8:15 -- This is the school where homeroom is not the same as first period, and all classes, including first period, rotate.
8:20 -- Today's rotation begins with sixth period. As it turns out, the two co-teaching periods are the first two classes in the rotation. Meanwhile, I've also subbed for this resident teacher before -- this was the multi-day assignment that I mentioned in my December 4th-6th posts. Unlike November and December, today the resident teacher is present in the classroom.
Last month, these seventh graders were working on solving equations, but now, they've moved on to solving inequalities. Today's lesson includes both two-step inequalities as well as inequalities with negative coefficients -- yes, the infamous "flip the inequality" rule.
The resident teacher demonstrates why inequalities need to be flipped by multiplying -3 < 3 by -2 and dividing 2 < 8 by -2. She then asks the students to write down a conclusion they can make from working on these examples. One guy writes as his conclusion, "Mathematics is complicated."
This seems like the perfect opportunity to mention today's New Decade's Resolution. Since today is Sixday in the Eleven Calendar, we look at the sixth resolution:
Decade Resolution #6: We ask, what would our heroes do?
And so I tell him exactly what I wrote on the blog -- "our heroes" are the inventors of much of the technology that we use everyday. The conversation goes just as I predicted on the blog -- he asks when we'll need to flip inequalities in real life, and I reply that much of the math taught between ages 10 and 20 are needed to get to the college-level math that's actually needed to create technology like computers, cell phones, and video games. And when he asks whether school would be better if only those who are interested in math could take second decade math and those like himself could stop taking math at age 10, I informed him that most of our heroes currently working at big technology companies probably didn't like math when they were in seventh grade either. Thus if we'd made math an elective past elementary school, higher technology still wouldn't exist.
Well, the resident teacher continues her lecture at this point, so our conversation ends here. Did this convince him to work hard at math? It's possible but doubtful -- if someone enters a math class believing that all math past the first decade is useless, a five-minute conversation is unlikely to convince him otherwise. The hope is that I at least get him thinking about why he must take math.
When class ends, he leaves the notes I helped him complete right there on his desk. This is one of those teachers who uses interactive notebooks, which is where he should have glued today's notes. I doubt that he even has his notebook today.
Indeed, if I could continue with this student, that's what I'd say to him next. Now that you know why math is important to the technology you enjoy -- and at any rate, you can't get out of it -- you can now work on how to be successful so you can earn higher grades. Step 1 is bring your notebook, and Step 2 is to glue your notes in everyday.
9:05 -- Sixth period leaves and first period arrives. This is the other co-teaching class.
In both classes, the resident teacher announces that she will introduce a new acronym, similar to PEMDAS, to help solve two-step equations and inequalities. As it turns out, the acronym is just SADMEP -- or PEMDAS backwards. It reminds us that in solving, we usually subtract or add (SA) first before dividing or multiplying (DM) on both sides.
Unfortunately, she admits that SADMEP appears to break down quickly:
x - 4 > 2
5
since we must multiply by 5 first before adding 4. Actually, the fraction bar is a problem for both SADMEP and the original PEMDAS. Back when I learned the Order of Operations, it was stated that the fraction bar is in fact a grouping symbol -- it groups together the numerator and denominator as if they were in parentheses.
Thus to solve this inequality using SADMEP, we multiply first because multiplication (M) comes before breaking up the implied parentheses (P) around the numerator.
Notice that it's possible to treat problems such as:
3(x + 1) = 6
as a SADMEP problem -- dividing (D) by three before breaking the parentheses (P). Of course, it's more likely to be taught using the Distributive Property. As for exponents (E), these would never appear in a seventh grade problem, especially since we don't wish to confuse middle school students with why "plus-or-minus" is needed with square roots.
9:50 -- First period leaves. Second period is the teacher's conference period, which leads directly into snack break. This is the Wednesday schedule, when school is out early for Common Planning and there are three classes before break.
10:40 -- Third period arrives. This is an eighth grade science class.
Back in my November 18th post, I wrote how the eighth graders were learning about electricity and magnetism -- which goes far beyond what I learned in middle school as a young student. Today's topic is also beyond what I learned back then -- waves and the reflection of light.
As usual, we compare this to the science I taught at the old charter school. This is a lesson that, of course, I never reached. An interesting idea would have been to teach the reflection of light right around the time that the reflection transformation is taught in Common Core Math 8. This would allow for an interesting connection between math and science.
11:25 -- Third period leaves and fourth period arrives. This is the first of two Math 7 classes.
These students are also solving inequalities. As special ed students they are slightly behind -- so far, they've only seen one-step inequalities. However, today's worksheet will introduce them to flipping the inequality with negative coefficients.
This is a tough lesson for any sub to handle, even a math sub like me. The students have already solved inequalities so far without needing to flip anything, and here comes this stranger out of nowhere telling them that suddenly they must flip the inequality. If I were a regular teacher, I'd try to save flipping the inequality for a non-sub day.
The lesson begins with going over last night's homework, then starting the main worksheet, to be glued into an interactive notebook as classwork. The first part is mostly review -- how to solve a one-step equation, how to graph the solution set, and so on. Then there are four one-step inequalities involving addition or subtraction. Only then do the multiplication and division inequalities begin, including a fill-in-the-blank box regarding what to do to the inequality if the coefficient is negative.
As I wrote in my November 18th post, this class is quite noisy. It takes extra time to correct the HW problems (partly because there was confusion regarding how many points each question is worth) as well as the review side of the classwork. I fear that we can't quite reach the inequality flip problems.
Some of the behavior problems include students throwing paper at each other and blaming each other, and cussing each other out. One guy asks to go to the restroom, but makes a scene as he leaves when he tries to figure out today's date (to fill out the sign-in sheet), and then he makes another scene upon returning when another girl tries to lock him out of the room. (On the other hand, that girl does answer some math questions out loud earlier in the period.)
Most of these are just minor annoyances, but when I'm desperately trying to reach the inequality flips, they become a pain. I try to repeat the sixth resolution and tell the students about heroes and the importance of being quiet to learn math -- but then the discussion turns into argument. And I fear that the argument crosses the line into yelling when one student becomes upset at the whole matter.
While this is going on, some of the stronger students try some of the problems on their own. I assume that they get the inequality flip questions wrong, since no one has taught them how to flip yet. When I do finally reach the flips, there are only a few minutes in class. The regular teacher suggests that I keep them a few minutes in for lunch, and I do, until the teacher next door dismisses them. By then, the damage has been done. I can't claim that any student in this class has learned when to flip the inequality -- and there's a quiz on Friday for which they now won't be prepared.
OK, so what exactly did I do wrong today? The purpose of the New Decade's Resolutions is to reduce arguments and yelling, but there certainly isn't any such reduction today.
We can look at some of the other resolutions for the answer. The first thing I should have done is go back to what I did back on December 4th-6th -- give a definite limit for going over HW. This reminds us of my ninth resolution (every second counts -- for me as a teacher) as well as my old New Year's Resolution on shortening the Warm-Ups. I probably enter this class not thinking of going over the HW as a Warm-Up, but in a way, it is.
Back in my November 18th post, I wrote that it's OK to extend the Warm-Up if the whole lesson is review, but not if there's something new to teach (like flipping inequalities). Therefore I should have stated a time limit for going over HW today. On December 4th-6th I allowed 15 minutes for the Warm-Up and HW combined, but since this is just HW, ten minutes should suffice -- perhaps even just eight minutes, one for each inequality on last night's HW. Shortening the HW time is especially important on a Wednesday, when each class is shorter.
I should remember that as soon as I see any Warm-Up or going over HW in a math lesson plan, I must automatically decide how much time to devote to these opening activities. I should remember to do so without thinking about it -- just as one remembers to ride a bike (fourth resolution). It's usually not necessary to go overboard with "T-minus" as I did from December 4th-6th (plus I wouldn't want to keep explaining what that is) -- I should just watch the clock and monitor the time.
The other thing to do is decide, in advance, whether the "keep students in a few minutes at lunch" punishment as recommended by the regular teacher should be whole-class or individual. Yesterday, my P.E. punishment was whole-class because I didn't know all the students' names. (They have names on their uniforms, but those are easily hidden under a jacket.) Thus sending them to their numbers helped me to identify students. (Note: Speaking of P.E., yelling at them isn't terrible when we're outdoors and it's the only way for students to hear me. It's yelling inside the classroom that I seek to reduce or eliminate.)
Today the students are in assigned seats, and so I preferred the individual punishment. The class has an aide, and she writes down some names to keep in a few minutes at lunch. But then she must leave to punish one student who keeps making faces at the door, so I don't see that list. (This is why she's not in the room when I get to the inequality flips -- otherwise she might have been able to quiet down the students without my having to yell.) Thus I switched to a whole-class punishment. This is wrong on my part -- I should have chosen one way or the other before class and stuck to it.
If I'd decided to make it a whole-class punishment, then I should give incentives throughout the lesson so that the students can leave on time without a punishment. I can choose in advance how far we must get on the worksheet in order to avoid punishment -- slightly past those first flip problems, of course. I can sing the "Solving Equations" song (or the inequalities version, which I first mention in my September 27th post) for each of the review problems and then call on a student to give the right answer. This works better because they now know exactly what's expected of them -- not reaching the selected problem means punishment time, just as "more than two balls on a half-court" meant punishment time yesterday.
(Note: I really do try to sing "Solving Equations" today in fourth period -- but then I feared that singing would just delay reaching the flips even more. Then again, I wrote in my November 18th post that this class didn't particularly like that song anyway.)
On the other hand, I think the individual punishments would have worked better today. I shouldn't have needed the aide's list to determine who gets punished -- those students are continuing to act out after she leaves, so I can just see with my eyes who needs to stay a few minutes extra. Most likely, I would have called them in pairs, since it's mostly pairs of students who are disturbing each other.
In the individual case, I have set times for each problem on the classwork. Going over the HW should take 8-10 minutes (as mentioned earlier), and the teacher suggests passing out tonight's HW with about ten minutes to go. This leaves the middle 20 minutes (of the 40-minute period) for the main part of the lesson. Halfway through this is when I need to get to the inequality flips.
There should be an individual good and bad list here. Each student who answers a question out loud from the classwork gets on the good list. If some students take longer and I fall behind the set pacing, then I just skip those questions (hence fewer get on the good list) to reach the flip questions at the correct time. I can just fill in the missing answers later. Based on the pacing, if there's enough time for me to sing "Solving Equations" (seventh resolution) at any point, then I do so.
And if I'd forgotten either of these classroom/time management plans, then it's better just to accept that we won't reach the flips than to yell at them. I can pass out tonight's HW and just tell them to skip the questions that require flips, since I'd rather them leave them blank than get them wrong and not understand why. (This sounds like a reward -- less HW -- for bad behavior. But I think of it as punishing myself for poor time management.) Then I inform the regular teacher that I failed to reach the flips and told the students not to do those problems -- she'll probably be more forgiving to me if I make a better effort to reach the problems in the next class.
Today's sixth resolution on heroes can still come into play. Notice that unlike yesterday's fifth resolution, today's resolution doesn't specifically identify the 1955 generation as heroes. In fact, I've written that hardworking students can be considered heroes as well. Thus this might have been a great opportunity to identify the quietly working students in the class as heroes, and tell the rest of the class that we can all avoid punishment if we ask ourselves, what would our heroic students do?
12:10 -- Fourth period finally leaves -- a few minutes late due to the punishment -- for lunch.
12:50 -- Fifth period arrives. This is the second of two Math 7 classes.
By this period, I make the corrections that I suggested above. I carefully time going over HW and every problem on the classwork in order to reach the flips by 1:10. I can see that a few of the students are starting to figure out which ones need to be flipped -- which is great, considering how tricky a concept this can be for beginners.
By the way, one girl says that the "one smart guy" in the class is absent today. I don't like this way of thinking -- the students should all be thinking of themselves as smart. I remember this girl from my November 18th assignment, and I'd been planning to show her something to inspire her the next time I see her -- which is today.
I have a notebook from my days at the old charter school. In it I kept notes regarding all the students I taught there. I tell this girl today about the one of the top seventh graders that year -- a hardworking student the first trimester, and how I hope she'd keep up the good work in second trimester.
Here's the kicker: the two girls -- the top girl at the old charter, and the girl I'm speaking to today -- have the same first name. (This is a positive example of someone who reminds me of someone else at the old school.)
Thus I'm following the sixth resolution here -- presenting the charter girl (who is now a high school sophomore) as a student hero. I know -- just having the same name doesn't make you alike (after all, both girls also share their first name with a famous singer), but I'm hoping that maybe I can inspire today's girl to be just a little more like the charter girl.
1:30 -- Fifth period leaves, thus ending my day. Fifth period goes the way fourth period could have gone, if only I had timed it better.
Lecture 3 of Michael Starbird's Change and Motion is called "Another Car, Another Crime -- the Second Idea of Calculus." Here is an outline of this lecture:
I. The second fundamental idea of calculus arises from a scenario involving another car, another crime. [To make a long story short, you're kidnapped and don't know where you are. All you have is a videotape showing the speedometer of the car and what time it is when that speed is reached.]
II. What information can we extract from the videotape?
III. Dealing with variable speeds involves doing a little at a time and adding them up.
IV. The exact distance traveled can't be found with any single division of the interval of time.
V. We can use the same analysis to find out where we were at any moment.
Starbird creates a chart for the kidnapping situation with the following:
Steady Speeds
1 mi./min, for 0 to 10 min.
2 mi./min, for 10 to 20 min.
3 mi./min, for 20 to 30 min.
4 mi./min, for 30 to 40 min.
5 mi./min, for 40 to 50 min.
6 mi./min, for 50 to 60 min.
Total dist. = (1 * 10) + (2 * 10) + (3 * 10) + (4 * 10) + (5 * 10) + (6 * 10) = 210 miles.
Now the professor considers a more realistic scenario where the velocity changes continuously -- the equation v(t) = 2t, where the velocity at time t is always 2t. He approximates this by assuming that the car is steady over small intervals of time:
v(t) = 2t
0 mi./min. for 0 to 0.5 min.
1 mi./min. for 0.5 to 1 min.
2 mi./min. for 1 to 1.5 min.
3 mi./min. for 1.5 to 2 min.
4 mi./min. for 2 to 2.5 min.
5 mi./min. for 2.5 to 3 min.
We traveled more than (0 * .5) + (1 * .5) + (2 * .5) + (3 * .5) + (4 * .5) + (5 * .5) = 7.5 miles.
Since this is an underestimate, Starbird repeats the calculation by using the maximum velocity over each interval (the right endpoints) rather than the minimum (the left endpoints):
v(t) = 2t
1 mi./min. for 0 to 0.5 min.
2 mi./min. for 0.5 to 1 min.
3 mi./min. for 1 to 1.5 min.
4 mi./min. for 1.5 to 2 min.
5 mi./min. for 2 to 2.5 min.
6 mi./min. for 2.5 to 3 min.
We traveled less than (1 * .5) + (2 * .5) + (3 * .5) + (4 * .5) + (5 * .5) + (6 * .5) = 10.5 miles.
The red car traveled more than 7.5 miles and less than 10.5 miles.
And of course the professor repeats the process, assuming that we took pictures of the speedometer every 0.1 minute rather than every 0.5 minute:
v(t) = 2t
0 mi./min. for 0 to 0.1 min.
0.2 mi./min. for 0.1 to 0.2 min.
0.4 mi./min. for 0.2 to 0.3 min.
0.6 mi./min. for 0.3 to 0.4 min.
0.8 mi./min. for 0.4 to 0.5 min.
...
5.6 mi./min. for 2.8 to 2.9 min.
5.8 mi./min. for 2.9 to 3 min.
We traveled more than (0 * .1) + (.2 * .1) + (.4 * .1) + (.6 * .1) +...+ (5.6 * .1) + (5.8 * .1) = 8.7 miles.
v(t) = 2t
0.2 mi./min. for 0 to 0.1 min.
0.4 mi./min. for 0.1 to 0.2 min.
0.6 mi./min. for 0.2 to 0.3 min.
0.8 mi./min. for 0.3 to 0.4 min.
1 mi./min. for 0.4 to 0.5 min.
...
5.8 mi./min. for 2.8 to 2.9 min.
6 mi./min. for 2.9 to 3 min.
We traveled more than (.2 * .1) + (.4 * .1) + (.6 * .1) +...+ (5.6 * .1) + (5.8 * .1) + (6 * .1) = 9.3 miles.
The red car traveled more than 8.7 miles and less than 9.3 miles.
And of course, Starbird tells us that we can keep taking smaller intervals of length delta-t:
v(t) = 2t
(v(delta-t) * delta-t) + (v(2delta-t) * delta-t) + (v(3delta-t) * delta-t) + (v(4delta-t) * delta-t) +...
Dist. traveled from 0 to 3 min. is 9 miles.
The word for the day is "integral," which the professor defines as "integrating a lot of pieces." And just like yesterday, he performs the integral for different intervals of time:
v(t) = 2t
time 0 to 0.5, distance is 0.25
time 0 to 1, distance is 1
time 0 to 1.5, distance is 2.25
time 0 to 2, distance is 4
time 0 to 3, distance is 9
And of course, we finally arrive at the position function p(t) = t^2.
Starbird concludes with the word for the day, INTEGRAL:
(v(delta-t) * delta-t) + (v(2delta-t) * delta-t) + (v(3delta-t) * delta-t) +..., as delta-t approaches 0.
Lesson 8-7 of the U of Chicago text is called "The Pythagorean Theorem." In the modern Third Edition of the text, Pythagoras appears in Lesson 8-6. In fact, Lesson 8-7 of the new version is on "Special Right Triangles," which don't appear until Lesson 14-1 in my old edition.
2020 update: Two years ago, I'd created special activities for the Pythagorean Theorem, which is one of my favorite lessons. Last year, I dropped them because my activity day is supposed to be Friday, not Wednesday. This year, I say, who cares that it's Wednesday? With so many of my favorite lessons this week, I'll now treat the first week after winter break as a special week full of activities.
And so this is what I wrote two years ago about today's lesson:
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.
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 during the year I taught at the old charter school, 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 that 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 the past, which aren't the same as the New Decade's Resolutions, but let's keep them in mind anyway]. 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.
That 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 [old] 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 [old] 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.
That 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 [old] 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 [old] 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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