Category Archives: Problems (math)

Ginger ale (also abbreviated list of Standards for Mathematical Practice)

We have some of these mini cans of ginger ale in the house this week. I am not sure where they came from; only that my wife bought them. Normally we only have sparkling water around, not pop (nor soda, nor…)

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So I’m looking at the can instead of grading like I should be and I notice the “25% fewer calories than regular ginger ales” claim.

And I think what any skeptical consumer ought to think. Sure fewer calories in the mini can. Duh.

Then I see this:

ginger.ale.2They have controlled for the size of the can. Nice. This one has 60 calories per 7.5 fl. oz. Regular ginger ales have 90 calories per 7.5 fl. oz.

I am briefly satisfied. And impressed.

But wait! 60 is 25% less than 90? ARGH!

Two possible explanations:

  1. 25% means at least 25%, and Seagram’s chose this nice simple number over the more complicated 33\frac{1}{3}%.
  2. It really is exactly 25%. But we know that calorie counts are rounded to the nearest 10 calories.

This second explanation leads to a sort of lovely task. How can we characterize the set of possible calorie counts for 7.5 fl. oz. of Seagram’s and of regular ginger ale so that, (a) the counts round to 60 and 90, and (b) one number is exactly 25% less than the other?

Extra credit: Which standards for mathematical practice are you using as you solve?

Double extra credit: Which of my abbreviated list of standards for mathematical practice (see below) are you using as you solve? And which was I using as I gazed at my can of ginger ale?

Prof. Triangleman’s Abbreviated List of Standards for Mathematical Practice.

PTALSMP 1: Ask questions. Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

PTALSMP 2: Play. See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

PLALSMP 3: Argue. Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

See also my Desmos graph of this relationship.

Betty’s Pies

In Two Harbors, on Minnesota’s North Shore of Lake Superior is a popular stopping point for vacationers: Betty’s Pies.

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This restaurant moves a tremendous amount of pie each year. How much? Well, I’m not really sure because there was a smudge on the fact sheet on my menu.

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Middle of the image: “Our bakers make…” (Please ignore the bold, oversized italic typo in the last line)

So how about it? How many pies do you think Betty’s Pies makes each year? Let me know how you think about it in the comments, OK?

Then you can click here to see the rest of Betty’s sign. Is it a good representation of how she cuts her pies?

Finally, you can view an unsmudged version of the menu to check your answer. (No cheating!)

Some more grocery store math

Problem 1

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.)

Problem 2

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?

Partitive fraction division

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 \frac{2}{3} of a lawn takes \frac{3}{4} 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 \frac{2}{3} \div \frac{3}{4}.

Click each image to see it full size. If you’re into this sort of thing.

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Common numerator fraction division [#algorithmchat]

My future elementary teachers explore the common denominator fraction division algorithm at the end of the semester. Reading their work got me thinking about common numerator fraction division, and about what sense I could make of the symbols that result.

I tried to keep my work neat so others could follow it. If this sort of thing amuses you (as it obviously does me), then you’ll want to take a few minutes with the larger versions of these images. If it does not amuse (and I cannot begrudge anyone this), then you’ll just want to move along; there’s nothing here for you today.

Page 1, in which I interpret the complex fraction that results from dividing across the fractions.

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