MathNotations

Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school


List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.

Comments:
(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"

(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number?

(3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?

(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!

(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in?

Posted by Dave Marain at 6:55 AM 6 comments

Labels: divisibility, middle school, reasoning, SAT strategies, warmup

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...

...Read more

Posted by Dave Marain at 9:19 AM 6 comments

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

[Did you think MathNotations was on hiatus? Actually, I've been working on a couple of investigations including an intro to the mathematics of circular billiard tables and the activity below -- hope you enjoy it...]

MathNotations has been invited to submit an article to Connect magazine. I'm considering something along the lines of the following investigation (the article would contain fuller explanations and additional teacher guidelines) and I would appreciate feedback particularly from middle school teachers. Feel free to suggest revisions, improvements, ...

If you have the time, as we approach the end of the school year, to implement some or all of the following, I would appreciate your observations. Also, what classroom organization (e.g., individual vs. small group) you used or what you would recommend. Thank you...

NOTE TO READERS OF MATH NOTATION: Your challenge is at the bottom!

CLOCK INVESTIGATION
Students are provided a handout with several clocks, numbered in the standard way from 1 through 12.

LEARNING OBJECTIVES/STANDARDS/TOPICS

• Divisibility concepts (remainders, lcm, factors)
• Repeating patterns (introduction to periodicity)
  NOTE: Later on, when students study the unit circle in trigonometry, they will encounter similar periodic behavior.
  
• Organizing data
• Developing effective communication - writing in mathematics
  

Part I
Place a marker at 3:00. This will be your START position. For the first part of this activity, you will be moving your marker FOUR hour-spaces in a clockwise direction from your starting point. So after your first move, you will be on 7:00. With your partner, record the results of each move up to 15 moves. You could of course mark it directly on the clock or you could make a table such as:

Start....3:00
Number of Move (N).................Position
1......................................................7:00
2......................................................11:00
...
15

Note: It's good experience for students to see that we often start indexing variables from zero, so instead of Start...3:00, one could start the table
0.....................................................3:00

Question 1: Try to answer the following without actually listing all the moves: What will the position of your marker be after 25 moves? 50 moves? 75 moves? 100 moves? Explain your reasoning or show your method.

Part II
Same starting point at 3:00, but this time you will move your marker FIVE spaces clockwise each time. Again, record the results of each move up to 15 moves.

Question 2: You should now have discovered that after 12 of these moves, you have returned to your starting point. Explain why at least 12 moves were needed (stating that you tried every move up to 12 isn't quite what we're looking for!).

Possible explanation (they may do better than this!): Starting position is repeated when the total number of hour-spaces moved is a multiple of 12. Since the the number of hour-spaces advanced after each move is also a multiple of 5, the position will repeat after 12 such moves. Note that 12⋅5 = 60 is the LCM of 12 and 5.

Question 3: Again, try to answer the following without actually listing all the moves:
What will the position of your marker be after 25 moves? 50 moves? 75 moves? 100 moves?
Explain your reasoning or show your method.

Question 4: In part I, you discovered that positions repeat after 3 moves. therefore, not all positions from 1 through 12 are reached. In Part II, you probably noticed that every location is reached. Explain both of these results in terms of divisibility.

Question 5: In both parts you started at 3:00. What results would be the same if you started from the 12:00 position? What results would be different?

Question 6: Devise at least one variation of your own for these clock problems. Extra points for most creative!
Sample: In addition to the obvious (changing starting position or number of spaces moved, you may want them to consider moving counterclockwise or changing the clock itself to 13 hours or some other variation).

Note: Students do not often consider generalizations (see challenge below) using variables to represent starting positions or the number of spaces moved each time. Middle schoolers may benefit from an introduction to such generalizations. I recommend only varying one of the parameters (either starting position or spaces). This would be appropriate for the prealgebra or more advanced student.

CHALLENGE TO READERS OF MATH NOTATION
Try to develop a general formula for the position of the marker after N moves given an initial position (S), number of hours on the clock (H) and the number of spaces moved (M). Also, an expression for the least number of such moves required to return to one's start position.

Posted by Dave Marain at 3:48 PM 0 comments

Labels: clock problems, divisibility, investigations, middle school, modular arithmetic, periodic behavior

Ok, so most normal people are not thinking about the significance of the digit '8' in 2008 the day before the Super Bowl. Sorry, but in this post there will be no predictions about the score, no 'over-unders', no boxes, no betting at all. You do have to admit that this is a great time for lovers of mathematics. People are actually interested in mathematical odds and chances of all kinds of weird number combinations occurring in the score on Sunday night. However, this post will focus instead on the number 8, the units' digit in 2008. The Super Bowl comments above will no doubt soon become outdated but the mathematics below will live on! Who knows, maybe the number 8 will turn out to have special significance on Feb 3, 2008? Remember, I said that here before the game!!

2008 is a special number for so many reasons, being divisible by 4 of course: Leap Year, Prez Election year, Summer Olympics and much more. In fact, 2008 is divisible not only by 4 but also by 8 itself. In the good ol' days, some students were even taught the divisibility rules for 2, 4 and 8:
Divisible by 2: If the 'last' digit is divisible by 2 (of course!)
Divisible by 4: If the number formed by the last TWO digits is divisible by 4
Divisible by 8: If the number formed by the last three digits is divisible by 8.

Let's demonstrate this for 2008:
2008 us divisible by 2 because 8 is divisible by 2
2008 is divisible by 4 because '08' is divisible by 4
2008 is divisible by 8 because '008' is divisible by 8

A little weird with those zeros and not particularly interesting, right? Anyone care to guess a rule for divisibility by 16? Interesting, but none of this is the issue for today....

BACKGROUND FOR PROBLEM/INVESTIGATION/ACTIVITY
Today, we are are interested in the squaresof numbers and their remainders when divided by 8. Notice that 4² is divisible by 8 but 6² is not. So we cannot say that the square of any even number is divisible by 8. What about the squares of odd numbers when divided by 8?
1² leaves a remainder of 1 when divided by 8
3² leaves a remainder of 1 when divided by 8
5² leaves a remainder of 1 when divided by 8
7² leaves a remainder of 1 when divided by 8

What is going on here? That's for your crack investigative team to decipher.

TARGET AUDIENCE: Our readers of course; Middle schoolers through algebra

PROBLEM/INVESTIGATION FOR READERS/STUDENTS
1. Discover, state and prove a general rule for the remainder when the square of an even number is divided by 8.
2. Discover, state and prove a general rule for the remainder when the square of an odd number is divided by 8.

Comments:
(1) These are well-known relationships and not very difficult questions. Just something to extend thinking about divisibility, remainders and the use of algebra to deduce and prove generalizations. Prealgebra students may be able to explain their findings without algebra!
(2) 'Discovering' or stating the rule for question (2) is transparent from the examples above. Instructors may prefer 'data-gathering' and making a table first. That is, have students develop a table for the squares of the first 10 positive integers and their remainders when divided by 8. Proving the result for the squares of odd integers is more challenging, even algebraically. Most will see the remainder when dividing by 4, but 8 is slightly trickier.
(3) Those who are more comfortable with congruences and modular arithmetic can approach these questions another way.

Posted by Dave Marain at 6:28 AM 4 comments

Labels: algebra, divisibility, investigations, proof, remainders

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