Wednesday, March 18, 2015

Section 10-3: Fundamental Properties of Volume (Day 134)

Today is the CAHSEE exam for math here in California, and indeed, I subbed at a school that was giving the CAHSEE today. I've discussed in great detail the PARCC exam, which Californians don't take. So let me discuss the CAHSEE exam that we do take here.

The CAHSEE exam hasn't changed much since the graduating class of 2006 became the first class required to take the test in order to graduate. It is based on old pre-Common Core California State Standards, which are notable for expecting eighth graders to take Algebra I. Here is a link to the Released Test Questions for the CAHSEE:

While the sophomores were taking the test, the juniors went to a meeting where they would prepare for the Common Core SBAC exam that we take here in California. The freshmen and seniors had an assembly about other school issues. Some people wonder whether the state should drop the CAHSEE exam and simply let the SBAC be the state's exit exam. On one hand, it would be one fewer test for the students to take, and students should be more motivated to take the junior-year SBAC if it's required for graduation. On the other hand, it may be more difficult to set up retakes for the SBAC if it's failed, since the computer exam can only be given during certain windows of time (otherwise it would be far easier to repeat a test on the computer).

Today's CAHSEE consists of 92 questions. One major difference between the CAHSEE and the PARCC exam -- besides the obvious fact that the CAHSEE is still a handwritten test while PARCC is given on the computer -- is that CAHSEE doesn't permit calculators on any part of the exam. The PARCC test has calculator and non-calculator sections, with the majority in the calculator section.

According to the link, these CAHSEE questions are divided into strands as follows:

Number Sense -- 14 questions
Statistics, Data Analysis, and Probability -- 12 questions
Algebra and Functions -- 17 questions
Measurement and Geometry -- 17 questions
Mathematical Reasoning -- 8 questions
Algebra I -- 12 questions

But this adds up to only 80 questions, yet there are 92 on the test.

The CAHSEE exam only tests up to Algebra I -- it doesn't actually test High School Geometry, which is the topic of this blog. That Algebra I is the highest class tested on the CAHSEE is a compromise between the pro-STEM crowd -- who wonder why the CAHSEE doesn't test any course higher than middle school as Algebra I was an eighth grade course in California -- and the non-STEM crowd -- who wonder why the CAHSEE forces students to learn Algebra I to graduate, thereby blocking those who aren't good at math from getting a diploma and pursuing their non-STEM career for which they don't need to know any math higher than arithmetic.

Notice that the strands "Algebra and Functions" and "Measurement and Geometry" don't refer to high school Algebra I and Geometry, but are actually Pre-algebra seventh grade topics. Remember, even the Common Core Standards for kindergarten contain strands for Algebraic Thinking, Measurement, and Geometry, so it's wrong to equate "algebraic thinking" with "Algebra I." Nonetheless, because this is a geometry blog, let's look at some of the geometry questions. These are from the Released Test Questions, not today's test, since it is illegal for me to post any actual CAHSEE test quetions here on the blog.

138. The width of the rectangle shown below is 6 inches (in.). The length is 2 feet (ft.). What is the area of the rectangle in square inches?

A 12
B 16
C 60
D 144

This is a straightforward area problem. The tricky part is noticing that the length is measured in feet, while the width is measured in inches. Since we are asked for the area in square inches, we should convert the feet to inches. The correct answer is D, 144. But notice that if a student forgets to convert the feet to inches, choice A is obtained. Choice C occurs when a student tries to calculate the perimeter rather than the area, and choice B occurs when a student makes both mistakes (failure to convert feet to inches and calculation of the perimeter rather than the area). I've noticed that when looking at multiple choice tests, often the correct answer for the question that the most students get wrong is D. Students are very often tricked when the three distractors precede the correct answer.

Even though this question is based on the seventh grade geometry standards, of course it occurs again in High School Geometry -- indeed, we just covered it in Section 8-3 of the U of Chicago text. So naturally, now we're going to look for surface area and volume questions that appear on the practice CAHSEE exam, since these are covered in the current Chapter 10:

128. One-inch cubes are stacked as shown below. What is the total surface area?

So clearly we have a surface area problem. Notice that in some ways, yesterday's Dan Meyer problem may prepare the students for this problem if it were to occur on today's CAHSEE -- a difficulty for students may be that the figure isn't a box, but is the union of two boxes. Now we check to see whether there are any volume problems:

134. The short stairway shown below is made of solid concrete. The height and width of each step is 10 inches (in.). The length is 20 inches. What is the volume, in cubic inches, of the concrete used to create the stairway?

Once again, notice that these surface area and volume questions are all based on the seventh-grade standards, in theory, the students shouldn't have to have seen these questions in High School Geometry in order to get them correct on the test -- even a sophomore currently enrolled in Algebra I having had no Geometry at all should still be able to pass the test. (Notice that most of the Algebra I questions are either first-semester or very early second-semester questions, and so a current Algebra I student can reasonably be expected to have seen those topics by mid-March.) The same should be true of an Integrated Math student -- but of course it won't be until next year when the first cohort of Integrated Math students reaches the 10th grade and the big test. Still, if the High School Geometry course repeats topics from seventh grade math that appear on the CAHSEE, then why not cover those lessons before the CAHSEE?

The practice CAHSEE lists the 7th grade California standards on which these questions are based. I notice the following standard:

2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

Those last three words "prisms and cylinders" imply that not only should Section 10-2 precede the CAHSEE, but so should Section 10-5, on the volumes of prisms and cylinders. But then looking at the actual Released Test Questions, the only three-dimensional figures that appear are boxes and their unions -- no prisms other than rectangular and no cylinders. So no matter what Standard 2.1 says, the guiding standard for the 3D questions appears to be Standard 2.3:

2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and volume is multiplied by the cube of the scale factor.

The last part of this standard is, of course, our Fundamental Theorem of Similarity -- which we've discussed in parts, since we covered similarity before area or volume.
Once again, I cannot post any test questions from today's test. But from what I observed today, there were two volume questions -- one was a box, the other the union of two boxes. So we see that these are in reality the only volume problems that appear on the CAHSEE.

Based on this, it may be a good idea to teach the surface area and volume of boxes together -- after all, the Dan Meyer lesson that we did yesterday already hints at volume even though it's officially a surface area question. Next year, we could do this lesson at the end of February, right after area, and then move on to pi and the surface area and volume of more complicated 3D figures at the start of March.

But some people might point out that this would confuse the students even more. Instead of doing all of the surface area formulas at once (as the U of Chicago does) and all of the volume formulas at once, as Dr. Franklin we'd keep going back and forth between surface area and volume. But another argument is that it's better to do all of the prism formulas at once, then all of the pyramid formulas, and finally all of the sphere formulas.

Anyway, today nearly four hours of the school day were devoted to the CAHSEE, two hours each of the two testing sessions of 46 questions each, with a snack break in between. After lunch, students attended only three classes, periods 2, 4, and 6, each only 39 minutes long.

And so I've changed my lesson plan for this week yet again. Today I will be doing Section 10-3 of the U of Chicago text, on the fundamental properties of volume, rather than Section 10-2, on the surface areas of pyramids and cones. Section 10-3 more naturally flows from yesterday's Meyer project, and is a better lesson to give during a 39-minute period, rather than force the students to learn the complicated pyramid formulas right after taking a hard math test in the morning. (Then again, it would have been still better to have done 10-3 before CAHSEE math, but it's too late now.) Then tomorrow's lesson can cover Section 10-5, which then naturally flows from both Sections 10-2 and 10-3 -- in 10-2 we have surface areas of prisms, and in 10-5 we have their volumes.

The cornerstone of Section 10-3 is a Volume Postulate. The text even points out the resemblance of the Volume Postulate of 10-3 to the Area Postulate of 8-3:

Volume Postulate:
a. Uniqueness Property: Given a unit cube, every polyhedral solid has a unique volume.
b. Box Volume Formula: The volume of a box with dimensions l, w, and h is lwh.
c. Congruence Property: Congruent figures have the same volume.
d. Additive Property: The volume of the union of two nonoverlapping solids is the sum of the volumes of the solids.

Just as we derived the area of a square from part b of the Area Postulate, we derive the volume of a cube from part b of the Volume Postulate:

Cube Volume Formula:
The volume of a cube with edge s is s^3.

And just as we can derive the area part of the Fundamental Theorem of Similarity from the Square Area Formula, we derive the volume part of the Fundamental Theorem of Similarity from the Cube Volume Formula:

Fundamental Theorem of Similarity:
If G ~ G' and k is the scale factor, then
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.

Technically speaking, if I were a regular math teacher here in California, all of the worksheets that I posted yesterday and today would be given to the students during the same 39-minute period. So obviously this is not realistic. Most likely, I would just tell the students to start the activity, and make sure that the students have learned about the surface area and volume of boxes at some point during this 39-minute lesson.

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