# Mindset List for incoming High School class of 2017

Happy first day of school! For us, today marks the first day of the school year and we’re welcoming students into our midst. What are the new kids (the freshmen) going to be like?

Each year Beloit College describes the incoming college freshmen class with its now famous “Mindset List.” I looked around and couldn’t find a high-school equivalent. So here’s one I came up with. This is a description of the incoming high school freshmen class (class of 2017). Note that all of descriptions on Beloit’s college freshmen mindset list apply also to high school freshmen. So here’s my own “high school class of 2017 mindset list.” Enjoy!

1. The Euro has always existed. So has Sponge Bob Square Pants. And Google, Inc. And the iMac. And Viagra.
2. Bill Gates has always been worth over $100 billion. 3. You can talk to them about the Sandy Hook shooting or the Virginia Tech massacre, but they won’t remember anything about Columbine, which happened the year they were born. 4. Star Wars Episode 1: The Phantom Menace is as much of an ‘old-school’ classic as any of the original Star Wars movies. The movies Fight ClubThe MatrixAmerican PieSaving Private RyanArmageddon, and The Sixth Sense also came out in the years they were born. 5. East Timor has always been a sovereign nation. 6. George W. Bush and Barack Obama are the only presidents that they really know. Clinton left office when they were just 2. 7. Exxon and Mobil have always been the same company. 8. Movies have always been reviewed by Ebert & Roeper .(Gene Siskel died the year they were born; Roger Ebert just died this past April.) 9. They won’t have any memories of John F. Kennedy Jr, Dr. Spock, Frank Sinatra, Roy Rogers, or Alan Shepherd, all of whom died just as they were being born. 10. They have always had their music in mp3 format and used mp3 players (invented in 1998). CD players? SO passé. 11. Seinfeld closed up shop before they were born. ——– For more events that happened in 1998 and 1999, visit the wikipedia articles. Please feel free to correct any of my above information or suggest additions! # Arithmetic/Geometric Hybrid Sequences Here’s a question that the folks who run the NCTM facebook page posed this week: Find the next three terms of the sequence 2, 8, 4, 10, 5, 11, 5.5, … Feel free to work it out. I’ll give you a minute. Done? still need more time? .. give up? Okay. The answer is 11.5, 5.75, 11.75. The pattern is interesting. Informally, we might say “add 6, divide by 2.” This is an atypical kind of sequence, in which it seems as though we have two different rules at work in the same sequence. Let’s call this an Arithmetic/Geometric Hybrid Sequence. (Does anyone have a better name for these kinds of sequences?) But a deeper question came out in the comments: Someone asked for the explicit rule. After a little work, I came up with one. I’ll give you my explicit rule, but you’ll have to figure out where it came from yourself: $a_n=\begin{cases}6-4\left(\frac{1}{2}\right)^{\frac{n-1}{2}}, & n \text{ odd} \\ 12-4\left(\frac{1}{2}\right)^{\frac{n-2}{2}}, & n \text{ even}\end{cases}$ More generally, if we have a sequence in which we add $d$, then multiply by $r$ repeatedly, beginning with $a_1$, the explicit rule is $a_n=\begin{cases}\frac{rd}{1-r}+\left(a_1-\frac{rd}{1-r}\right)r^{\frac{n-1}{2}}, & n \text{ odd} \\ \frac{d}{1-r}+\left(a_1-\frac{rd}{1-r}\right)r^{\frac{n-2}{2}}, & n \text{ even}\end{cases}.$ And if instead we multiply first and then add, we have the following similar rule. $a_n=\begin{cases}\frac{d}{1-r}+\left(a_1-d-\frac{rd}{1-r}\right)r^{\frac{n-1}{2}}, & n \text{ odd} \\ \frac{rd}{1-r}+\left(a_1-d-\frac{rd}{1-r}\right)r^{\frac{n}{2}}, & n \text{ even}\end{cases}.$ And there you have it! The explicit formulas for an Arithmetic/Geometric Hybrid Sequence:-). (Perhaps another day I’ll show my work. For now, I leave it the reader to verify these formulas.) # Friday fun from around the web Here are two fun mathy things that came through my feed today. Many of you have probably already seen today’s math-themed xkcd: I’ll definitely have to share this ever-so-helpful guide with my BC calc students :-). And I also saw this today [on thereifixedit], which delighted the mathematician in me: Happy Friday everyone! # Yay for King Arthur Flour! Three cheers for King Arthur Flour for running a pi-day special! (HT: Kelly Chase) Also, while you’re getting an early start on pi day, check out this most recent numberphile video! # Lego Price Statistics Do you ever get the feeling that Lego Bricks are becoming more expensive? When we were kids, boy, it felt like they were cheaper, right? I mean, the biggest sets were$150 at most. I have a HUGE Lego collection, and it definitely seems like Legos back in my day were more affordable.

Trouble is, that’s not really true. It turns out that Lego bricks have actually gotten cheaper, by almost every measure you can think of (weight/number of pieces/licensed sets). Check out this incredibly thorough post on Lego Price statistics over time. The article is entitled, “What Happened with LEGO” by Andrew Sielen. It’s very thoughtfully done.

[ht: Gene Chase]

# Half-your-age-plus-seven rule

Looking for a great application of systems of linear inequalities for your Algebra 1 or 2 class? Look no further than today’s GraphJam contribution:

You might just give this picture to students and ask THEM to come up with the equations of the three lines.

There’s also a nice discussion to be had here about inverse functions, or about intersecting lines. And there might also be a good discussion about the domain of reasonableness.

Here are the three functions:

$f_{\text{blue}}(x)=x$

$f_{\text{red}}(x)=\frac{1}{2}x+7$

$f_{\text{black}}(x)=2x-14$

This is especially interesting because I never think of the rule as putting boundaries on a person’s dating age range. Usually people talk about it in the context of “how old of a person can I date?” not “how young of a person can I date?” Or rather, if you’re asking the second question, it’s usually phrased “how young of a person can date me?” (All of these questions relate to functions and their inverses!) But in fact, the half-your-age-plus-seven rule puts a lower and and upper bound on the ages of those you can date.

As far as reasonableness, is it fair to say that my daughter who is 1 can date someone who is between the age of -12 and 7.5? I don’t think so! I’m definitely going to be chasing off those -12 year-olds, I can already tell :-).

For my daughter, the domain of reasonableness might be $x\geq 18$!

# Happy e day!

Well, as you can see by the count-down on the right side of this blog, e day has arrived! I mentioned it in most of my classes today, even though it’s not as well-celebrated as π. I think we should change that. It’s my opinion that e is at least as good, if not a more important constant, than π.

I’m biased though. Today is also my daughter’s first birthday. I love that she was born on e day! Awesome! 🙂

Happy Birthday, Ruthie Chase!

# Friday tidbits

Happy Friday! Hope everyone has their kids registered for the AMC next week. If you haven’t already subscribed to the AMC problem-a-day from the MAA, you should! It’ll keep you sharp :-).

Here are a few nice things seen ’round the web recently:

• The Scrambler, by Dan Meyer & co. Here, Dan challenges us to analyze a classic carnival ride, and asks us to predict where you end up at the end of the ride. And by Dan & “co”, I mean “comment” folks who have generated lots of fun solutions and applets. Dan made a great interactive version here, too.

• And finally, this lengthy article “Reflections on mathematics and Democracy” by Lynn Arthur Steen is well worth the time [ht: Gene Chase]. He thoughtfully discusses the need for math education among the citizenship. Is “usefulness” to the democracy the highest goal of secondary math education? Do we aim to create quantitatively literate citizens? Or do we put them on the Calculus track and prepare them for college-level STEM careers? Does teaching “quantitative literacy” even count as Mathematics with a capital M? This is obviously something I’ve been thinking a lot about recently. Here are a few of my favorite excerpts:

Ten years ago I addressed the first question posed to this panel in Mathematics and Democracy—a collection of essays from a variety of professionals both inside and outside mathematics.4 (These essays are available for free downloading on the MAA website.) The chief message of this volume is that the mathematics taught in school bears little relationship to the mathematics needed for active citizenship. That mathematics we called quantitative literacy (QL) to contrast it with traditional school mathematics which, historically, is the mathematics students needed to prepare for calculus.

Mathematics and quantitative literacy are distinct but overlapping domains. Whereas mathematics’ power derives from its generality and abstraction, QL is anchored in specific contexts and real world data. An alternative framing of the challenge for this panel is to ask whether perhaps QL might be a more effective approach to high school mathematics for all.

What we forget, however, is that when NCTM initiated its standards work, most mathematics teachers did not actually believe in the goal of teaching mathematics to all high school students. Whereas now we argue about how much and what kind of mathematics to teach in high school, three decades ago debate centered on who should learn high school mathematics. At that time, the curriculum was designed to efficiently sort students into those who were capable of learning high school mathematics and those who were not. So between grades 7 and 9, somewhere between one-third and one-half of the students were placed in a course called General Math—an enervating, pointless review of arithmetic.

Another decade has passed, and our ambitions are now much higher: a common core for all, with everyone emerging from high school ready for college. In one generation, the political view of high school mathematics has progressed from something only some need (or can) learn to a core subject in which all students can and must become proficient. That’s quite a rapid change in ends, which has been matched by a major change in means. The very idea of a common curriculum enforced with common assessments was all but unthinkable back in the 1980s.

# Stellar application of the FTC

Here’s a “stellar” application of the Fundamental Theorem of Calculus, created by one of my students. It is a stellated icosahedron made up of 30 individual pieces of paper, all of which have the FTC printed on them. Doesn’t it just make you smile? 🙂