Question

Dear Mitch,

I am an eighth-grade math teacher at a large public high school in a suburb of Chicago. My question is this: Every Friday I like to give one extended word problem that I have the students work on in small groups, and I am looking for one that will impress upon them the importance of unit changes. Year after year, I have found that they get some of the easiest questions wrong because they forget to do the unit changes properly, such as converting inches to feet, or converting feet to yards. Can you recommend the sort of problem I am looking for?

Sincerely,

Mr. Hammersmith
Teacher of one unit at a time

Answer

Dear Unit teacher,

I think I have just the problem for your class. In fact, it is one of my favorites, so good luck with it!

MRS. FLUTZ's CAKE:

Mr. Flutz read in Hampton Pampton Magazine that the best dessert to serve at a child's birthday party when the child was turning eight was a perfectly cut, one-inch cube of pound cake elegantly placed in the center of a large white plate, one for each child. They could be decorated with a few sprinkles, but this wasn't necessary.

Mrs. Flutz invited twenty-eight children to her son's party, but wasn't sure if all of them would be able to attend. Mrs. Flutz also wondered if some of the children might come with a brother or sister, and she certainly did not want to run out of cake for any child who attended. Finally, of course, she realized that some children might ask for second helpings, or even more. She did, however, decide that she would not offer any dessert to the parents who stayed, because she knew from experience that once the parents become too comfortable, a child's party can go on longer than it needs to.

Mrs. Flutz didn't mind having a little extra food left over after the party, but she did not want to be caught short. She realized that if each child who was invited attended, and each one of those children brought one brother or sister, that would add up to fifty-six servings. Then, if each of the fifty-six asked for seconds, that would be over one hundred servings! And some might ask for more! Mrs. Flutz did not have time to do the math, so, to play it safe, she ordered four cubic yards of pound cake and was careful to cut them into perfect one-inch cubes before the children arrived.

Eleven of the children who were invited came to the party, and three of them brought one brother or one sister. So, all together, there were 14 children at the party. Each child had one serving of the cube-cake, and no one asked for more.

How many cubic inches of pound cake did Mr. Flutz have remaining after the party was over?

Now, the most important part of this question is your estimation.

Please pick one of the following answers as your guess BEFORE you pick up your pencil to work it out:

A) about 100

B) about five hundred

C) about one thousand

D) about fifty thousand

E) about two-hundred-thousand

NOTE: It is important to either print this page out and circle the one you guess BEFORE you use your pencil, or at least write down on a scrap of paper the answer you would select from these choices BEFORE you pick up your pencil.

The answer will probably surprise you!

First, we take a moment to reread the problem, this time to go a level deeper than Mrs. Flutz's deceptively straightforward story offers on its first reading.

And here in this case it comes down to one simple fact: There are several unit changes slipped into the wording. In fact, not only are there probably more than you think, but each one of them matters more than you might at first think.

Okay, now, let's look. The problem is about measurement, yes, but a very special type, VOLUME. What this means is three dimensional stuff, which has to be measured with 3 dimensions. So the cube has to have its height, its width, and its depth measured, each separately, and then multiplied together to include everything inside (h x w x d). We're lucky here, because a 'cube' is the easiest of the volume shapes to measure, because the word cube means that its height is the same as its depth, which is the same as its width.

By the way, you should sense (or at least you will after some experience with this type of problem) that the author of the story thinks that he or she is about to show the reader how much more of a thinker he is than we are. Ha!

First look at the diagram.

Notice we take a cubic yard and draw a few lines on it so we can see it in feet.

QUESTION: Why? Why'd we switch to feet instead of staying with yards or going all the way to inches?

Answer: Because the answer choices are presented in inches (cubic inches), it's often easiest to work our way down from one unit to another one step at a time (one unit-change at a time).

So three feet times three feet times three feet equals 27 cubic feet. So together there are 27 smaller cubes in each cubic yard. (Notice there are 3 square layers stacked up, each with nine cubes).

Next, as in the diagram, we enlarge one of the 27 cubes (it doesn't matter which one we pick, they're all equal!) So, using the same volume formula again, a cubic foot is twelve inches by twelve inches by twelve inches.

Formula: length x width x depth (or 'height', instead of 'depth', depending upon which way you're looking at the object you're measuring)

L x W x D =

12 inches x 12 inches x 12 inches = 144 x 12

 144
 x12
-----
 288
144
-----
1728

And since we have 27 of these blocks in each cubic yard, we have 27 groups of 1,728 cubic inches in each yard (or 1728, 27 times). So we multiply 1728 times 27:

And we get 46,656 cubic inches in each yard.

And recall that Mrs. Flutz was wise enough to buy 4 of these yards. So we have that number 4 times:

46,656 x 4 = 186,624 cubic inches

So, we have 186,624 altogether, minus the 14 dessert cubes which the children gobbled up, and we have 186,610 left over. (And that should give them plenty to enjoy over the weekend!)

-- Mitch