When It Can't Be Wrecked

We're getting some mileage out of this lately.  Today, I have a new problem to add to the pile of those that foster the process of hypothesis wrecking.

I posed the question with a rubric.

Can a unit fraction always be written as the sum of two unique unit fractions?

Rubric
5: Precise proof that demonstrates all cases (abstract, general rule)
4: Reasonable argument that demonstrates some cases (numeric, gives examples)
3: Gut level or weak argument
2: Does not present an argument

1: No evidence of understanding

Students played around with a few unit fractions and after a few minutes we had a couple of them.

Shortly, we had a student come up with an hypothesis:

which was soon followed by another student example:

Uh-oh, that doesn't fit the pattern.

"Does this example wreck our hypothesis?"

This led to a nice conversation on whether this new example and our hypothesis can coexist.  It was interesting to see how many students initially thought the hypothesis was wrecked.

We tested a few more examples and shared results--all confirming our hypothesis.

Then I asked, "So where does this put us on the rubric?"

And a student asks, "What has to happen for a 4 to become a 5?"

In other words, when does a numeric (quantitative) argument become abstract _[1]?

Had to pause.  This one is worth it.  So we discussed simple example:

I quickly came up with the question and answers 1, 3, and 4.  At lunch I added 2, which really added to the conversation for 6th period.

Which answer provides the stronger argument?  Most saw 4 as the strongest and agreed 1 was the weakest.  But very few saw 2 on the same level as 4.  Then one student says, "I see that 2 and 4 are similar but 4 is just kinda strung out."

Yep, the kid has a feel for brute force vs. elegance.  Love it.

By the end, we agreed that 2 and 4 were more abstract and 3 was more quantitative. What about 1?

Well, 1 was what they would've considered a great answer a few months ago.

[1] This is what prompted my question about SMP 2 on Twitter. 

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.  

Fostering the Hypothesis Wrecking Mindset

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

Ashli Black:

@dcox21 @cheesemonkeysf I love the 'see if you can wreak' it mentality. It's not something ppl naturally do and so important
— Ashli (@Mythagon) April 5, 2014

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.

 

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. 

The Farming [Thing]

I called this a Project. It's not.  It's more of a problem-y kind of performance task learning opportunity assessment of  for of for? learning that hits close to home.  Literally.  We live in a huge agricultural area and kids don't know what an acre is.  Anything that gives students a chance to wrestle with the fact that a piece of land can't have dimensions of 20 acres x 20 acres, is a win.  Anything that allows me to answer the question "What's an acre-foot?" by doing this, is a win.

In this project  problem's first iteration, I was focused on the skills of equation writing, line graphing and solving mixture and work problems.

In the second iteration, I was less focused on the skills and more interested in having students explain what each component of an equation represented, why we'd want that equation and how graphing inequalities made sense.  We got to discuss why understanding the problem makes sense--kids tried to hire crews to prune cotton.  For you city-slickers out there--you don't prune cotton. It doesn't grow on trees.  Students had to sign up via Google form to interview with me as they finished a task.  I did something north of 175 interviews for one class that year.

This year, I've changed it a bit more.  They are no longer tasks, they're constraints.  There are fewer of them and they don't specifically tell kids what to do.  Before, I told them to create inequalities and graph them. Now, I'm removing some of the scaffold.  They get to decide what tools they want to use.  Before, I did this project after we had done systems, mixture and work problems.  This time, we have only done systems. They're going to have to work through the mixture/work stuff.

That's been the highlight--the mixture problems.  I have a few students who went straight for that constraint and have been on a mission to figure out how to make sense of it.

Today, one boy asked, "Mr. Cox, how accurate to I need to be?  I'm accurate to the trillionth, but I can't get it to be exactly 36%."

I said, "How accurate do you think you need to be?  We're killing weeds, not sending someone to space."

So, with all that, here's the updated version complete with dynamic answer key.

When Perseverance Pays Off

Our high schools are committed to taking the integrated path with the first three courses and since the middle schools will be teaching these courses, I'm part of the team building the units.  I've been piloting the curriculum by the folks from Utah and, for the most part, I like it.  I'm particularly enjoying the learning cycle they employ:  Develop Understanding, Solidify Understanding and Practice Understanding, mostly because it's pretty easy to discuss with the majority of teachers.

This task was at the beginning of a learning cycle.

Source: Mathematics Vision Project

I had a student, H,  come to me before class and say, "Mr. Cox, I spent like three hours on problem 3 last night.  I couldn't quite get it."

During class, students worked with their groups and started presenting solutions.  As I approach H's group, she gives a high-five to the student next to her.

Me: "What's that about?"

H: "We figured it out!  I get it now."  Then she shows me her solution.

"Feels nice, huh?"

"Yeah, I think I'm gonna cry."

Me too, H.  Me too.

The Student Rubric

We are currently working on a performance task where students have to gather data, apply a line of best fit, determine a rate and then make a prediction.  It's been a task to help students shift their thinking from right/wrong to more/less.  In other words, I don't want them to see their understanding as binary--I get it; I don't get it.  I want them to see their understanding as something that falls on a continuum.

When doing something like finding a line of best fit, I think it's less important to discuss what the line looks like and more important to discuss why a particular line is best. This leads us to the descriptors we've been using to discuss both sides of the same coin:

Concept and Precision

5: Strong concept; Precise

4: Strong concept; Somewhat precise

3: Problem with concept; Somewhat precise

2: Problem with concept; Lacks precision

1: No attempt

Through a few discussions with different classes, the top three descriptors have evolved into something like this.

5: Precise answer with precise method

4: Estimate backed by reason

3: Estimate

Then I walked by a student and noticed the self-assessment she was doing.

How's that for kid friendly?

The Plan

I blogged about the template I'm using.  Most of the activities we have done have focused on a particular piece. We did two quick activities focusing on making a plan.  Before sending students outside, they had to submit their plan for peer review.  If another group could read their plan and understand what was going to be done, then I signed off on it.

Day 1: How Far?

Question:  How far is it from the first tree to the last tree (ie. point A to point B)?

Rules: You can take a pencil, paper and clipboard outside with you.  Nothing else.  

                 

Different groups were able to tell me the distance from one tree to the other using units like:

• Jose's feet


• Jasmines (not her feet, but her)


• Clipboards


• Brandon's longest stride

A few groups made adjustments to their plans once they got outside and saw how tedious it would be to try to walk a heel-to-toe straight line.  We had quite a few groups decide to measure the distance from the first tree to the second and then just multiply.  This led to a couple of really good conversations that went something like this:

Student: "Mr. Cox, we are going to measure from the first to the second then multiply by the number of spaces."

Me: "Will that work?"

"Yeah. Because the spaces are the same."

"How can you be sure? "

"Because look at them..."

"Yeah, I want them to be the same too.  That'd be really helpful, huh?"

Now they have dilemma: do they go and measure the distance between each tree or just measure the entire distance from the first to the last?  (wait, that's the same thing...which makes it a doublemma) 

Oh, you want my help?  Lemme show you how Google Earth can help you out here. 

Day 2:  How high?

Question: Come up with two different methods for finding the height of the building.  

Rules: Don't climb up there. 

Some of the methods:

• Ask Chuck.  (Turns out Chuck, our custodian, had a copy of the elevations.)


• Take a picture of Cameron next to the building and see how many Camerons to the top. 


• All kinds of crazy uses of a meter stick. 


• Count how many bricks in a foot and then count the total bricks.  (3 bricks and spaces = 1 foot.)

"Ask Chuck" allowed us to discuss the importance of trustworthy sources of information.  And Chuck is awesome.  He'd throw out all kinds of crazy numbers and see if kids would bite.

Takeaways

• A well thought out plan makes jobs easier 


• Sometimes we need to adjust our plans


• Assumptions need to be investigated


• We can use some tools in ways we've never imagined (eg. cell phone camera, Google Earth)


• Some sources aren't trustworthy

Middle School Modeling: Integrated Math/Science

or  My Apologies to the Scientists, Polya and All the Modeling Teachers Out There

I decided to go with a process rather than specific content in this class.  I know stuff is going to be on the test and we need to cover it.  But, I also know that my students will one day leave and go be anything but a scientist or a mathematician.
So I settled on asking students  to question, think, plan, model/analyze and tell people about what they did. That's it.
Everything we did this semester followed this template.  I found the following questions/directives to be helpful when turning students loose on a problem.

1. What's the problem?
I think we call this "inquiry", but I really don't know anymore.  Does it count if I give the question?

2. What do you think the answer's going to be?
Props to Dan for making a guess be an explicit part of the lesson plan.  Something I should've been doing 10 years ago but somehow didn't.

3. What smaller questions will you need to answer first?
This is tough.  Students live in circular argumentation.  I mean, c'mon kid, give me at least a spiral argument once in a while.  The name of this blog should mean I have some grasp on the importance of questions, but I've never explicitly asked students to break larger questions into smaller manageable questions nor realized how badly students need help with this.

4. What's the plan for answering the smaller questions?
Two big take away here for students:
1.  a good plan = good data = good analysis
2.  plans change

5. Go do the plan. (ie. get your data)
See #4

6. Make sense out of the data.
This was the sweet spot.  How can math be used to turn data into an answer?  Kids are getting the hang of this and it's fun to watch.

7. Answer your question. 
Cross check the answer with the guess.

8. Tell someone about it.  
I use the word "presentation" very loosely here.  This was anything from a write-up to a group presentation to an informal interview after an activity.


None of this is new.  But, for some reason, it seemed new.  The first few activities we did would focus on a particular piece (I'll blog about these--this year. Promise.).  The challenging part was to keep from over-planning.  Not because I'm that kind of teacher, but because the more I planned, the less students had to.  And, well, #4.  Oh, and time.  It takes a lot more time to have students make the plan and we have bells.

On Problem Solving

Phil Daro:

The American teacher looks at a problem they're going to use in a lesson and asks themselves, "how can I teach my kids to get the answer to this problem?"  The Japanese teacher asks, "What's the mathematics they're supposed to learn from working on this problem? How can I get them to learn that mathematics?"

If you want better answers, ask better questions.

Did I Get It?

The Prompt

Given: The green, red and blue points are collinear. What is the dimension of the blue square if the green and red squares are 4x4 and 7x7 respectively?

Applet's Here

After a few minutes, B comes up and says, "Mr. Cox, can you check this out. I think I've got something."

She shows me her diagram.

And her results.

Yep, kid.  You've got something.

Hypothetically...

What if we could design a project-based class for middle school students that integrated math and science-- that provided context for math and gave instructional minutes back to science?  What if this class capitalized on the natural correlation between the Practices For K-12 Science Classroom and the Standards for Mathematical Practice.  What if students were asked to design investigations, used modeling to move from the concrete to the abstract and then presented their findings for peer review?  What if they used technology to take snapshots of their learning along the way and kept a running journal of the process?

What if...

On Teaching By Learning

One of the things that I have learned over the years is to let go of any preconceptions I have about how a problem should be solved.  I have methods I prefer, but my students need to develop their own.  Never has this been more obvious than it was today.

We are beginning to play around with non-linear functions and so I gave my class the following problem:

You're going  to build a garden and need to build a fence around it.  If you have 120' of fencing, how would you set it up in order to have the biggest garden?

...or something like that.

I didn't specify rectangle although I figured most kids would default to that.  I didn't mention the barn in the back of the property that could be used as one of the sides.  I just kept the prompt as simple as possible so I could see where they took it.

Many assumed a square right off the bat. And one group felt pretty ambitious and looked up some shapes in their school planner only to settle on the typical decagon.  (Good luck calculating the area of that one, fellas.)

So, I'm walking around and seeing groups all proud of themselves by defining x as one side and y as the other and they're arguing about whether 2x + 2y = 120 or x + y = 60 is the better equation.  I ask a few questions like, "so how will that equation help you determine the largest garden" to which they reply, "we'll find the maximum."

"Mmm-kay," I say as I mutter to myself, "I'll be back..."

Then I walk up to a group of three boys who usually push the envelope when it comes to creative problem solving and I see the equation x + y = 60.  I think to myself, "oh, no not you too."

Then I look a little closer and I see this other thing they're working on.

xy > 900


I double take.

"Wait, what?"

Before any of them even address my incredulous look, one of them says, "let's go put it in GeoGebra."

"So how'd you come up with that?"

"Well we figured that x + y = 60 tells us how much fencing we have.  And since a square would give us 900 ft², we want to know if there is an area out there that's greater than 900."

"Uh, yeah.  That's, uh, yeah, that's exactly how I'd do it."

Not really.  This is what I really said.

These kids used a system.  Not in a million years would I have ever considered using a system to solve this problem and three 13 year-olds set me straight.

Man, I love this job.

The Timeline of Awesome

Friday August 12, 2011

Kate poses a great problem.

Thursday, November 17, 2011

Dan asks a great question.

To which I responded something like, "yeah, prolly, but it'd take a bunch of brute force."



Saturday, November 19, 2011

I forward it to the GeoGebra Forum.

Sunday, November 20, 2011

Raymond responds.  

This flow of information absolutely amazes me.  I mean, I loved the question after Kate posted it.  In fact, I immediately created an applet and had used the problem with my advanced class early in the first quarter.  They struggled a bit with it, but then when Dan asked about highlighting the squares and doing some of the counting, things changed.

I consider myself to be a little better than average when using GeoGebra, but Raymond is a freaking Jedi. Take a look at his stuff.  He takes an applet that I thought would require a number of tedious steps and bangs it out using 6 steps--and within 24 hours.  That's ridiculous.

The applet is here.

Well, Since You Asked

I had a copy of John Van de Walle's book on my desk the other day when a student asks, "Mr. Cox, what are you reading?"

"Oh, this book on how to teach math.  It's pretty interesting."

"Oh, yeah?  What's so interesting about it?"

"Well, I like this chapter on Teaching Through Problem Solving.  It mentions three different ways problem solving can be taught:  Teaching for problem solving, Teaching about problem solving and Teaching through problem solving."

"What's the difference?"

"Well, teaching for problem solving is when students will learn a certain set of skills and then later be asked  to solve a problem using those skills.  Teaching about problem solving is when students learn about particular strategies for solving problems.  And teaching through problem solving is when students are given the problem first and then they figure out how they want to solve it.  It's kind of the opposite of teaching for problem solving."

"Hmm. So teaching for problem solving means the teacher shows us how to do stuff first?"

"Yeah, pretty much."

"They don't think we can think for ourselves?  That's kind of offensive."

Ticket Prices

The other night, Dawson and I were doing some math together and we ran across a problem that asked to interpret a scatterplot for the average movie ticket prices for the past 10 years. We poked around online until we found this image:

http://www.the-numbers.com/market/

We used the information to come up with an average price and dropped it into Excel and came up with a best fit line and all. Personally, I find Excel clunky, but since we could use it to quickly calculate average price, we went with it. But it just so happens that I'm doing linear relationships with my 7th graders, so I get a two-fer with this one.

Step 1: Calculate average ticket prices and created a scatter plot.

This went pretty quickly, but one group mistook the raw number for the average ticket price. For example: from 2003 to 2004 the revenue went down, but so did the number of tickets sold. The raw numbers led this group to believe that the prices went down. After a quick conversation, they realized that if the number of tickets sold decreases as well, the price can actually increase.

Step 2: Predict the price of a ticket in 2020 and justify answer.

Most groups used the trend of the graph to predict, but one group actually calculated the average rate of change from year to year and came up with $.24/year. They used this to extrapolate a price in 2020.

Step 3: Decide what type of curve best fits the data.

Two camps on this one: Those who thought it was linear and those who thought it would be like "the graph we got when we did compound interest." Nice work kids.

Note: We will explore that exponential thingy but we ran out of time today.

Step 4: What line would best fit the data?

They were creating a group graph so they took a meter stick and just plopped it down where they thought the line should go.

Step 5: Estimate the equation of that line.

We played with this applet yesterday, so students had a pretty good idea how to use the graph to predict what the equation should look like. I was pleasantly surprised at the fact that all groups decided the rate of change was somewhere between $.22 and $.25 and all said the initial condition was $4.34.

Step 6: Let GeoGebra work her magic.


Add the ordered pairs and use the "Best Fit Line" tool to calculate the regression.

Step 7: How good was our guess?

Insomnia

I went to bed with this problem swimming around in my mind. I kept trying to figure out how to construct in GeoGebra. I told it to be quiet so I could sleep.  It wouldn't.

Problem:

Given two squares, deconstruct and reconstruct as one square. 

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Double click to open in a new window.

It's almost 1am.  I'm going to bed now.

I Love This Problem

I love simple problems with simple proofs.  And I love GeoGebra even more for making it possible for my students to use their induction to help the proof along.

Problem:

Find the shortest path from point C to point D that touches line AB.₁

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


¹ This problem is one of the many reasons I'm glad I finally got around to reading this.

WCYDWT: Projectiles

The camera angle messes with the perspective a bit, but I still find these images interesting. Does it mess with the perspective so much that it ruins the problem or does it just lead to another discussion on how things aren't always as they seem. I'm thinking that I'll show the video, but make digital and hard copies of the photos available to students.

Question #1


Projectile Question (YG) from David Cox on Vimeo.

Question #2

Projectile Question (PG) from David Cox on Vimeo.

Answer #1

Projectile Answer (YG) from David Cox on Vimeo.


Answer #2


Projectile Answer (PG) from David Cox on Vimeo.

Next up:
Create something that helps kids see vertical and horizontal motion independence.

Mixed Meta Four

If you haven't read about the Wolverine_[1] (here and here) or Sam's Exasperating Problem, you need to get your priorities straight go read 'em now. Sam's wondering if the problem can be scaled down so a precalculus class can handle it.  I see the problem and think, "GEOGEBRA!  I CAN USE GEOGEBRA!"  Mr. H beat me to it.  (If you haven't seen his applet, I question your dedication to the cause go ahead, we'll wait.)

I love this!  We are always looking for ways to iterate problems and extend them, but there's nothing to extend with this problem.  It's all ready for the wolverine wrangler to do his stuff.  I'm looking for the guy who can make this wolverine sit and quit bearing its teeth so my 8th graders can pet it for a second.  GeoGebra does this.  Mr. H's applet makes this problem accessible to an 8 year old.  In fact, my son was so mesmerized by the animation that I swear I heard him muttering, "Heffalumps and Woozles.  Heffalumps and Woozles."  Heck, I found a strange urge to put on some Pink Floyd myself.

Can you imagine starting a problem in middle school and finishing it with calculus?  That's how beautiful (that's right I said it!) this problem is. Why can't we let these younger kids see the beauty of the wolverine without actually having to be the one to handle it?  I can see posing the problem, setting the kids up with GeoGebra (with minimal prerequisites) and turning them loose.  They'll see the pattern, make a conjecture and inductively decide the answer.  Show the applet which demonstrates the first 360 cases and inevitably, the question will be:

Why?! 

Now, talk about storytelling. The table's been set for the sequel that the kid's gonna have to wait a couple of years to see.  precalculus kids can actually calculate the answer and the trilogy will be complete once they have the tools to actually prove that for n chords, the product is n+1.  This problem can span four years. At least. 

_[1] Apologies if I misused the metaphor.