As usual, the English learner class has an aide who speaks Spanish, and so there's definitely no need for "A Day in the Life" today. These students watch a movie that's definitely familiar to both me and the readers of this blog -- McFarland USA. I described this movie on the blog in several posts back during the last week of November. The class that watched it after Thanksgiving (around the time of the State Cross Country Meet) was a Spanish class, so we watched it in Spanish with subtitles in the same language.
But today is an English class, so we watch it in English with subtitles in the same language. We begin the movie just as we see the Diaz brothers work in the field for the first time. Since it's a double-block we essentially finish the movie -- the bell rings right after McFarland is announced as the winner of the State Meet.
In between the two periods is tutorial. I'm able to help one of the students -- an English learner who remains between his two classes -- on his Algebra I assignment, inequalities in two variables. It's tricky due to the language barrier, so I hope that my explanation helps him a little bit. Fortunately, there's another guy in the same class, so we look at his graphs for comparison.
In fifth period, the juniors must do a pre-write for their upcoming essay on what exactly it means to be American. Before, there's a Warm-Up on racism and prejudice. (Today is the 100th birthday of Jackie Robinson, a baseball player who faced more than his share!) It's a district assessment -- and yes, that was one of the issues during the LAUSD teacher strike. In the LAUSD, district assessments are supposed to be reduced 50% as part of the settlement -- but once again, the district where I'm working isn't LAUSD.
Lesson 10-2 of the U of Chicago text is called "Surface Areas of Pyramids and Cones." In the modern Third Edition of the text, surface areas of pyramids and cones appear in Lesson 9-10.
This is what I wrote last year ago about today's lesson. Notice that I spent much of that post comparing the U of Chicago text to three other math texts (and I decided to preserve this discussion):
One of the texts was published by Merrill, the others by McDougal Littell. I ended up purchasing the latter, which is dated 2001. I actually recognize this text from when I spent one month in an advanced seventh grade math classroom back in 2012. Geometry is covered in Chapters 8 through 10. Chapter 8 covers points, lines, polygons, transformations, and similarity. The transformation section covers reflections and translations (but not dilations in the similarity section), but of course, this is an old pre-Common Core text, so transformations aren't used to define congruence. Chapter 9 is officially called "Real Numbers and Solving Inequalities," but the real numbers portion of the chapter segues from square roots to the Pythagorean Theorem and to the Distance Formula.
That takes us to Chapter 10. As it turns out, much of Chapter 10 of this seventh-grade text matches up with the same numbered chapter of the U of Chicago geometry text. Here are the sections:
Section 10.1: Circumference and Area of a Circle
Section 10.2: Three-Dimensional Figures
Section 10.3: Surface Areas of Prisms and Cylinders
Section 10.4: Volume of a Prism
Section 10.5: Volume of a Cylinder
Section 10.6: Volumes of Pyramids and Cones
Section 10.7: Volume of a Sphere
Section 10.8: Similar Solids
It's often interesting to see how much surface area and volume appears in pre-algebra texts. Wee see that this text gives all of the volume formulas, while only the cylindric solids have their surface areas included in the text. But let's keep in mind that this text was specifically written for the old California state standards that we had before the Common Core.
The final chapter, Chapter 12, of this text is on polynomials. This chapter actually goes a bit beyond the seventh grade standards -- most notably, Section 12.5 is "Multiplying Polynomials" and actually teaches the FOIL method of multiplying two binomials. I was only in the classroom that taught using this text for a month, but I was told that the honors class would cover Chapter 12 around the start of the second semester, with the rest of the chapters taught in numerical order. (Non-honors classes would not cover Chapter 12 at all.) The next section, Section 12.6, may also seem a bit advanced for a pre-algebra class -- "Graphing y = ax^2 and y = ax^3" -- but it appears in the 7th grade standards.
If we compare this to the Common Core Standards, we see that much of Chapter 10 of the McDougal Littell text corresponds to an eighth grade standard in Common Core:
CCSS.MATH.CONTENT.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
This is to be expected. The Common Core Standards are based on Algebra I in ninth grade, while the California Standards were based on Algebra I in eighth grade. So many eighth grade Common Core Standards must have been seventh grade standards in California.
Before we leave the McDougal Littell text, let me note that Section 4.3 is on "Solving Equations Involving Negative Coefficients." For comparison purposes, let's look at the McDougal Littell Algebra Readiness text in more detail:
1. Expressions, Unit Analysis, and Problem Solving
2. Fractions
3. Decimals and Percents
4. Integers
5. Rational Numbers and Their Properties
6. Exponents
7. Square Roots and the Pythagorean Theorem
8. Equations in One Variable
9. Inequalities in One Variable
10. Linear Equations in Two Variables
The purpose of Algebra Readiness was to prepare students for Algebra I. Therefore, as we can see, there is very little geometry in this text compared to the McDougal Littell Math 7 text. The only geometry that appears is in Chapter 7, with even less geometry content than Chapter 9 of the Math 7 text (as the Distance Formula doesn't appear in the Algebra Readiness text). Area and volume are nowhere to be seen in the Algebra Readiness text (except the appendix, "Skills Review Handbook").
In many ways, Algebra Readiness was more like a Common Core 7 text than Common Core 8, as Common Core 8 contains more geometry (and even a little more algebra) than the Readiness text.
Now leave MacDougal Littell and continue with the Merrill text:
I didn't purchase the Merrill Pre-Algebra text, so I don't recall how old the text is. But I glanced at it and noticed that all of the equations that appear in Chapter 10 of the U of Chicago Geometry text also appear in this text, with the exception of the equations involving a sphere. That is, the surface area formulas of all cylindric and conic solids appear in this text. This is unusual since, as we've seen, neither the CAHSEE nor the Common Core Standard expect students to learn the more complex surface area formulas before high school Geometry. Since today's lesson is Lesson 10-2 of the U of Chicago text, which is on surface areas of pyramids and cones, I want to discuss what I remember about the Merrill lesson on these surface areas.
Both Merrill and the U of Chicago give the lateral area of the pyramid as the sum of the areas of its triangle lateral faces. But only the U of Chicago gives the formula for a regular pyramid, which it defines in Lesson 9-3 as a pyramid whose base is a regular polygon and the segment connecting the vertex to the center of this polygon is perpendicular to the plane of the base. The formula for the lateral area of a regular pyramid is LA = 1/2 * l * p.
But now we must consider the surface area of a cone. The Merrill text does something interesting here, as it considers the area of the net of the cone. We cut out the circular base and a slit in the lateral region, and then flatten this lateral region. What remains is a sector of a circle. Then the Merrill text simply gives the area of this sector as pi * r *s (where s, rather than l, is the slant height) without any further explanation.
The U of Chicago text, meanwhile, gives a limiting argument for the surface area of the cone, as its circular base is the limit of regular polygons as the number of sides approaches infinity. But there is Exploration Question 25, where the Merrill demonstration is done in reverse -- we begin with a sector of a disk and fold it into a cone.
But neither tells us why the area of the sector (and thus the lateral area of the cone) is pi *r * l. Let me give a demonstration of why the area of the sector is pi * r * l.
We begin with the area of a circle, pi * R^2. The reason why I used a capital R is to emphasize that the radius of the circle that appears in Question 25 is not the radius r of the base -- indeed, it's easy to see that the radius of the circle becomes the slant height l. So the area of the circle is pi * l^2 -- that is, before we cut out the sector. We want to fit the area after we cut it.
Let's recall another formula for the area of a circle given by Dr. Hung-Hsi Wu: A = 1/2 * C *R -- and once again, R = l, so we have A = 1/2 * C * l. But neither one of these gives us the circumference or area of a sector. If we let theta be the central angle of a sector, we obtain:
x = theta / 360 * C
L.A. = theta / 360 * A
= theta / 360 * 1/2 * C * l
For lack of a better variable, I just let x be the arclength of our sector. But here I let L.A. be the area of the sector, since these equals the lateral area of the cone we seek. The big problem, of course, is that we don't know what angle theta is for the cone to have a particular shape. But we notice that we can simply substitute the first equation into the second:
L.A. = 1/2 * theta / 360 * C * l
= 1/2 * x * l
And what exactly is the arclength x of our sector? Notice that once we fold the sector into a cone, the arclength of the sector becomes the circumference of the circular base of the cone! And this we know exactly what it is -- since the radius of the base is r, its circumference must be 2 * pi * r:
L.A. = 1/2 * (2 * pi * r) * l
= pi * r * l
as desired. QED
I incorporate this demonstration into my lesson.
2019 Update: Today is no longer an activity day. The old worksheet mentions a quick activity based on the Exploration question that I just mentioned above. This can now be considered a Bonus question, with this year's new activity to appear tomorrow.
Let's continue with the next proposition in Euclid:
This proof should be easy to modernize:
Given:
Prove: Plane P | | Plane Q
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Point G in plane Q so that 2. Proposition 11 from Monday (construction)
3. H in Q so that
K in Q so that
4.
5.
6.
7. Plane P | | Plane Q 7. Proposition 14 from yesterday (a form of Two Perpendiculars)
Euclid's original proof should be simple enough for high school students to understand without the need to convert it to two columns. All that's need is to replace phrases such as "therefore the sum of the two angles...is two right angles" with "Perpendicular to Parallels," for example.
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