The Other Golden Ratio

You’re a math geek. I know that to be true, because you’re reading this blog. And I also know that when you hear golden ratio, you think of this:

Or this:

But there’s another, different — and, dare I say, better? — golden ratio that may be even more important to learn. Especially if you’re one of the mathy folks to whom this adage applies:

Wherever you find four mathematicians, you’ll likely find a fifth.

As Homer Simpson says, “It’s funny ’cause it’s true.”

Like many classroom teachers, I’m often ready for a cocktail on Friday afternoons. And like those teachers, I don’t want to spend a lot of time thinking about it; I don’t want to rummage for a recipe; I just want to relax and have a drink. But, I also don’t want to have the same cocktail every Friday; variety is the spice of life.

So, what’s a boy to do?

Simple. Follow the advice from the folks in the Food Hacks division at Wonder How To, who claim that the following ratio will yield a delicious cocktail every time:

2 : 1 : 1 :: alcohol : sour : sweet

Right now, you’re probably scratching your head and thinking, “Can it really be that simple?”

I’m here to tell you, friends — it is.

For instance, 2 parts tequila, 1 part lime juice, and 1 part triple sec? That’ll get you a tasty margarita.

And 2 parts bourbon, 1 part lemon juice, 1 part simple syrup? None other than a classic whiskey sour.

If you combine 2 parts tequila, 1 part rhubarb liqueur, and 1 part malic acid — it’s what gives green apples their tartness — then you’d be getting close to a drink they call The Scarlet Lantern at Bar Congress in Austin, TX. (They also mix in black cardamom-strawberry shrub. Keep Austin weird, eh?)

Now that you know about the other golden ratio, here’s what you need to do: Organize your liquor cabinet into two parts, hard alcohol and sweet mixers. Then, make sure you keep a couple of sour mixers in your fridge. When you get home on Friday, just grab a bottle from each side of the liquor cabinet and one more from the fridge, pour, and — voila — instant happiness.

Wanna get a little crazy? Find your favorite cube-shaped random number generator, give it three rolls, then choose the appropriate item from each column in the table below.

Die Roll	Alcohol	Sour	Sweet
1	Tequila	Lemon Juice	Simple Syrup
2	Vodka	Lime Juice	Triple Sec
3	Rum	Grapefruit Juice	Cointreau
4	Rye Whiskey	Strawberry Shrub	Gran Marnier
5	Bourbon	Bloody Mary Mix	Honey-Ginger Syrup
6	Mezcal	Dry Cider	Grenadine

Personally, I’m hoping for 6-3-5, which is kind of like a Mezcal Paloma, sort of like a Honey and Smoke, but not really similar enough to be either one. So, I guess I get to name it. And given what I’ve heard about the fat-burning properties of honey and grapefruit juice, I’m going to call this newly minted beverage the Weight Watcher.

Now we just need to come up with names for the other 215 combinations. I’ll get started on that right away… soon as I finish this drink.

How Would You Answer These Questions?

The following is one of my all-time favorite assessment items:

Which of the following is the best approximation for the volume of an ordinary chicken egg? 0.7 cm3 7 cm3 70 cm3 700 cm3

The reason it’s one of my favorites is simple: it made me think. Upon first look, I didn’t immediately know the answer, nor did I even know what problem-solving strategy I should use to attack it.

I used estimation, first assuming that the egg was spherical — and, no, that is not the start of a math joke — and then by attempting to inscribe the egg in a rectangular prism. Both of those methods gave different answers, though, and not just numerically; each led to a different letter choice from above.

Not satisfied, I then borrowed a method from Thomas Edison — I filled a measuring cup with 200 mL of water, retrieved an ordinary egg from my refrigerator, and dropped it into the cup. The water level rose by 36 mL. This proved unsatisfying, however, because although choice B is numerically closer to this estimate than choice C — only 29 mL less, compared to 34 mL more — it was five times as much as B but only half as much as C. For determining which is closer, should I use the difference or the ratio?

It was at this point that I decided the answer doesn’t matter. I had been doing some really fun math and employing lots of grey matter. I was thinking outside the box, except when I attempted to inscribed the egg in a rectangular prism and was literally thinking inside the box. And, I was having fun. What more could a boy ask for?

On a different note, here’s one of the worst assessment questions I’ve ever seen:

What is the value of x?

3 : 27 :: 4 : x

I can’t remember if it was a selected-response (nee, multiple-choice) item, or if was a constructed-response question. Either way, it has issues, because there are multiple possible values of x that could be justified.

On the other hand, it’s a great question for the classroom, because students can select a variety of correct responses, as long as they can justify their answer.

The intended answer, I’m fairly certain, is x = 36. The analogy is meant as a proportion, and 3/27 = 4/36. (Wolfram Alpha agrees with this solution.)

But given the format, it could be read as “3 is to 27 as 4 is to x,” which leaves room for interpretation. Because 27 = 3³, then perhaps the correct answer is 64 = 4³.

Or perhaps the answer is x = 28, because 3 + 24 = 27, and 4 + 24 = 28.

Don’t like those alternate answers? Consider the following from Math Analogies, Level 1, a software package from The Critical Thinking Company that was reviewed at One Mama’s Journey.

If this analogy represents a proportion, then the correct answer is $10.50, but that’s not one of the choices. Instead, the analogy represents the rule “add $1,” and the intended answer choice is $10.00.

What amazing assessment items have you seen, of either the good or bad variety?

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.

Russell Westbrook

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.

Oscar 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

Oreos, Ratios, and the Perfect Cookie

Okay, first things first. What do you call the following shape?

I call it a pill. My sons call it a racetrack. But is there a formal name for a shape formed by a rectangle with a semicircle attached to each end? If not, I feel like there should be. Place your suggestions in the Comments.

Until I hear a better suggestion, I’m gonna keep callin’ it a pill.

The following trivia question is the reason I ask.

How many pills appear around the circumference of the trademarked design on an Oreo^® cookie?

What’s that, you say? You didn’t know that there were little pills along the edge of each wafer on an Oreo cookie? Then you, my friend, need to pay a little more attention.

Because there aren’t just some pills around the circumference. There are 96 of those little buggers, and each of them has a rectangle with a length-to-width ratio of approximately 3:2 between two semicircles.

See for yourself.

Ironically, the ratio of 3:2 brings me to the main reason I’m writing today.

The original Oreo represented good design: a single layer of vanilla cream filling trapped between two crisp, chocolate wafers. But it always felt lacking to me. If only it had just a little more cream, then it would be perfect. A potential solution arrived in 1974, when Nabisco released the Double Stuf variety — two chocolate wafers with twice as much filling¹ as its predecessor. Yet the Double Stuf teetered too far in the opposite direction. It was too sweet.

Which brings me to the delectable treat that I discovered today: the Triple Double Oreo, which is running a strong campaign for the title of World’s Best Cookie. My wife describes it as “the Big Mac of cookies.” Not two but three chocolate wafers with a thin layer of cream filling between each pair. And the pièce de résistance — one layer of vanilla cream filling, the other chocolate.

Now that’s what I call intelligent design.

It absolutely nails the ratio for wafer:filling.

	Original	Double Stuf	Triple Double
Wafers	2	2	3
Filling2	1	1.86	2
Ratio of W:F	2	1.08	1.5

The chart above makes it all clear. The ratio is too high in the original, too low in the Double Stuf, and just right in the Triple Double. Indeed, the Triple Double Oreo is the Little Bear’s porridge of the cookie world.

This reminds me of Reese’s Peanut Butter Cups. While I haven’t quantitatively analyzed the peanut butter to chocolate ratio, qualitatively I would say that the original had a little too much peanut butter, the Big Cup was disgusting with far too much peanut butter, but nirvana was captured with the peanut butter to chocolate ratio in Miniature Reese’s Peanut Butter Cups. (Insert smacking lips sound here.)

So if you read this blog and wonder why I’m so hyper sometimes, now you know. I consume an unholy amount of refined sugar.

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¹ There is some debate about the actual amount of filling contained in a Double Stuf Oreo cookie. Although a spokesperson for Nabisco claimed that the cookies indeed contain twice as much filling as a regular Oreo, a math class in upstate New York experimentally found that Double Stuf cookies contain only 1.86 times as much cream filling as a regular Oreo. As I generally trust unpaid high school students more than money-grubbing corporate types, I’m using 1.86.

² The numbers for “filling” are relative to the amount of cream filling in a regular Oreo^®.

Ratio Celebration

While flipping through the Big Book of Zany Activities from Kidsbook^®, my sons came across the following challenge:

 Can you make 25 or more words from the following word?
CELEBRATION

Alex put pencil to paper immediately and beamed as he wrote the word ratio.

It made me wonder if I could find 25 or more math words within CELEBRATION. Sure enough, I could. My list of 42 words is at the bottom of this post, though admittedly, a rather liberal view of what constitutes a math word is required, and there are even a couple of proper nouns on the list. But even removing the ones that were iffy, I think I still met the requisite number. 

Speaking of ratio, you might be interested in the discussion of the No Slope… Ratio! joke. Though probably not, so here are some other ratio jokes.

A trainer at the gym was asked, “How can I calculate my body fat ratio?”
The trainer responded, “Well, if you have a body and you have fat, the ratio is 1:1. If you have two bodies, the ratio is 2:1. And so on.”

Yo momma is so fat, her ratio of circumference to diameter is 4!

Math Words Within CELEBRATION:

1. Abel
2. Ace
3. Acre
4. Arc
5. Bar
6. Bi
7. Bin
8. Bit
9. Caliber
10. Cantor
11. Cent
12. Center
13. Coil
14. Coin
15. Cone
16. Election
17. Eon
18. Era
19. Lie
20. Line
21. Linear
22. Loci
23. Lone
24. Once
25. One
26. Orb
27. Orbit
28. Net
29. Nil
30. Rate
31. Real
32. Recent
33. Table
34. Tail
35. Tare
36. Teen
37. Ten
38. Tic
39. Tie
40. Tile
41. Tree
42. Trice