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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• 1. Xander  |  June 18, 2018 at 12:10 pm

$3 : 27 :: 4 : 256$, obvs. 😛

• 2. venneblock  |  June 18, 2018 at 12:33 pm

Sorry, X, I don’t accept answers without justification…

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