And speaking of juggling lessons around different chapters, I myself made changes on the blog both two and three years ago. In those years, I combined the Pythagorean Theorem with the Distance Formula in Chapter 11, and my worksheets reflect this.
After yesterday's short post, today's post will be long and full of cut-and-paste from various old posts of mine. Let's start with what I wrote two years ago about my worksheet:
Lesson 8-7 of the U of Chicago text is on the Pythagorean Theorem, and Lesson 11-2 of the same text is on the Distance Formula. I explained yesterday that I will cover these two related theorems in this lesson.
The Pythagorean Theorem is, of course, one of the most famous mathematical theorems. It is usually the first theorem that a student learns that is named for a person -- the famous Greek mathematician Pythagoras, who lived about 2500 years ago -- a few centuries before Euclid. I believe that the only other named theorem in the text is the Cavalieri Principle -- named after an Italian mathematician from 400 years ago. Perhaps the best known named theorem is Fermat's Last Theorem -- named after the same mathematician Fermat mentioned in yesterday's lecture. (We discuss some other mathematicians such as Euclid and Descartes, but not their theorems.)
It's known that Pythagoras was not the only person who knew of his named theorem. The ancient Babylonians and Chinese knew of the theorem, and it's possible that the Egyptians at least knew about the 3-4-5 case.
We begin with the proof of the Pythagorean Theorem -- but which one? One of my favorite math websites, Cut the Knot (previously mentioned on this blog), gives over a hundred proofs of Pythagoras:
http://www.cut-the-knot.com/pythagoras/
The only other theorem with many known proofs is Gauss's Law of Quadratic Reciprocity. Here is a discussion of some of the first few proofs:
Proof #1 is Euclid's own proof, his Proposition I.47. Proof #2 is simple enough, but rarely seen. Proofs #3 and #4 both appear in the U of Chicago, Lesson 8-7 -- one is given as the main proof and the other appears in the exercises. Proof #5 is the presidential proof -- it was first proposed by James Garfield, the twentieth President of the United States. I've once seen a text where the high school students were expected to reproduce Garfield's proof.
So far, the first five proofs all involve area. My favorite area-based proof is actually Proof #9. I've tutored students where I've shown them this version of the proof. Just as the Cut the Knot page points out, Proof #9 "makes the algebraic part of proof #4 completely redundant" -- and because it doesn't require the students to know any area formulas at all (save that of the square), I could give this proof right now. In fact, I was considering including Proof #9 on today's worksheet. Instead, I will wait until our next activity day on Friday to post it.
But it's the proof by similarity, Proof #6, that's endorsed by Common Core. This proof has its own page:
http://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtml
Here is Proof #6 below. The only difference between my proof and #6 from the Cut the Knot webpage is that I switched points A and C, so that the right angle is at C. This fits the usual notation that c, the side opposite C, is the hypotenuse.
Given: ACB and ADC are right angles.
Prove: BC * BC + AC * AC = AB * AB (that is, a^2 + b^2 = c^2)
Statements Reasons
1. ADC, ACB, CDB rt. angles 1. Given
2. Angle A = A, Angle B = B 2. Reflexive Property of Congruence
3. ADC, ACB, CDB sim. tri. 3. AA Similarity Theorem
4. AC/AB = AD/AC, 4. Corresponding sides are in proportion.
BC/AB = BD/BC
5. AC * AC = AB * AD, 5. Multiplication Property of Equality
BC * BC = AB * BD
6. BC * BC + AC * AC = 6. Addition Property of Equality
AB * BD + AB * AD
7. BC * BC + AC * AC = 7. Distributive Property
AB * (BD + AD)
8. BC * BC + AC * AC = 8. Betweenness Theorem (Segment Addition)
AB * AB
I mentioned before that, like many converses, the Converse of the Pythagorean Theorem is proved using the forward theorem plus a uniqueness theorem -- and the correct uniqueness theorem happens to be the SSS Congruence Theorem (i.e., up to isometry, there is at most one triangle given three side lengths). To prove this, given a triangle with lengths a^2 + b^2 = c^2 we take another triangle with legs a and b, and we're given a right angle between a and b. By the forward Pythagorean Theorem, if the hypotenuse of the new triangle is z, then a^2 + b^2 = z^2. (I chose z following the U of Chicago proof.) Thenz^2 = c^2 by transitivity -- that is, z = c. So all three pairs of both triangles are congruent -- SSS. Then by CPCTC, the original triangle has an angle congruent to the given right angle -- so it's a right triangle. QED
Interestingly enough, there's yet another link at Proof #6 at Cut the Knot, "Lipogrammatic Proof of the Pythagorean Theorem." At that link, not only is Proof #6 remodified so that it's also an area proof (just like Proofs #1-5), but, as its author points out, slope is well-defined without referring to similar triangles!
Now, my original worksheet included the Distance Formula as well, but this year, we're waiting until Lesson 11-2 -- which is where distance belongs in the test. So I decided to keep only the second worksheet -- which contains exercises but no proofs -- and include an activity I posted two days later, which gives the proof of the Pythagorean Theorem as given in the text. This is essentially Proof #4 from the link above.
I didn't write much about the activity two years ago, since I posted it in the middle of the 2016 MTBoS Blogging Initiative -- and mentioned several other lessons (including science lessons!) in the same post:
Date: About a year ago [that is, 2015]
The Lesson: Today I am posting a Pythagorean Theorem Activity from last year.
My Teaching: I've never taught this lesson in a classroom, but I had back when I was once a math tutor myself. I enjoy letting the students visualize exactly why the Pythagorean Theorem holds.
Student Responses: Whenever I have students try this out, it's always easier for them to complete the puzzle with the a^2 and b^2 squares than with the c^2, since it must be placed at an angle.
The worksheets that I'm posting refer to this Pythagorean Theorem activity. I'm hoping that I can teach this in the classroom soon.
By the way, there's still no word of a 2018 Blogging Initiative -- and by this post, I strongly believe that there won't be one this year. (Then again, there's no reason that the initiative must be in the month of January. It could be as late as December and still be called the "2018 Blogging Initiative"!)
And so two years ago, I began by posting the Common Core similarity-based proof and then the U of Chicago area-based proof two days later. This year, I wrote that I would stay true to the U of Chicago version, so that's what I'm doing today.
But last year, recall that that Pythagorean Theorem also appears in the eighth grade standards. And so I'm using the rest of this post to discuss how I taught -- or, as usual, how I failed to teach -- the Pythagoran Theorem last year.
Notice that even though the Common Core tells us to use similarity to prove Pythagoras in high school Geometry, similarity isn't mentioned in the eighth grade standard:
CCSS.MATH.CONTENT.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
CCSS.MATH.CONTENT.HSG.SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Then again, the Illinois State text suggests using similarity to prove Pythagoras in eighth grade. And so I tried to give the project in the text in which the connection from similarity to the Pythagorean Theorem is made. But this project failed for three reasons:
- The previous lessons on similarity and dilations were cancelled due to science lessons -- and arguments about science lessons. (See last week's Epiphany post and read under the section labeled "January 2017.")
- The day on which this Illinois State lesson was scheduled turned into "finish the extra credit project on Hidden Figures" -- also caused by a domino effect of poor previous lessons.
- The Illinois State project wasn't designed well, anyway. Students were supposed to form similar right triangles on Geoboards -- and somehow that would lead to their discovering the Pythagorean Theorem. But I fail to see, for example, how students would see what the hypotenuse of a right triangle with legs 3 and 4 is. Even building a square on the Geoboard with that hypotenuse as a side doesn't make it obvious that it's 5.
Traditionalists, of course, don't like the use of projects to teach math anyway. But in this case, I think that some projects are much more effective than others. The next day, I used the area-based project that I'm posting today, and that was followed by the traditional lesson. This is what I wrote last year about today's lesson:
In all classes, I end up continuing the lessons I started yesterday. For the eighth graders, this is the Pythagorean Theorem lesson. I think that I did an okay job explaining the theorem -- and I did use my lesson from last year where students use a puzzle to prove the theorem -- but many students are confused due to the classroom being too loud during the lesson. (This is what necessitated the visit by the instructional assistant in the first place.)
Let's think back to the New Year's resolutions I posted in my last two winter break posts. How could I have taught the Pythagorean Theorem lesson better?
I want to keep the sixth resolution in mind:
6. If there is a project-based curriculum such as Illinois State, then implement all components of it.
And "all components of it" includes badly designed projects such as the Geoboards. Of course, last year I wrote that I should have preempted all math projects with the much better science projects, but let's assume that we're writing about this current year -- if I had remained in the classroom, I assume that science wouldn't have been a factor.
The proper pacing plan is to teach one standard a week, beginning with 8.NS1. Skipping over short weeks, Benchmark Testing, and so on, it's possible to reach 8.G8 (the Distance Formula) before my old school gives the SBAC -- if not, hopefully at least 8.G6, when the Pythagoras is introduced.
Last year, I began 8.G1 by teaching translations the first day, then reflections the second day, and then rotations the third day. This won't work if we follow the pacing plan -- where only one day is devoted to the traditional lesson, with other the days for projects, learning centers, and so on.
But I do see a loophole where I can teach the transformations one day at a time even with only one traditional lesson per week:
Week of 8.G1 -- reflections
Week of 8.G2 -- rotations
Week of 8.G3 -- translations
Week of 8.G4 -- dilations
Week of 8.G5 -- follow as written
And so the first week, I introduce my students to reflections -- and then in the Student Journal, we cover only the questions on reflections. Last year, I started with translations as these are the easiest for students to understand. But notice that translations lines up perfectly with 8.G3, since this is about the coordinate plane -- we like to perform translations with coordinates. Standard 8.G4 is about similarity and so it's the perfect time to teach dilations. Then we follow 8.G5 as written and teach Triangle Sum, Parallel Consequences, and AA Similarity.
When we reach 8.G6, we follow the sixth resolution and use similarity to teach Pythagoras. It's possible to create an activity that's just like the area-based project I posted today, yet is based on similarity rather than area.
Take a standard sheet of paper and cut it along its diagonal. This, of course, divides the paper into two right triangles. Now take one of the triangles and cut it along its altitude to the hypotenuse. Now there are three right triangles -- and notice that these are the three similar right triangles that appear in the similarity-based proof!
To make this proof easier for the students to understand, we'll begin by using numbers -- suppose the legs of a right triangle are 3 and 4, then what is the hypotenuse? We might as well use c for the hypotenuse, as usual. So even before we cut, we label the sides of the paper 3 and 4. (Notice that the sides of a standard sheet of paper, 8 1/2 * 11, are already nearly in a 3:4 ratio.)
After we cut, we label all three triangles with legs 3 and 4 and hypotenuse c. Of course, the triangles are merely similar, not congruent, so they can't all be 3-4-c right triangles. So instead, we draw a box (or blank) next to each 3, 4, and c, to be filled in by a constant for each triangle. Indeed, we anticipate this activity by emphasizing, during the Week of 8.G4, that we can multiply all three sides by the same constant (a dilation!) and obtain a similar triangle.
Notice that none of these triangles is considered the original 3-4-c triangle -- all three triangles must be multiplied by some constant. This is to avoid fractions -- for example, if the largest triangle were simply 3-4-c, the other two would have fractions in their lengths. The same would happen even if we chose the smallest triangle to 3-4-c -- we'll eventually find that the other lengths are fractions.
Small triangle: 3( )-4( )-c( )
Medium triangle: 3( )-4( )-c( )
Large triangle: 3( )-4( )-c( )
Now we start moving triangles and comparing their sides. We'll see that the short leg of the medium triangle is congruent to the triangle leg of the small triangle. The former is labeled 3( ) and the latter is labeled 4( ). Since 3 * 4 = 4 * 3, let's fill in the blanks with 3(4) and 4(3):
Small triangle: 3( )-4(3)-c( )
Medium triangle: 3(4)-4( )-c( )
Large triangle: 3( )-4( )-c( )
Recall that within each triangle, all the blanks must be labeled the same (otherwise the triangles aren't similar), so we write:
Small triangle: 3(3)-4(3)-c(3)
Medium triangle: 3(4)-4(4)-c(4)
Large triangle: 3( )-4( )-c( )
Notice that we won't actually multiply anything until we have to! The next thing we notice is that the hypotenuse of the small triangle c(3) equals the short leg of the large triangle, and the hypotenuse of the medium triangle c(4) equals the long leg of the large triangle. This suggests that the blanks in the large triangle should be filled with c:
Small triangle: 3(3)-4(3)-c(3)
Medium triangle: 3(4)-4(4)-c(4)
Large triangle: 3(c)-4(c)-c(c)
Now we move the pieces around to form the original rectangle. We now see that the short leg of the small triangle 3(3) and the long leg of the medium triangle 4(4) add up to the hypotenuse of the largest triangle c(c):
3(3) + 4(4) = c(c)
9 + 16 = c(c)
c(c) = 25
c = 5
This proof easily generalizes -- change all the 3's to a's and all the 4's to b's, and we instantly obtain the Pythagorean Theorem. Notice that in avoiding multiplication until it was necessary, the only numbers we had to multiply are a^2, b^2, and c^2.
And so this is how I keep the sixth resolution -- we follow the basic framework of the Illinois State text while modifying the projects in order for them to make sense for the student. The students can keep the Illinois State textbook open while using a worksheet to complete the project.
Of course, I haven't addressed the elephant in the room -- the first resolution. Students aren't learning anything unless they are quiet during the lesson. My eighth graders would talk the whole time regardless of whether I'm giving a project or a traditional lesson. Typically the Pythagorean Theorem is one of the easier lessons in the eighth grade curriculum, yet my students learned nothing. Not only did they forget the theorem by the time of the unit test or SBAC, but they would have forgotten it for that night's homework or the next day's Warm-Up, not even being able to do the first step.
This is the perfect lesson to introduce dry-erase packets, set up so that students can calculate either a, b, or the hypotenuse c. But it's important to make sure that the students are learning and not just doing the minimum to complete the project or traditional lesson while talking instead of thinking.
And "all components of it" includes badly designed projects such as the Geoboards. Of course, last year I wrote that I should have preempted all math projects with the much better science projects, but let's assume that we're writing about this current year -- if I had remained in the classroom, I assume that science wouldn't have been a factor.
The proper pacing plan is to teach one standard a week, beginning with 8.NS1. Skipping over short weeks, Benchmark Testing, and so on, it's possible to reach 8.G8 (the Distance Formula) before my old school gives the SBAC -- if not, hopefully at least 8.G6, when the Pythagoras is introduced.
Last year, I began 8.G1 by teaching translations the first day, then reflections the second day, and then rotations the third day. This won't work if we follow the pacing plan -- where only one day is devoted to the traditional lesson, with other the days for projects, learning centers, and so on.
But I do see a loophole where I can teach the transformations one day at a time even with only one traditional lesson per week:
Week of 8.G1 -- reflections
Week of 8.G2 -- rotations
Week of 8.G3 -- translations
Week of 8.G4 -- dilations
Week of 8.G5 -- follow as written
And so the first week, I introduce my students to reflections -- and then in the Student Journal, we cover only the questions on reflections. Last year, I started with translations as these are the easiest for students to understand. But notice that translations lines up perfectly with 8.G3, since this is about the coordinate plane -- we like to perform translations with coordinates. Standard 8.G4 is about similarity and so it's the perfect time to teach dilations. Then we follow 8.G5 as written and teach Triangle Sum, Parallel Consequences, and AA Similarity.
When we reach 8.G6, we follow the sixth resolution and use similarity to teach Pythagoras. It's possible to create an activity that's just like the area-based project I posted today, yet is based on similarity rather than area.
Take a standard sheet of paper and cut it along its diagonal. This, of course, divides the paper into two right triangles. Now take one of the triangles and cut it along its altitude to the hypotenuse. Now there are three right triangles -- and notice that these are the three similar right triangles that appear in the similarity-based proof!
To make this proof easier for the students to understand, we'll begin by using numbers -- suppose the legs of a right triangle are 3 and 4, then what is the hypotenuse? We might as well use c for the hypotenuse, as usual. So even before we cut, we label the sides of the paper 3 and 4. (Notice that the sides of a standard sheet of paper, 8 1/2 * 11, are already nearly in a 3:4 ratio.)
After we cut, we label all three triangles with legs 3 and 4 and hypotenuse c. Of course, the triangles are merely similar, not congruent, so they can't all be 3-4-c right triangles. So instead, we draw a box (or blank) next to each 3, 4, and c, to be filled in by a constant for each triangle. Indeed, we anticipate this activity by emphasizing, during the Week of 8.G4, that we can multiply all three sides by the same constant (a dilation!) and obtain a similar triangle.
Notice that none of these triangles is considered the original 3-4-c triangle -- all three triangles must be multiplied by some constant. This is to avoid fractions -- for example, if the largest triangle were simply 3-4-c, the other two would have fractions in their lengths. The same would happen even if we chose the smallest triangle to 3-4-c -- we'll eventually find that the other lengths are fractions.
Small triangle: 3( )-4( )-c( )
Medium triangle: 3( )-4( )-c( )
Large triangle: 3( )-4( )-c( )
Now we start moving triangles and comparing their sides. We'll see that the short leg of the medium triangle is congruent to the triangle leg of the small triangle. The former is labeled 3( ) and the latter is labeled 4( ). Since 3 * 4 = 4 * 3, let's fill in the blanks with 3(4) and 4(3):
Small triangle: 3( )-4(3)-c( )
Medium triangle: 3(4)-4( )-c( )
Large triangle: 3( )-4( )-c( )
Recall that within each triangle, all the blanks must be labeled the same (otherwise the triangles aren't similar), so we write:
Small triangle: 3(3)-4(3)-c(3)
Medium triangle: 3(4)-4(4)-c(4)
Large triangle: 3( )-4( )-c( )
Notice that we won't actually multiply anything until we have to! The next thing we notice is that the hypotenuse of the small triangle c(3) equals the short leg of the large triangle, and the hypotenuse of the medium triangle c(4) equals the long leg of the large triangle. This suggests that the blanks in the large triangle should be filled with c:
Small triangle: 3(3)-4(3)-c(3)
Medium triangle: 3(4)-4(4)-c(4)
Large triangle: 3(c)-4(c)-c(c)
Now we move the pieces around to form the original rectangle. We now see that the short leg of the small triangle 3(3) and the long leg of the medium triangle 4(4) add up to the hypotenuse of the largest triangle c(c):
3(3) + 4(4) = c(c)
9 + 16 = c(c)
c(c) = 25
c = 5
This proof easily generalizes -- change all the 3's to a's and all the 4's to b's, and we instantly obtain the Pythagorean Theorem. Notice that in avoiding multiplication until it was necessary, the only numbers we had to multiply are a^2, b^2, and c^2.
And so this is how I keep the sixth resolution -- we follow the basic framework of the Illinois State text while modifying the projects in order for them to make sense for the student. The students can keep the Illinois State textbook open while using a worksheet to complete the project.
Of course, I haven't addressed the elephant in the room -- the first resolution. Students aren't learning anything unless they are quiet during the lesson. My eighth graders would talk the whole time regardless of whether I'm giving a project or a traditional lesson. Typically the Pythagorean Theorem is one of the easier lessons in the eighth grade curriculum, yet my students learned nothing. Not only did they forget the theorem by the time of the unit test or SBAC, but they would have forgotten it for that night's homework or the next day's Warm-Up, not even being able to do the first step.
This is the perfect lesson to introduce dry-erase packets, set up so that students can calculate either a, b, or the hypotenuse c. But it's important to make sure that the students are learning and not just doing the minimum to complete the project or traditional lesson while talking instead of thinking.
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