Friday, April 8, 2016

Activity: The Three-Body Problem (Day 136)

Today I subbed in a seventh grade math class. So naturally I have several things to say in this post about my day as a sub.

First of all, notice that I've discussed the Glencoe California Math texts for sixth and eighth graders so far, but never the seventh grade text (though ironically I posted about over seventh grade texts.) So here is the listing for the seventh grade text.

1. Ratios and Proportional Relationships
2. Percents
3. Integers
4. Rational Numbers
5. Expressions
6. Equations and Inequalities
7. Geometric Figures
8. Measure Figures
9. Probability
10. Statistics

If I were teaching this class, it would be easy for me to make this into a pacing plan. With ten chapters, I would shoot for one chapter per month. If Chapter 1 is covered in August -- especially if it's an Early Start school -- then Chapter 10 would be finished in May. This also has the advantage of placing Chapter 8, on Measure Figures, in March. Seventh grade is when students first learn about pi, so why not teach it in the month that includes Pi Day?

The one tricky thing is that the volume break that most modern texts have is between Chapters 4 and 5, rather than the more natural 5 and 6. This is because every two chapters make up a unit. Unit 3, consisting of Chapters 5 and 6, is our Pre-Algebra unit, and so the text authors don't wish to place the semester break during that unit. But waiting until second semester to start Chapter 5 makes it more difficult to cover Chapter 8 by Pi Day and Chapter 10 by the SBAC.

Anyway the students ought to be around Chapter 9 by now. So which chapter were the students really covering today? It was Chapter Two, on percents -- and furthermore, it appears that they just barely started the chapter, since they were still turning in Chapter One Test corrections! The class didn't appear to be a special ed class (though one period did have an instructional aide), so I can only imagine what the class covered in the first semester, to finish Chapter 1 not until March and Chapter 2 not until April.

Second period was "math lab" class, and the proper math classes were periods 3-6. As often happens, the teacher leaves a stack of Pizzazz worksheets for the students to complete. On one side of the worksheet, the students convert fractions into percents. The first five problems are:

1. 1/2
2. 1/5
3. 1/10
4. 1/8
5. 1/3

On the other side, the students calculate percentages of fixed amounts. The first five problems are:

1. 21% of 68
2. 85% of 36
3. 8% of 144
4. 3% of 720
5. 2.5% of 55

The teacher suggests that I demonstrate the first 3-4 questions from each side of the Pizzazz worksheet on the board, while the students write them down on a separate sheet of paper before passing out the actual worksheet. Then after I check for understanding, only then do I pass out the worksheets for guided practice. The main problem is that the seventh graders don't take the assignment seriously until they see a real worksheet.

I keep modifying the lesson throughout the day until I find a way to get through to the students. By the time I reach sixth period, here's what I come up with: I play a special version of my game. After the "guess my age" and "guess my weight" questions, the third question is the first from the worksheet, to convert 1/2 into a percent. This isn't a true-false question, but it's fairly easy -- the only reasons any group would get it wrong is that they converted to the decimal 0.5 instead of a percent, and otherwise they are simply not paying attention (or misbehaving). So it serves the same purpose as my usual true-false question.

The next two questions, 1/5 and 1/10, are given in race format. I would have stopped right there, except that I knew that 1/8 would be tricky for the students, and 1/3 still trickier. So I continue to ask these questions in race format, to see who already understands them.

Then it's time for guided practice. I differentiate by letting the groups who are ahead in points work on their own, while I walk up to those groups who are behind to figure out why. If they are having trouble with the conversion, I show them how to divide to find the answer.

The teacher has informed me that the students will have trouble with the 21% of 68 side. So I show the students how to do this problem and make everyone copy the answer -- this serves as the second true-false question in the game. Then the next two questions are given in race format. Afterward, I return to guided practice and have the students work on this side on their own. I continue to award points to those groups who answer questions on this more difficult side.

By the time I implement this format in sixth period, many of the groups are motivated to earn the extra points. But I feel that I don't spend enough time helping out the struggling groups -- ironically, I believe I've helped the weaker students better in the earlier periods than in sixth (especially fourth, when the aide is able to assist several students).

In fact, in many ways, today's class is a perfect example of why I lean more towards the progressive side and away from the traditionalists in the higher grades, especially as a sub:

-- Traditionalists say that students learn better as individuals, yet I always play a group game. This is because it's easier for me to keep track of a dozen groups of three or nine groups of four than 36 individuals whose names I don't know. I would have taught an individual lesson today if I could count on the students to behave.
-- Traditionalists say that students learn better when they take notes from a lecture, yet I end up avoiding notes today. This is because students don't take notes seriously, especially since I am only a sub in the class. I would have had the students take notes today if I could count on them to behave.
-- Traditionalists say that students learn more when they are given new material, not spending weeks on review, whether it's at the beginning of the year or just before Common Core testing, or reviewing the wrong answers on a previously taken test. I would present more new material if I could count on the students to remember the old material.
-- I even deviate from the teacher's lesson plan if doing so is more likely to maintain order. Today's teacher requests that I collect several papers (including the aforementioned test corrections) before starting the lesson. I would have done so if I could count on the students to behave while doing so.

And in many cases, I tried to follow the lesson plan to the letter in third period, only to have the students fool around loudly the whole time, including making airplanes out of the papers they were supposed to be taking notes on. By sixth period I deviate from the lesson plan, and the students work harder during this period, even though still a bit on the loud side.

Today is an activity day. Since the last lesson I covered here on the blog is Lesson 13-5, I will give an activity based on that lesson.

Notice that Lesson 13-5 is called "Tangents to Circles and Spheres," but so far, we've seen many circles but very few spheres. Well, this next set of questions, which comes from the Exploration section of Lesson 13-5, refer to three very large spheres:

23. The drawing of the earth, sun, and moon in this lesson is nowhere near scale.
a. Draw a figure of the earth and the moon to scale using a tracing of the circle drawn at the left as the earth.
b. How far away would the sun be on the scale of your drawing and what would be its diameter?

24. Draw the relative positions of the moon, earth, and sun during an eclipse of the moon.

25. What is an annular eclipse?

What do any of these questions have to do with tangents, you may ask? Well, one of the examples of tangents in the text involves the three bodies -- the earth, sun, and moon. In particular, we consider a solar eclipse, when the moon and sun appear from the earth to be the same size. Then the common tangents to the moon and sun come very close to intersecting on the earth. The text writes:

If you ignore the earth in the drawing, then the drawing looks somewhat like a sphere and its size-change [dilation -- dw] image. This means that there are proportions. These proportions can be used to calculate the radius of the sun.

Example: It is known that the moon is about 240,000 miles from the earth, the sun about 93,000,000 miles from the earth, and the moon's radius is 1080 miles. Estimate the radius of the sun.

Solution: First draw a picture. Here M and S are the centers of the sun and moon, N and T are points of tangency of a common tangent. The distances EM = 240,000, ES = 93,000,000, and MN = 1080 are known. ST is the radius of the sun.

Because MN and ST are radii, they are perpendicular to line ET. Since angle E is common to triangles EMN and EST, by AA Similarity, triangle EMN ~ EST. And so,


Substituting for these lengths,

240000/93000000 = 1080/ST.

Simplify the fraction at the left to 24/9300 before using your calculator! Solving this proportion, ST = 418,500, quite close to the actual value of about 432,000 miles.

These are the values we must use to solve the three problems in this activity:

23. The circle given in the text as representing the earth has a radius of about 0.3 inches. Using this value, here are the other distances relevant to the problem:

a. EM (Earth-moon) = about 18", MN (moon's radius) = about 0.08" (or 2 mm)
b. ES (Earth-sun) = about 7045" (or 587'), ST (sun's radius) = about 32.7". The question actually asks for the sun's diameter, so that's about 65.4" (or 5' 5.4").

If we want, we can simplify this problem just by saying the radius of the earth is 0.396 inches rather than 0.3 inches. Then EM is now 24", MN is now 0.108", and so on.

24. During a lunar eclipse, the earth is now between the moon and sun. Here is an image that I found on the web, from "Mr. Eclipse."

25. An annular eclipse occurs when the sun appears to be an "annulus," or ringHere is an image that I found on the web, also from "Mr. Eclipse." As he explains, the orbit of the moon around the earth is not a perfect circle, but is an ellipse instead. So EM is not a constant 240,000 miles, but could be as great as 252,000 miles. Then the comment tangent lines of the moon and sun intersect outside the earth rather than inside. So the moon isn't large enough to cover all of the sun.

As usual, I can't see this activity being too popular with traditionalists. There is too much drawing and not enough calculating for their taste -- and besides, this activity ends up being more about similarity and scale factors than the properties of tangent lines. But again, I would avoid activities if I could count on the students to work hard, especially at the beginning or end of the week.

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