Sometimes It's an Accident

We are learning about angles in grade 7.  Yesterday, I asked students to draw 10 different angles (at least 3 acute and at least 3 obtuse) and measure them with a protractor.  Historically, I've been really bad at teaching students how to use a protractor, but they made their best pass at it.

Today, I drew random angles around the classroom on our whiteboards and was going to ask for volunteers to walk up with their protractors and measure them at the board.  But before doing so, I went around the classroom asking the class to classify the angles as acute, obtuse or right.  When we came to a right-is angle, the class was divided; some said acute, some obtuse and a few said right.

"Ok, so what do we do?"

Sam picks up his protractor, holds it out in front of his face, closes one eye and peers through the hole at the bottom of the protractor.  I stepped back and watched what he was doing.  He was peeking at the vertex into the hole, while lining up one of the rays with the guides at the bottom of the protractor.  He then says, "Mr. Cox, it's pretty much 90 degrees."

Now, I loved this for reasons.  1) This kid invented a hack for doing a little better than estimating 2) The entire class understood what he was doing and started using his hack and 3) I had never thought of doing this before.

But that isn't even the good part.

I had drawn three straight angles that had a second ray breaking it into two supplementary angles.  We argued a bit about whether there were two or three angles shown.  Everyone eventually agreed that there was an acute, obtuse and straight angle represented.  Then we got to the measures.  Glad to say, the pairs they measured were all supplementary.  Then came the two students who made mistakes (on purpose).  We discussed how some students use the wrong numbers on the protractor, but if they classify the angle first as acute or obtuse, that helps them know which number to use.

That wasn't the good part either.

I wanted to start a conversation on supplementary pairs and I was going to use the drawings of straight angles broken into two supplementary angles that were on the board.  But then I thought about Sam and his protractor hack.  Change of plans.

"Ok, take a look at your protractors and look at the pairs of numbers.  What do you notice?"

We created a list of numbers.

170   10

160   20

150   30

140   40

130   50

120   60

110   70

100   80

 90    90

Then I added one more entry:

34   ?

"Without using your protractor, make your best guess about the number that should be paired with 34."

We had three answers. 

154   146   156

Argument 1:  "I think it's 154 because 150 and 30 are paired together.  Since we added 4 to the 30 to get 34, we need to add 4 to 150 to get 154."

Quick check of the class to see who understood the argument. I was careful to let them know that saying they understood the argument was different than agreeing with it.  They understood. 

Argument 2:  "I disagree with 154 because on one side the numbers are increasing and on the other they're decreasing.  34 is between 30 and 40, so our answer needs to be between 150 and 140.  So, I think it's 146."

Homework:  Who do you agree with and why?  If you think the answer is something different, make an argument.

Can't wait until tomorrow. 

I Had My Doubts

Dawson is the oldest.  Aidan is the musician/artist.  Bohdan is the baby.  And Jabin is, well, Jabin.

But my second son, Nevan just finished his first semester of high school.  He hadn't been in a classroom since the end of his second grade year and he hadn't done traditional math course work in years.  Instead, he did a lot of interesting math problems.

So, I has a little hesitant when the local high school recommended he take math 2 and chemistry as a freshman based on a single placement test.

Our conversation went a little like this:

Me:  Nevan, I know they say you're ready for these classes, but remember, you're being asked to take math 2 and there's a lot of content in math 1 that you haven't done.

Nevan:  I know Dad, but I want try it.

Me: Ok, that's good enough for me.

I mean, he wanted to challenge himself.  And even though I have reservations about acceleration in general, I knew Nevan would have support at home.   So, we decided he could give it a shot on the condition that he could always move into a math 1 class if he felt overwhelmed.

Yeah, that didn't happen.

Suffice it to say, he has done very well in both his math 2 and chemistry classes so far.

The takeaway for me here isn't that my wife and I have done anything extraordinary.  The takeaway is that my son has learned how to learn.  He reads everything that he can get his hands on, is curious and is willing to take risks.

The fact that he had some content gaps didn't matter because he was willing to do what he needed to do in order to fill them.

This really makes me wonder about all of the time we spend worrying about "gaps" in student's knowledge.  I realize that claiming this as a mindset win would be shortsighted as Nevan has advantages that most of my students don't.  However, it's a clear indication that habits of mind matter.  They matter a lot.

Amplifying Student Voice

Recently, I had a student "teach" me how to solve a Rubik's Cube.

This experience found its way into our next staff meeting.

This student and I recreated the entire scenario.  He had a cube and I had a cube.  This time I had an audience of my peers, not his.  His instructions were faster than I could follow and I got lost a few times.  He said "move top left", I went right.  He said "bottom right", I went left.  He whispers to me, "now I know how you guys feel.  This is hard."  I couldn't look at him because I was dialed into my failure.   I began to feel flushed and was tempted to just give up and tell the staff, "well, you get the point."  We didn't quit and I'm glad because the tension in the room was important.  This is the same tension our students feel when right answers matter and they don't know them.

I juxtaposed this experience with a visual pattern.  I gave very simple instructions for the staff to demonstrate what figure 100 would look like.  This wasn't a math activity at this moment; it was an opportunity for individuals to describe what they see and understand.  There was no "right way" to describe the 100th figure.  You want to draw a picture? Go ahead.  Use a table?  Sure.  How about a verbal description?  Of course.

The math notation or vocabulary wasn't necessary for everyone to enter into the task, however it could prove useful for explaining to someone else.

There are so many layers to this experience for me.

As we were going through the process of trying to solve the cube, I was incredibly frustrated.

My "teacher" was telling his story without considering mine.  He shared his connections and ignored mine.  He gave many instructions and kept going assuming I heard them and responded appropriately. I didn't.

This is where we fail our students.  We assume we have a shared understanding/experience with our students.  We don't.

A staff member later told me she was frustrated because she wasn't sure what connections I wanted her to make.  But then she said, "Then I realized, that was the point.  We needed to make our own connections."

This is what Max means when he says 2 > 4.  Or Dan when he suggests we cast students as the hero.

At least, that's what I think.

"Figures Never Lie...

...But Liars Always Figure"

I remember a professor saying this to class many years ago.  It stuck with me.

Good

Hey, look!  Since 2009, unemployment rates are going down. Wow, let's graph a regression line and marvel at that negative slope.



Un-good

But wait, over the same period of time, labor force participation has also been on the decline.



How do we help our students make sense of this?

When The Activity Isn't Enough

I love the learn by playing nature of activities like Marbleslides.  In fact, I just visited a classroom yesterday where kids were digging in.  It was interesting to watch as students engaged in this environment.  It was fascinating to try to understanding their thinking.

If we walked into 100 classrooms where students were learning about graphing lines in slope-intercept form, we'd find more than our fair share of lessons where some sort of direct instruction is happening.  We'd likely hear academic vocabulary, see a formula for finding slope and probably even a general equation like y = mx + b.

I'm not against those things.  However, I'm for giving students an experience that can be precisely described by knowing those things. Activities like Marbleslides do this.

The activity isn't enough.

Here are four different students who are all engaged in the same activity.  Consider the following questions:

What do you notice?
What questions would you ask this student?
What could you have offered this student prior to starting this activity?


Student 1

Student 2

Student 3

Student 4

Here's what I see.

Student 1 is WAGging like crazy.  These are just random guesses. No adjusting or learning from feedback.  If this student achieves success, it'd be like a blind squirrel finding an acorn.

Student 2 is an answer chaser.  I mean literally, look at the guesses.  Once this student sees which part of the equation to adjust and the line moving in the right direction, the adjustments are incremental.

Student 3 is a strategic thinker.  Slope? Nah, don't need it.  y-intercept? Yeah, that's the stuff.  Let's trap the answer and close in on it.

Student 4 is engaged, believe it or not.  This student is paralyzed by options.  Just waiting for the correct answer to pop into the brain.


So, how do you respond to each student?

Lessons in Pedagogy With Papa Frank

 AMORIS LÆTITIA 261:

Obsession, however, is not education. We cannot control every situation that a child may experience. Here it remains true that “time is greater than space”. In other words, it is more important to start processes than to dominate spaces. If parents are obsessed with always knowing where their children are and controlling all their movements, they will seek only to dominate space. But this is no way to educate, strengthen and prepare their children to face challenges. What is most important is the ability lovingly to help them grow in freedom, maturity, overall discipline and real autonomy. Only in this way will children come to possess the wherewithal needed to fend for themselves and to act intelligently and prudently whenever they meet with difficulties. The real question, then, is not where our children are physically, or whom they are with at any given time, but rather where they are existentially, where they stand in terms of their convictions, goals, desires and dreams. The questions I would put to parents are these: “Do we seek to understand ‘where’ our children really are in their journey? Where is their soul, do we really know? And above all, do we want to know?”

 When I first read this, I couldn't get it out of my head.  One sentence in particular, stood out.

In other words, it is more important to start processes than to dominate spaces.

As a father of five boys, the struggle with finding that line between holding on and letting go is real. Fortunately, my wife and I have always approached parenting through the lens of "we are preparing them to leave."  However, it doesn't make the struggle any less difficult.

It didn't take very long for my thoughts to extend to education in general.  So much of what we do dominates student spaces instead of helping them start processes.  And even if we "start processes," they're all too often processes we, the adults, determine to be important.

How do we help students determine their own processes?

How do we help them strengthen their own voice?

How often do we pretend to help students start a process when really we're just masking the ridiculous game of "guess what I'm thinking" that we'll publicly reject, but privately use as a default setting?

Questions?  I have many.  Answers? Not so much.

But that's my process.

"What I See Doesn't Matter...

"... All that matters is how you see it."

This is such a difficult thing for students to believe.  But I try to say this in some variation every day to my students.  

Grace nails the sentiment here.

points out diff btwn "show your work" as window into S thinking vs as backwards engineering what the teacher wants me to write

— grace a chen (@graceachen) November 3, 2016

 Once students begin to believe that the way they see something is the currency, then our job is to simply help them refine their communication so their audience can understand them.  Only then does the syntax of mathematics matter.

"Help me understand you."

"Help me see what you see."

These are the things we should say more often.

Pretend I'm Not Here

Yesterday we worked on this pattern. 

By the end of the period, we had two different rules.

n + n + 5     or      (n - 2) + (n - 2) + 9

Today we had to decide whether or not these two rules were equivalent.  We had a brief discussion about the different ways students could make their argument:  numerically, visually, symbolically or verbally.  I asked each student to choose a method they preferred and spend a few minutes constructing an argument.  The plan was to then have them pass their journal around the group and have their partners help them make their arguments more convincing.  

As I circled around the classroom, I noticed the work of a particular student who doesn't yet have the confidence I believe will eventually show up.  I stopped and asked him about his work. 

Me: So, tell me about what you have going on here?

Student:  ...

Me:  What type of argument are you trying to make here?

Student: Numbers. 

Me:  Ok, so what numbers are you choosing?

Student:  I chose 55.

Me:  Does it work for both rules?

Student:  Yes. 

Me:  Now that I'm sitting here with you and hear you explain, I can totally understand what you're trying to say.  

Me:  Let me ask you something:  Do you think that if you ripped this page out of your journal and left it for me to read after class, I'd be able to understand your argument?

Student:  No, I don't think so. 

Me:  Can you treat this as a rough draft and try to convince me as if I wasn't here?

Student:  Yes. 

Me:  Ok, great.  I'll come back and check in a bit. 

After a second pass around the class, I come back to this:

I asked if I could have his permission to take a picture of both and show it to the class.  We'd keep it secret if he wanted, I assured him.  When I projected the first iteration, other students tried to explain his thinking.  When I showed the work of the "second student", we all agreed it was much easier to follow the thinking.  Then I said, "This is the same kid."

Class:  "Wait, WHAT?!  

The coolest part of this was that when I wouldn't say the name of the student, many of his classmates said, "It's obvious Mr. Cox.  Look at him."

He was beaming. 

Making Connections

One of the things I really appreciate about the CCSS New California State Standards is that we want students to make connections across domains and grade levels. And, while some may disagree with me here, I appreciate the transferrablilty of the standards for mathematical practice. Things like attending to precision, constructing viable arguments, critiquing the reasoning of others, looking for and using structure, and problem solving in general all play in other content areas life.

Any chance I get to make a connection to another area, I do it.  I read an article a while ago by Hung Hsi Wu where he treated a variable as a pronoun.  It made sense to me.  It makes sense to my students, so we go with it.

We are about to dig into equations and expressions, but I really can't stand how textbooks approach this.  You get to see maybe one or two simple expressions that may be tied to a context, but then a million exercises with expressions so complicated, there's no way a kid can tie it to anything that matters.

So here's where pattern problems come in.  Fawn has done a tremendous service for us.  I'm also really digging Dudamath lately because I can be more intentional with the patterns I put in front of my students. Seriously, if you haven't played around with this site, go there now.  It's pretty amazing.

We've done a few pattern problems and we are getting the hang of doing the generalization, but writing an expression has been more difficult.  So, here's where Wu helped.

Our morning announcements just mentioned our volleyball team won yesterday, so that provided a nice context.

Maria played volleyball.
Shanay played volleyball. 
Teresa played volleyball. 
Jan played volleyball. 
Jill played volleyball.

I wrote these sentences on the board and asked if they could write one sentence that captured the essence of all the others.

"Maria, Shanay, Teresa, Jan and Jill played volleyball."

became

"Maria and her friends played volleyball."

which eventually became

"She played volleyball."

Right next to the sentences, I wrote the following expressions:

3 + 1
3 + 2
3 + 3
3 + 4
3 + 5

and it didn't take long for us to settle on some version of 3 + n.

The groups then went to work on today's pattern problem.  The use of some sort of variable when trying to describe a rule made it's way into their work.  Many of the groups are still in progress, but movement was made today.  Let's see how it goes tomorrow.

Building Fraction Sense

The struggle here is real.

These two samples are from the same student. Makes me think about what we prioritize, and about who else is hiding. pic.twitter.com/ePCfiEVVWK

— Kent Haines (@KentHaines) September 13, 2016

A student has no idea where to place a fraction on a number line (because fractions aren't numbers, of course) but can convert to a mixed number like a champ.

My attempt to help out:




This applet gets at the heart of the things I've enjoyed working on lately.  The initial estimate offers very little help, but as the student progresses through, they have more references which allow the revisions to become more precise.  When my students worked with this applet, there were audible groans when I asked them to lower the lids on their computers as well as exclamations of "I got it!" when they moved closer to 0% error.

Here's a GeoGebra book that goes from estimating fraction to addition to multiplication.  I'm still working on division, but that should drop soon.

When Your Good Friends Don't (But Should) Get Along

I swear I'd give CalcDave's left arm (you're right handed, right Dave?) in order to be able to embed a GeoGebra applet into a Desmos activity.  I mentioned it, once or seven times, but that's right about the time the Desmos customer service director seems to drive into a tunnel.

I mean, I don't hate this activity or anything.

But I hate this slide.

Stupid Math Notation

Sometimes students show a misconception that makes me pause and wonder how we can continue without clearing this up.

Sometimes the misconception isn't their fault.

Take the "-" symbol for instance.  Are we talking about subtraction?  Negative numbers?  How about "the opposite"?  Or inverse; maybe it's inverse.

I gave students this number line today with the prompts.

1.  Tell me everything you can about the number P.
2.  Show where -P is on the number line. Tell me everything you know for sure about -P.

They did very will with the first prompt.  Lots of responses like:

"P is on the negative side."
"P is a negative number.  It's between -2 and -3."
"P is probably about -2.7 because it's closer to -3 than it is to -2."

Ok, I'm loving this.  Then they drop the hammer on me.

"-P is negative."
"-P is also on the negative side."
"-P has a negative sign in front of it so it's also negative."

What are your first steps when you encounter thing like this?  

The (Selfish) Reason I Keep Teaching

Pick a teacher's blog.  Go ahead, pick one.  Go through the archives and you'll likely find a post talking about vocation or calling or some other noble reason to enter the profession.  You'll also find some variation of the phrase "I don't teach subjects; I teach children." These are all true, but I don't think they get at why I teach.  I mean, I'm no Mr. Shoop and, while, I do think there's  satisfaction to be found in helping others, I'm not quite ready to side with the Tribbianian philosophy of good deeds. What I am willing to admit is that one of the things that keeps me teaching is a little selfish.

Let me explain.  When I was in high school, I took one of those aptitude tests.  The results of that test told me I should either be a teacher, a  youth pastor or, yes you guessed it, a cab driver.  At first, I was thinking, "Cab driver?  What's that about?"  But as I thought about it, these three career paths have something in common:  people.  So, then why teaching?  I'm going to try to impact people no matter what I do.  So why teach?

That brings me to the selfish reason:  Teaching is a case study in why people do what they do.  I'm really interested in that.

Dan recently asked about the motivation for moving away from the text book when lesson planning. I think this gets at why I'd rather do my own thing even though I didn't realize it when I first responded.  I want to know why kids do what they do, and most textbooks can only expose what they do.  If I make my own activity, I can ask the questions the way only I ask them. It's my way of starting the conversation with my students.

Today was a great example of this.  We were working through a Desmos activity where kids had to model sums using a number line.  (I really wanted them to be able to sketch on an interactive graph, but, whatever, can't have everything.)  But this activity exposed two really important misconceptions. One I was very aware of and the other I had never considered.

Misconception #1

I've seen this one before.  Often times students count the numbers and not the spaces.  Ok, got it.  I know how to deal with this. 

Misconception #2

At first glance I thought I had this one pegged too.  Students are just stating the length of the segment.  Through the discussion, however,  it came out that a significant number of students said the blue segment represented positive three because it was on the positive side of the number line.  

In 20 years, I've never seen that.  

It led to a nice chat about direction and location and how these can influence the value of a number.  

I don't have this conversation locked down.  And that's why I want to come to work on Tuesday. 

That and I want to see if Desmos has those sketchable interactive graphs yet.  

Math Don't Break

Integer operations are always an interesting endeavor with 7th grade students because they come pre-loaded with so many rules.  So. Many. Rules.

We've been talking about making our own rules, so we have this sequence of products and I ask students to discuss what patterns they notice.

-3 (3) = -9

-3 (2) =  -6

-3 (1) = -3

-3 (0) =  0

-3 (-1) = ??

Stuff we noticed:

"It starts with a -3 every time."

"It goes down by 1."

"It changes by 3."

I zero in to the apparent contradiction in going down by 1 and changing by 3 so we can clean up the language a bit.  This starts an nice little exchange about whether or not going from -9 to -6 is an increase or decrease.  We conclude it's actually an increase.  I have to remember to take my time here because this isn't an insignificant point:  Kids seem to think in absolute value.  

So what comes next? 

I wrote down everything I heard.  

"3".  "-3".  "4".  "-4".  

"Wow!"  I say.  "We've got a great argument about to happen.  This is awesome!  So many different opinions.  So which is it?"

Some minds change when groups start to discuss.  The students who thought 4 or -4 were thinking of sums and not products.  That leaves 3 or -3.  

"Ok, so which is it?"

If I had a dollar for every time a student said "A negative times a negative is a positive" followed by "because my teacher told me", I'd have all the dollars.  

But then Isaac offers a reason worth looking at. 

"I think it's -3, because positive 3 times positive 1 is positive 3, so negative 3 times negative 1 is negative 3."

So I write the following on the board:

(pos) (pos) = pos

(neg) (neg) = neg

We talk about this pattern Isaac. has noticed.  "Does this work for you all?"

Jordan speaks up, "I don't think so.  It has to be positive three so that it doesn't break the pattern."

"Which pattern is that?"

"The pattern goes from -9 to -6 to -3 to 0.  It's increasing by 3 each time so the next answer has to be 3."

"Why would that be so?" I ask. 

Then Vanessa chimes in.

 "Because math don't break."  

Strategy vs. Procedure

I really want to focus on students being mindful of their process.  What they are doing is important, but they really need to know why they're doing it.  We've been doing daily exercises, How Many Squares?  that are based on Michael Fenton's activity, How Many Peaches?

We usually highlight different student strategies and have spent some time developing a continuum of strategies that looks something like:

counting --> grouping/adding --> skip counting --> multiplying --> writing/evaluating math expressions

This student's particular strategy generated a nice conversation.

I asked whether or not students thought this was a strong strategy.  Responses were less than enthusiastic so it was time to move a little.

Me:  Alright, if you think this is a strong strategy stand on this side of the room;  if you think it's not move to the other.

It was 31-2 in favor of the strong.  So I walk over to the "not strong" side and make my case.

Me: It can't be a strong strategy because the answer is 84 and this student said it was 76.

About half the class moves to my side.  I figure it was an even split on who was convinced by the "right answer" argument and who was convinced by the "I'm your teacher" argument.

Two students on the strong side raise their hands.

Student 1:  I think it's still a strong strategy because he probably just made a mistake.

Me:  Probably?  Where does that fall on our argument continuum, gut level, some reason or convincing reason?

Student 1:  Some reason.

Me:  Ok, great.  Can anyone take it to the next level?

Student 2:  I think it's still a strong strategy because he just counted 11 instead of 12 across the top.  He still multiplied right, but he just used the wrong numbers.  Everything else was good.

Yeah, that'll play.

From the Gut to the Head

Keeping in mind that we often get what we measure, I started from day 1 talking to students about an argument continuum.

Gut Level Answer

We're all pretty good at this one.  Offer an answer, but when asked why we .   This is often a student's default, especially if they're used to an answer getting culture.  


Answer With Some Reason

This is a step above the shrug, but isn't entirely satisfying.  I'm ok with students being in this area for a bit--"I think because " even if isn't completely convincing.  

Answer With Convincing Reason

I'm not really pleased with the wording on this one, but the gist is that we are looking for a student's thinking to be able to stand the test of peer review.  Does it convince others?  Can others use your process and arrive at the same conclusion?  If so, then we'll call this good.

I think this is something that I've had in my mind for as long as I've been teaching, but being more explicit about it with students has been beneficial.  I hear things like "show your work"  which has morphed into "show your thinking"  and I think they both are trying to get at the same thing.  Unfortunately, I think students usually interpret these in a quantitative way that amounts to something to check off the list.  Did I write a number of things down because teacher asked me to?  Yep, so let's move on.

As students begin to look at the quality of their work, we all win.

Don't Call it a Comeback...

...I haven't been here in (what seems like) years.

The past couple of years have been a whirlwind of change.  Full time math to full time elective to elective/part time math coach and now finally...

One section Math 7, three sections of electives and afternoon math coach.

Oh, and five of my 17 kids are now 17, 14, 11, 9 and 6 years old.  The older two are a senior and freshman (respectively) in high school while the younger three are still reaping the benefits of having an amazing mother who is willing to donate herself to homeschool.

It seems like a lot has changed since this blog was more active, but I hope to catch up with you all.

It Was a Simple Question...

...until it wasn't.

Features of Functions is the unit and the problem focused on the following graph.

Typical questions like:

1. What is f(2)?
2. For what values, if any, does f(x) = 3? 
3. What is the x-intercept?
4. etc.

 Groups were working well together and I asked them to write their agreed upon answers on their easels.  As we looked around the room, everyone agreed until we got to #6. 

      6.  On what intervals is f(x) increasing?

Naturally, everyone said the function increased on the interval [-4, 6]. Everyone except Group 7.  They're always contrary like this.  Probable just stirring the pot a little.  Just pat them on the head and move on.  

Except J says, "Mr. Cox, I stand by my answer."

"Wait, what? Do you know who I... ahem, tell me more."

"Well, since the function starts at -4, it's not increasing yet.  And since it ends at 6, it stops increasing."

R chimes in, "Ok, J.  I see your point and I'd be willing to say the interval is (-4, 6] because at -4 it hasn't started increasing, but at 6 it's been increasing and then stops.  Maybe we include one and not the other."

This took us on an interesting discussion about what we really mean by rate of change, increase and decrease; how our interpretation is influenced by our left-to-right reading convention; and how many points we actually need to identify a rate of change.  We talked about instantaneous rate of change and how you can actually have a "slope" using one point.  

But I still have questions.  J was looking at the endpoints of the functions as if they were a relative maximum and minimum.  They aren't included in the increase interval because the rate of change is actually 0 at those points.  Was he correct to think this now?  Was this simply a really good wrong answer?  Should he be considered correct on the argument alone?  

What say you?

Sizing Up Diagonals

Question: How can we find the diagonal length of any rectangle?

Part 1:  On a sheet of paper, draw the following rectangles:

            A.  3 x 4

            B.  5 x 12

            C.  6 x 8

            D. 8 x 15

Part 2:  Draw the diagonal for each rectangle. 

Part 3:  Measure the length of each diagonal and record it in the table below.

Rectangle	Side 1	Side 2	Diagonal
A	3	4	5
B	5	12
C	6	8
D	8	15

Part 4: Use the side lengths of the rectangle, addition, multiplication and square root to write an expression that equals the diagonal length. 

Rectangle 1 would look like this:

Be sure that your equation works for each of the rectangles. 

Part 5:  Write a general equation that could be used for any rectangle. 

Part 6: Draw a new a x b rectangle.  Use your equation from part 5 to predict what the diagonal length should be. 

Part 7: How accurate was your prediction?  Explain.

Commentary

This task asks students to notice a special relationship between the sides of rectangle and its diagonal, which they can later apply to the sides of a right triangle and its hypotenuse.  It is the numeric counterpart to task Sizing Up Squares, which looks at the same relationship geometrically.  Students will draw rectangles of given side lengths, measure the diagonals, and then find an equation that relate the sides to the diagonal using specified operations: addition, multiplication, and square roots. Once students have an equation that works for the initial data, they will write a general equation, and test this hypothesis against other rectangles of their choosing.

This task allows students to construct right triangles indirectly (in the context of rectangles and diagonals), observe the relationship between their side lengths, and then "play" with the numbers and use inductive reasoning to discover the Pythagorean relationship. 

Teacher Note:  (1) Units are not specified here so there is flexibility in how students construct the rectangles.  If graph paper is used, students may need guidance with regard to measuring the diagonal lengths.  For example: if using the unit of the graph paper to construct the sides of the rectangles but using a ruler to measure the diagonals, they will need to divide the ruler measurements by the size of the grid to ensure that the numbers being compared are in the same unit.  Alternatively, they could use a strip of the same graph paper as the "ruler," with the unit of measurement being the side length of each square on the paper. (2) Use of calculators may facilitate students' experimentation with the numbers in Part 4. 

Well, since you asked...

We've been looking at the volume of prisms, cylinders and cones this week.  I had students working on a project where they had to build one of each with equal heights and widths/diameters.  The idea is to explore the volume of each and see how the eventual formulas will relate to one another.

Then, Jacob traces can on his paper and cuts out the circle.  He cuts a radius and begins rolling the paper (as if he's cut out different sized sectors) to make different cones.  He comes up and says, "Mr. Cox,  I think the cone that is almost flat has the highest volume because the tighter I roll the paper, the less stuff I can fit in it."

Me:  What if the circle is flat?  What's the volume then?

J: There isn't any volume.

Me:  So then when does the cone go from 'flat' to having the most possible volume?

J:  ...

Me: ...

J: What do you mean?

Me:  Maybe there's some kind of sweet spot where the volume gets bigger then starts to get smaller.

J:  Let me think about that.

At this point, I was with Jacob.  I didn't really know what the volume did as the cone changed.  But we were both interested.

The next day, Jacob comes in and says, "Mr. Cox, I thought about what you were saying and I think you're right, there has to be some kind of sweet spot."

So, we sit down and go to work.

I'm thinking about how to model this thing and Jacob enlisted the help of a friend to gather data.  They're cutting sectors from a circle and making cones.  Jacob has dibs on 30, 60, 90, ... degrees and Armando has 15, 45, 75, ...

Our first bit of trouble came when Jacob said, "I can find the radius of the cone's base, but I'm having trouble getting the height because of this..."

Wish these rulers came with a bubble level. 

But, we figured out that the Pythagorean Theorem was a nice work around.

Now, does our data match the model?

It took a while, and thanks to CalcDave for cleaning things up, but this is a pretty cool function.

 Desmos graph is here.


We're estimating the maximum to be about 66 degrees.  And because my calculus is a little rusty, I'm thankful for the folks at WolframAlpha.

This particular function is using a circle with radius = 3.1. 

The function is a little dense at this point, but Jacob was dialed in as we talked about it.  The idea that these crazy expressions really just amounted to V_cone = ⅓πr²h blew him away.