MathNotations

Do you remember the problem I posted a couple of days ago at the bottom of one of my updates:

What is the greatest possible number of Saturdays in a string of 100 consecutive days?

Well, here's a new feature that I hope will work. Click "Read more" and, hopefully, the answer and solution(s) will appear! If it doesn't work, then you will see the entire post.
Let me know if this works by posting a comment or emailing me (dmarain at geemail dot com)!



Answer: 15

Suggested Solutions
To maximize the number of Saturdays it is logical to start with 1 as the first Saturday, then the next Saturday will be day #8, then day #15, and so on. Each term of this sequence can be described by the expression 7a+1, that is, the positive integers which leave a remainder of 1 when divided by 7. The largest multiple of 7 less than 100 is 14x7 = 98, thus our sequence of Saturdays proceeds: 1,8,15,22,...99. Note that the first term 1 is actually 7x0+1 and the last term 99 = 7x14+1, for a total of 15 Saturdays.

Students should also recognize that if a sequence can be described by a linear function of the form s(n) = kn+b, then the sequence is arithmetic and we can apply the well-known formulas for arithmetic sequences. Thus 99 = 1 + (n-1)7 leading to our result of n = 15. Here n represents the number of terms of our sequence starting with a value of 1.

Posted by Dave Marain at 9:13 AM 4 comments

Labels: arithmetic sequence, more, patterns, SAT strategies, SAT-type problems

A simple warmup for Grades 3-12?? Can one problem really be appropriate at many levels?

Would 3rd, 4th or 5th graders guess the obvious answers 0,0 or 2,2 provided they understand the meaning of the terms sum and product? Do youngsters immediately assume the two numbers are different? Children at that age are thinking of whole numbers, however, what if you allowed them to try 3 and 1.5 (with or without the calculator)?

For middle schoolers: After they 'guess' the obvious integer answers, what if you were to ask them: "If one of the numbers is 3, what would the other number be?" If one of the numbers were 4? 5? -1? -2? Is algebra necessary for them to "guess" the other number? Would a calculator be appropriate for this investigation? Would they begin to realize there are infinitely many solutions? What if you asked them to explain why neither number could be 1...

For Algebra students: If one of the numbers is 3, they should be able to solve the equation:
3+x = 3x; they can repeat this for other values including negatives as well. They should be able to explain algebraically why neither number could be 1. Let them run with this as far as their curiosity takes them!

For Algebra 2, Precalculus and beyond: See previous ideas. Should they be expected to solve the equation x+y = xy for y obtaining the rational function y = x/(x-1)? Analysis of this function and investigation of its graph may open new vistas for this 'innocent' problem about sums and products. Does this function really make it clear why 0,0 and 2,2 are the only integer solutions?

The original question is well-known. At any level, I would recommend that they be allowed to explore and make conjectures before more formal analysis. High schoolers enjoy coming up with 0,0 and 2,2 as much as 8 year olds! Modifying it and asking probing questions as students mature mathematically is the challenge for all of us. Have fun with this 3rd grade question!

Posted by Dave Marain at 8:33 AM 0 comments

Labels: arithmetic sequence, investigations, pedagogy, warmup

[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!

Posted by Dave Marain at 6:01 AM 16 comments

Labels: arithmetic sequence, combinatorial math, discovery learning, investigations, middle school math, patterns

Here are two standardized type questions that students sometimes struggle with. Those who do well on these kinds of questions know the key is to understand the basics of arithmetic sequences. The 2nd question is a bit more sophisticated. What changes did I make to complicate the picture?


1. The median of a set of 20 consecutive integers is 14.5. What is the mean of the first 10 of these?

2. The mean of 98 consecutive odd integers is 44. What is the greatest of these numbers?

Notes:
BIG IDEAS for ARITHMETIC SEQUENCES:
(a) MEAN = MEDIAN!
(b) MEAN = MEDIAN = (LAST + FIRST)/2
(c) N = (L-F)/D + 1
Can you guess what these variables represent?
Hint: This is a variation on a well-known formula for arithmetic sequences.

Posted by Dave Marain at 7:10 PM 5 comments

Labels: arithmetic sequence, mean, median, standardized tests

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!

Posted by Dave Marain at 5:30 AM 10 comments

Labels: arithmetic sequence, combinatorial math, geometric sequence, middle school math, sequences