Wednesday, February 14, 2018

Lesson 10-9: The Surface Area of a Sphere (Day 109)

Lesson 10-9 of the U of Chicago text is called "The Surface Area of a Sphere." In the modern Third Edition of the text, the surface area of a sphere appears in Lesson 10-7.

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Find the cylinder's volume, if the volume of the cone is 4 2/3 cm^2.

(The right cylinder and cone have the same base and height, the latter of which is 2 cm.)

Well, this question fits the current Chapter 10 of the U of Chicago text like a glove. Lessons 10-5 through 10-7 of text provide us the formulas we need to answer this question:

V_cylinder = Bh
V_cone = (1/3)Bh

First of all, notice that Pappas gives the volume of the cone in square centimeters. Let's assume that this is a typo, and so we rewrite the question as:

Find the cylinder's volume, if the volume of the cone is 4 2/3 cm^3.

Now we know that the common height of both figures is 2 cm, so we could use the cone volume formula to find the area of the base -- but why should we? We see that Bh is the same for both figures, so we can think of this as:

V_cone = (1/3)V_cylinder

We substitute in V_cone = 4 2/3 and solve:

4 2/3 = (1/3)V_cylinder
14 = V_cylinder

This reminds us of Exploration Question 22 in Lesson 10-7, which directs the students to fill a cylinder with three cones full of dirt or sand. It doesn't matter what the actual height of the cone is -- all that matters is that they have the same base and height. Even the fact that it's a right cylinder is irrelevant for this problem. Unfortunately, we didn't reach Question 22 because I linked to an activity page based on Question 23 instead.

We wouldn't want students to waste time calculating the area of the base. Yes, we can easily calculate it to be 7 cm^2. But our students might make a mistake, especially with the fractions, and obtain something other than 7 cm^2 for the base and hence something other than 14 cm^3 for the volume. It would be even worse for the students to attempt to find the radius of the base, especially if they think of the cone volume formula as (1/3) pi r^2 h. This would introduce rounding errors, both in dividing 7 by pi and by taking the square root of this. Students should perform as few calculations as possible, multiplying (4 2/3) by 3 to get 14 cm^3. So the answer is 14 -- and of course, today's date is the 14th.

That's right -- today is February 14th, Valentine's Day. But today is also Ash Wednesday. This is the first time since 1945 that Valentine's Day and Ash Wednesday fell on the same date.

Recall that some of the more obscure holidays like Ash Wednesday are important in churches where it's observed as a fasting day. Catholics, for example, are supposed to eat only one large meal and two small meals today -- with no red meat, of course. This might make it difficult to celebrate Valentine's Day, when couples might want to eat a steak dinner and lots of candy. On the other hand, Ash Wednesday is largely ignored by Protestants. Again, you're more likely to pay attention to the liturgical calendar when it controls what you can and cannot eat.

It turns out that in years like 1945 and 2018 when Valentine's Day and Ash Wednesday coincide, another secular-religious holiday pair falls on the same date -- April Fool's Day and Easter. (Oops -- I mentioned religion in this school year post. To be safe, let's add a "Calendar" label, since I allow myself to mention religion in Calendar-labeled posts.)

OK, let's get back to the Lesson 10-9. But there are two problems here. The first is that both two and three years ago, I rearranged the lessons so that Lesson 10-9 was taught close to Easter. (This was close to -- but not exactly on -- April Fool's Day both years.) And so my old Lesson 10-9 worksheet made a reference to Easter and "Spring Spheres." So instead of the holiday worksheet, today I'm posting an alternate activity based on both Exploration Questions 22 and 23 from Lesson 10-9.

This is what little I wrote two years ago about today's activity:

-- From the U of Chicago text: calculate the surface area of the earth. Then compare the area of the United States and other countries to that of the entire earth.

The second problem today is that this is a nine-lesson chapter. Just as we did with Chapter 8 last month, today we must begin our review for the Chapter 10 Test. Tomorrow, Day 110, will be the Chapter 10 Test itself, and Friday, Day 111, will be Lesson 11-1.

Both two years ago and three years ago, I rearranged the lessons. Three years ago, I added two lessons from different chapters to the Chapter 10 Test. Then two years ago, I dropped not only the extra lessons, but Lessons 10-8 and 10-9 on spheres as well. On review day I posted the first worksheet from the previous year, which left out all of the non-Chapter 10 questions as well as some, but not all, of the sphere questions.

This is what I wrote two years ago about today's worksheet. Again, I referred to the Easter holiday, which was a huge part of why I changed the chapter order from 2015 to 2016:

You may notice that today's blog entry is called "Review for Chapter 10 Test." At this point you're probably wondering -- how can there be a Chapter 10 Test already? After all, we haven't covered the surface area or volume of a sphere yet!

The problem, of course, is that next week is spring break. Chapter 10 is long, and the Easter holiday ends up splitting the chapter. This is actually a domino effect caused by Pi Day falling on Monday -- I wanted to cover pi -- part of Lessons 8-8 and 8-9 -- on Pi Day Monday, so we didn't start Chapter 10 until Tuesday. So that ended up pushing back Lessons 10-8 and 10-9 on the sphere.

We know that the formulas in Chapter 10 are hard for students to remember -- that is, after all, why the U of Chicago text devotes a full lesson, 10-6, just for remembering formulas! So imagine how much harder the formulas will be to remember when we have a week of spring break separating the start of Chapter 10 from its end!

And so I decided to declare this week to be the end of the unit and give a test this week. This is the same rationale for the Early Start Calendar -- we want to test the students before they have a chance to forget the material over the vacation weeks.

Returning to 2018, we are now following the U of Chicago order, and today is Valentine's Day, nowhere near Easter. And so this will be a pure Chapter 10 Test. I'm posting the old review worksheet from last year, which contains some sphere questions, but fortunately no non-Chapter 10 Questions.

If you want, you could add some more review questions. In fact, today's Pappas problem is actually a great question to ask our students. Let's see how many of them try to find the area or radius of the base and how many realize that they only need to triple the cone volume.

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