Personal Responsibility vs. Learning?

Yesterday I had a few students absent and we did a lot of examples involving multiplying binomials, factoring and solving quadratics by completing the square. It was one of those lessons that “just happened.” I had one idea I wanted to nail down and it kinda morphed into a bunch of examples. I made up most of the examples on the fly because I was just gauging their reaction and taking what they gave me. So believe me when I say, “they wasn’t the pertiest lookin’ notes ya ever did see.”

Apparently, they were effective, though. The countenance of the class went from chin-on-hand-it’s-Friday-I’m-tired-here-we-are-now-entertain-us to thank-you-sir-may-we-try-another-cuz-this-is-some-cool-stuff-and-I’m-gettin’-it.

Its tough to reproduce lessons like that so I exported the notes to .pdf and emailed them to the absent students.

I just received this email from one of the recipients:

“Thanks for the notes! They will really help. I do have one question though; did you have to take time specifically out of the lesson to take the pictures or some other program that did them for you? I’m asking this because I think that if you did this every time we learned something new and posted it on your website[s], it would be a good resource.”

So I told him I slaved over my computer all of 30 seconds to export and email as an attachment. Which leads me to my question:

I have always taken a “students gotta take responsibility for their notes and review them regularly” kind of approach which has prevented my from exporting and posting the chicken-scratch covered slides from class. But if posting them is going to help them learn, should I care about the personal responsibility they take on (or don’t take on) in regards to their own note taking?

Whatdaya think?

Note: if you’re interested in what a spur-of-the-moment-ugly-as-heck-yet-equally-effective lesson in my class looks like on static slides, hit me up in the comments and I’ll update the post with a link. I’m posting this from my phone and won’t have access to the notes until Monday.

Triangle Centers Lab

The other day I made up a triangle centers lab for my 8th graders. 

Here is how it went:

Day 0: Homework for tonight is to make a triangle larger than your hand out of some material heavier than paper.  Cardstock or cardboard are ideal.

Day 1: Open GeoGebra and get to work.  Kids got after it.  Some slowed themselves down by not reading directions very well.  The nice thing about GeoGebra is that it’s easy to erase. 

Day 2: Most finished the lab and went onto extension activity.  Those who didn’t finish had a difficult time managing time.  They could do the work, but staying focused was the issue. 

Extension:  Now that you know how to find the circumcenter and incenter, construct an inscribed and circumscribed circle using only compass and straight edge.  These students haven’t done anything with a compass, so I offered a 6th point (assignment was worth 5) for those who could figure out how to do the constructions on their own.  If they chose to look up the “how to” of constructions, they then would have to prove that the method of angle bisecting works. 

David came up with his own extension.  He asked, “why does the centroid allow you to balance the triangle?” 

“Nice question.  Now go away and come back with an answer.”  He’s figured out that the three medians divide the triangles into six smaller triangles with equal area and that would account for equal weight distribution from the centroid.  He can see it in GeoGebra, but is working on a formal proof.  Brandon tried to backdoor me with a proof by contradiction, “well the six triangles have to have equivalent areas because if they weren’t, the large triangle wouldn’t balance.”

Don’t try to beat me at my own game, son. 

Chris’ reflection:  “Hey Mr. Cox, you just kinda gave us a test without giving us instructions.”

“Yeeeeaah kinda, huh?

For Next Time:  Stamp each page after students have demonstrated the correct constructions.  Then allow them to go to the next page.  Take a little more time discussing the difference between “drawing” and “constructing.”

Speaking Mathanese

Kids butcher the Mathanese language.  I’m just sayin’.  We have all these kids who speak text just fine.  It seems to me that Mathanese should be right up their alley.  All we are doing is taking a bunch of words and converting it to symbols.  Should be easy, right?  Not so much. 

I find that kids have a tough time translating algebraic expressions to English and vice versa.  Am I alone? 

Yeah, didn’t think so. 

One of the things that I have been trying to focus on this year is to convey to students the universality of the things they are learning.  For example, cause/effect in language arts becomes input/output in math.  Conflict resolution is the same as problem solving.  Language arts has expressions and sentences, so does math.  Scientific method can compare to making a conjecture in geometry, testing it out and then using inductive logic to arrive at a conclusion (read: rule). 

So what happens when you tell them to translate: the product of 3 and the sum of x and 2?

You get: 3x+2, right? 

Not quite. 

Well I figured we needed to develop a mashup of English and Mathanese; Mathglish, if you will.  Here is what we came up with:

English to Mathanese:

This should read: The product of 2 and the sum of the product of 4 and x and 3.

Mathanese to English:

The key this time was to allow the mashup.  I live in a rural area where the Spanish speaking population is very large.  Many of my kids speak and understand Spanglish.  I have never done it this way before and the kids nailed it. 

How do you do it?

Update:  Just did a quick check for understanding 2nd period and  26/28 kids circled the bases.

Joining the Fray

I’m sorry, I couldn’t help myself.  I can’t let Sean Sweeney have all the fun in class (see here and here).  I do, however, have to credit Sean with giving me the push I needed to actually do this with my class.  Thanks!

The Setup

I told my students that before the Fray was “The Fray”, they were called The Phray and their lead singer was a math teacher.  He wrote a song called “Solve to Save Your Life” but when they were signed they changed the name of the band and made some adjustments to the song on account of “math songs don’t make the top 40, baby.” It took some searching to find the archive of the old song, but I did it.  I also told them that OneRepublic had a song called “Rationalize.”  We’ll see if that one surfaces.

So with no further ado: The Phray performing their hit single, “Solve to Save Your Life.”

If you want the lyrics.

We’ll be releasing the official video soon. 

Reflection: This really isn’t my thing.  I was debating whether or not to scrap the whole thing even though my 4 yr. old can now solve equations.  The thing that really hit me was that me leaving my comfort zone allowed some of my students the freedom to do the same.  I made some connections with kids where I may not have otherwise been able to.  I also learned that playing guitar for 1.5 hours over 3 periods may cause tendonitis.  Advil anyone?

I’m Telling Ya, Lesson Plans are Overrated.

This year, I have kind of introduced equation solving to my 7th graders very informally.   One way I have done this is by giving  them a few balance equations like this:

It seems like it takes the edge off when the variable isn’t there.  But today one of our warmup problems was: 5x + 1 = 2x + 7.

I have been amazed at how many of my students have been willing to attack equation solving by using a guess and check table.  I’ve never taught it that way, but some kids have just taken to it.  After today, I may start to encourage it.  One kid noticed that when you let x=1, the right side is greater than the left side.  But if you let x=10, the left side is greater.  When the balance of power shifts, you know that the answer is between your last two guesses.  Of course, typical guess and check strategy.  But the thing I like about it when dealing with these linear equations is that they are beginning to think in terms of linear systems and how the point of intersection acts as a dividing point between which equation has greater value.  They’re teaching me something. 

But Brandon took the cake.  He says, “Mr. Cox, you can tell the left side is going to be 6 because 5+1=6 and the right side is going to be 9 because 2+7=9.”

“What does x have to be for that to be true?”

“X=1.  But as we make changes to x, the other one is growing faster.”

“How fast is it growing?”

“The left side is growing by 5 and the right side is growing by 2.  So eventually, we know that the left side is going to be greater than the right side.”

“Yeah.  So when are the 1 and the 7 important?”

“Only at the beginning.”

It took all the self control I could muster to keep from talking about initial condition or rate of change at this point.  I’m glad I didn’t because I think I would have ruined an authentic learning moment for this kid.  The thing I wanted to encourage the most in him was the fact that he looked for patterns and then asked questions to help make sense of those patterns. 

One warmup which I expected to spend 5 minutes on turns into 20 minutes of slope, y-intercept, linear systems and problem solving strategies all because a few students took an approach I’ve never taught. 

Another example of the kids re-writing the lesson plan.

Chandler Saves the Day

You ever have a lesson that you thought was going to go pretty well only to have it fall flat?  Yeah, that happened today. 

My 7th graders have been going over quadratics for the past couple of weeks.  I have been out quite a bit on school business, so the progress has been slow, but very rewarding.  Last week, students discovered that if you change the value of “a”, it has an effect on how fast the parabola grows.  I then had them graph a bunch of parabolas whose line of symmetry was the y-axis only to have a student ask,

“Can we move the parabola left or right?”

“Well since you asked…”  So I did what any responsible teacher would…I had them graph a bunch of parabolas whose vertex sat on the x-axis which led to the next question,

“Can we move them up/down and left/right?”

So Friday, we are graphing parabolas in the form y =a(x-h)^2 +k and they are getting it.  This is stuff I couldn’t do until Algebra II with my high school students and these 7th graders are just crushing everything I throw at them. I would even give them a vertex and  a second point and they were giving me an equation because they figured the stretch factor using the second point.  Things are looking good and I am thinking:

Man this is just toooo easy…

Yeah, I know, pride comes before the fall.  Which is what started to happen today.  I have had a planning block.  Now that we have graphed a bunch of parabolas in vertex form, where do I go from there?  Do I start dealing with standard form? Do I show them how to expand (x-h)^2 in order to arrive at standard form?  I am still not sure what the ideal path would be.  But being the “try anything once” kind of guy I am, I figured that since I have already had them:

• Graph quadratics organically (area vs. radius; area vs. side length)
• Graph quadratics with a not equal to 1.
• Graph quadratics with vertex on y axis.
• Graph quadratics with vertex on x axis.
• Graph quadratics in the form y = a(x-h)^2 +k

…then I would focus in on what was necessary to graph a parabola:  vertex and stretch factor.  If they could identify the vertex and a stretch factor, they can graph anything, right?  So today I wanted them to graph a bunch of parabolas in standard form, look for the line of symmetry and recognize the relationship between a,b and the line of symmetry.  I didn’t expect them to necessarily “discover” that the line of symmetry is x = -b/(2a), but I figured that if we graphed enough of them, we might start to notice some patterns.  Once we have the line of symmetry down, then we could start looking at x intercepts which would lead us to factoring and completing the square as well as quadratic formula.  (If my sequencing on this is bad, please save me from myself. )

This is where it started to go bad.  GeoGebra is a great program, but it doesn’t save a weak lesson.  Kids were all over the place with their parabolas and we were getting lines of symmetry like x=.7923496, which wasn’t going to help at all.  I was about to put us all out of our misery and jump ship when Chandler says, “Mr. Cox come here, I think I found something.” 

She had about 10 parabolas that all had the same line of symmetry.  She was making adjustments to the a and b values and recognized that the c value had no effect on the symmetry. So rather than aborting the mission, we just changed course. 

“How about choosing a line of symmetry and keeping the vertex on that line?”

They got right to it.  Tomorrow they are going to come to class with five different a,b,c values and the corresponding line of symmetry.  We will see where it goes. 

Note to the reader:  Quadratics are not a 7th grade standard and these kids will go through it on a deeper level as 8th graders.  So, I am not worried about “finishing” this with them.  I have the flexibility to let concepts marinade for a while.  Usually, I would have just followed the pacing of the book, but I have become very dissatisfied with that.  I am pretty sure that I want to continue from linear relations right into quadratic relations and that graphing is a good gateway to all the other skills that go along with quadratics.  I am just not sure how one skill will best lead into another.  Any suggestions?

Note to self: Quit gettin’ ahead of yourself and be sure to see the lesson through the eyes of a student rather than your own. 

Oh yeah, and thank Chandler.

Upgraded to Pandemic

At least in my own head. No, I can’t shut it off! Not sure I would want to even if I could.

I am getting gas the other day and this shot is screaming at me:

So today, I put the slide up and the kids immediately start talking about slope.  I like that they thought slope, but slope isn’t really going to do much when it comes to fences. So I asked them:

“What would you want to know if you had to build the fence?”

They caught on pretty quickly that one would want to know the length of each board. 

“Alright, then tell me the length of each board.”

“But, Mr. Cox, we don’t have enough information.”

“What do you want to know?”

“We need to know how long the shortest board is.”

Done.

“Alright, now tell me how long each board is.”

“We can’t. We need more information.”

So I had them discuss with their groups what information they had to have in order to figure out how long each board was.  Once they had an exhaustive list, they were to write it on their group’s easel.  We quickly came up with the following:

• The length of the next board.
• The length of one more board.
• The width of each board.

So do we need the next board, or will any board do?  We eventually settled on any other board.

It didn’t take long before students had listed all the board lengths.  Many used the rate of change and then added the increase to each board to find the length of the next.  But it did’t take much prodding for them to realize that having an equation would be nice.  We came up with y = 2.5x + 36 fairly quickly.  The interesting discussion came about when I asked what x represented.

“X is the number of boards.”

“Okay, so go up to the board and point to board #1.”

Which board do you think they pointed to? (You guessed it, the one labeled 36″. )

“So, plug 1 into your equation and check it out.  Tell me if your equation works.”

You would have thought that I asked them to stand in the corner of a round room. But once the “but this equation haa-aas to work” wore off.  They realized that it wasn’t the equation’s fault.  It was how they defined x.  Board 0 is important because we aren’t actually counting boards, we are counting the number of increases. 

Reflection: I think that the lesson went really well, but it was very telling how many students wanted to impress with their knowledge of the vocabulary as opposed to just looking at the problem and asking the obvious questions.  They were trying to be mathematicians rather than someone who just needs to build a fence.  Next year I want to do a better job of introducing concepts a bit more organically as opposed to “here are the rules, here are some examples, let’s get to it.” Students are much more engaged when the information is given a little at a time.  It keeps them from answer chasing and allows them to think a little.  It may take a bit longer to deliver the lesson, but the benefit of having kids think about the math is priceless.  

Questions: What else could I have done with this image? 

 

 

Intro to Quadratics: 7th Grade Style

Can I just say that I love middle school kids.  I mean, sometimes keeping them on the same page is like trying to herd a bunch of cats, but I love them.  Now that testing is over, it is time to start preparing my 7th graders for the wonderful world of quadratics.  We have been doing a bunch of activities on linear relationships and the results have been pretty good.  My kids have a pretty firm grasp of the following:

• Slope and rate of change mean the same thing.
• If the rate of change remains the same, then we have a line.
• The initial condition is the y intercept.
• If the initial condition is 0, then we have a direct variation.

They like lines; they are comfortable with lines.  It is time to take them out of their comfort zone.  So here is how it went down:

We had already done some activities on linear relationships like:

• Farenheit vs. Celcius
• cm vs. inches
• km vs. miles
• start height vs. rebound height for a bouncing ball on concrete between 75 and 80 degrees with no wind resistance. (Alright, we didn’t control the experiment that much, but they still saw that the ball rebounded to about 70% the original height.  Daniel learned that baseballs don’t bounce very high when you drop them from 1 meter and the seams mess up the bounce.)

The latest installment was to have them bring in a few circular items and then find the relationship between radius vs. circumference.  This led quite nicely into, ‘Well, since you have some circles here, you may as well calculate the areas too.  Graph those compared to the radius and see what you get.”

It was interesting to see how many kids tried to force it into a linear relationship. 

Angel recognized that “choosing a bunch of circles with around the same radius doesn’t tell us much, huh, Mr. Cox.” 

“Nope, next time we may want to expand our sample space.”

Regardless, by the end of the activity, they understood that sometimes we have relationships that are “curved” or “non-linear.”  Fareen recognized that we get a “half of a parabola.” 

So from there we do some work with the side length vs. area of a square.  Hey, we may as well start at the beginning, right?  But it was the simplicity of the exercise that produced the magic that I never saw in 10 years of teaching high school kids the same thing. 

“Hey, Mr. Cox, it isn’t a line because the slopes don’t stay the same.”

“Yeah, so what?”

“Well the areas increase by 3, then 5 then 7.  But the increases all increase by 2.”

“Okay, so what does that mean?”

“The rate of change has a rate of change.”

At this point I get goosebumps.

This is when Abel, asks: “What will happen if we cube x?  What happens to the rate of change then?”

Couldn’t pass this one up, so we drew up a chart and did it.  The kids concluded that for a cubic: the rate of change of the rate of change has a rate of change. Oh, and the number of times we have to check the rate of change tells us what the exponent is.

Moral of the story: Don’t assume that your lesson objective is the right one.

or, Sometimes it is better to follow the herd of cats.

Dear Dan,

You are really starting to make me angry.  I was content to just do my job, help as many kids as I could, do some extra stuff to help them make it connect to their reality.  And then at 4:00pm shut it down, go home, kiss my wife, play with the kids, have a nice dinner, watch a little TV and go to bed.  But no!  You just had to start in with all this “using pictures to help kids learn math” stuff.  You couldn’t leave well enough alone.  When I couldn’t get Graphing Stories to work, you couldn’t just say “sorry dude, I am not sure why those chapters won’t play.  Better luck next time”… no you had to send me a copy and not even take the reimbursement I offered.  Who do you think you are?

Man I can’t even go to Target with my family without trying to take a picture of something.  You know how disruptive that is?  You know how hard it is to hold the baby, push the cart and snap a pic at the same time?  I am looking into having an extra arm grafted onto my torso.  You think my insurance will cover that?  Nooooo! And it is all because of YOU! 

The worst part is that I have these video cameras lying around my classroom and a blue screen in the library so I had to skip lunch the other day and take some footage that has resulted in some stills like these:

 

 

 I mean, look at that.  How can they not see that the ball is accelerating as it falls?  I don’t want that.  I want them to depend on me to tell them that gravity is an acceleration.

I'm not mad at you for this one, I actually think it's pretty cool.

What’s even worse, is I have all this raw video footage and I have no friggin’ idea what to do with it. What am I supposed to do, have students graph the height of the ball vs. time and realize that there are some relationships that aren’t linear?

My students are even getting into the taping.  They look so cute  and happy throwing the ball back and forth, but little do they know that one day this concept of math actually helping them  to interpret the world around them can consume them.  What’s next?  Are we going to start treating mathematics like a humanity and sit around discussing it as if it were a piece of literature or work or art?  Don’t you know that math is only supposed to be important  8:20-2:55 in Room 405 from September to June?  Don’t you know that math is supposed to be a set of rules that we force our kids to memorize until April 22 and then they aren’t supposed to think about it again?  Get with the program, will ya?

And no textbook?  C’mon, man!  What are you thinking?  Those things were all written by people who really care about making math matter to our children. Don’t you know that the more a student sits in front of a textbook, the more they learn?  I saw some research done by an independent agency Houghton, McGraw, Holt and Littell that says students can actually teach themselves with these things.

In closing, you have got it all wrong.  Kids want to have subjects forced upon them.  They want to be told the rules, they want to mindlessly copy exactly what the teacher says and does and they especially want to ask questions like, “I don’t get it.” 

Inquiry?  Yeah, right!

Sincerely,

David

p.s. Can I make a cameo in Graphing Stories vol. 2?

p.p.s Can you all help me package this up into a series of lessons?

Confessions of a 13 Year Veteran

Ok, I admit it.  My lesson planning skills suck!  It just hit me the other day. I am reading all your blogs and seeing all the cool things you are doing in your classrooms, and all I can ask myself is, “Why don’t I think of stuff like that?” 

 I think I know why.  I cut my teeth on CPM (another post on CPM coming soon) and all of my lessons were pre packaged.  All I needed to learn was to ask good questions and get out of the way.  That suited my style very well.  By the time NCLB came around and we dropped CPM, my school was already going lock step with pacing guides and common assessments.  Neither one of these approaches allowed for much innovation nor did they require a bunch of thought.  It didn’t help that I became the varsity baseball coach in my first full year of teaching and spent way more time planning practice than I did lessons.  The real problem was that I didn’t really know how much I had bitten off until recently. 

Now that I am teaching middle school math as the high school guy brought in to handle all the GATE kids, I am not only responsible for my own classes but for setting a tone and pace for an entire department.  I feel more accountability now than ever before.  It has been in these last three years that I have really started to ask reflective questions regarding not only my practice, but good practice in general.  I also have access to tools that I didn’t even know existed when I was at the high school. 

So here is a snapshot of where my lessons were compared to where they are.  Feel free to take your shots and help me make this better.  This lesson is stolen  borrowed adapted from a lesson in the April 2009 issue of Mathematics Teaching in the Middle School. 

What I would have done before:

I don’t know if I would have even bothered to re type the lesson, I may have just photo copied it and given it like this:

I would have passed out this handout and told the students to represent each savings plan as a: graph, input/output table, algebraic expression and verbal expression.  My students would have muddled through it and most would have just jumped through another hoop. 

What I did today:

I toggled between these four slides and asked students to pick out what information they thought was important.  It was interesting that some students felt it necessary to try to copy the information while the rest of the class was willing to observe and jot down what they thought was essential info.

 

Questions students came up with:

• Does the chart represent the amount of money Diana makes each week or the total she has saved?
• Will she continue to save at this rate? “Yeah, look at the “dot, dot, dot.”

Teacher: “What can you tell me about Yoni?”

Class: “She has $300.”

Teacher: “How much will she have in 3 weeks? 2 years? 100 years?

Class: “$300. $300. $300.”

A few students were quick to point out that we only needed to focus on two or three key points in order to make a generalization of Michael’s situation.  They immediately zoomed in on (0, 30) and (10, 80).  However, I did have one group who decided to focus on the last point (20, 130). They decided that Michael averaged $6.50 per week.  But when they were asked if their trend would continue, they quickly realized that they had failed to consider the original amount of Michael’s savings.

Chandler, one of my 7th graders decided to ask, “Does x represent weeks?”  Beautiful, that info was on the next slide. 

The question of the day had to come from Paul: “So, if the only point we know for sure in Michael’s graph is (30, 80) wouldn’t he have more money if he actually started at (0,0)?  In his mind he was seeing this…

…which led to an interesting discussion on how slope represents the amount of money saved per week. 

My students are starting to catch on to how things work in my class because as soon as these slides came up:

I asked, “What is my next question?” To which they all responded:

Once they had made the decision on which represenation they prefered, I had them represent the other three savings plans the same way.

At first the justifications for “why” one representation was preferred over another were weak at best, lazy and unthoughtful at worst:

• An input/output table is more organized.
• It’s easier to see the data.
• I like it better because it is better and less worse than the others.

Okay, that last one was mine, but you get the point.  Being the father of four boys, I can smell an “alright get off my back, already” answer a mile away, I continued to prod.  Eventually we had answers like this:

• I prefer the chart because it is easier to see which numbers go together; on the graph, you have to work a little more to see which numbers are related.
• I prefer the graph because I can actually see how the different points relate.  The slope helps me see how fast someone is saving.
• I prefer the verbal expression because it helps me understand the overall situation better.

We eventually got to the discussion on “Who has the most money right now?”

Hands fly upright around the same time I hear: “Yoni! Danny!” 

Then Adrian pipes up, “When is now?”

Gotta love that kid!