Category Archives: Common Core

Circles Part 2: Angle Relationships


Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments.  This post will look at using visual information from central angles and inscribed angles.

Pi Charts

Students sometimes lack intuition for the measure of something.  Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises.  Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.

Non-Central Angles

At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees.  Built with this intuition in mind, we look at a triangle created by inscribed angles.

The next day we get to see the formula that collapses 3 ideas down to 1.

Dynamic Angles in…

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Start – Stop – Continue 1/31#MTBoS

Just read about a blogging challenge for the month of July.  I tried this with #MTBoS30 and only got up to 12.  This time around I’m going to divide and conquer across four blogs I have various levels interactions with:, DailyDesmos, and #ggbchat.

For post 1 of 31, the theme is a derivative from others and it focuses on the past, present and future goals with 3 items for each of the Start, Stop, and Continue theme.


  • Desmos API:  I am so excited about this one.  I’m a huge fan of @geogebra and @desmos (and pretty much any other dynamic math visualization tool).  After an open invitation from Chris Lusto, I’m excited to learn from others.
  • Books: I read one book this summer so far.  Looking forward to the next.  There’s something about the raw nature of a book that balances out my passion for and interaction with technology (my wife would probably say it leans more toward addiction).
  • Cross Curricular: Late in the year this last season of school, I spoke with a science teacher about integrating geogebra applets into a physics setting.  There’s too many overlaps with math and science NOT to exploit the potential collaboration opportunities. With the CCSS Standards for Mathematical Practice we are also seeing an increased focused in constructing arguments, organizing evidence, and making sense of problems.  These type of frameworks lend to collaboration with Humanities.  This conversation of cross curricular collaboration is too far overdue.


  • Driving (as much as possible): My car, a lovely Buick that has passed from my grandparents, to my great aunt, and now onto me, is nearing it’s end.  I only live 6.5 miles from work.  There is also a Super Target less than a mile away.  I like to ride my bike, and I feel like I don’t show it the love that it deserves.  Time to stop driving (when possible) and start riding more.
  • Frustrations with Growing Pains in CCSS: There is plenty of argument and frustration with the changes in education.  Progress and growth doesn’t jive well with those who have established systems in place.  Education is a continual evolution that I’ve learned to embrace.  Those that resist this change often take plenty of shots at new ideas.  I will concede that new ideas without proven track records can be a gamble.  However, I feel that the mantra, “If it ain’t broke, don’t fix it” has little place in education.  I feel it’s better to apply a growth mindest and look at education as “Don’t knock it till to try it.”  Learning that something doesn’t work is still learning, and that should be our focus, learning.
  • Playing Candy Crush: Level 140 has been stuck on my phone for a month.  Seriously, why do I continue.  I’m done.


  • CCSS: it’s not that I have to re-learn math, or teaching, or learning.  This label is probably overused if nothing else.  I look at recent transitions in education, especially in math, and am glad for the increased coherence and creativity.  My most recent ambition is learning more about the progressions.
  • Geogebra:  Recently a group of colleagues and I started a #ggbchat on twitter.  I’ve only been using this software for about a year, but the potential has only grown the more interactions I have with it.  I plan to get more organized with my work, especially in ways that makes the applets more user friendly for students.
  • #MTBoS: OHHHH, EMMMM, GEEEEE.  If you’re reading this post, hopefully you’re already aware of the gold mine that exists out there on the net.  Get plugged in, buckle up, and try not to blink.  You will be overwhelmed, and it will be awesome.

So now, your turn: What do you plan to Start, Stop, and Continue?

Talk Some Sense into It #MTBoS30 12 of 30


People joke with me and say, “You must do calculations in your head for fun.”   My instinct is to deny this accusation, but it probably is true.  Math educators run through numbers all the time.  We play numbers games in our head like would you rather, without prompt or encouragement to do so.  I see a fraction that could be written in different way, so I try.  I see a complex shape that can be broken into parts, so I do.  These instincts were fueled by years or even decades of me talking numbers in my head.

I should never assume that this is normal, even though it is one of my life’s missions to make it so for the general population.  The CCSS Standards for Mathematical Practice says to make sense of problems and persevere in solving them (among other things).  This idea of having a sense for numbers has been discussed by some awesome educators like Fawn Nguyen, and Sadie Estrella, and it’s been the hot topic of journalists and bloggers alike.  I’m in the camp that says, “Feeling comfortable with manipulation of numbers directly supports comfort with manipulating the abstract.”

So today we were talking circles, and proportions of circles.  Instead of just giving a formula…



…we developed our own.

It started with a spreadsheet.

then a geogebra applet

then into paper practice.



Students shouldn’t be given a formula, and then be expected to make sense of the abstract values without having developed a sense for the concrete values.

Math educators need to talk numbers with students daily.  It just makes sense.

Interior Angles with Triangles (and Circles) #MTBoS30 9 of 30

The foundation:

The are only a few givens in geometry, building on those we derive many other patterns theorems.  One theorem I often prefer to focus on visually is the sum of the interior angles of a convex polygon.  There’s different ways to approach the process, and most of them refer to creating a fan of triangles inside the polygon:


Early on in most geometry studies, we learn that the sum of the interior angles of a triangle is equal to 180, building off of this we can use the above animation to calculate the sum of the convex hexagon: (180)(4)=720.

The Detour

There’s often the student who wants to divide the hexagon up differently.  The student wants to draw lines criss crossing all over the shape.



The Brakes:

Too often I would have totally passed up this opportunity and said, “That’s not how we do it, so it can’t work.”

Taking The Long Way isn’t an Error:

Thankfully, this year we saw something different.  There were still triangles inside, and with a few more lines added, the picture had only triangles inside.


Once we have all triangles, we can just count those and multiply by 180.



(16)(180)=2880.  That can’t be right, can it?

From here we get a better definition for what is an interior angle of a hexagon.  It’s an angle inside the hexagon, but still attached to the edge of the shape.  Somehow we need to get rid of the angles that are inside the hexagon but not touching the edge of the shape, like these red circles:



There are 6 of these circle sets of angles, each of them having a value of 360 degrees.  We just need to discount those (aka subtract).


This kind of process also builds a more flexible understanding of how to decompose a geometric shape in multiple ways.  I encourage all math teachers out there to try this next time a student chooses to slice it up however he/she wants.

By the way, thanks to @justinaion  for inspiring this with his post earlier today.

Outlier for Too Long #MTBoS30 8 of 30

This was me:


We are doing a unit on statistics in Integrated Math 1.  So far, I’m loving it (way more than I thought I would).

When I thought of basic statistics appropriate for High/Middle school, my mind wandered toward mean, median and mode.  I could work with these concepts decently enough.  I even understood how they were similar yet different.  Only now that statistics is a larger part of the CCSS curriculum am I taking it more seriously (better late than never).  We (myself and other integrating math 1 teachers in my district) are doing a unit that includes the basic descriptive statistics.  Now we’re working our way through linear regression and my appreciation for the content is increasing, in a concave up sort of way.  I’m see the big picture, or at least starting to.

First, we had covered the basics of descriptive statistics at the beginning of the school year.  We included some fun activities getting data from various things the students were involved with.  Recently we reviewed the content courtesy of some awesome practice via Khan Academy.  Yesterday and today the topic was correlation.  We focused on developing intuition and applying such insight toward predictions.  We used a great activity from @yummymath to try and make a prediction for how much the lifetime gross of Amazing Spiderman 2 will be.  We were able to also incorporate @desmos into the work to get some more pretty graphs.  Early next week our content team plans to continue with this topic going further and using some data from the students in the classroom.

The awesomeness today came from multiple students having the conversation about the strength of the correlation in a data set was only so-so because there were a few outliers that weren’t close to our guess for a line of best fit.  One student even used language like, “it’s not that strong cause it varies too much off the line.”  I didn’t prompt them to do it.  Nobody did.  They came up with half or more of the academic language without me defining it for them.  The students covered nearly all, if not all, of the CCSS SMPs with very little explicit direction on my part.

I feel slightly ashamed to not have had this appreciation for statistics before.  #facepalm  If you’re not including statistics and probability as a large part of your math curriculum, please ask yourself, “Why not?”  I consider myself a math geek and now I’m gaining a better overall understanding of how Stats ties in.  The support and opportunity it provides with math modeling and critiquing the arguments of others is invaluable.  I used to think that statistics was too “fuzzy” for me.  Not anymore.

New School Math, Old School Games #MTBoS30 4 of 30

This is Anthony (aka best bud):

Anthony Lopes

We were playing this:


And we were talking about these:

Standards for Mathematical Practice

We focused on SMP 1 (making sense and persevering) as well as 7 (find /use structure).  Mostly 7.  Anthony and I are from the Sega and Nintendo  generation.  Beyond a tech ninja, volleyball all-star, and just an awesome guy, Anthony is a video game connoisseur in its truest form. There have been many a late night where we would stay up around the context of a video game, but it’s not just because we have to get that next achievement, or finish the level. Instead it’s the conversation.

I always enjoy our discussions that break down the elements of engagement and strategy of a game. As he and I were reminiscing on the design of a game as old as DuckTales, his wife Maira simply commented, “I don’t see it.” I immediately pictured the student in the classroom with the blank stare that says, “I don’t get it.” Learning is about structure. Those that can master the game identify and then manipulate this structure. We noted how a game often has predefined mechanics, and it is up to the user(s) to learn and then apply said mechanics. If you are in the world of education you should be seeing the correlation at this point. I’m not saying that video games = learning. What I am saying is the design structures built into gaming could teach us a lot about engagement and learning.  One could find plenty of discussion on this following the twitter feed #gblchat or #gamefication.

Anthony and I discussed how clever design of an interactive experience allows the user to identify relationships of objects, and then progressively learn more about such relationships in order to use them toward advancement. One of the games that is entirely dependent on this learning process is Portal.  A player has to identify how to use a simple set of tools in varied combinations to accomplish tasks.  Questions like,

  • Does it matter the order in which I travel through these rooms?
  • Is there another way I can use this tool?
  • What will happen if I ______?
  • Why is that platform there when clearly I can’t get to it right now?
  • I can see an item through the window, but can’t reach it yet.  When will I come back to it?
  • Why do I need to get that item?
  • Can I use the items in combination such that an entirely new outcome is possible?

These are questions that a user intuitively asks and rarely articulates.  Such analyses happen so fast that most hardly bother to even out words to the thoughts.  The amateur simply play point and shoot while looking at the flashy colors and listening to the cool sounds.  In similar fashion the amateur student just comes with the simple tools and plays the game of school on Easy.  He/she might maintain low scores along the way but it doesn’t matter because you can often reset the level or maybe even purchase a power up for $.99.

The gamer/student progressing in his/her skill and performance takes advantage of the clues hidden within the game and exploits them.  The learning experience appears more like a puzzle worth solving, and the experience becomes self guided at times.  Those that excel in school aren’t better at memorizing formulas and passages.  The strong student is the one that can identify structure and use it.

For some teachers and learners though, this game of school is out of date, too simple to play, and tedious like no other.  We as educators need to take a bigger interest in the design of our game.  Not just the standards and outcomes, but the structure that guides the path of the user.  So where do we start?  I suggest following some tips from Dan Meyer.

Really, you should.  GO now and read it.



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