On the news: A very literal pie chart

A very literal pie chart can be seen 20 seconds into this news clip.

A Review of WuzzitTrouble: an app for math education

Most apps and computer games that claim to assist with the development of mathematical knowledge only focus on rote memorization. There’s certainly a place for rote memorization, but I’ve been very disappointed with the paucity of games that encourage mathematical creativity beyond, say, immediate recall of the times tables.

Enter WuzzitTrouble, a new app that was developed by Keith Devlin, a professor of mathematics at Stanford and one of the great popularizers of mathematics today. An introduction to WuzzitTrouble can be seen in this promotional video:

One minor complaint about WuzzitTrouble is that the first few levels are so easy that it’s easy for children to low-ball the game… in much the same way that the first few levels of Angry Birds are utterly easy. (My other complaints is that the game only assume one user, so that a parent can’t play the game without affecting a child’s settings.) However, the level of difficulty does eventually increase. Here’s another promotional video showing how to solve Level 1-25:

Here’s a sampling of some of the higher levels. Remember that the wheel has 65 steps along the circumference, as shown in the above picture and videos.

Level 2-5: Using cog wheels of size 5 and 9, pick up keys at 23 and 36 and prizes at 27, 45, and 55.
Level 2-15: Using cog wheels of size 5, 7, and 9, pick up keys at 11, 16, and 21 and prizes at 32 and 42.
Level 2-25: Using cog wheels of size 5, 9, and 16, pick up keys at 24, 48, and 59; prizes at 11 and 37; and avoid a penalty at 64.
Level 3-3: Using cog wheels of size 3, 4, and 5, pick up keys at 7, 17, and 27 and prizes at 12 and 22.

In the words of their promotional materials:

At InnerTube Games, we set out to design and build mobile casual video games and puzzles that can attract and engage a large number of players, yet are built on fundamental mathematical concepts and embed sound mathematics learning principles.

We start with one simple, yet powerful observation. A musical instrument won’t teach you about music. But when you pick up an instrument and start playing – badly at first – you cannot fail to learn about music. And the more you play, the more you learn. In fact, using that one instrument, you can go all the way from stumbling beginner to virtuoso concert performances. It’s the music that changes, not the instrument. In modern parlance, the instrument is a platform. And (well designed) platforms are good for learning because they make the learning meaningful and put the learner in charge.

InnerTube Games does not build video games to “teach mathematics.” Rather, we build instruments which you can play, and we design them so that when you play them, you cannot fail to learn about mathematics. Moreover, each single game can be used to deliver mathematical challenges of increasing sophistication.

Our vision for learning design is to build the game around core mathematical concepts and practice so it looks and plays like the familiar casual games on the market. As a result, you won’t be able to see the difference by playing the first few levels, or by watching someone else play. It’s the educational power under the hood that makes our games different.

We’re not making a secret of the fact that our games are math-based. It’s not “stealth learning;” it’s a form of learning through action that the brain finds natural, having much in common with what educational researchers call embodied learning.

Wuzzit Trouble is our first puzzle to reach the market. It is built around the important mathematical concepts of integer partitions–the expression of a whole number as a sum of other whole numbers–and Diophantine equations. At the easiest levels of the puzzle, these provide engaging practice in basic arithmetic, leading to arithmetical fluency.

But that’s just the start. Integer partitions and Diophantine equations are major areas of mathematics, still being worked on today by leading mathematicians.

Freeing the Wuzzits won’t take you into those dizzy realms—at least in the initial release, which comes loaded with puzzles aimed at the Elementary and Middle School levels. But as you progress, you will face challenges that increasingly require higher-order arithmetical thinking, algebraic thinking, strategy design and modification, optimization, and algorithm design, all crucial abilities in today’s world. Getting three stars can require considerable ingenuity.

As you attempt to free each Wuzzit and maximize your score, you will be developing and applying valuable conceptual, analytic thinking skills that sharpen your mind—all without lifting pencil to paper.

As educators and former educators, all of us at InnerTube are very aware of the importance of learners meeting agreed standards. In its initial release version Wuzzit Trouble provides natural learning in the following areas of the US Common Core Curriculum:

*Grade 2, Operations & Algebraic Thinking #2

*Grade 2, Number & Operations in Base Ten #2, #8

*Grade 3, Operations & Algebraic Thinking #1, #4

*Grade 4, Operations & Algebraic Thinking #5

*Grade 6, Number System #5, #6

But we don’t want anyone to play our game purely to hit those Common Core markers. We want you to play it because it’s fun and challenging. Improvement in those CC areas comes automatically. Just like learning music by playing a musical instrument!

The analogy that I prefer is playing basketball. When young children are first learning to play basketball, there’s a place for learning how to dribble, how to pass, how to shoot free throws, etc. (These are analogous to learning how to add, subtract, multiply, and divide.) But children don’t just learn skills: they also go out and play. That’s where the WuzzitTrouble app fits in: it offers children a chance to just play with mathematics and enjoy it.

More references:

http://profkeithdevlin.org/2013/09/03/the-wuzzits-free-at-last/

Review: Wuzzit Trouble

Venn diagrams

Sources: http://www.xkcd.com/668/

http://www.xkcd.com/747/

http://www.xkcd.com/773/

Graphing a function by plugging in points

I love showing this engaging example to my students to emphasize the importance of the various curve-sketching techniques that are taught in Precalculus and Calculus.

Problem. Sketch the graph of $f(x) = x^5 - 5x^3 + 4x + 6$ .

“Solution”. Let’s plug in some convenient points, graph the points, and then connect the dots to produce the graph.

$f(-2) = (-32) - 5(-8) + 4(-2) + 6 = 6$
$f(-1) = (-1) - 5(-1) + 4(-1) + 6 = 6$
$f(0) = (0) - 5(0) + 4(0) + 6 = 6$
$f(1) = (1) - 5(1) + 4(1) + 6 = 6$
$f(2) = (32) - 5(8) + 4(2) + 6 = 6$

That’s five points (shown in red), and surely that’s good enough for drawing the picture. Therefore, we can obtain the graph by connecting the dots (shown in blue). So we conclude the graph is as follows.

Of course, the above picture is not the graph of $f(x) = x^5 - 5x^3 + 4x + 6$ , even though the five points in red are correct. I love using this example to illustrate to students that there’s a lot more to sketching a curve accurately than finding a few points and then connecting the dots.

Word Problems

Handling insightful questions from students

Source: http://www.xkcd.com/803/

Rejection regions

Sage words of wisdom that I gave one day in my statistics class:

If the alternative hypothesis has the form $p > p_0$ , then the rejection region lies to the right of $p_0$ . On the other hand, if the alternative hypothesis has the form $p < p_0$ , then the rejection region lies to the left of $p_0$ .

On the other hand, if the alternative hypothesis has the form $p \ne p_0$ , then the rejection region has two parts: one part to the left of $p_0$ , and another part to the right. So it’s kind of like my single days. Back then, my rejection region had two parts: Friday night and Saturday night.

Finding the equation of a line between two points

Here’s a standard problem that could be found in any Algebra I textbook.

Find the equation of the line between $(-1,-2)$ and $(4,2)$ .

The first step is clear: the slope of the line is

$m = \displaystyle \frac{2-(-2)}{4-(-1)} = \frac{4}{5}$

At this point, there are two reasonable approaches for finding the equation of the line.

Method #1. This is the method that was hammered into my head when I took Algebra I. We use the point-slope form of the line:

$y - y_1 = m (x - x_1)$

$y - 2 = \displaystyle \frac{4}{5} (x-4)$

$y - 2 = \displaystyle \frac{4}{5}x - \frac{16}{5}$

$y = \displaystyle \frac{4}{5}x - \frac{6}{5}$

For what it’s worth, the point-slope form of the line relies on the fact that the slope between $(x,y)$ and $(x_1,y_1)$ is also equal to $m$ .

Method #2. I can honestly say that I never saw this second method until I became a college professor and I saw it on my students’ homework. In fact, I was so taken aback that I almost marked the solution incorrect until I took a minute to think through the logic of my students’ solution. Let’s set up the slope-intercept form of a line:

$y= \displaystyle \frac{4}{5}x + b$

Then we plug in one of the points for $x$ and $y$ to solve for $b$ .

$2 = \displaystyle \frac{4}{5}(4) + b$

$\displaystyle -\frac{6}{5} = b$

Therefore, the line is $y = \displaystyle \frac{4}{5}x - \frac{6}{5}$ .

My experience is that most college students prefer Method #2, and I can’t say that I blame them. The slope-intercept form of a line is far easier to use than the point-slope form, and it’s one less formula to memorize.

Still, I’d like to point out that there are instances in courses above Algebra I that the point-slope form is really helpful, and so the point-slope form should continue to be taught in Algebra I so that students are prepared for these applications later in life.

Topic #1. In calculus, if $f$ is differentiable, then the tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ has slope $f'(a)$ . Therefore, the equation of the tangent line (or the linearization) has the form

$y = f(a) + f'(a) \cdot (x-a)$

This linearization is immediately obtained from the point-slope form of a line. It also can be obtained using Method #2 above, so it takes a little bit of extra work.

This linearization is used to derive Newton’s method for approximating the roots of functions, and it is a precursor to Taylor series.

Topic #2. In statistics, a common topic is finding the least-squares fit to a set of points $(x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)$ . The solution is called the regression line, which has the form

$y - \overline{y} = r \displaystyle \frac{s_y}{s_x} (x - \overline{x})$

In this equation,

$\overline{x}$ and $\overline{y}$ are the means of the $x-$ and $y-$ values, respectively.
$s_x$ and $s_y$ are the sample standard deviations of the $x-$ and $y-$ values, respectively.
$r$ is the correlation coefficient between the $x-$ and $y-$ values.

The formula of the regression line is decidedly easier to write in point-slope form than in slope-intercept form. Also, the point-slope form makes the interpretation of the regression line clear: it must pass through the point of averages $(\overline{x}, \overline{y})$ .

Engaging students: The area of a circle

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 comes from my former student Dale Montgomery. His topic, from Geometry: the area of a circle.

History

Archimedes was the mathematician who we attribute with finding the area of a circle to be Where r is the radius and π is the ratio of circumference to diameter of a circle. (Note that Archimedes was not the first to find the area of a circle, but was the first to find π). I would really like to start the class with something along the lines of introducing Archimedes supposed final words “Do not disturb my circles.” And then go into the death of Archimedes and the mystery surrounding his tomb, such as the account of Cicero and the fact that no one knows where the tomb is now. Cicero said that his tomb had a sphere inscribed in a cylinder, which Archimedes considered to be his greatest mathematical proof. From there, the class should have great interest in what is going on. And we can talk about the fact that the area of a circle is the same as the area a triangle with the same base as the circumference and the same height as the radius.

Rorres, Chris. “Tomb of Archimedes – Illustrations”. Courant Institute of Mathematical Sciences. Retrieved 2011-03-15.

Culture

http://newsfeed.time.com/2013/02/02/are-crop-circles-more-than-just-modern-pranks/

I would show this article in class, most likely passing it out to read. I would ask if they thought it was a prank, and then give them a similar picture as presented in the article but mapped out with radiuses. Then I would say that the average person could do so many square feet of crop’s per hour. If it gets dark at 9 pm and the sun comes up at 6 am, could a person pull a prank like this?

After we discussed how to find the area of a circle I would have found one that it was impossible for one person to do. Then I would display this youtube video.

Seeing that there were 2 people working on it could display that it is possible for it to be a hoax. I like this because it gives the students a way to analyze information that they are given. Does it make sense for these things to be aliens? Not really, so let’s find other explanations. It both introduces the concept and teaches some critical thinking skills.

Applications

You could apply the area of a circle to the diameter of a pizza. When you order pizza you order things like an 8 or a 12 inch. These are diameters and do not give the best idea of how much pizza you are actually getting. You can even include this lesson with a pizza party or something similar. This would easily get kids excited since it is something that most kids like, and they would have the possibility of getting pizza afterwards.

Finger trick for multiplying by 9

I’m constantly amazed at the number of college students who, through no fault of their own, simply were never taught this simple trick for multiplying by 9 when they were kids.

Why does this trick work? In the picture, if the left pinkie is brought down, there are nine fingers to the right that are up (corresponding to $9 \times 1$ ). If the second finger is lowered and the first is raised, that’s equivalent to adding $10$ (since there’s one additional finger in the “tens” part) and subtracting one (since there’s one less finger in the “ones” part). In other words, changing the lowered finger by one digit (pardon the pun) is like successively adding $9$ , and successively adding $9$ is the same as multiplying by $9$ .