Recall that Chapter 13 of the new edition corresponds to Chapter 14 of the old edition (since the old Chapter 13 has been split up into different chapters). The new Lesson 13-1 is the last lesson of the old Chapter 12, while the new Lesson 13-2 is on the Angle Bisector Theorem -- a theorem that doesn't appear in the old text (but occasionally appears on the Pappas calendar).
This is what I wrote last year about today's lesson:
Lesson 14-2 of the U of Chicago text is on lengths in right triangles -- specifically, those lengths that are related to the altitude and involve the geometric mean.
Geometric Mean Theorem:
The geometric mean of the positive numbers a and b is sqrt(ab).
(Note: This may sound like a definition, but actually the U of Chicago defines geometric mean to be the number x such that a/x = x/b, so we need a theorem to get the geometric mean as sqrt(ab).)
Right Triangle Altitude Theorem:
In a right triangle:
a. The altitude of the hypotenuse is the geometric mean of the segments dividing the hypotenuse.
b. Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.
In this lesson, I give the proof of the Pythagorean Theorem based on similarity, but this time I gave the proof in the book, which mentions the geometric mean. Let's look at the proof -- as usual, with an extra step for the Given:
Given: Right triangle
Prove: a^2 + b^2 = c^2
Proof:
Statements Reasons
1. Right triangle 1. Given
2. a geometric mean of c & x, 2. Right Triangle Altitude Theorem
b geometric mean of c & y
3. a = sqrt(cx), b = sqrt(cy) 3. Geometric Mean Theorem
4. a^2 = cx, b^2 = cy 4. Multiplication Property of Equality
5. a^2 + b^2 = cx + cy 5. Addition Property of Equality
6. a^2 + b^2 = c(x + y) 6. Distributive Property
7. x + y = c 7. Betweenness Theorem (Segment Addition)
8. a^2 + b^2 = c^2 8. Substitution (step 6 into step 7)
It is uncertain whether this is the proof that Common Core intends the students to learn, or whether my earlier proof that avoids geometric means suffices.
Actually, since posting this last year, I've decided to check both the PARCC and SBAC released test questions for those related to the proof of the Pythagorean Theorem. There were a few questions that required use of Pythagoras, but none directly related to the proof. Of course, some people lament that there aren't very many proofs on the Common Core tests.
Speaking of the Pythagoran Theorem, I've been thinking last week about how I presented it to my eighth graders last year:
Monday: Coding
Tuesday: Illinois State project
Wednesday: Traditional lesson on the theorem
Thursday: "Learning Centers" (actually the traditional lesson continued)
Friday: Test
which too many students failed. But as I think about it more, we can see how the students were set up for failure with only two days between introduction and assessment. (As it turned out, even the original project failed because it was on the day that the Hidden Figures movie extra credit was due, and many students focused on finishing that rather than the project.)
It might have been better to teach the theorem on Tuesday, and then the students can apply that knowledge to the project on Wednesday. (And it would have been even better to have the traditional lesson on Monday, but I had no control over the coding teacher's schedule.) The problem is that Illinois State, on its pacing guide, insists that the project be given before the traditional lesson. This in itself leads back to the traditionalist debate, which I'm not getting into in today's post. Another problem is that this might work for sixth and eighth grades, but not for seventh grade, as I had less time with them.
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