Engaging students: Ratios and rates of change

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Avery Fortenberry. His topic, from Algebra: ratios and rates of change.

In this viral YouTube video a man asks his wife the question “If you are traveling 80 miles per hour, how long does it take to travel 80 miles.” The wife overthinks the question and instead of trying to calculate how long it would take using the information of 80 miles per hour and how that they were going to travel one hour, she tries to think of how quick the tires are spinning and estimating the speed using her speed in running. The couple later goes on to talk on the Comedy Central show Tosh.0 where the wife explains the reason she was confused was that she had not slept well the night before and she was stressed with just finishing her finals. This video stresses the importance of making sure people understand that 80 miles per hour means you travel 80 miles in one hour.

The history of a rate of change is interesting when you consider the history of calculus itself. An important concept of calculus is finding derivatives, which is finding the rate of change or slope of a line. Calculus’s discovery was credited to both Isaac Newton and Gottfried Leibniz who both published their work around roughly the same time. This caused a dispute between the two men and they both accused the other of stealing their work. While both contributed much to the world of mathematics, it was many of Leibniz’s concepts of calculus that we still use today such as his notation dy/dx used for derivatives. Despite that Leibniz died poor and dishonored while Newton had a state funeral.

One of my favorite websites is khanacademy.org. This website has helped me from when I was in high school all the way to now it is still helping me understand concepts I may not have fully understood in class. It is a valuable resource to use when teaching about rates of change because there are countless videos over rates of change and slope and derivative that explain in detail all the concepts of it. Also, it has multiple practice problems that help you practice and study for an exam. I even used it for this project to help refresh my memory on rates of change and I was also looking at its word problems to help think of a word problem on my own for the A1 section of this project. Khan Academy also teaches you by reviewing all difficult steps in problems so that you can understand all the concepts.

Resources:

https://www.youtube.com/watch?v=Qhm7-LEBznk

http://www.uh.edu/engines/epi1375.htm

www.Khanacademy.org

Another poorly written word problem (Part 8)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

On its face, problems 11 and 12 don’t look so bad. For #11, the appropriate inequality is

$1400 + 243 + w \le 2000$

$1643 + w \le 2000$

$w \le 357$

For #12, the inequality is

$7 + g \le 15$

$g \le 8$ .

These indeed are the answers that the textbook is expecting. However, both answers are wrong because both $w$ and $g$ have to be positive. So the answers should be $0 \le w \le 357$ and $0 \le g \le 8$ . Which would be no big deal — except that these problems appeared before compound inequalities were introduced. (Notice that problems 7 through 10 only contain a single inequality.)

So, in a nutshell, the correct answers for these problems require skills that students have not yet learned at the time that they would attempt these problems.

Another poorly written word problem (Part 7)

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

Based only on how the questions are worded, should the answers to #53 and #54 be

$5x - 10 < 6x -8 \qquad \hbox{and} \qquad x + 20 < 4x - 1$ ?

Or should they be

$5x - 10 < 6(x -8) = 6x - 48 \qquad \hbox{and} \qquad x + 20 < 4(x - 1) = 4x -4$ ?

My answer: I have no idea. An argument could be made for either interpretation. And if a problem can be read two different ways by reasonable readers, then it should never be published in a textbook.

Another poorly written word problem (Part 6)

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

This one makes my blood boil. According to its advocates, the whole point of the Common Core standards was to increase the rigor in secondary mathematics. However, this one is SIMPLY WRONG.

The textbook does correctly note that the proper definition of a function is a set of ordered pairs. The “correct” answer, according to the textbook, is answer G — the plotted points do not match the ordered pairs.

However, answer H is also wrong. The textbook would have students believe that order is important when listing the elements of a set. However, order is not important — the domain of $\{-3, 1, -1, 3\}$ is the same as $\{-3, -1, 1, 3\} or$ latex \{3, -3, -1, 1\}$. This is standard mathematical notation — in an ordered pair (or ordered $n-$ tuple), the order is important. For a set, the order is not important.

Specifying that the domain is $\{-3,-1,1,3\}$ and the range is $\{2,5,8,11\}$ does not uniquely determine the function. In fact, there are 24 different functions that have this domain and range (where we distinguish between the range of a function and its codomain).

In other words, in trying to be clever about properly defining a function and showing different representations of a function, the textbook promotes a misconception about sets… which makes me wonder if the textbook’s attempt at trying to be ultra-careful about the definition of a function is really worth it.

Another poorly written word problem (Part 5)

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

This one really annoys me. The area is less than 55 square inches, and so the appropriate inequality is

$\frac{1}{2} (5)(2x+3) < 55$

$5(2x+3) < 110$

$2x + 3 < 22$

$2x < 19$

$x < 9.5$

However, part (c) asks for the maximum height of the triangle. But there isn’t a maximum possible height. If the height was actually equal to 9.5 inches, then the area would be equal to 55 square inches, which is too big! Also, if any height less than 9.5 is chosen (for the sake of argument, say 9.499), then there is another acceptable height that’s larger (say 9.4995).

Technically, the problem should ask for the greatest upper bound (or supremum) of the height of the triangle, but that’s too much to expect of middle school or high school students learning algebra.

This problem could have been salvaged if it had stated that the area is less than or equal to 55 square inches. However, in its present form, part (c) of this problem is unforgivably awful.

A Group of American Teens Are Excelling At Advanced Math

I really enjoyed reading this article from The Atlantic: http://www.theatlantic.com/magazine/archive/2016/03/the-math-revolution/426855/

If you have about 10-15 minutes to spare, I highly recommend it.

Your Plan is Foiled

I normally prefer the phrase “by using the distributive law” instead of the more colloquial “by FOIL,” but I’ll make an exception in this case.

Seventy Times Seven

I normally hate these math-is-hard jokes, but this one made me laugh. Forgiving somebody is harder than computing $70 \times 7$ ?

I don’t know who created this cartoon; I’ll be happy to credit it if anybody knows.

Mathematics that Swings: The Math Behind Golf

From the YouTube description:

Mathematics is everywhere, and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.

New England Patriots Cheat At the Pre-Game Coin Flip? Not Really.

Last November, CBS Sports caused a tempest in a teapot with an article with the sensational headline “Patriots have no need for probability, win coin flip at impossible rate.” From the opening paragraphs:

Bill Belichick is never unprepared. Or at least that’s the perception. When other coaches struggle with when to use timeouts or how to manage the clock, the Patriots coach, almost effortlessly, always seems to make the right decision.

Belichick has also been extremely lucky. The Pats have won the coin toss 19 of the last 25 times, according to the Boston Globe‘s Jim McBride.

For some perspective: Assuming the coin toss is a 50/50 proposition, the probability of winning it at least 19 times in 25 tries is 0.0073. That’s less than three-quarters of one percent.

As far as the math goes, the calculation is correct. Using the binomial distribution,

$\displaystyle \sum_{n=19}^{25} {25 \choose n} (0.5)^n (0.5)^{25-n} \approx 0.0073$ .

Unfortunately, this is far too simplistic an analysis to accuse someone of “winning the coin flip at an impossible rate.” Rather than re-do the calculations myself, I’ll just quote from the following article from the Harvard Sports Analysis Collective. The article begins by noting that while the Patriots may have been lucky the last 25 games, it’s not surprising that some team in the NFL was lucky (and the lucky team just happened to be the Patriots).

But how impossible is it? Really, we are interested in not only the probability of getting 19 or more heads but also a result as extreme in the other direction – i.e. 6 or fewer. That probability is just 2*0.0073, or 0.0146.

That is still very low, however given that there 32 teams in the NFL, the probability of any one team doing this is much higher. To do an easy calculation we can assume that all tosses are independent, which isn’t entirely true as when one team wins the coin flip the other team loses. The proper way to do this would be via simulation, but assuming independence is much easier and should yield pretty similar results. The probability of any one team having a result that extreme, as shown before, is 0.0146. The probability of a team NOT having a result that extreme is 1-0.0146 = 0.9854. The probability that, with 32 teams, there is not one of them with a result this extreme is 0.9854³² = 0.6245998. Therefore, with 32 teams, we would expect at least one team to have a result as extreme as the Patriots have had over the past 25 games 1- 0.6245998 = 0.3754002, or 37.5% of the time. That is hardly significant. Even if you restricted it to not all results as extreme in either direction but just results of 19 or greater, the probability of one or more teams achieving that is still nearly 20%.

The article goes on to note the obvious cherry-picking used in selecting the data… in other words, picking the 25 consecutive games that would make the Patriots look like they were somehow cheating on the coin flip.

In addition the selection of looking at only the last 25 games is surely a selection made on purpose to make Belichick look bad. Why not look throughout his career? Did he suddenly discover a talent for predicting the future? Furthermore, given the length of Belichick’s career, we would almost expect him to go through a period where he wins 19 of 25 coin flips by random chance alone. We actually simulate this probability. Given that he has coached 247 games with the Patriots, we can randomly generate a string of zeroes and ones corresponding to lost and won con flips respectively. We can then check the string for a sequence of 25 games where there was 19 or more heads. I did this 10,000 times – in 38.71% of these simulations there was at least one sequence with 19 or more heads out of 25.

The author makes the following pithy conclusion:

To be fair, the author of this article did not seem to insinuate that the Patriots were cheating, rather he was just remarking that it was a rare event (although, in reality, it shouldn’t be as unexpected as he makes it out to be). The fault seems to rather lie with who made the headline and pubbed it, although their job is probably just to get pageviews in which case I guess they succeeded.

At any rate, the Patriots lost the coin flip in the 26th game.