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?

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Fractional Eggs

I search for new recipes at allrecipes.com all the time. This morning, a search yielded a delicious recipe for pumpkin pancakes, which sounded like the perfect breakfast for a crisp fall morning.

One of the things I love about allrecipes is the ability to customize the number of servings. The default number of servings for the pumpkin pancake recipe was six, but I could adjust it to four, a more appropriate number for our two-adult, two-child family:

So I did. And as you’d expect, each item in the ingredient list was reduced to ⅔ its previous amount. Sort of. Two cups of flour was reduced to 1⅓ cups. One cup of pumpkin puree was reduced to ⅔ cup. But 2 teaspoons of baking powder was reduced to 1¼ teaspoons, and 1 teaspoon of cinnamon was reduced to ¾ teaspoon.

The reduction in the number of servings was 33⅓%, yet the range of reductions in the ingredients varied from 25% for salt (from 1 teaspoon to ¾ teaspoon) to 50% for ground ginger (from ½ teaspoon to ¼ teaspoon).

But I get it. It’s not typical for most kitchens to contain a spoon that measures ⅙ teaspoon. So there’s clearly some part of the algorithm that completes the conversion but then finds a “nice” fraction that’s in the right neighborhood. Fair enough.

But what the hell’s going on here?

Is it really better to display ⅝ egg instead of ⅔ egg? Couldn’t the algorithm recognize that fractional eggs just aren’t all that common and leave it as a whole number?

My guess is that the programmer is one of the folks to which this statement alludes:

5 out of 4 people aren’t very good with fractions.

That joke represents one-fifth of my favorite fraction jokes. Here are the other four:

Why won’t fractions marry decimals?
They don’t want to convert.

I’m right 4/5 of the time. Who cares about the other 10%?

There’s a fine line between a numerator and a denominator.

Sex is like fractions. It’s improper for the larger one to be on top.

If you find a store that sells ⅝ egg, please let us know about it in the comments.

Teasing Out Some Math Jokes

Let’s do a quick warm-up before jumping into this post.

A boy leaves his house headed for school, walking at a rate of 4 miles per hour. Ten minutes later, his sister leaves the school headed for home, walking at a rate of 3 miles per hour. Assuming they travel the same route (just in opposite directions), which one will be nearer the school when they meet?

This problem is based on the first puzzle that appears in Mathematical Teasers by Julio A. Mira. As best I can tell, this book is out of print, and probably for good reason. Written in 1970, it contains no semblance of political correctness. For instance, the image below appears at the beginning of Chapter 1:

Click to Enlarge Image

Hmm… a pigtailed coed in a skirt sitting on a desk, tickling the chin of a math professor? There is no doubt an editor in 2013 would prohibit such an image from appearing in a publication; quite honestly, I’m even surprised an editor allowed it 43 years ago.

My copy of this book was obtained when the National Council of Teachers of Mathematics cleaned out their staff library. Inside the front cover, an insert states that it was a review copy “sent with the compliments of Barnes and Noble, Inc.,” and a stamp on the insert reads:

RECEIVED
MAY 26 1970
NCTM

Despite its political incorrectness and outdated contexts, my five-year-old sons have been enjoying the puzzles in the book. But they aren’t just puzzles. Like the warm-up problem above, they are jokes, in the sense that the punch line (answer) is unexpected. The following are a few of my favorites.

1. A man with $50 in a bank account withdraws $20, leaving $30. He then withdraws $15, leaving $15. Then $9, leaving $6. And finally $6, leaving $0. The sum of his withdrawals is 20 + 15 + 9 + 6 = $50, as expected, but the sum of the remainders is 30 + 15 + 6 + 0 = $51. Where did the extra dollar come from?
2. If it takes 3 minutes to boil an egg, how long will take to boil a dozen eggs?
3. How many cubic inches of dirt are in a hole that measures 1 ft. × 1 ft. × 1 ft.?
4. A man purchased a pair of shoes that cost $25 and gave the shop owner a $100 bill. After the man left with the shoes and his change, the owner took the $100 bill to the bank, where he was told that it was counterfeit. What was the total loss to the owner?
5. Every day, Johnson’s cat would climb 11 feet higher in a tree that is 63 feet tall. But every night, the cat would climb back down 7 feet. How many days would it take her to reach the top of the tree?

No answers will be posted. Y’all can attempt to reach consensus in the Comments section.

Most Eggs-Cellent Math Jokes

You may have received the following advice about dividing fractions without thinking about it:

Ours is not to reason why; just invert and multiply.

Similarly, don’t waste your time trying to figure out why I’m posting a bunch of jokes about chickens and eggs. I can’t explain it. Just enjoy them, and please don’t analyze me.

How do you teach math to a chicken?
Show it lots of egg samples!

Why do chickens hate school?
They don’t like eggs-aminations!

Who tells the best math jokes on the farm?
Comedi-hens!

How can you drop an egg six feet without breaking it?
Drop it from seven feet!

Why did the chicken go to school?
To get an egg-ucation!

Why do chicken coops have only two doors?
Because if they had four doors, they’d be sedans!

And a joke about the smartest chicken I know…

A chicken walks into a bar. “I’d like a burger and a beer,” he says to the bartender.

“Oh, my God!” the bartender says. “You can talk!”

“Well, look at that,” the chicken replies. “Your ears work!”

“But, you’re a chicken!” the bartender says.

“Ah, I see your eyes work, too,” the chicken says. “Now, can I have my burger and beer?”

“Certainly,” the bartender says. “Sorry about that. It’s just not every day that I see a talking chicken. What are you doing around here?”

“I’m working at the university,” the chicken says. He goes on to explain that he’s helping a professor with research on representation theory and integrable systems, but the bartender clearly has no idea what he’s talking about. So, the chicken enjoys his burger and beer and leaves.

A little while later, the owner of the circus comes into the bar. The bartender says, “You’re the owner of the circus, right? Well, have I got an act for you! I know this chicken who talks, reads, and drinks beer!”

“Sounds great!” says the circus owner. “Have him give me a call.”

The next day, the chicken returns to the bar. The bartender explains that he thinks he can get the chicken a great job at the circus.

“The circus?” asks the chicken. “You mean the place with the big tent, animals, lion tamers and trapeze artists?”

“Yeah!” says the bartender. “The owner would love to hire you!”

“Why?” asks the chicken. “What use would he have for an algebraist?”