## Posts tagged ‘proportion’

### Math and the NBA All-Star Game

How awesome was Anthony Davis last night? In a word: very. He set a new All-Star Game scoring record with 52 points, adding 10 rebounds and 2 steals.

(Disclosure: I’ve liked A.D. since he was one-and-done at Kentucky. But it wasn’t until last night that I bought an Anthony Davis jersey:

But as much as I like A.D., I couldn’t help thinking that there needs to be an asterisk next to this new All-Star scoring record. If you only look at points, sure, 52 > 42, so Davis scored more points last night than Wilt Chamberlain scored in the 1962 All-Star Game. But that’s only a part of the mathematical story.

First, let’s talk **scoring percentage**. In 1962, the final score of the game was 150‑130, meaning that Chamberlain accounted for 15.0% of all scoring. The final score of last night’s game was 192‑182, meaning that Davis accounted for 13.9% of all scoring. Chamberlain gets the nod, but only slightly, and I’ll admit it’s not insignificant that Davis only played 31 minutes last night, while Chamberlain played 37 minutes in 1962. So, maybe this is a push.

But let’s consider **shooting percentage**. Last night, both teams combined for 55.5% shooting, whereas in 1962, they managed just 43.8% shooting. Perhaps the all-stars from 50 years ago just didn’t shoot as well as players today? Actually, that’s somewhat true: The league FG% for 1961‑62 was 42.6%, the league FG% for 2016‑17 (so far) is 45.6%. But the all-stars last night were 9.9% above the league average, whereas the all-stars in 1962 were just 1.2% above their league average, suggesting that the defense in New Orleans was negligible at best. Which brings me to my next point…

Let’s talk **defense**. Maybe the combined 374 points that were scored last night doesn’t convince you that defense was nonexistent. Then how about this: In the 1962 game, there were 62 personal fouls. Last night, there were only 16. Even more stark, though: In 1962, all-stars shot 95 free throws during the game; last night, they only shot 8. That’s not a typo, and it’s a pretty clear indication that no one was making much effort to contest shots.

Davis played a great game, but it doesn’t feel right that he unseats Chamberlain, given the circumstances. Not to mention, Chamberlain played a more complete game — shooting 73% from the field, grabbing 24 rebounds, and adding 1 assist.

This brings me to my final point, **proportions**. The teams last night scored 1/3 more points than their 1962 counterparts, and if you take away that extra third from Davis, he’d have ended the night with 39 points. So if an asterisk is good enough for Maris’s 61 and Flo-Jo’s 10.49, then it ought to be just fine for Davis’s 52, too.

But it is what it is. Congratulations, Anthony Davis.

Looking at the math of basketball is something I get to do quite a bit these days. Discovery Education has formed a partnership with the NBA, and we’re creating a collection of “problems worth solving” using NBA stats and highlight videos. Wanna see some of what we’ve done? Check out **www.discoveryeducation.com/NBAMath**.

### Deliver Us Not Into Bad Math

What better way to celebrate National Pizza Day than sharing this sign, which hangs in our local Pizza Hut:

Admittedly, I’ve never been very good with proportions, but even I know that

.

Yet, that’s what’s implied by the statements for $3 and $5 in the sign. Further,

- For $1, you can feed 4 children for 1 day. That’s a
**daily rate of 25.0¢**per child. - For $3, you can feed 2 children for 7 days. That’s a
**daily rate of 21.4¢**per child. - For $5, you can feed 1 child for 30 days. That’s a
**daily rate of 16.7¢**per child.

Will the real **price per child per day** please stand up?

And then I took a look at that last statement — that $10 can feed a classroom for a day — and it really blew my mind. Daily rates of 16.7 to 25.0¢ per child imply that classrooms have 40 to 60 students. I don’t know where these hungry students are, but maybe there should be a secondary campaign to reduce class size?

Though let’s be honest. What really seems to be needed here is an entirely new campaign:

### Is It Phi Day Yet?

There are several pressing matters that needs to be resolved.

Let’s use the following poll to resolve the first…

Don’t be swayed by the giant.

The second matter concerns the date of Phi Day.

As if it weren’t enought that the contrivance known as Pi Day is celebrated on 3/14, simply because the three digits of that date agree with the first three digits of π. Now the folks at www.phiday.org say that Phi Day should be celebrated on June 18, since 6/18 are the first three digits after the decimal point of φ, the golden ratio. What’s next? Are we gonna say that *e* Day should be celebrated on 7/18, since *e* = 2.718? Please.

But wait, there’s more. The folks (or should I say *folk*, since I think it’s just one guy) at www.goldenratio.org have proposed that Phi Day should be celebrated on October 31 in the Northern Hemisphere and May 6 in the Southern Hemisphere. The convoluted calculations for these dates can be found in this white paper.

Let’s settle this once and for all. The Golden Ratio divides a line into mean and extreme ratio, so Phi Day ought to divide the year into mean and extreme ratio, too.

The golden ratio divides a line segment into two parts, *a* and *b*, such that

For a non-leap year, then, we are looking for the date that divides a 365-day year into the golden ratio.

This formula yields

For leap years, which contain 366 days, the result is

In non-leap years, the 139th day of the year is May 19; in non-leap years, the 140th day is May 19. Consequently, a rather satisfying result occurs: the same date can be used for Phi Day in leap and non-leap years.

Further, it’s nice that the date occurs during the school year. One last chance to have a math party before summer break. And while a standard cake pan measures 9″ × 13″, you are strongly encouraged to use an 8″ × 13″ cake pan to concoct a treat for this special day.

So there you have it. An official MJ4MF declaration:

Whereas, phi represents the golden ratio, which divides a length into the ratio 1:1.618; and,

Whereas, the 19th day of May divides the year into the golden ratio; therefore, be it

RESOLVED, that Phi Day henceforth shall be celebrated on May 19; and be it further

RESOLVED, that there shall be no further discussion of this matter.

We’ll use the results of the poll above to determine whether it should be pronounced National Fee Day or National Fie Day.

### Blown Out of Proportion

In preparing a workshop about proportional reasoning for the 2011 NCTM Annual Meeting, I came across the following from “Learning and Teaching Ratio and Proportion: Research Implications” by Cramer, Post, and Currier, which appears in *Research Ideas for the Classroom*, edited by Douglas T. Owens. The authors discuss the following problem, which they presented to a class of pre-service elementary teachers:

Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

The authors claimed, “Thirty-two out of 33 pre-service elementary education teachers in a mathematics methods class solved this problem by setting up and solving a proportion: **9/3 = x/15**.”

I wanted to be surprised. Sadly, I was not.

Participants in my workshop (all current middle or high school math teachers) were asked to solve the same problem. Several incorrect answers were suggested, among them 3, 15, 27, and 45. Less than 40% of the attendees obtained the correct answer, 21.

This is frustrating, as proportional reasoning is extremely useful in analyzing real-world phenomena. In fact, it’s even applicable to language arts, as evidenced by the following graphic from my presentation:

The teachers in my workshop aren’t the only ones who have difficulty with proportional reasoning. Students have endless trouble, too…

Teacher: Today, we will discuss inverse proportion. Here’s an example: “If it takes 6 days for 2 men to finish a task, how long will it take 3 men to complete the same task?” The number of men needed is inversely proportional to the number of days required. Consequently, 3 men will be able to complete the task in 6 × 2/3 = 4 days.

Student: Oh, I see! I think I’ve got a real-life application of this. If it takes 6 hours for 2 men to hike to the top of a hill, then it will only take 4 hours for 3 men to hike to the top!