Wednesday, April 16, 2014

Dirty Triangles

I've been out for a couple of days--let's just say that I can think of better ways to drop 10 pounds--so, I'm in a really special frame of mind today.  While I was out, I left a few distance/rate/time problems for students to solve.  Upon my return, I was asking students about the problems and many students had similar responses.

S:  "This is easy, you just use the Dirt Triangle."

Me:  "The what?"

S:  "The Dirt Triangle."

Me:  "Hmm. I don't know what that is."

S:  "Look, Mr. Cox it's like this...


"...You cover up the one you're looking for and if the other two are next to each other, you multiply.  If one is above the other, you divide."



Me:  "Really? That's strange.  I never learned the Dirt Triangle. I learned...


The Turd Triangle





S1: "No, that won't work. That's not what he[1] told us."

S2: "He said it didn't matter how we wrote it."

Me: "So which is it; does one work or are they the same?  Make your case and be ready to defend it."


Helping students develop a turd detector one day at a time.



[1] Students picked up the triangle in another class.  They said that the formulas were given early on and explained.  However, many were still missing problems so the triangle was introduced.  

Monday, April 7, 2014

Fostering the Hypothesis Wrecking Mindset

Hypothesis wrecking is not natural.  I think a few of you have summed it up quite well.

Ashli Black:


Dan Meyer:
The fact that you are supposed to wreck your own conjecture. Your conjecture isn't something you're supposed to protect from your peers and your teacher as though it were an extension of your ego. It's supposed to get wrecked. That's okay! In fact, you're supposed to wreck it.

Kirsten (1st Period):
It's easier said than done. 
 
We've grown accustomed to math that does the following:

1. Teacher asks question
2. Student answers question
3. Teacher evaluates answer while student moves on about her day


Hypothesis wrecking requires a different model -- one that asks students to take a look in the mirror and give constant self-evaluation.  It also depends on problems that lend themselves to establishing this mindset. These aren't always easy to find, though. I've found that the problems with a really simple prompt tend to work best.  Here is a list of  problems I've used:

1.  The Diagonal Problem
     
2.  Pick's Theorem
       
3.  Doodle Math

4.  The Locker Problem

5.  The Handshake Problem

6.  Tilted Squares (or Pythagorean Theorem disguised)
        I don't actually use this lesson, but I liked how the "tilt" of the square was defined as x/y which makes             data gathering quite nice.

7.  How many ways?

8. Pile Pattern Problems
        a. Fawn's Visual Patterns
        b. GeoGebra Book we're working on

9.   Avery's Edges, Vertices, and Faces

10. The Painted Cube

11. Math Without Words

12.  Add the numbers 1 to n.





 

Friday, April 4, 2014

Instead

You know what?

Instead of having to teach things like perpendicular bisectors and systems of equations, I just wish we could do things like this.


Hypothesis Wrecking and the Diagonal Problem

We've been doing more problems lately where students can gather data and look for patterns.  Today's installment is via the Diagonal Problem which I think I first saw via Kate.

I'm noticing that more kids are gaining confidence in looking for patterns, forming hypotheses and then seeing if they can make the hypothesis fail.  The phrase that seems to be gaining ground when it comes to hypothesis testing is "wreck it"-as in "Oh, you think you have a rule?  See if you can wreck it."

This diagonal problem is nice because a lot of students seem to zero in on special cases. For example, an n x n (or I just call them squares) rectangle has a diagonal that passes through n squares.  There have also been some nice attempts at nailing down rules for odd x odd and even x even rectangles.  We're finding that special cases don't lead us right to a general rule, but the information can be useful.  


I've put together a flow chart that seems to be helpful. 
Some students get caught in the Do research-->do you see a pattern?--> Do research loop others are making it to the hypothesis before being kicked back to research. All are having to come face to face with their impatience.  Some are owning it.  

There are a lot of mistakes being made.  There's some frustration.  There's arguing.  There's collaboration. 

There's learning. 





Friday, March 28, 2014

Desmos Art: Fries


The Assignment

1.  Draw a design using only straight lines. 
2.  Snap a photo of your drawing. 
3.  Import your photo into Desmos
4.  See if you can duplicate your drawing using linear functions.  

Assessment

We looked for two things: challenge and precision. We got anything from a simple right triangle to a box of fries.  Anything with 3 or more lines intersecting at the same point proved to be very challenging; 2 lines fairly challenging; no intersections...not so much.  

Precision was key.  They could fool themselves as long as the grid, axes and labels were turned on.  But once those things went away, it was just lines drawn (with pencil or graphs) by students.  

I created a filter that contained the words "shared a graph with you" that dumped the graphs students emailed directly into a folder labeled "Desmos Art."  I used SnagIt to take screenshots and highlight areas of the graphs that needed feedback and sent the screenshots back to students.  It was a pretty nice workflow, actually.

Student Feedback
Most students noted that they started out thinking this assignment was really hard--that they couldn't do it. Then they got their first line to match. The second line was easier then the first; third easier than the second , and so on. Domain restrictions turned into range restrictions for vertical lines and some learned really quickly that it was easier to restrict the range for really steep lines.  

The perseverance I saw in students makes this one a keeper. 



Tuesday, February 25, 2014

Science: Completing the Square

Veritasium
"If you think that something is true, you should try as hard as you can to disprove it.  Only then, can you really get at the truth and not fool yourself."
This video could not have come at a better time.  In fact, Derek sums up in about 4:40 what's taken me a semester to convey to my students.

Last week's quiz asked students to investigate a parabola given these three points.

Naturally, many assumed that (-5, -3) was the vertex.  However, upon close examination, they should have realized that the rates of change between the points wouldn't allow that.  So, the next day, we set out on an exploration.

"Ok, open up Desmos, plot the points and find a quadratic that fits these points."

It didn't take long before a student came up with y = x2 + 8x + 12. We all verified it and then I suggested we try something else.

"Enter the function y = a(x - h)2 + k and make sliders for a, h and k.  Now find a function that fits."

Soon we had y = (x + 4)2 - 4.  Students loved this form because of the obvious horizontal and vertical shifting that was going on.

Problem
Is y = x2 + 8x + 12 the exact same function as y = (x + 4)2 - 4?  If so, is there a way we can take a quadratic in standard form and re-write it in this magical form?

Research
We spent the better part of a period graphing functions in standard form and then matching them in vertex form.

"Now, look at all the different functions you have.  Write down what you think is going on here.

Hypotheses
Note: To this point, all quadratics have been a =1. 


Their job over the next day is to determine which hypotheses (if any) they'd like to accept.

I'll keep you posted.