Monday, August 29, 2016

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."  




Thursday, August 25, 2016

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.




Monday, August 22, 2016

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.





Wednesday, August 17, 2016

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.

Thursday, January 22, 2015

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?



Tuesday, August 12, 2014

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.
RectangleSide 1Side 2Diagonal
A345
B512
C68
D815

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. 

Friday, May 9, 2014

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 Vcone = ⅓πr2h blew him away.