## Posts tagged ‘math’

### Our Library’s Summer Math Contest

Every summer, our local library runs a contest called *The Great Big Brain Game*. Young patrons who solve all of the weekly puzzles receive a prize. The second puzzle for Summer 2017 looked like a typical math competition problem:

Last weekend, the weather was perfect, so you decided to go to Cherry Hill Park. When you got there, you saw that half of Falls Church was at the park, too! In addition to all the people on the playground, there were a total of 13 kids riding bicycles and tricycles. If the total number of wheels was 30, how many tricycles were there?

First, some comments about the problem.

- I dislike using “you” in math problems. I believe it’s a turn-off to students who can’t see themselves in the situation described. There are enough reasons that kids don’t like math. Why give them another reason to shut down by telling them that they went somewhere they didn’t want to go or that they did something they didn’t want to do?
- Word problems are not real-world just because they use a local context, and this one is no exception. This problem attempts to show an application for a system of linear equations, but true real-world problems don’t have all the information neatly packaged like this.
- Wouldn’t the person posing this problem already have access to the information they seek? That is, if she counted the number of kids riding bikes and the total number of wheels, couldn’t she have just counted the number of bicycles and tricycles instead? It has always struck me as strange when the (implied) narrator of a math problem wants you to figure out something they already know.

All that said, this was meant to be a fun puzzle for a summer contest, and I don’t mean to scold the library. I don’t know that I’d use this puzzle in a classroom — at least, not presented exactly like this — but I love that kids in my town have an opportunity to do some math in June, July, and August.

Now, I’ll offer some comments on the solution. In particular, the solution provided by the library was different than the method used by one of my sons. Here’s what the library did:

Imagine that all 13 kids were on bicycles with 2 wheels. That would be a total of 26 wheels. But since 30 wheels are needed, there are 4 extra wheels. If you add each of those extra wheels to a bicycle, that’ll create 4 tricycles, leaving 9 bicycles. So, there must have been 4 tricycles at Cherry Hill Park.

And here’s what my son did:

If you can’t see what he wrote, he created a system of two equations and then solved it:

2a + 3b = 30

a + b = 13a + 2b = 17

13 – b + 2b = 17

b = 42a + 12 = 30

2a = 18

a = 9

That’s all well and good. In fact, it’s perfect if you want to assess my son’s ability to translate a problem and solve a system of equations. But I have to admit, I was a little disappointed. What bums me out is that he went straight to a symbolic algorithm instead of considering alternatives.

I think I know the reason for this. This past year, my son was in a pull-out math program, in which he studied math with someone other than his regular classroom teacher. In this special class, the teacher focused on preparing him to take Algebra II in sixth grade when he enters middle school. Consequently, students in the pull-out class spent the past year learning basic algebra. My fear is that they focused almost exclusively on symbolic manipulation and, as my former boss liked to say, “Algebra teachers are too symbol-minded.”

A key trait of effective problem solvers is flexibility. That type of flexibility comes from solving many problems and filling your toolbox with a variety of strategies. My worry — and this isn’t just a concern for my son, but for every math student in the country — is that students learn algorithms at the expense of more useful problem-solving heuristics. What happens when my son is presented with a problem that can’t be translated into a system of linear equations? Will he know what to do when he doesn’t know what to do?

The previous pull-out teacher said that when she presented my sons with problems that they didn’t know how to solve, their eyes would light up. They liked the challenge of doing something they hadn’t done before. I’m hopeful that this enthusiasm isn’t lost as they proceed to higher levels of mathematics.

### Russell, Robertson, and Ratios

In the NBA, a **triple-double** happens when a player has a double-digit total in three of the five categories (points, assists, rebounds, steals, and blocks) in a game. Triple-doubles are very rare; on average, one has been recorded only once every 27 games since 2003. So far this season, there have been 111 triple-doubles throughout the entire NBA — and **Russell Westbrook** has 41 of them.

In 1961-62,** Oscar Robertson** set a record that Westbrook is about to break. That year, Robertson recorded 41 triple-doubles in 80 games. Westbrook recorded his 41st triple-double of the current season in just 78 games. When two fractions have the same numerator, the one with the smaller denominator is larger. Consequently, 41/78 > 41/80, so Westbrook’s accomplishment exceeds Robertson.

But ratios can be used to make the point even more dramatically. In the early 1960’s, pro basketball games were played at a faster pace than they are today. In 1961‑62, the average game featured 126.2 possessions, meaning that Robertson typically had more than 60 tries to grab a rebound, make an assist, or score some points. By comparison, there have been an average of just 96.4 possessions per game during the current NBA season, meaning that Westbrook generally has fewer than 50 attempts per game to improve his stat line. So another ratio — the comparison of points, rebounds, and assists to number of possessions — also leans in Westbrook’s favor.

Who knew that either of these guys were such fans of math?

At Discovery Education, we’ve been having a lot of fun writing basketball problems based on real NBA data. Check out a few problems at http://www.discoveryeducation.com/nbamath, and get a glimpse of the NBA Analysis Tool within Math Techbook^{TM} by signing up for a free 60-day trial at http://www.discoveryeducation.com/math.

#mathslamdunk

### College Basketball Round-Up

The sheepdog returned to the farmhouse and told the shepherd, “All 200 sheep have been returned to their pens.”

“200?” asked the shepherd. “But we only have 196 sheep.”

The dog replied, “Well, yeah, but you know I like to round up.”

Rounding up has been a topic of conversation in college basketball this week.

Marcus Keene, a guard for the Central Michigan Chippewas, scored 959 points in 32 games this season, giving him a points-per-game (PPG) average of 30.0.

Sort of.

Technically, his average is 29.96875, just shy of the highly coveted 30 points-per-game mark that’s only been attained by a few dozen players in NCAA history. Since 1981, only 8 players have reached 30 PPG, most recently Long Island’s Charles Jones in 1996‑97.

But the controversy swirled this week because Keene didn’t actually average more than 30 points per game. He was one point shy. His lofty accomplishment was nothing more than smoke-and-mirrors due to round-off error, or so the critics say.

Per-game statistics are used to compare players with one another, because totals can’t be compared for players who have played a different number of games. And let’s face it, no one wants to get into the habit of comparing per-game stats to seven decimal places. The NCAA reports all per-game statistics to the nearest tenth, and the truth is that Keene’s PPG average would be reported as 30.0, 30.00, 30.000, and 30.0000 if rounded to tenths, hundredths, thousandths, and ten-thousandths, respectively.

It’s been a good year for math and basketball. Anthony Davis can have an asterisk for his record-setting 52 points in the NBA All-Star Game because no one played defense; and now Marcus Keene can have an asterisk for his 30.0 points-per-game average.

In related news, it was reported that 53% of men say that they will watch the NCAA Division I Men’s Basketball Championship (aka, “March Madness”). And just to prove the men are the dumber sex, 61% of them admitted that they’ll watch while at work. Simple math says that 32.3% of men will watch the tourney at work. Which means that if you’re a man with two friends who don’t like basketball, then you’ll be the one killing office productivity next Thursday.

### 2 Good 2 Be True

I was eating a bowl of shepherd’s pie at the Irish pub in our neighborhood. A man walks up to my table and asks, “What’s your favorite number?”

“Uh, 153,” I respond.

“And 153 × 2 is 306,” he says, then hurriedly scurries away.

He approaches another table, asks another patron for her favorite number, and again multiplies it by 2. He does this over and over, popping from table to table, annoying customer after customer. Eventually, the manager notices this eccentric behavior and approaches the man.

“Sir,” says the manager, “You can’t keep interrupting people’s dinners by asking them for a number and then multiplying by 2.”

“What can I say,” he responds. “I love Dublin!”

A little while later, the gentleman at the table next to me says to his companion, “I know a sure-fire way to double your money.”

This piqued my interest, so I leaned over to eavesdrop on his advice.

“Fold it in half,” he said.

Dismayed that you’ve read this far and have only heard two terrible jokes? Well, buck up, because your fortune is about to change. I can’t help you double your money, but I can help you get twice as much for it.

Perhaps you’ve been wanting a copy of **Math Jokes 4 Mathy Folks**, but just haven’t pulled the trigger yet. Well, now’s the time. Robert D. Reed Publishers is offering a BOGO special for MJ4MF, so now you can buy a copy for yourself at regular price and get another for the special math geek in your life **at no charge**!

**http://rdrpublishers.com/blogs/news/yes-math-is-fun**

And check this out.

- If you buy 2 copies, you’ll get 2 additional copies absolutely free!
- If you buy 3 copies, you’ll get 3 more at no cost!
- Buy 4 copies, and 8 copies will be delivered to your door!
- And if you buy 50 copies? Why, you’ll have 100 copies arrive to your home, office, or post office box for the exact same price!
- If you want
*n*copies, you’ll only pay for*n*/2 of them!

Folks, this is a linear relationship that you’d be foolish to ignore!

### NBA, Discovery, and the Math of Basketball

Last week, Discovery Education and the National Basketball Association (NBA) announced a partnership in which real-time data from stats.nba.com will flow into Math Techbook, and students will use that data to solve problems.

How cool is that?

Eighty students from John Hayden Johnson Middle School in Washington, DC, participated in the event, which was emceed by Hall-of-Famer “Big” Bob Lanier and made silly by Washington Wizards mascot G-Wiz.

The event received a lot of press coverage, and as you may have heard, **there’s no such thing as bad publicity**. But one of the articles quoted me as saying:

It’s not like a beautiful, traditional math problem.

That is **not** what I said. I am absolutely certain that I have never used the words *beautiful* and *traditional* in the same sentence. Well, perhaps when referring to a wedding dress or an Irish cottage, maybe, but certainly not when referring to a math problem.

I was also quoted as saying:

It’s going to be messy, for sure.

That is, in fact, one of the things I did say. Because by definition, good math problems are messy. For this project, our writing team created problems that don’t have one right answer. For instance, one problem asks students to **generate a formula to predict which players should be on the All‑NBA 1st Team**. Should they use points and rebounds as part of their formula? If so, how much weight should they give to each? And should there be a deduction for the number of turnovers a player has? All of that is up to the student, and it’s certainly possible that more than one formula would give reasonable results. (If you don’t believe me, do a search for NBA Efficiency, TENDEX, Thibodeau, VORP, or New SPM to get a sense of some formulas currently used by professional statisticians.)

A microsite with a four sample problems is available at **www.discoveryeducation.com/nbamath**. To see all 16 problems and to experience the NBA Math Tool, you’ll need to login to Math Techbook; sign up for a 60‑day trial at **www.discoveryeducation.com/mathtechbook**.

I’m ecstatic about the problems that our writing team — which includes folks who love both math and basketball, like Brenan Bardige, Ellen Clay, Chris Shore, Shauna Hedgepeth, Katie Rhee, Jen Silverman, and Jason Slowbe — has created. One of the simpler problems they’ve written, meant for middle school students, is to **determine which player should take a technical free throw**. It’s not a hard problem, but students get to choose which team(s) to examine and how to use free-throw data to make their choice. With the NBA Math Tool that we’ve created, which includes FTM and FTA but not FT%, one possible formula is **=ROUND(100*FTM/FTA,1)**, which will display the free-throw percentage to one decimal place of accuracy — though there are certainly less sophisticated formulas that will get the job done, too, and students could bypass formulas entirely by using equivalent fractions.

But a different article said that the “questions may look something like” this:

Andrew Wiggins is making 49.1% of his two-point shots and 52.3% of his threes. Which shot is he more likely to make?

Actually, we would **never** ask a question like this in Math Techbook, either as part of this NBA project or otherwise. By the time students start working with percentages in middle school (6.RPA.3.C), we expect that they already understand how to compare decimals (4.NF.C.7). Though basic exercises are included in the service, most problems — and especially those based on NBA data — exist at a greater depth of knowledge.

But what we might do is ask students to use proportions to make a prediction.

As you know, basketball announcers and sportswriters make predictions all the time. They talk about players being “on pace” to score some number of points or to grab a certain number of rebounds. In fact, the Washington Post recently prophesied that Steph Curry will hold the NBA’s all-time three-point record before the next presidential election.

During the first part of the event at Johnson Middle School, the students set out to make a prediction:

How many assists will John Wall finish the season with?

John Wall recently set the Wizards franchise record for assists, so the context was timely.

To solve this problem, students explored the **NBA Math Tool**, which now resides inside Math Techbook. This tool allows students to analyze both NBA and WNBA stats. Students considered data for the Washington Wizards:

Row 6 shows that John Wall had 98 assists through 11 games. Good information, to be sure, but it led to more questions from students than answers:

- Some players on the Wizards have played 13 games. How many games have the Wizards played so far this year?
- How many games will John Wall play this year?
- How many games are in an NBA season?

Looking at team data in the NBA Math Tool, students learned that the Wizards have played 13 games so far this year. And one student knew that every NBA team plays 82 games in a season. Good info… but now what?

One approach is to set up a proportion with the equation

which yields the number of games (*g*) that we can expect John to play this year (69), and then the equation

can be used to find the number of assists (*a*) that we can expect John to record (602).

But the eighty students in the gym were sixth- and seventh-graders, and they weren’t ready for algebraic equations. Instead, they attacked the problem by noting that Wall had 98 assists through the first 13 games, so they estimated:

- He should have about 200 assists through 25 games.
- He’d have about 400 assists through 50 games.
- He’d have about 600 assists through 75 games.
- That’s 7 games shy of a full 82‑game season, and Wall should have about 50 assists in 7 games.
- So, we can expect him to finish the season with about 650 assists.

My role at the event was to lead students through the solution as a group-problem solving activity; and then, to work with them in the media center on the free-throw problem described above. It was an incredible day! I got to co-teach with Ivory Latta, point guard for the Washington Mystics:

I got to meet some incredible people, including current players, former players, and NBA executives:

But most importantly, I was finally able to let the world know about this amazing project, which my team has been working on for a year.

The NBA slogan is,** This is why we play**. But today I say, **this is why we work**: to develop rich curriculum resources that are fun, relevant, and powerful in teaching kids math.

### Math Millionaire Quiz

It’s hard to believe that *Who Wants to Be a Millionaire* has been on the air since 1999, isn’t it? Even harder to believe is the number of math questions that have been missed by contestants.

In this post, I’m going to share five questions that have appeared on *WWTBAM*, followed by a brief discussion. If you’d like to solve them before reading the discussion, or if you want to share the quiz with friends or students, you can download it:

Three of the five questions were answered incorrectly by contestants. In one case, the contestant polled the audience and received some bad advice. If I hadn’t put this collection together, I’m not sure I would’ve been able to identify which ones were answered correctly. So maybe that’s a bonus question for you: **Which two questions were answered correctly?**

I’m unquestionably biased, but I always feel like the math questions on *WWTBAM* are easier than questions from other disciplines. Then again, maybe a history major would think that questions about Eleanor of Aquitaine are trivial. But take these non-math questions:

- In the children’s book series, where is Paddington Bear originally from? (Wait… he’s not from England?)
- What letter must appear at the beginning of the registration number of all non-military aircraft in the U.S.? (Like most things, it’s obvious — once you know the answer.)
- For ordering his favorite beverages on demand, LBJ had four buttons installed in the Oval Office labeled “coffee,” “tea,” “Coke,” and what? (Hint: the drink wasn’t available when he was Vice President.)

My conjecture is that non-math questions generally have an answer that you either know or don’t know, but math questions can be solved if given enough time to apply some logic and computation.

Perhaps you’ll disagree after attempting these questions.

**1. What is the minimum number of six-packs one would need to buy in order to put “99 bottles of beer on the wall”?**

- 15
- 17
- 19
- 21

**2. Which of these square numbers also happens to be the sum of two smaller square numbers?**

- 16
- 25
- 36
- 49

**3. If a euro is worth $1.50, five euros is worth what?**

- Thirty quarters
- Fifty dimes
- Seventy nickels
- Ninety pennies

**4. How much daylight is there on a day when the sunrise is at 7:14 a.m. and the sunset is at 5:11 p.m.?**

- 9 hours, 3 minutes
- 8 hours, 37 minutes
- 9 hours, 57 minutes
- 8 hours, 7 minutes

**5. In the year she turned 114, the world’s oldest living person, Misao Okawa of Japan, accomplished the rare feat of having lived for how long?**

- 50,000 days
- 10,000 weeks
- 2,000 months
- 1 million hours

**Discussion and Answers**

**1.** Okay, really? Since 16 × 6 = 96, one would need 17 six-packs, **B**.

**2.** This is the one for which the contestant asked the audience. That was a bad move… 50% of the audience chose A, but only 30% chose the correct answer. Since 25 = 9 + 16, and both 9 and 16 are square numbers (9 = 3^{2}, 16 = 4^{2}), the correct answer is **B**.

**3.** It’s pretty easy to calculate $1.50 × 5 = $7.50. The hard part is figuring out which coin combination is also equal to $7.50. Okay, it’s not *that* hard… but it took Patricia Heaton a lifeline and more than 4 minutes.

**4.** My question is whether daylight is officially defined as the time from sunrise to sunset. Apparently, it is. That makes this one rather easy. From 7 a.m. to 5 p.m. is 10 hours, and since 11 and 14 only differ by 3 minutes, we need a time that is 3 minutes less than 10 hours: **C**, final answer.

**5.** Without a doubt, this is the hardest of the five questions. Contestants aren’t allowed to use calculators, so they need to rely on mental math. Estimates will do wonders in this case.

- Days: 114 years × 365.25 days/year ≈ 100 × 400 = 40,000 days
- Weeks: 114 years × 52.18 weeks/year ≈ 120 × 50 = 6,000 weeks
- Months: 114 years × 12 months/year < 120 × 12 = 1,440 months
- Hours: 114 years × 365.25 days × 24 hr/day ≈ 40,000 × 25 = 1,000,000 hours

Only the last result is close enough to be reasonable, so the answer must be **D**.

What’s amusing is that the contestant got the correct answer, but for the wrong reasons. For instance, he estimated the number of weeks to be 50,000, not 5,000. He then used that result to say, “It can’t be 50,000 days, because it’s about 50,000 weeks.” That’s using a false premise to arrive at a correct conclusion. On the other hand, I wonder how well I’d be able to calculate in front of a national audience with $25,000 on the line. Regardless of how he got there, he correctly chose **D**, to which host Terry Crews said, “You took your time on this. You worked it through. It’s what we all need to do in life sometimes. And that’s how you *win the game*!”

Should I ever become a question writer for *Millionaire*, I’d submit the following:

**Which of the following are incorrect answers to this question?**

- B, C, D
- A, B, C
- A, C, D
- A, B, D

### Math of the Rundetaarn

As we were exiting the Rundetaarn (“Round Tower”) in Copenhagen, Denmark, I noticed a man wearing a shirt with the following quotation:

Find what you love, and let it kill you.

The only problem is that the shirt attributed the quotation to poet Charles Bukowski, when apparently it should have been attributed to humorist Kinky Friedman. For what it’s worth, my favorite Friedman quote is, “I just want Texas to be number one in something other than executions, toll roads, and property taxes.” But this ain’t a post about Kinky Friedman, or even Charles Bukowski. So, allow me to pull off the sidewalk and get back on the boulevard.

Whoever said it, the quotation hit me as drastically appropriate. I suspect that **math will someday kill me**… likely as I cross the street while playing KenKen on my phone, oblivious to an oncoming truck. As I exited the Rundetaarn, I was thinking about all the math that I had seen inside — much of which, I suspect, would not have been seen by many of the other tourists.

The Rundetaarn, completed in 1642, is known for the 7.5-turn helical ramp that visitors can walk to the top of the tower and, coincidentally, to one helluva view of the city. That leads to Question #1.

Along the outer wall of the tower, the winding corridor has a length of 210 meters, climbing 3.74 meters per turn.

What is the (inside) diameter of the tower?

Above Trinitatis Church is a gift shop that is accessible from the Rundetaarn’s spiral corridor. The following clock was hanging on the wall in that little shop:

I have no idea who the bust is, but the clock leads to Question #2.

What

sequence of geometric transformationswere required to convert a regular clock into this clock?

And to Question #3.

Do the hands on this clock spin

clockwiseorcounterclockwise?

And to Question #3a.

What is the “error” on the clock?

A privy accessible from the spiral corridor in the Rundetaarn has been preserved like a museum exhibit. Sadly, I have no picture of it to share, but a sign next to the privy implied that the feces deposited by a friar would fall 12 meters into the pit below.

That leads to Question #4.

What is the

terminal velocity of a depositwhen it reaches the bottom of the pit? (Or should that be “turd-minal velocity”?)

The first respondent to correctly answer all of these questions will earn inalienable bragging rights for perpetuity.