*Mathematics and the Physical World*, "Explanation Versus Description," introduces the reader to the birth of modern science.

"Nor should it be considered rash not to be satisfied with those opinions which have become common. No one should be scorned in physical disputes for not holding on to the opinions which happen to please other people best." -- Galileo

And there's yet another name that needs no further introduction. As we see, Kline credits the 17th century Italian physicist Galileo Galilei as the father of modern science:

"Descartes and Fermat [Pierre de Fermat, mentioned several times on the blog before -- dw] created one of the key mathematical toolds for the develoment of modern science. Galileo Galilei created modern science."

Kline describes how Galileo radically changed the way people thought about the world. Ever since the days of Aristotle, educated Europeans assumed that there were only four elements -- air, fire, earth, and water -- and that objects fell because each element was seeking out its natural place, which for solid objects (earth) was the ground.

But Galileo challenged these notions. Kline writes:

"Galileo decided that he would seek descriptions of

*how*things worked rather than explanations of

*why*they worked or what purpose they served."

Although Kline introduces us to some of Galileo's early experiments, his most famous experiment of all will have to wait for tomorrow's chapter.

Today is Day 154 according to the blog calendar, which is based on one of my districts. According to the other district, today is Day 160. At the school where I will teach in the fall, today is Day 152.

Question 11 of the PARCC Practice Exam is on rotations -- but a different sort of rotation:

Question 11 of the PARCC Practice Exam is on rotations -- but a different sort of rotation:

11. Each of the two-dimensional figures shown will be rotated 360 degrees about the respective line, creating a three-dimensional figure.

Drag the appropriate two-dimensional figure to identify the correct representation of the resulting three-dimensional figure.

(Here are the 2D figures: a circle not intersecting the axis of rotation, a semicircle whose diameter lies on the axis, a rectangle whose length is on the axis, and a right triangle whose leg is on the axis. Here are the 3D figures: a torus, a cone, a cylinder, and a sphere.)

There was a question like this on last year's practice PARCC. Here are the correct matches -- the circle creates the torus, the triangle generates the cone, the rectangle creates the cylinder, and the semicircle generates the sphere.

There are several things going on in this question. First of all, this is the only problem on the PARCC where transformations are performed in 3D. Recall that just as the mirror of a reflection is a line in 2D and a plane in 3D, the center of a rotation is a point in 2D and a line (axis) in 3D. This is why we are given a line to rotate about, rather than a point.

Second, our rotations are actually producing a new solid. This is called a

Only one solid of revolution appears in the U of Chicago text, and I mentioned it a month and a half ago -- Lesson 10-7, Question 15. In this case, a right triangle is rotated -- its axis is the line containing one of its legs -- to form a cone. But this problem tells us that the solid is a cone -- the students' task is to find its volume, if the length of the other leg and the the hypotenuse are given. If the given figure had been a rectangle, then the solid would have been a cylinder rather than a cone. I wonder whether Question 11 on the PARCC would be more palatable to traditionalists if the dimensions of the rectangle were given and students had to find the volume of the cylinder.

*solid of revolution*. Notice that tomorrow, many Calculus BC students will be struggling to calculate the volumes of solids of revolution as they take their AP test. And as we found out last year, identifying solids of revolution appears on the PARCC Geometry test.Only one solid of revolution appears in the U of Chicago text, and I mentioned it a month and a half ago -- Lesson 10-7, Question 15. In this case, a right triangle is rotated -- its axis is the line containing one of its legs -- to form a cone. But this problem tells us that the solid is a cone -- the students' task is to find its volume, if the length of the other leg and the the hypotenuse are given. If the given figure had been a rectangle, then the solid would have been a cylinder rather than a cone. I wonder whether Question 11 on the PARCC would be more palatable to traditionalists if the dimensions of the rectangle were given and students had to find the volume of the cylinder.

**PARCC Practice Test Question 11**

**U of Chicago Correspondence: Lesson 10-7, Volumes of Pyramids and Cones**

**Key Theorem: none**

**Common Core Standard:**

CCSS.MATH.CONTENT.HSG.GMD.B.4

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

**Commentary: The only question relevant to solids of revolution is Question 15, and it rotates a triangle to form a cone, rather than a rectangle to form a cylinder. Another U of Chicago section that may be relevant to the first part of the above standard is Section 9-4, which is on "Plane Sections" (cross-sections). On this blog, we jumped around Chapters 9 and 10, so Section 9-4 may not have been adequately covered.**
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