A Fruit Roll Up weighs 0.5 oz & is a 12.5 by 11 cm parallelogram.
A Fruit by the Foot weighs 0.75 oz & is rectangular. One dimension of this rectangle is 2.2 cm. What is the other dimension?
(Be sure to state your assumptions, and any other information you draw upon in your solution.)
There are now Cheez-Its BIG. They claim to be “Twice as Big” as ordinary Cheez-Its. One serving of regular Cheez-Its consists of 27 crackers and weighs 30 grams. One serving of BIG Cheez-Its also weighs 30 grams.
(A) How many crackers should one serving of BIG Cheez-Its contain?
(B) How many does it contain?
As promised, more notebook pages on fraction division. This is based on the work I did a while back on trying to write authentic partitive division problems with fractional divisors. (As I wrote that last sentence, I reminded myself what a bizarre niche market I am trying to occupy on this here blog.)
I settled on situations involving fractional values of unit rates, such as the following.
If of a lawn takes of an hour, how much can I mow in one hour?
Before we begin, remember that if the problem were about 2 lawns in 3 hours, we would easily and naturally divide by 3. Only the numbers have changed, so the mathematical structure remains the same and we need to find .
Click each image to see it full size. If you’re into this sort of thing.
I was feeling guilty the other day about spending $7.00 on a gallon of cider. We are at the end of cider season here in Minnesota. It’s a seasonal treat that doesn’t come cheaply.
But we are on a budget (community college salaries only go so far!) and I was feeling bad about the expense-guilty for giving in to the sweet, sweet temptation of delicious fresh cider.
And then my daughter asked for a juice box.
There they were, side by side…a $7.00 gallon of cider and a $2.39 pack of eight 4.23 boxes.
And suddenly I didn’t feel so guilty anymore.
Unit rates can have that effect sometimes.
Here’s another gem from the good folks at Common Core:
If 5/2 and 100/T are viewed as unit rates obtained from the equivalent ratios 5:2 and 100:T, then they must be equivalent fractions because equivalent ratios have the same rates.
This is but one gem in 15 pages of Talmudic exegesis on ratios and proportional reasoning at sixth and seventh grade.
Wu on rates expressed as percents
[W]hen the National Mathematics Advisory Panel reviewed two widely used algebra textbooks to determine their “error density” (which was defined a the number of errors divided by the number of pages in the book), it found that one had an error density of 50 percent and the other was only slightly better at 41 percent. (p. 4)
What I think he is saying
- All rates are expressible as percents. A Toyota Prius’s mileage, for example could be expressed as a “mileage density” of 5100 percent. Or conversely, as a “gasoline density” of 2 percent.
- My argument sounds more impressive if I support it with an abstract measure. Saying, “There are half as many errors as pages,” or even, “On average, every other page contained an error,” is much less impressive than “an error density of 50 percent”.
- American textbooks have too many errors and should be more carefully produced.
I actually agree with this last one.
The error density business is garbage intended to mislead though. The takeaway message for a casual reader is “50% of what’s in a typical math textbook is wrong!” I’m not sure whether it’s just bad writing or intentional deception.
But Wu’s gotta be careful here.
Yesterday’s Daily Wu counted two errors (a mistaken “no more than” when it should be “more than” and a false claim that the only possible number system using 10 digits is our Hindu-Arabic decimal system).
Wu now has a minimum error density of 18%. Good thing he wrote a long article.