3-Act Math Images - Circular Farms

I really like this satellite image from NASA of center-pivot irrigated farm in Kansas - it looks like hundreds of pennies to me.

What questions could we ask with this image?

Image: Dalhart Texas, NASA Visualization

Some of my ideas:

• How much land goes unused in this image because the fields are planted as circles?
• If you could add smaller center-pivot sprinklers in the corners of the fields (and therefore, smaller circles) to cover some of the unused land, how many sprinklers would you need to cover over 98% of a field?
• What proportion of this land is covered in dark farms, and in light colored farms?
• Assuming irrigation costs were the same for both fields, how much more money could a farm make that planted in rectangular plots instead of circular plots of wheat?
• Which setup would best maximize use of the land - one big center pivot sprinkler (one circle on a square plot, two sprinklers that do a half rotation (two hemispheres on a square plot), or four smaller sprinklers that do full rotations (four circles on a square plot)?

Is it Tech-Ready? 3D Solids Town Project

A large part of my job as self-appointed edtech evangelist in my building is seeking out opportunities for my colleagues to take something they are already doing that works well and take it to an even higher level with technology enabling them to learn more, share more, or show more.

Last week I was inspired by a 3D solids project my mentee and another Geometry teacher did earlier in the semester. I did a Google Sketchup project last year to assess 2D area, and one of the greatest tools in Sketchup is its ability to easily create 3D solids, so I asked Colleen and Emily if we could have a chat about their project.

Our discussion was centered on these 4 questions:

• What were you assessing with this project?
• Why did you choose to assess those goals this way?
• How do you feel your project went?
• Would using the technology bring added value to this project?

So is this project tech-ready? As far as simply evaluating ability to calculate surface area, volume, and write linear equations, I think it translates nicely, however (and I think this was a surprise to Emily), you do lose the physical, tangible skill reinforcement of measuring a physical object if you take this project into Sketchup. With the emphasis we're trying to place on graduating kids that have and/or are ready to gain technical skills, I think it might be worth it to keep this as a paper-based project, or at most provided Sketchup as an option.

I have really enjoyed seeing these projects on the walls the past month, and I'm impressed at the evolution it represented in instructional design from the last project these Geometry classes. It was a good integration of some Algebra concepts that are sometimes hard to put in the Geometry curriculum, particularly in this unit. However, since this is an ed tech AND math blog, let's explore some ways we could go further next year.

What are some things we could do to keep the tangible skills but still enhance the learning experience with this project?

• We were talking about having kids check the workability of their word problems by having their classmates solve them - you could generate a rich feedback loop having kids post their problems on Google Classroom, a student blog, or Voicethread and require kids to respond and give feedback to a set number of their classmates. You could do the same on chart paper in the room, I think, but moving that opportunity online keeps the feedback more easily accessible (instead of a giant piece of chart paper), and opening up the number of people who could comment on any one student's problem.
• Have students decide the surface area or volume they want for each 3d solid and then create and print the net for the solid themselves using image editing software. Even MS Paint would work. :)
• Students chart the lines of their streets on an online graphing utility like Desmos, and just enter the placement of their physical 3d solids on their poster as coordinate points. This option would keep a map-making element to the project with a tool more approachable (and accessible) than Sketchup. 

What other ideas do you have, readers? Do you think the physical measuring skills are not worth the limitations they place on the type of product the kids can make and share? Is there something else we can use technology to enrich and enhance the experience while keeping the physical manipulatives?

The Infinite Chocolate Trick, Tangrams, and Area on the Coordinate Plane

With 2 months to go in this semester, it'll be time this week to begin the work on the last big project in my Applied Math class where the kids use tangrams to study area, the distance formula, and linear equation writing. Usually I hook the kids with a day of playing tangrams to practice the spatial reasoning, but before that, I played this video of "The Infinite Chocolate Trick," where the user supposedly breaks off a chocolate bar, moves the pieces around, and winds up with an extra piece of chocolate, which if its true, would mean extra surface area just materialized out of nothing.

I didn't know how much the chocolate video would pull them in, but I'd declare it was a huge success! Kids were jumping out of their seats to come to the smartboard and try to explain their hypothesis on where the chocolate was coming from, several theories were floating around the room, and kids were asking me to play the clip over and over so they could try and catch what was happening. It set up so well for the essential question, someone actually REMEMBERED it today without me even asking. :)

For the next layer of the learning I started class with a Do Now of a triangle, circle, and rectangle on a coordinate plane, but with the relevant sides laying horizontal or vertical so the lengths could be determined just by counting units.

After we reviewed the solutions, I had to students focus on the example projects above the whiteboard and asked, "Looking at the projects up there, why do you think we started with these specific problems today?"

A student jumped right into, "So that we can prove that area of a shape is the same even if you move the pieces around..." Granted, that wasn't really why we started with that specific Do Now (I was setting up the distance formula), but I was glad that EQ stuck with her!

The official essential question for this unit is:

"How can you prove that the area of a given polygon is constant, no matter how its parts are arranged?"

Practically, the kids will be thinking, "Can I show that the area of my tangram goose (or sailboat, candle, fox, etc) is the same as the square my shapes came from?"

This project covers:
Area on the coordinate plane
Distance formula
Linear equation writing
Parallel and perpendicular lines/equations
Spatial reasoning
Using tools appropriately (measuring with a ruler, setting up their grid paper)

Here are some examples of previous years' work:

Instruction/example handout

An example of a kid who had all the required elements, but it WASN'T pretty.

If you've listened to either of the podcast episodes about our statistics based project based learning unit, you'll remember I was very frustrated with how little buy in my students showed behind what I thought was a highly relevant, meaningful project. This one is less relevant, so I wanted to make an extra effort to start with a strong hook and have an experience to draw from when the kids seemed to be lost on the connection or purpose of our work in this unit.  

Special Right Triangles - Really?

What's in the list you keep (internally or physically) of things we still traditionally teach in our math courses that just feel "wrong" in 2014?

Memorizing formulas is probably one of my least favorite things, and I know I have that in common with my students, so wherever and whenever possible! I like to teach them the concept on a pattern level or with strategies that have more than one application. In other words, if the only reason for me to teach it is to maybe get lucky and steal a question or two on a test, then I usually won't stress it.

I was having a good week with my Applied Math class. The kids have been more or less focused recently, and we were all getting along. Kids love Pythagorean Theorem, for some reason. Right up there with "y=mx+b!" It's hard to find a kid who at least doesn't think they know how to use it. Taking advantage of those good vibes, I would have rather rolled on into circles, but the ugly special right triangles lesson stood in my way.

I went through the trouble of having them use Pythag the night before to actually solve the SRTs, and my first example showed them again how they can just use Pythag on any right triangle to find a third side. Good feelings were still flowing, and then our first question from the book had a 60 degree angle and ONE side. "Try and do THIS ONE with Pythagorean Theorem," it seemed to taunt me. I took the bait and walked the class through the relationships. And one by onem my confident students fell to the wayside, and my focused, eager students started to check out. 

So, why? Why invest time in a chunk of knowledge that isn't necessary if you know  Pythagorean Theorem or basic trig ratios?

Here's a note sheet I plan to share with my students tomorrow to articulate the alternatives again.

Student Tips for Simplifying, Multiplying, or Dividing Radicals

To wrap up the lesson today, I had my students try to process their learning with an exit slip giving two tips for simplifying, multiplying, or dividing radicals.

Some of them referred to some notes we had written yesterday, some referred to something I'd said in the last couple days, and some used a kernel they'd picked up during class today that helped it click for them. We had a successful day, I think.

The "or dividing" is awkwardly  tacked onto the end of that heading because after I wrote "2 tips for multiplying or simplifying radicals," someone piped up, "Can we do dividing, too?" You win, students - its your class. :)

Argument Writing in Math - the Oldest "Fad" in Instruction

Either you are a long-time teaching vet and you've said this or you've heard one say something like it - "Common core is just a next the fad. We've seen things like this come and go just like NCLB did."

It's true that a lot of the content and practice standards we see in the common core may only be shuffled content from the SHOW-ME standards and course-level objectives we already had in Missouri. To think that the rest is only based on the latest fads and research, however, is to have misread the document and dishonor the history of mathematics. 

Look at this from the standards for mathematical practice (my favorite part of common core).

If you're not careful, getting your kids to construct arguments can feel like another add-on to curriculum that rarely gets anything taken away. There are probably going to be enough shifts within the content standards that actually do represent changes to your classroom (for example, descriptive statistics in algebra) for you to get bent out of shape about the things that should be there anyway. 

If constructing arguments in our math classes is the reaction to a call for "rigor and relevance" and just another fad, then I want to remind you that it's perhaps the oldest "fad" in instruction.

If you're familiar with our old friend Euclid, then you know that the whole purpose of his work was to build arguments proving the geometric ideas of his contemporaries. It's commonly agreed that Euclid did not make up the ideas from Elements, but that instead, he was the first to curate and examine what was already being discussed. 

Aren't our students working in a similar environment? There's little to be "discovered" in planar geometry at this point, but we can still do well engaging students in the same work of critiques and proof gathering as Euclid and great mathematicians have practiced for millennia. 

Is the idea of argument and proof new to the current generation (post NCLB) of math students? Perhaps. Should we go back to teaching two column proofs the way we did it fifteen to twenty years ago? I don't happen to think so. I mean, there's a reason that Geometry instruction became more formulaic in this century - two column proofs are hard, and the rigidity of their structure turns off kids who are already struggling to find relevance in their work. I prefer flowchart or paragraph proofs because flowcharts have more cross-curricular appeal (logic and systems planning for coders), and paragraph proofs are more familiar (and therefore approachable) to the non-fiction writing our students already do in their other classes. Which students should do this? Everyone. Keeping proofs only in honors classes or with the "good" kids is prejudiced and demeaning to "regular" students. Will it take more work getting "regular" students to think critically? Probably. I think the way you package argument writing and proof to your students has a huge determining factor in their attitude toward it and belief that they can do it. 

Present it as a trial or a chance to argue with you (or each other). Kids love debates.

Argument Writing in Math: Getting Started

Are you an English teacher prepping a training on argument writing for the teachers in your building/district (and you want to be applicable to those STEM folks)? Are you a teacher that just went through an argument writing training that left you wanting more? Are you a math teacher that wants to integrate more logic, reasoning, and writing into your course? (Let's be friends!) Whoever you are, I wanted to share a few thoughts and resources to get you started on your argument writing (and thinking) in a math class. If you'd like more of a primer on argument writing in general, check out this post I wrote after I practiced some argument "writing" with my 3 year old daughter last year. (Teaching Argument Writing for Preschoolers...and anyone else!)

The most natural application of argument writing is in proofs, which most often come up in Geometry, and should also usually be used in Algebra (but rarely are in my experience). To make a common core connection, standard for mathematical practice #3 requires students to "Construct viable arguments and critique the reasoning of others."

Most students experience with proofs (and perhaps you remember your own) is with column proofs like this, for proving things about angles, line segments, or shapes.

Have I induced any terror sweats yet?

But proofs can also be written in paragraph form, which is where the English training and Hillcock can be applied. Reasons are the warrants, the statements are evidence, and claims will be what must be proved. "Prove that angle A is a supplementary angle," or "prove that the lines defined by y=2x+3 and y=2x-25 are parallel"

But...I don't even know how to explain what a proof is! 

Watch this adorable TED-Ed video introducing and explaining the basis and application of mathematical proof.

Want some more? Here's a resource from Berkeley on mathematical logic
"First, a proof is an explanation which convinces other mathematicians that a statement is true. A good proof also helps them understand why it is true. The dialogue also illustrates several of the basic techniques for proving that statements are true.

Table 1 summarizes just about everything you need to know about logic. It lists the basic ways to prove, use, and negate every type of statement. In boxes with multiple items, the first item listed is the one most commonly used. Don’t worry if some of the entries in the table appear cryptic at first; they will make sense after you have seen some examples.

In our first example, we will illustrate how to prove ‘for every’ statements and ‘if. . . then’ statements, and how to use ‘there exists’ statements. These ideas have already been introduced in the dialogue." - from Introduction to Mathematical Arguments (http://math.berkeley.edu/~hutching/teach/proofs.pdf)

A couple more resources for your classroom:

1. "Making arguments with equations, figures, and images" - (http://wacillinois.wordpress.com/2014/04/22/making-arguments-with-equations-figures-and-images-writing-in-stem/)

As soon as you have students start writing more in math class, some of them will start trying to write out EVERYTHING. The point of this post is that sometimes mathematical symbols are still most appropriate

2. "Developing argument writing in math using crime scene investigations" -(http://teacherleaders.wordpress.com/2012/12/15/developing-argument-writing-in-math-using-crime-scene-investigations/)

This blog post from a teacher directly integrates an argument writing text by George Hillocks, Jr., Teaching Argument Writing, Grades 6-12 (with a bonus handout!) as a strategy for students to attack math word problems

Supersize the Floorplan Area Project

Designing a floor plan to have students calculate and study area is pretty standard fare for a high school Geometry course, but the further we get into the CAD-era of drafting, the further it gets from relevance.

The only way I've ever seen this done is on grid poster board or with grid paper. It's good for modeling how someone might draft up an informal plan at the kitchen table or your workbench, but its completely unlike what "real" architects, engineers, and designers do at their desks. Your students might get some exposure in the career and technical ed department with the heavy tools from Autodesk, but there's no reason you can't get everyone a quick exposure from these free tools.

Google SketchUp

My students are in the middle of a project right now designing a water park for a fictitious town in southwest Missouri using SketchUp to lay out the 2D geometry, pulling up into 3D shapes, wrapping it with colors or images, and adding dimensions. You can add some pizzazz to your plan from the extensive 3D warehouse gallery of items contributed by SketchUp users.

I'm not mandating this for our purposes, but you can also add any text you want for identifying parts.

When you're done you can export your SketchUp to 3D printers or as images to be embedded anywhere else on the web.

How to get it:
SketchUp is available for download here for your personal use, but its software, so you'll have to wait for someone with admin privileges to install it for you at school. (I always hate coordinating this, so I stay away from locally installed apps whenever possible).

You can also run it with a browser plug-in from this link. I haven't tested that yet, but I imagine it might get slow at times. On the positive side, your students will be able to use it at home without having to install it on their own.

Autodesk Homestyler

Homestyler is marketed directly toward consumers wishing to design for their personal home or office. Therefore, there's less to offer to the "power" user, but its easier to pick up and use for your students that might be wary toward the technology aspect of your redesigned project.

My favorite parts of Homestyler:

• When adding doors/windows/appliances to your model, beyond the generic options are brand name designs that you could specifically purchase at your favorite home improvement warehouse. 

• Adding a budget or cost layer to your project because as simple as keeping a window open to homedepot.com or lowes.com

• Exporting images AND interactive walkthroughs 

• Users have a choice of several nature backdrops to have peeking behind any windows, and can even choose a setting to have the sun shine through. The point is to get a better feel for how a paint color would look in the afternoon sun of your actual home, but for your students' fake floor plans, it just adds an extra pop.

How to get it:

Not only is Homestyler free, but its also a web app, which means all your students need is the url to finish on their own at home. Launch the web app here. (requires Adobe Flash)

CONFUSION WARNING: There is also a Homestyler app in the iOS app store that has a completely different interface and functionality. The app directs the user to take a picture of the room they want to style and then add elements layered onto the image of that space. From a consumer standpoint, its a lot simpler than having to measure and layout your space like you do on the web app, but for the purposes of your floor plan area project, its useless.

Honorable Mention: Free, online tools that may also work for your project

RoomSketcher - looks a lot like Homestyler

Roomle

Placepad

"Lying" to Your Students for Good - Crafting a Story to Engage Your Students

I'm lying to my students this week, and I think its okay.

Let me tell you about the town of Oklaville, MO...

If you're not from Missouri, you may not know, but "Oklaville, MO" is a completely fictional town that I created last week to hold a fake competition for designs for a community aquatic park. I created a relatively simple website in iWeb (but probably could have used Google Sites as well) and hosted it on Dropbox using these instructions.

This "design a water park project" is the marriage of something I've been wanting to do since February (use Google SketchUp for designing and area) and fixing a project that is never really as good as I want. The prompt for the design actually comes from my textbook -

The last two years when I had my students do this project, I gave them a clean sheet of 1cm x 1 cm graph paper, and it usually came back to me in a pretty rough draft that hopefully was not totally wrinkled. They were uninspiring to say the least, but maybe that wasn't completely the students' fault. Is there really a lot of sense in taking pride in a fake water park on a piece of paper that literally no one else but the teacher will see? I wanted them to look better for the sake of work ethic and pride in work, but honestly, I was kidding myself.

When the situation arose again this year, I knew it was time for something different. I just the last couple weeks spent many hours designing a website for my friend trying to launch a side-career in Gospel music, so I had design heavily on my mind.

Take a second to go back and peruse that site if you haven't yet - its three simple pages, but they look pretty legit, right? I included a "real" seal, an address for city hall, and a couple of examples. If you take 5 seconds to do a Google Map search for that address, or even Oklaville itself, you'd know it was fictional, so when/if my students get to that point, I'll easily fess up and tell them much of what I'm writing here.

Is it deceptive? Probably. It may be risky when I tie so much of my relationship building to honesty, but I'm really counting on the relationship we can build during this project as a serve as Google SketchUp mentor.

Is this for you? I don't know, but a colleague of mine pointed out all of the times she lies to her students when they ask her if something they do is going to be for a grade. We've probably all done that several times, but for what? A single piece of paper, 10 minutes worth of work? When you create worlds for your students' work, they can get real skills out of it.

Chuck's Tips for Lying To Students (for Relevance)
1. Take time to pay attention to the details. 
It's kind of like The Matrix - your students won't know things aren't real unless they are really looking for it.

2. Prepare multiple sources of "reality."
Besides the three pages I created for the website, I also registered a new Gmail address and crafted an official looking memo for another copy of the "proposal requirements"



3. Be non-chalant about everything
I showed this to all of my students today, but I pitched it briefly to just one student on Friday and was able to get him excited about it all weekend.

4. Have a real prize, reward, or recognition ready
Of course, there isn't a real Oklaville City Council waiting to vote on my students' submissions, but that isn't any reason why I can't make a big deal out of the best submission. Have a panel of your colleagues come in to judge once you let the students in on the secret. One student asked if he could have a medal if he wins. Sure!

5. Have fun.
The whole reason you're doing this is to bridge a gap between worksheets, bookwork, and teacher-driven assessment to something that is more authentic, more engaging, and more fun. Be creative in the setting you create, and cherish the enthusiastic conversations you have along the way.

QUIZZES ON MONDAY, 2/9/09

The Geometry Quiz will cover lessons 8.4 and 8.5, Area of a Regular Polygon and Area of a Circle. Students will be able to use the formulas note organizer we made in class today on the colored paper.

Algebra 1's quiz is over Elimination by Addition and Subtraction and Elimination by Multiplication (5.3 and 5.4). Students will be able to use their Solving Systems note organizer we have been working on all chapter.

If you have any last minute questions this weekend, shoot Mr. Baker an email or post it to the forums! Good luck!

Welcome to the New Semester!

Welcome back (or for the first time) to Mr. C Baker's math class. This semester in Geometry we're going to be covering topics from chapters 6, 8, 9, 10, 11, and 12.

Of course, your ol' friend the conjecture will be making a return, as well as investigations and even a little bit of constructing. Don't forget everything you learned about triangles, quadrilaterals, and other polygons!

As always, remember one of the hallmarks of Geometry study - how do you know what what see is true?