Sunday, March 22, 2009

The String of 100 Saturdays Problem -- READ MORE!!

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.


Anonymous said...

The "Read more" worked for me.

I think my students would only be confused by using algebra in this problem. 100/7=14R2 , so you have 14 full weeks (each of which contains a Saturday) and two extra days. If your string of consecutive days started with Friday or Saturday, then one of the extra days will also count, giving a maximum of 15 days.

Dave Marain said...

Yur explanation is cleaner and simpler and definitely more for that age group. I was really writing he solution for the Algebra student and beyond and to reinforce the idea that arithmetic sequences are always described by linear functions. I've observed over the years that students often don't relate the two!

The other big issue is the logical one of dividing and then deciding what to with the remainder or those counting procedures where students often make the classic "off by one error!" I also refer to this as a fencepost or mile marker problem. You know the type: You enter at mile marker 45 and exit at mile marker 65. How many miles did you go? How many mile markers did you see (assuming one marker each mile)?
Students would rather just memorize some rule like "Add 1" or not!

The arithmetic sequence formula takes the guesswork out of the problem which makes it useful for standardized tests and problem-solving in general.

Joshua Zucker said...

Just wanted to echo Dave's comments:

The relationship arithmetic sequences to linear functions is an important and oft-neglected connection. I hope more teachers will make it!

I also have spent way too much energy fighting the "just add one" fencepost errors. I wish students would think instead! My method is to encourage them to transform their sequence to 1, 2, 3, ... and see how many they have in that way. So how many mile markers from 45 to 65? Subtract 44 from each and you have 1, 2, ..., 21, so there are 21 markers. No adding 1 necessary, unless your list originally started with 0.

(This method has the added advantage of reinforcing the idea of transformations, which is another big emphasis for me.)

Dave Marain said...

The transformation approach is brilliant! I guarantee some of our readers here will use that with their students immediately. Thank you...

The linear function or arithmetic sequence approach of course was to get at those "equally spaced" sequences which are not "consecutive".

After all the ways I used to explain the "add 1", I finally decided to show them a variation on the standard arithmetic sequence formula:

The number of terms equals
The "add 1" is already in the formula!

To motivate this, I would choose an example like 3,7,11,15,19,23 and place these on a number line, heavily emphasizing the dots on the board. I approached the formula logically, one step at a time using the idea of distance:
(1) 23 - 3 = 20, the length of the segment or total distance from end to end.
(2) 20 divided by 4 = 5. "Ok, boys and girls, what is '4' and why am I dividing by it?" "Now, what does the result '5' have to do with the number line model? Yes, the number of SPACES!" (Usually, somebody saw this. If not, I would say, "Is FIVE the number of 'dots'? No, then what is it!").
(3) "So, why do we have to add 1?" Explanation: "There is a one-to-one match between the "dots" and spaces, but there will be one dot left over at either end!" They wouldn't say that of course, in those words, but they could "see" it.

Even before the full linear discussion in algebra, middle schoolers can handle this explanation. In the end they can memorize the formula! By the way, one could approach the general arithmetic sequence using your transformational approach. Very cool, Joshua!!