRIME (strikethrough the letter I, circle the last letter 'E.')
This may help students recall that 1 (the letter I) is not defined to be a prime number; further, there is only one (E)ven prime. Lame yes, but the lamer the better.

5. F)M Some students still listen to their favorite FM station.
This is to help them recall that a (F)ACTOR 'goes into' a (M)ULTIPLE. Thus, 4)12 suggests that 4 is a factor of 12, while 12 is a multiple of 4. Ok, stop groaning!

6. (From one of my outstanding Algebra teachers E.S.):
Permutations are Picky
Combinations don't Care (about order).

7. 'If you're Y's, you go to the top' or RISE rhymes with Y's (and things that RISE always end up on TOP).
These silly statements may help them recall that, in the formula for slope, the y's are in the numerator.

Now it won't be hard to top these, so go ahead...

Posted by Dave Marain at 7:01 AM 5 comments

Labels: math mnemonics, primes, SAT-type problems

[Important Note: Thanks to 'e', Question 1 below has been modified to 'five' 2-digit numbers with 12 factors. Please read the comments for follow-up.]

[The ideas and problems today are suitable for grades 6-12.]
Those into number theory know many basic principles that help them solve problems involving factors that seem arduous at first. One extremely useful formula that mathletes are taught early on and SAT students should know is the following (the abstract form hides how easy it is, so get to the example quickly!):

The FUN-damental FACTORING RULE: (I coined this silly name so don't quote me!!):
If the positive integer N = p₁^e₁p₂^e₂p₃^e₃...p_n^e_n then
N has (e₁+1)(e₂+1)(e₃+1)...(e_n+1) positive integer factors!

The p_i's are distinct primes and the e_i's are positive integer exponents.
Note: From this point on, whenever the term factor is used, it refers to a positive integer factor.

Ok, we need an example fast!

Example: How many positive integer factors does 48 have?
Solution: First we need to write the prime factorization of 48 = 2⁴3¹
[For larger numbers, writing the prime factorization is more problematic and a computer program or a calculator like the TI-89 would be useful].
ADD ONE to each of the exponents and MULTIPLY: (4+1)(1+1) = 5 x 2 = 10. Voila!
Verify: 1,48; 2,24; 3,16; 4,12; 6,8. Ten, indeed!

The explanation of this very handy rule involves some basic combinatorial thinking since EVERY factor of 48 (similar argument for N, in general) can be written in the form 2^a3^b where a could be 0,1,2,3, or 4 and b could be 0 or 1. Thus, there are FIVE (4+1) possibilities for a and TWO (1+1) possibilities for b. By the multiplication principle, there would be 5 x 2 ways to form different factors of 48.

Ok, so here are some examples (not very challenging) for middle schoolers and on up:

1) Using our FUN-damental Rule above, find the five 2-digit positive integers which have exactly TWELVE distinct factors.
The object is not to list every number from 10-99 and count factors!
Extras: Explain why a 2-digit number cannot have more than 12 factors. What would be the smallest integer that has more than 12 factors?

2) How many factors does 2007 have?
[Easy, once you have the prime factors, but it's always fun finding them for each new year or showing it is prime. Students better know why 2007 is NOT prime!!].

3) SAT-type (easy using above rule): If N = pqr, where p, q and r are distinct primes, explain, without listing or plugging in numbers, why N has exactly eight factors.
Then list the eight factors in terms of p, q and r.

There are endless variations and applications of the FUN-damental Rule. I'll leave it my readers to suggest really 'wicked' ones!

Posted by Dave Marain at 5:21 AM 9 comments

Labels: factors, middle school, multiplication principle, primes, SAT-type problems

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!

Posted by Dave Marain at 5:11 AM 6 comments

Labels: combinatorial math, middle school, primes

PLS READ THE COMMENTS TO SEE HOW THE LESSON FARED OVER TWO DAYS. DO YOU THINK I HAD TO MODIFY IT A LITTLE OR A LOT TO MAINTAIN THEIR INTEREST?

Thanks to jd2718 for motivating me to include the following famous number-theory conundrum. You can find many references to it on the web but I remember it from the classic text by Hardy and Wright. Give this as a challenge bonus problem or as an activity for your middle or high schoolers. Revise and modify it to make it appropriate for your students. I've tried to turn this into something students can at least attempt, but you could probably make it much more user-friendly...
Remember: My goal is to provide meaningful activities for ALL of your students, not only for honors or accelerated classes. The challenge is to modify them for students who struggle in math. The real test comes when I implement today's problem in my 'skills' class to which I frequently refer. That is my plan -- I'll let you know if it worked or was a disaster! Pls note that these activities also provide practice for those open-ended types of questions that now frequently appear on standardized and state assessments.
Finally, I hope you enjoy these challenges, but my primary target audience is students. I am really interested in reading students' reactions to these.

MISSION IMPOSSIBLE?
Now, boys and girls, we know that, despite all attempts by the most famous mathematicians, no one has yet devised a formula for primes. At one time, it was thought that 2^p-1 would always be prime if p is prime:
2^2-1 = 3; 2^3-1 = 7; 2^5-1 = 31; 2^7-1 = 127 -- all primes. But alas, 2^11-1 - 2047 = (23)(89). Imagine the disappointment!
But I think I found a method that will produce primes EVERY time! If you can do all parts below and prove I'm wrong, you get 5 bonus points. You'll get at least one point for doing part (a).

Ok, I decided to start with my favorite prime 41, since that's the age I once was!
41 + 2 = 43
43 + 4 = 47
47 + 6 = 53
53 + 8 = 61
61 + 10 = 71
71 + 12 = 83
...
Here's your mission:
(a) I listed the first 6 and they're all prime. You can see the pattern, right? Keep the sequence going until you reach 40 numbers. Check that they are all prime by researching a list of primes on the web. So, am I right? Am I famous now?
(b) My method is easy to see but harder to describe algebraically. Here's one way:
To get each number in the list after the first, you can see that I added the next even number to the preceding term. SHOW that the following recursive description produces the first 6 terms:
a(1) = 41; a(n+1) = a(n) + 2n, n = 1,2,3,... where the notation a(n) refers to the nth term. Then explain why this will generate all the terms.
(c) This one is harder but I'll give you a hint: Devise a polynomial that will generate this sequence of numbers when n = 1,2,3,....
Hint: Think of a quadratic like n^2 +.... Just remember, when n is replaced by 1 it has to yield a value of 41!
(d) Show that my amazing sequence 'blows up' when you reach the 41st term. Why? There goes my million dollars from the Clay Institute!
Extension: Is there any other sequence like this? What's special about 41? Happy web-questing...

Posted by Dave Marain at 5:03 AM 13 comments

Labels: math challenge, number theory, primes, recursive functions

Important Update: Day 10 - received a reply today from Jennifer Graban of the National Math Panel - will be posting her entire statement and my reply by Monday.

Update to Activity Below: This lesson was implemented today with my group of 9th graders. Math is generally a struggle for them. Do you think they completed it in 40 minutes? If not, then how much? Do you think this activity engaged them or they lost interest after a few minutes? Do you think anyone identified what kinds of numbers are 'unsummable'? Before I tell you, I'd be interested in your best guesses!

The following investigation enables the student to explore concepts in factoring, primes, composites, odd vs. even, consecutive integers, averages, median, pattern recognition, arithmetic series, generalization and proof just to name a few ideas!

It may appear to be written for middle schoolers but it can modified for grades 9-12 using algebraic methods (particularly sequences and series) and more sophisticated reasoning. Students may discover new ideas I never imagined when I wrote it. The basic idea of this question is very well known. What might make it different is the journey you and your students will be taking. Ok, if you're not an educator, enjoy the ride (even if it may be simplistic!).

Students should work in pairs. Calculator for checking sums is optional. Allow one period for this, however, additional time may be allocated for further investigation outside of class. This problem is about much more than making an organized list! How would you modify it to make it better? Richer? More suitable for younger children? Older children? What questions might you ask to guide them through it when they appear to be 'stuck'? Is it better not to say anything and let them struggle with it? Is there a place for this kind of discovery? Is it worth all the effort and time 'lost'?

You and your partner are trying to unlock the secret of the 'UNSUMMABLES'!

The number 5 can be expressed as a sum of two consecutive positive integers: 5 = 2+3
Similarly, 6 can be written as a sum of three consecutive positive integers: 6 = 1+2+3
22 can be written as a sum of four consecutive positive integers: 22 = 4+5+6+7
9 = 4+5 but it can also be written as 2+3+4
Ah, but no matter how hard you try, the number 8 cannot be written as the sum of 2 or more consecutive positive integers (try it!!). The number 8 is one of the mysterious unsummable numbers!

(a) In a table format, express each of the integers from 5 through 35 as a sum of 2 or more consecutive positive integers if possible. If it is not possible for some integer, call it unsummable! If you are able to find more than one way to sum a number, that's even better.

(b) Write at least 5 observations and conjectures, i.e., what did you notice and what do you think will always be true. We'll start you off:
We noticed that every odd number can be ________________________.
Note: Think about primes, composite numbers, factors, ...

(c) How many unsummable numbers did you find? What did you notice about these numbers? Can you unlock their secret? A special prize if you can explain WHY they are unsummable!

Posted by Dave Marain at 10:40 PM 11 comments

Labels: patterns, primes

Tomorrow's problems focus on sequences and are of varying levels of difficulty. Although #4 may be more appropriate for Algebra 1/2 students, middle schoolers should be able to handle the others. Again, read the comments later in the evening for the answers, comments and solutions. There were some profound ideas expressed about today's questions particularly that innocent-looking quadrilateral problem with the 60 degree angles!

Posted by Dave Marain at 1:05 PM 9 comments

Labels: logic, number sense, primes, problem-solving, sequences