Tag Archives: algebra

So much prep work, and I got it (partly) wrong…

I know I messed this up.

My pH videos came out clean and distraction free. I am satisfied with their quality. I spent two hours getting 20 minutes of footage and have edited it down to tidy, manageable packages.

But I know I didn’t get a really important one into the mix.

See if you can guess what’s missing…

I have the pH of water, of 100 ml of water with 1 ml of orange juice concentrate, of 100 ml of water with 10, then 20, then 100 ml of concentrate. And I have the pH of straight concentrate.

So what’s missing? Orange juice made correctly-in a 3:1 ratio of water to concentrate. I have 1:1 and 5:1 but not 3:1.

Now I understand that the pH difference between 5:1, 3:1 and 1:1 is virtually unnoticeable. But that’s not the point. The point is that when I do this lesson on Wednesday I know one of the student questions in the “What can we ask here?” part will be about the pH of properly made juice. And it’s a less compelling answer to have me say “It’s between these other two” than to see it play out on screen (this is the whole point of the effort, after all, right?)

Too late for this time around. Lesson learned.


Sometimes it IS that easy

Both here and elsewhere, I have been part of a lively and robust discussion of the role of the real-world in mathematics instruction. One concern that is often expressed of instruction that is based in real-world problems is that of time and efficiency.

But sometimes it is really easy to turn the tables and make the standard word problem the motivational setup. An example.

In my College Algebra class, I started class with the following image the other day.

from http://www.ally.com 3/14/2011

And I asked What is there to wonder about here? What questions can we ask? Here is what my students came up with.

  • What is APY?
  • What is the difference between Rate and APY? And what does this have to do with interest?
  • How much does the principal grow over time in response to the rate?
  • How does the amount of the deposit affect the growth over time?

These are precisely the questions I would hope for. But they asked them, not me.

In order to get to the answer to the question about the difference between APR and APY, we have to consider this one:

If you offer a 1.54% APR, how should you go about compounding interest monthly?

I have no idea how to get my students to the place where they can ask this question, so I asked it myself. Their ideas:

  1. Figure a year’s worth of interest in dollars, then divide that by 12 and add to the account each month, and
  2. Divide the interest rate by 12. Apply this new interest rate each month.

Brilliant! Next year, I’ll use (1) as part of a homework assignment-how do we run the computations when money gets added to account midway through the year? It gets very tricky very quickly.

But (2) is how we have agreed to run the computations in the financial world.

And it’s what accounts for the difference between APR and APY.

Now they were ready to hear about and derive interest formulas and to do some computations on their own.

A small bit of preparation put my students in a much better frame of mind for the material.

Sometimes it is that easy.

But not always.

All right all you algebra teachers…

So I have this idea.

My College Algebra students don’t get logarithms. I’ve got a bunch of ideas for improving the mathematical flow of our study of logarithms this semester, mostly having to do with lots of “find the exponent” tasks.

And I also want to provoke them to ask some questions of their own about logarithms. No small task, I assure you. Logarithms are abstract beasts.

So I’m thinking about where in the real world one encounters logarithms. The Richter scale and pH are my go-to examples. The Richter scale, while interesting (and of-the-moment), doesn’t suggest to me ways to bring it into the classroom to make things problematic for my students. I’m unsure how to get them wondering about the Richter scale.

So I’m going the pH route.

Here’s the setup: Orange juice concentrate. We have various mixtures of orange juice (which is acidic, so low pH). We will examine the pH of several of these mixtures and the question will be What is the pH of the concentrate?

So today I got some pH strips from our Chemistry department (props to Sara the CLA by the way-very helpful!) and bought some grapefruit juice in the College Store and mixed it up with water in various concentrations, testing each one with my strips.

Surprised by my results, I ran some computations.

So here’s the question for all you algebra teachers. Answer from your gut, not from computations. You have 10 seconds after reading the question to formulate the answer.

I can find it algebraically, and I know you can too, so no need to show off. Commit yourself to an answer.

(Readers who are not algebra teachers will find the answer in the comments section, but not a worked out solution.)

The question

My grapefruit juice has pH=4 and my water has pH=7. I mix them in a 50-50 ratio. What is the pH of the resulting mixture?

Remember, answer from your gut. Then calculate.

stay tuned

The task goes into full production tomorrow. Because of the surprising (to me, anyway) results of my grapefruit experiment, I now know for sure that pH strips are not good enough for my task. I need a pH meter and I’ll need to make special arrangements in the Chem Lab to make it happen.

Update: Special permission in chemistry obtained. Storyboard made. Filming scheduled for Thursday afternoon.

The end of word problems redux

Dan Meyer linked to and quoted from my screed on the end of word problems the other day. This led to some robust discussion on his blog, which I now take the opportunity to reply to.

jg writes:

the lion’s share of the difficulties isn’t the silliness of the problems, no matter how silly they are – it’s the illogical, poorly defined, and trite mental worlds that most folks live in! _That’s_ our challenge!

One of the most useful pieces of educational research comes from the folks at the University of Wisconsin. Cognitively Guided Instruction (CGI, about which I have written a couple of times) demonstrated quite definitively that young children think quite a bit about mathematics-especially early number concepts, and addition and subtraction. What they don’t do is think like adults. This they need to be taught. But it’s not “procedure first then word problems”. It’s “procedure is an abstraction that follows from students’ informal ideas that come from their interaction with the every day world”.

waldo writes:

I tell the kids pure math is strength training for the brain. Just because you never do bicep curls while in a “real sports” situation, it doesn’t mean that doing those curls is irrelevant to your sporting ability.

This is such a tempting analogy but we really don’t have a shred of evidence that it’s so. What I learned from reading Thorndike, the godfather of transfer, is that practice on certain skills makes one better at those certain skills and on nothing else. We have never been able to demonstrate that learning thing A makes you better at thing B. Even so-called “near transfer” is unsubstantiated. What waldo points to is “generalized transfer”-a harder nut to crack and completely unproven to exist.

Now there are lots of other reasons to study pure mathematics. And I have no objection to the idea at all. Mathematics, like poetry, is a beautiful achievement of human intellect for instance. But the cans of peaches problem is not one from pure mathematics. And it’s not from applied mathematics either. It is a word problem. A clever little puzzle, to be sure, but not a useful representation of the subject.

Mark Schwartzkopf writes:

I’m not sure what could be meant by “thinking mathematically”. Translating words into algebraic expressions, without the need to understand the situation fully IS thinking mathematically.

The emphasis was intended to be on thinking, not on mathematically. Recall the Johnson quote was, “students having difficulty can learn basic procedures even if they are unable to reason out a problem.” I was objecting to the author’s satisfaction with getting students to apply procedures without reasoning (thinking).

And I’m not so sure I agree that “translating words into algebraic expressions…IS thinking mathematically.” Especially when the words are written with this translation being the only purpose. If “of” always means “multiply”, am I really thinking mathematically when I rewrite “1/2 of 3/4” as “1/2 x 3/4”?

Schwartzkopf continues:

Before 1500 or so, the science of math was developing at a snails pace. It was extremely hard to think about. So much so that people would have to travel to other countries in order to learn the arcane skills of multiplication and division. At this point, math and algebra texts were pretty much exclusively word problems; algebraic notion had not been invented yet. As the mathematical community began to develop the means of translating word problems into algebraic notation, math became way easier, and began to develop at a faster and faster rate.

It is absolutely the case that algorithms and memorized procedures free up the human mind to tend to other, more important matters. I have no issue with algebraic symbolism, nor with its use in K-12 classrooms. My beef is with curriculum that offers students little of intellectual value and little in the way of honesty about the actual uses of the subject.

A visual launch-probably interesting only to CMP teachers

Connected Mathematicsa 6th through 8th grade mathematics curriculum-is designed with a teaching model in mind: Launch-Explore-Summary. In my professional development work, I help teachers to understand the teaching model in general, and I demonstrate effective techniques for particular problems.

For a few summers now, I have been demonstrating a Launch for a problem in the 8th grade unit Frogs, Fleas and Painted Cubes. I came up with the idea after leaving the middle school classroom, so I have not used it with students. But every summer, teachers ask where they can find written instructions for it and I have had to say that they do not exist.

My summer teaching partner has reported to me that it has been effective in her classroom, so I am now motivated to write it up and share it widely. In what follows, I will assume that the reader has access to the Student and Teacher Editions of the unit. The problem in question is Problem 2.1 of Frogs, Fleas and Painted Cubes.

The original launch

In the original problem, students consider the effects of a land swap in which a square piece of land is swapped for a rectangular one with the same perimeter. So a 5 unit by 5 unit square, might be swapped for a 3 unit by 7 unit one. Students record their information in a table and look for a pattern in the relationship between the areas of the two pieces of land.

In designing my alternate launch, I wanted to achieve two ends: (1) increasing access for visual learners, and (2) increasing the generality of the results.

(1) was important to me because I have been working in my own teaching on visual representation in mathematics, and because the problem asks us to represent something geometric in a table. It seemed like a natural place to increase the visual components of the problem.

(2) is important as a teaching principle. The original problem asks students to consider a variety of squares, but to always change each dimension of the square by 2 units. Thus a 5 by 5 square becomes 3 by 7, and a 6 by 6 square becomes 4 by 8. But there is nothing special about 2 units. A more general pattern emerges if we consider different numbers of units.

The new launch

For this problem, students will work in groups of 3 or 4. Each group receives a bunch of colored (say pink) inch grid paper and a bunch of white inch grid paper. After setting up the context as in the Student and Teacher Editions, have each student draw a square (with whole number side lengths) on a sheet of the white grid paper. Students should coordinate to make sure that each student has a square of a different size from their groupmates.

Each group is assigned a number from 1 to 6, repeating numbers if there are more than 6 groups.

Using the pink grid paper and the group’s assigned number, each student draws a new rectangle. Say my group’s assigned number is 3. Then I will transform my original square (say it was a 5 by 5 square) into a new rectangle by increasing one dimension by and decreasing the other by 3. My new rectangle would be 2 by 8. This preserves the perimeter. What does it do to the area?

To investigate this question, each student tries to cover his/her original white square with his/her pink rectangle. Students will need to cut these pink rectangles apart in order to cover as much as possible of the white square.

Then each group puts their covered squares onto a large poster paper and these are displayed around the room, in order of the assigned numbers (see below).

Now the class is ready to do the mathematical work of investigating the relationship between the assigned number, the size of the original square, and the area of the pink rectangles. Some possible observations that students might make and questions they might pose include:

  • It does not matter what size square we start with; the difference between the area of the original square and the area of the pink rectangle is the same within each group.
  • The group’s assigned number matters-as the assigned number increases, so does the difference between the areas of the original square and the pink rectangle.
  • The area of the pink rectangle is always less than the area of the original square. We can see this because we never quite cover the white with the pink.
  • Several of these images show a white square peeking out from behind the pink. Is it always possible to rearrange the pink so that a white square peeks out from behind?
  • When will it be possible to arrange the pink so that it is in the shape of a square (with whole-number side lengths)?
  • etc.

After some work observing patterns and asking questions, now students should be ready to make the table in the text. However, students should alter the table to match their group’s assigned number, and they should conjecture how the tables of groups with other assigned numbers should look.