I Like Triangles

Last night, I asked if anyone could point me back to this fantastic animated factorization visualization. (h/t @calcdave)

Now, I'm kicking myself for not thinking to use this in the first weeks of the school year.  Talk about some Fake World math doing a number on pseudo engagement strategies.

I started the animation at the end of each period and walked out to greet students as they walked in.  Once everyone got settled, I walked back in the room and each time the dots would circle up, I'd yell, "PRIME!"

"Alright, I'm up 1-0. PRIME!, man I'm smoking you guys."

Kids caught on really quick and started looking for the circled numbers.  In fact, it took many students a while to realize that the applet literally said "prime."

We started out by looking at the patterns and how each number was represented visually.

But, next I said, "You know what, I really like triangles. What is the smallest number that will give us a triangle?"

This one's easy. 

"Alright, what's the next number that will give us nothing but triangles?  Write your guesses on your easel."

Guesses were about 50-50 between 6 and 9. 

"Alright, what about the next one?"  

Still guesses were a little sporadic.  But by the time we got to 81, most students thought they figured out a pattern.  From 243 on, we were at about 100%. 

After about 10 minutes of doing this and discussing our results, I put up this slide.

There was a nice discussion on clarifying our question.  Three different student offerings illustrated the idea of First Idea; Best Idea. 

Student 1:  At what stage is each triangle?

Student 2:  How many triangles are in the green circle?

Student 3:  How many dots are there in each circle?  *Boom*

Now get to it and be prepared to justify your answer.  

The first student said there were 9, 27, 81 and 243 dots.

"Ok, great. So how did you do that?"

"Well, the green circle has 9 dots, then I multiplied by 3 to get the red. Multiplied by 3 again to get the blue and then by 3 again to get the black."

"Alright, so let's press on this idea a little."

I know what question I want to ask, but I just bit my tongue until a student speaks up.

"How can you be sure that there are 9 dots in the green circle?"  *There it is*

"I estimated."

Here's where it gets good. 

 From across the room, I hear, "It looks like there are 9 dots in the green circle, but we have to look past that."

Wait, what?

"Yeah, we can't trust the picture because the dots are too small.  We know there are 6,561 dots on the whole page and there are three black circles of dots.  We have to start there."

So, Dave, keep this link handy, I'm sure I'll be asking for it again next year.  

First Idea; Best Idea...

...and the Worst Idea

Creating a Culture of Questions was, by far, the most popular post on this blog until someone somewhere starting linking to the post on Exponent Rules.

I think a natural follow up to the Culture piece would be with regards to establishing a classroom culture where feedback is given and accepted.


The First Idea is the Best Idea and the Worst Idea

The first time students hear this, I usually get, "Gosh, that's mean."

But we discuss how the first person who puts forth an idea holds the best idea as there is nothing to which we can compare it.  But using the same logic, this idea should be the worst.  This assumes the flow of ideas that should follow.

I think this encourages two important things:

1.  "If I go first, it doesn't matter that my idea isn't fully formed."  This student has established a floor on which each other student can stand and/or build.

2.  "I can take someone's idea and help them make it better."  The real work is done by the first follower.  This student chips away at any imperfections and helps the first student refine her idea.  Subsequent students then follow suit.


What's this look like?

Yesterday, we trying to determine the equation between the points below and students wanted the y-intercept.

Students were using what they knew about slope to find other points and had to wrestle with the fact this particular line doesn't have a lattice point for a y-intercept.  Once we were finished, I asked students to write down any questions they had.

Student 1:  "I have a comment."

"Ok, what is it?"

Student 1: "No matter which points we choose, the slope simplifies to the same thing."

"Can you turn your observation into a question?"

Student 1: "Will that happen all the time?"

Now here is where it happens.

"I can misunderstand [Student 1]'s question, can we make this more precise?"

Student 2: "Will the slopes always simplify to the same thing?"

Student 3: "Will the slopes between two points always simplify to the same thing?"

"Are we only using two points?"

Student 4: "Will the slopes between three points always simplify to the same thing?"

Student 5:  "Will the slopes between any two pairs of points always simplify to the same thing?"

Student 6: "Are the slopes between any two pairs of points always equal?"

"Are we really talking about any 4 points here?"

Student 7: "Are the slopes between any two pairs of points on a line always equal?"

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.

The Real Flip

If we can get students to flip their thinking from this:

If I know the rules, then I can do the math. 

to this:

If I do the math, I can know the rules.

Then we've won.

I'm Bringing Multiple Choice Back

So here's the idea:

One problem with multiple paths to solution.  Students connect as many skills as they can to the problem.  I listed eight possible skills two of which wouldn't necessarily apply to the problem. Students had to assess themselves on the skills they demonstrated. 

Question of the day:  "Mr. Cox, is it possible to use all of these skills?"

Answer to Question of the day:  "It's possible that some of the skills don't apply."

For this first iteration, I used the standard Ticket Problem. 

Below are samples of student work. 

As an exercise for the reader:

1) What are your thoughts on this process?  

2) How did each student do?

Let me know in the comments. 

Student A

Student B

Student C

Student D

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?

Adventures in Pedagogy: Four Zero

Aidan was having trouble with subtracting numbers that required renaming renamingborrowing, umm, taking away more than you have. He was simply taking the bottom smaller digit from the larger regardless of which one was the subtrahend or minuend on top or bottom.

He picked up some sort of rule along the way and was obviously misusing it. So he decided to get a little creative.

Take a look.


- Posted using BlogPress from my iPhone

The Data

or How Fast Does Google Think We Drive?

I picked this question up on Twitter a while back and really liked it. So, after we worked on making a plan, I thought it would be good to look at data.  Lots of data. And then coming up with ways to make sense out of it.

Process

So, we took to Google.  Students went to Maps, entered starting location and destination (I didn't realize that I'd need to explain that you can't drive from California to China, but, whatever.) and then entered their data into a Google Form.  We did this over five different classes and got a lot of "stuff" to sift through.

We dumped the data into GeoGebra and then took a look at a few different perspectives.  It's interesting to see how the data changes as we look at trips of different distances.

The applet below will really give you a good picture.




Takeaways

• Unit rates are valuable.  
• When points don't line up perfectly, sometimes we can use a line to help us answer questions. 
• As soon as we have a line we like, the actual data points can kind of get in the way. 
• You can't drive to China. 

These are 7th graders and they have some experience with linear relationships.  However, that experience has been limited to "the number in front of the x is the slope and the other number is the y-intercept" kind of stuff.  It really threw some kids that their line of "best fit" may not have been the same as everyone else's.  We are doing this very informally at this time.

Questions

I know that the formal process for determining a linear regression is pretty involved, but does it have to be for a proportional relationship?  That is, if we know a relationship (like distance : time)  is a proportion but the data doesn't line up exactly, is it appropriate to simply average the distance:time ratios to determine a "rate of best fit?"

When informally drawing a line of best fit for a proportional relationship, should (0,0) always be the starting point?

Questions? Yeah, I've got questions.

or Modeling Problems

My electric bill is a mystery.  I started looking into how the bill is actually calculated and found some interesting stuff.  

So Cal Edison coming through with some lesson potential. twitter.com/dcox21/status/…
— David Cox (@dcox21) May 11, 2013

@dcox21 oh you Californians and your complicated pricing schemes.
— Dan Anderson (@dandersod) May 11, 2013

@dandersod I prefer to call it "social engineering."
— David Cox (@dcox21) May 11, 2013

@dcox21 Hi David, please feel free to contact us at CCOeChannel@sce.com or call us at 800-655-4555 if you have questions.
— SCE (@SCE) May 12, 2013

I don' think SCE appreciated the "social engineering" comment. 

So, I decided to turn this into a 3 Act lesson.  Except, their price doesn't fit my model that I modeled after their model.

  

What am I missing? 

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.

CCSS 8: Unit Building

At the end of February, representatives from grade levels K-8 spent two days unpacking the CCSS and clustering them into units of study. My previous experience with unpacking standards became a process of identifying "essential" standards which assumed the existence of non-essential standards.  Those standards that didn't make the cut were ultimately ignored in favor of those that were most heavily tested essential.
We have the same provider leading these new sessions, so I was a little worried we would end up looking for content to cut rather than incorporate.  So far, that hasn't been the case.  Obviously, there are certain topics that will require more focus (eg. linear relationships as opposed to exponents) but the goal has been to see how and where these supporting standards fit with the focus standards.
Our team has come up with the following units of study.  We haven't reached the point where I can discuss specific activities/tasks, but I'd like some feedback on the pedagogy that motivated the clustering and sequencing.

Transformational Geometry

Use the coordinate plane to discuss transformations, congruence and similarity.  Use dilations as an application of the ratios/proportions work done in grades 6 and 7.  Use a graph as a tool to describe proportional relationships.

Data Analysis

Use bivariate data to create scatter plots which can then be the jumping off point for informal line of best fit (where the line may have an initial value other than zero) and an introduction for a future defining of function/non-function.

Linear Relationships

Graphing, graphing and more graphing.  Take the informal line-of-best-fit and formalize the definition.  Allow math to be it's own context.  Graph systems of equations and look for common point (read: solution to system).

Equations

Use the work done in graphing systems to motivate more abstract symbolic manipulation required for solving linear equations.

Exponents

This was a tough one.  Expressions with integer exponents and scientific notation seemed like an island unto themselves.  We are still working on finding a place that fits nicely for these ideas.

Functions

Use the informal introduction to functions and formally define a function.  Look at linear and non-linear functions.  Compare functions using different representations (ie. graph vs. table vs. equation vs. verbal).

Pythagorean Theorem

I think this one speaks for itself.

3D Geometry

Problem solving involving the volume of cones, cylinders and spheres.
We are trying to move from the concrete/informal to the abstract/formal while allowing students to explore these ideas while creating their own formal definitions.  I'm particularly interested in the sequence that runs from Data Analysis to Functions (note: Exponents look to be a unit that can be dropped in and our school calendar lends itself to having that unit kick off the second semester) as it may receive the most push-back from our high school colleagues.  Traditional textbooks usually go the route of
Functions-->Equations-->Graphing-->Applications so we're going to have to have solid rationale.
No one pushes back better than you all.  I'm counting on that.

Do you see what I see?

Who knew that Volkswagen could nail the state of Math Ed in :32?