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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The pH of concentrate

introduction

I set this up and critiqued it in this blog before teaching it. The basic idea is that I needed a way into logarithms with my College Algebra students.

The questions

I have two question videos. I debated which to use in class and ultimately chose the first option below.

Option Number 1

In this option, we measure the pH of water and the pH of pure orange juice concentrate.

Option Number 2

In this option, we measure the pH of water and the pH of 100 ml of water mixed with 1 ml of orange juice concentrate.

I imagine that Option 1 will prompt questions such as, “What is the pH of regular orange juice?” and “What if we mix it 50-50?”

I imagine that Option 2 will also prompt the question, “What is the pH of the straight-up concentrate?”

In fact, this last question is the one that motivated this project. I was expecting (as will my students) that the difference between the pH of juice and the pH of concentrate will be quite large.

answers

Juice (20 ml)

This actually doesn’t demonstrate juice (see post where I regret this oversight in my data collection). But it’s pretty close. Real OJ should be 3:1 water to concentrate. This is 5:1.

50-50 (100 ml)

10 ml

1 ml

Pure concentrate

…I got it (partly) wrong, addendum

I lamented the other day that I didn’t capture a question I was sure students would ask about my pH videos.

And today they didn’t ask it.

But they asked other stuff I had not anticipated.

I taught the lesson today. I chose the video in which I measure the pH of water and the pH of pure concentrate. We clarified what we had seen (including the lovely question Did the color of the orange juice concentrate (orange) affect the resulting color of the pH strip (also orange)?)

Then I asked them what there was to ask or wonder about here. Their questions:

1. How are logarithms used in the calculations of pH?
2. How much water do we need to add to the concentrate to get a pH of exactly 7 (in the video, the water measures pH=7.2)
3. What is the concentration of hydrogen ions in the orange juice concentrate?
4. How does the mixing of pH work? Does 1 unit of pH 1 mixed with 1 unit of pH 2 give us 2 units of pH 1.5?

Two of these questions were from students who have come to office hours many times, but who have never piped up with a question or answer in a whole class session. They are always engaged and on task, but never vocal.

I hadn’t thought of adding water to concentrate. I had thought of adding concentrate to water. So I had to tweak question 2 into one closer to something my premade answer videos could answer. To wit:

our question

How much concentrate do I need to add to 100 ml of water in order to make the pH exactly 7? (Recall that the water starts at 7.2)

Student guesses

In pairs, students wrote their answers on slips of paper I had provided, then held them up all at once. Their guesses ranged from 3 ml to 300 ml. The median was approximately 7 ml.

We watched the 10 ml answer video and I asked whether anyone wanted to change their guess. They all did.

We watched the 1 ml answer video. And students revised their guesses again.

Then I restated the definition of pH and demonstrated solving for the hydrogen ion concentration in the orange juice concentrate (question 3, recall).

On Friday, we’ll do some pH computations to verify/approximate the experimental results.

On the way out of class, a student left her final guess on my desk.

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.

Continued…

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.