Diagrams, week 7

We are studying logarithms in College Algebra.

We began with fact families. Many reps; some started with the exponential fact, some with the log fact, some with the root fact.

We expressed our heights as logarithms, base 10, of our heights in inches, and we brought a tall and a short student to the front of the classroom to stand next to each other. Given the log height of the tall student, we predicted the log height of the short one and were surprised to find that the difference was less than one-tenth of a log-unit (whatever that is).

We came to terms with this difference by noticing that the tall student, while much taller than the short student, was not exponentially taller. The tall student wasn’t even twice as tall-never mind ten times as tall.

I gave the definition of pH as the negative log of the hydrogen ion concentration. We considered the consequences of this goofy definition, and what it says about possible values on the pH scale. We puzzled over why pH stops at 14, when there should be no theoretical limit on the upper value of a negative logarithm of a positive ratio, and we wondered why 7 is considered neutral.

Then we watched the video in which I measure the pH of orange juice concentrate and of water (with an imperfectly calibrated pH meter-so don’t give me grief about the precise values involved here). I reminded students of our height-measuring surprise and summarized our knowledge of the acidity of water and of orange juice with the diagram below:

Finally, I asked them to predict the pH of a 50/50 mixture of water and orange juice concentrate.

Nearly all of them averaged the two pH values (of course), getting 5.7 or some adjusted value nearby.

We watched that video and saw that the result is nearly indistinguishable from the original pH of the orange juice concentrate.

We were surprised.

But someone noticed that the difference of roughly 3 in the pH values means that the hydrogen ion concentration is 10,000 times as much in the orange juice concentrate as it is in the water.

Just like our tall and short students had a big difference in height, but small difference in log heights, our two substances must have an enormous difference in hydrogen ion concentration to account for the sizable difference in pH values.

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Can’t get enough of logs!

You know what fact families are? In elementary and middle school, this is increasingly common terminology to express relationships among basic number facts. A sample fact family:

 

Fact family thinking is intended to encourage connections between operations and their inverses. Addition and subtraction fact families work the same way as multiplication and division fact families:

 

I have written a lot about logarithms this past year and I keep wondering about ways to get students’ minds wrapped around the relationships that logarithms are intended to express, and to focus less on the notation.

Here is a fact family for an exponential relationship.

Some things I notice:

1. We’re not used to thinking about roots, logs and exponentials all at the same time. Maybe we should be.
2. Exponentiation is not commutative, so we only have three facts in the family, not four. That seems useful to know.
3. Off the top of my head, I can’t think of another non-commutative binary operation (besides those listed out above), so I can’t test the proposition that:
4. All non-commutative binary operations will necessarily have TWO inverse operations in their fact families.