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?

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Anticipating student struggles…

UPDATE: Results from the class described have been added at the end of the post.

Putting my money where my mouth is tonight.

I have had some saucy things to say about our adopted Calc text. In particular, that it has not been written from a perspective that is empathetic with students’ states of mind.

So tonight we’re proving the area of a circle formula as an introduction to trigonometric substitution. There’s a rabbit-out-of-the-hat moment in which the substitution in question gets introduced:

x=r•cos (theta)

I anticipate that this will be a challenging point for students. So we’ll stop there and collect responses to the question, What does this make you wonder, question or think about?

Here’s what I anticipate they’ll say:

• Why r?
• Why cosine (instead of, say, sine)?
• Why theta?
• Is this like u-substitution?

As always, I am prepared to be schooled.

I know they’ll have ideas I never thought of. Can’t wait to see what they are.

Post mortem

Sue writes in the comments:

I don’t start with the strange substitution. I start with a triangle. For me, the triangle I draw explains the substitution I decide to use. Then it’s all about reasoning, instead of memorizing (?) 3 very strange substitutions.

This is spot on.

Students asked about the theta, they asked about the r. They didn’t make any connection to u-substitution. Lots of them worked on triangles. Students thought about vectors in physics and they thought about the unit circle.

In contrast with Sue’s suggestion, and in contrast with where my students’ minds were, our text talks about this:

In general, we can make a substitution of the form x=g(t) by using the Substitution Rule in reverse. To make our calculations simpler, we assume that g has an inverse function; that is, g is one-to-one…This kind of substitution is called inverse substitution.

And the triangle Sue suggests? That comes in at the end of the technique as a tool for changing variables back from theta to x. So we finish with where students’ minds are, but we start with an abstract, backwards version of something they’re not even thinking about to begin with.

And we wonder why students don’t become better problem solvers in their college courses, and why they don’t develop the kinds of critical thinking skills we would like them to. And why students don’t read the textbook. They don’t read the textbook because it’s not speaking to them.

Questions from middle schoolers IX: Length

Can you have a negative length?

Preliminary confidential to the KMS middle schoolers who asked this question: I am continually impressed by your active minds; it was a delight to meet you and be your teacher for a day recently. And you should be very grateful for the teacher you get to have every day. Ms. O cares deeply about you, about keeping your minds active and about your success. Thank her for that now and again in 10 years when you realize the impact she has had on your lives.

Back to the question: Technically, no. “Length” is always positive. As is “Area”. As is “Speed”.

But if you are willing to expand your mind and imagine a world with negative lengths, you don’t cause any problems. But you will have to be ready to consider negative areas. And negative speeds.

In Physics, they deal with this by the difference between speed and velocity. Velocity can be negative, as it includes direction. Forward velocity is positive; backward velocity is negative. Speed is the absolute value of velocity. Whether my velocity is 10 mph or -10 mph; whether I am running forward or backward, my speed is 10 mph (positive).

And this requires a negative version of distance. In Physics, this is displacement. Displacements can be positive or negative. Distance is the absolute value of displacement.

In Calculus, it can be useful to imagine a world in which area (and length) can be negative. It’s not usually encouraged, but it causes no problems and in fact can make some concepts more intuitive.

How can negative areas and lengths be intuitive? Well, you’re an eighth grader and you thought of them-shouldn’t a college student be able to also?