During the year, I pointed out that many texts begin with a lesson on inductive reasoning, which often entails completing number patterns. I like this sort of lesson at the start of the school year, but the U of Chicago text unfortunately doesn't contain such a lesson. I found today's activity in a different text that I mentioned earlier this week -- Michael Serra's Discovering Geometry.
Lesson 1.2 of the Discovering Geometry text is on Number Patterns. In this lesson, students will use inductive reasoning to find patterns in sequences of numbers, letters, and names.
I decided to create my worksheet from a variety of questions from the text. I began with some simple sequences where students had to find the next two terms. Notice that Exercise 5 is the Fibonacci sequence -- a nod to Fawn Nguyen, who gives her Geometry students a worksheet on that famous sequence on the first day of school.
Speaking of Fawn Nguyen, yes, many of my opening activities are based on Nguyen's. As it turns out, one of her best known lessons is on patterns -- except these are visual, not numerical, patterns:
Of course, many of the visual patterns lead to numerical patterns. For example, pattern #2 -- which looks very much like yesterday's building activity -- lends itself to the sequence 1, 3, 5, 7, ..., 85 as the number of rooms in each building. Nguyen makes her students find the 43rd term -- my worksheet only asks for the next two, which could be as far as the eighth and ninth terms.
Exercises 7 and 8 on my worksheet come from neither Serra nor Nguyen. Instead, they come from the texts used by the students I tutored. Those students enjoyed trying to figure out the patterns in these lists of names. I know that I've spent so recent posts about future presidents, but these lists are all about past presidents. Exercise 7 refers to George Washington, John Adams, Thomas Jefferson, and James Madison, so the correct answer is another James (Monroe), another John (Quincy Adams), and then Andrew (Jackson).
Exercise 8 looks similar, but this time it refers to money -- George Washington ($1), Thomas Jefferson ($2), Abe Lincoln ($5), and Alexander Hamilton ($10). So the correct answer is Andrew (Jackson again, $20), Ulysses (Grant, $50), and then it's all about Benjamin (Franklin, $100). As it turns out, one can actually extend this sequence. I was recently watching old 1960's episodes of the game show Let's Make a Deal on the new BUZZR channel, and often the host Monty Hall would offer contestants $500 bills (William McKinley) and $1000 bills (Grover Cleveland). There was even an episode when Monty showed a contestant an extremely rare $5000 bill that the bank had allowed him to show on that episode only -- had the contestant won it, she would have received a check for $5000 as the bill would have to be returned to the bank. James (Madison, not Monroe) was on the $5000 bill, so the sequence would continue Benjamin, William, Grover, James. It may be a good idea for teachers to give the related number sequence 1, 2, 5, 10, 20, 50, ..., as a hint.
Exercises 9 and 10 come from the Investigations section in Serra. This gives students the opportunity to work with a partner during the first week of school. Finally, Exercises 11 and 12 are two of the four conjectures that are part of this lesson. Notice that I wrote them just as Serra did, with the full equations written as 51 + 85 = 126. Perhaps I should have just written 51 + 85 = and required the students to calculate the sum. This is because I wanted to stick to the way Serra presents them and besides, I wanted the students to have fun during this activity making the conjectures, not having to calculate the sums. Of course, teachers who prefer the students to do the addition can simply white out the answers before copying.
The worksheet was getting long, so I stopped here. but notice that there are still many problems left in this section in the original Serra text. For the benefit of those who don't own the Serra text, let me reproduce Exercises 31 and 32:
Quatros: 4, 108, 60, 52, 36, 144
Not Quatros: 2, 29, 106, 18, 15, 22, 6
Which are Quatros? 86, 737, 42, 72
Semirps: 2, 13, 11, 23, 53, 97, 71, 47
Not Semirps: 15, 25, 209, 21, 190
Which are Semirps? 123. 67, 51, 27
Notice that "Quatros" are simply multiples of four. The word "Quatro" comes from the Latin word for four -- and we'll see that root later on in Geometry when we cover quadrilaterals. As it turns out, the modern Portuguese word for "four" is quatro. A few other Romance languages pronounce the word for "four" identically to the Portuguese, albeit with a slightly different spelling.
As for "Semirps," any nerd -- or even a dren -- can see that "Semirp" is "primes" spelled backwards. I do find it a bit awkward that the text pluralized "prime" to "primes," reversed it as "Semirp," then pluralized it again to "Semirps." Then again, one advantage to calling them "Semirps" rather than "emirps" is that the extra s- may trick readers into thinking about the prefix semi-, which is Latin for one-half -- especially right after seeing the Latin root for "four" in the previous question.
I was also considering including the first two sequences from the "Improving Reasoning Skills" section, which contains some bonus problems:
1. 18, 49, 94, 63, 52, 61, ...
2. O, T, T, F, F, S, S, E, N, ...
3. 4, 8, 61, 221, 244, 884, ...
I was able to figure out the first one, and I'd seen the second one before, but the third question stumped me -- and I suspect that it will stump our students as well. (The second one in the sequence is as easy as One, Two, Three!)
One of my favorite websites when considering number sequences is the On-Line Encyclopedia of Integer Sequences:
The OEIS is one of the oldest sites on the Internet. Notice that it was first created in 1964 -- long before the Internet existed as we know it! Back in the 1960's, users had to submit queries by sending it a primitive form of e-mail. Nowadays, of course, it is web-based like most other sites.
Many of the sequences in Serra's text are entries in the OEIS. Here they are:
180,360,540,720 (Notice the geometrical interpretation -- sum of the angles of an n-gon!)
1,3,6,10,15,21 (triangular numbers)
1,3,4,7,11,18 (Lucas numbers, similar to Fibonacci)
1,3,7,15,31,63 (sometimes called Mersenne numbers)
2,6,15,31,56,92 (given by the polynomial that generates these, (n+2)*(2*n^2-n+3)/6)
3,12,48,192,768 (there are some signed sequences in the database, but here the signs were ignored)
In fact, the only sequences I didn't enter were the ones containing letters or fractions, since this is in fact an integer sequence database.
Here's a sequence related to one of the lists of dead presidents that I entered:
Finally, here are the answers to the bonus questions:
18,46,94,63,52,61 (but I really did figure this one out before entering it into the OEIS)
4,8,61,221,244,884 (too hard for me, no problem for OEIS)
2,3,6,1,8,6,8 (too hard for me, no problem for OEIS)
The last sequence did stump the OEIS, however. Neither one of us figured out the sequence:
Maybe you can figure it out, then try submitting it to the OEIS! Unfortunately, the OEIS has been swamped with submissions for months. Still, you can see why I enjoy the OEIS as a handy resource for integer sequences.
Let me wrap up this post by reminding you, the readers, that all of these problems come from my own version of Serra's text -- the first edition. My copy of the U of Chicago text is also the first edition. I only know about the contents of the newer editions by reading other websites such as the following link to a Minnesota classroom:
Assuming that each bullet point corresponds to a chapter, we see that the biggest difference between the old and new versions of the U of Chicago text is that the old version delays translations and rotations to Chapter 6, while the new gives all of the isometries in Chapter 4. I observe that the new Chapter 5 is called "Congruence Proofs" -- but I wonder what figures the text is proving congruent and how, considering that "Congruent Triangles" isn't until Chapter 7 (just as in the old text). Indirect proofs have been moved up from Chapter 13 to Chapter 11, with the new Chapter 13 being about circles -- the old Chapter 15. Trigonometry -- the old Chapter 14 -- has been dropped altogether.
I hope you enjoyed today's activity. Now you have three choices as to what to cover the first week of school -- and one can cover one, two, or all three activities during the first week.