Switching Things Up, Need Help: Geometry is on the Horizon

For the past seven years, since I started teaching, I have been teaching calculus. When I started, I had 8 students in my class. Now I have 36 students. I’ve had to shift how I’ve thought about the course tremendously, and I’ve undergone a dramatic transformation in the content I teach (it is non-AP) and in the style in which I teach it. Seven years with a class is both a blessing and a curse. And honestly right now, I’ve reached the end of my usefulness for the course. I’m spinning my wheels. The only way I would be able to do a better job with it is to leave it for a few years and come back to it with a fresh pair of eyes.

And luckily, I have the opportunity to try something new. Next year, I will be giving up Calculus to teach an Advanced Geometry course for the first time. In fact, it’s the first time I’ll have ever taught geometry at all.


When I first began teaching, I was scared of geometry. Partly because as a student in high school, I found geometry to be uninteresting. It certainly didn’t have the elegance of algebra, at least the way I was taught it. Partly because I realized in that course — more than any other course — you as a teacher really have to focus on hard things. If you want kids to be able to do a proof of any kind (two-column or not), you are really teaching intuition building and connection making. Which is tough, and daunting for any new teacher, and this is why I recoiled at the thought. Now, years later, I see this as such an exciting challenge.

Right now, I am not anywhere about how to teach this course. And in fact, I’m only teaching one section and the other teacher is teaching three sections. But he’s very open to really revitalizing the course. So now we’re in exciting territory. Before I go bananas on scouring everything out there, I thought I’d crowdsource.

For any of you geometry teachers out there, if you have time to answer one or two (or all!) of these questions in the comments, I’d be ever so grateful!

1) What are your favorite geometry teaching resources — both online and offline? I’m talking books, websites, applets, manipulatives, whaever?

2) What are your favorite math teacher blogs that focus on geometry?

3) Is there a lesson you absolutely could not imagine teaching Geometry without?

4) Do you teach the course with a connective thread? Like: We are studying space and the properties inherent in space as we build space? Or: We are studying exactitude –and in particular, how we define mathematical entities so they yield uniquely understandable creatures? Or: We are studying “measurement” (in the vein of Paul Lockhart’s book).

5) I’m concerned that our kids lose a lot of their Algebra I skills when they take geometry. The other teacher and I have talked about putting coordinate geometry front and center from the beginning to help with this. Do y’all do anything else that helps keep their algebraic skills sharp, and maybe even push them forward?

6) Anything else? Problem solving? Sangakus? Geogebra use? Things you throw out because you feel strongly it’s only taught because it’s always been taught? Incorporation of Euclid’s Elements or math history? Graphic-design-y projects? Math art?

UPDATE: WOW, everyone, thank you so much for your resources and advice and for taking the time to type out so much great stuff. Now I’m genuinely THRILLED and CHOMPING AT THE BIT to get started re-learning geometry (and then teaching it). I am going to sort through things this summer!



  1. a colleague at my old school used the chocolate fix logic puzzles (http://www.thinkfun.com/chocolatefix) and had students solve and explain in a one-on-one setting outside of class during a unit on proofs. she also did a graphic design/infographics project that built off their study of area. she’s no longer in the classroom but her twitter handle is @BreeAGroff if you want to reach out!

    1. These logic puzzles are based on Meta-Forms (http://www.amazon.com/FoxMind-5512646-Metaforms/dp/B0015MC2TO), with Meta-Forms having a few additional options for puzzle types and conditions that Chocolate Fix doesn’t have.

      I agree that Discovering Geometry is a very good book. I used to teach the quadrilateral and area material from Connected Geometry, then that material became part of CME Project’s Geometry book. Considering your style you might even like the old Moise & Downs book, which has deep proofs, lots of three-dimensional work, and a formal postulate and theorem perspective. It’s a little too formal for my taste but it was popular with honors groups. CME Project Geometry does very well with the “experience before formality” concept. You’ll find several of the geometry problems you liked from PCMI in that book, along with some from “Fostering Geometric Thinking”.

      I highly recommend using a textbook, and sticking to it fairly cleanly the first time through. It will be very difficult to write a coherent, connected curriculum from scratch if you’ve never taught the course before!

      Biggest advice: make sure you are teaching geometry and not algebra. There are many opportunities for a geometry course to devolve into algebra-land, such as crazy proportionality diagrams, weird equations involving angle relationships, and the Power of a Point theorem. These can all be valid geometry topics, but it’s common to see these topics done to death with deep algebra. I warn you of this specifically because it is exactly what I did the first time I was asked to teach Geometry. All the books I mentioned above do a good job of making sure students are learning geometric reasoning, instead of just contexts for more algebra problems.

      I do NOT recommend starting with coordinate geometry, since students almost immediately lose their skills about Euclidean properties. Two lines are parallel if they are in the same plane and don’t intersect; they happen to also have the same slope in coordinate geometry.

      Good luck!

      1. I agree with Bowen that for many reasons you should hold off on coordinate geometry and spend considerable time (at least a semester) working without coordinates before introducing them. Students come in with an idea that math IS algebra and coordinates prevents them from thinking about the geometric idea. I just downloaded Lockhart’s book which I had not seen before. I’ll begin reading it tonight, but it looks like he takes that same approach of FIRST thinking about the geometry and then in part two investigating coordinates.

        CME is obviously a great choice. I also like James Tanton’s 2-volume set on Geometry (http://www.jamestanton.com/?page_id=214). It’s written prior to the common core and thus take the approach of assuming S-A-S and using that to prove rigid motions are isometries instead of vice-versa, but Tanton gets to the heart of what the big ideas of geometry are.

        One challenge for any course which focuses a great deal on proof and formal definitions–and I agree with Bowen on experience before formality–is why become formal at all. If one has an intuitive idea of what it means to be “congruent” why do we need a formal definition? Why do we need to prove that a side of a triangle is longer than the sum of the other two sides, isn’t that obvious? Even animals know the shortest way from point A to point B is to go straight there. Even statements which are not at all obvious (like the Pythagorean theorem), why bother to prove them. Can’t we trust all those teachers and the internet who have told us that is true?

        There are several answers one can give to address these natural questions. One I’ve heard is that it helps develop logical thinking. That might be true, but even if it were, why not just do ken-ken and logic puzzles. Two responses I like better are (1) Thinking carefully and rigorously about things where we do have intuition can help guide us into how to think about generalizations where our intuition is no longer so strong. For example, what would a generalization of a cube look like in 4 dimensions (or higher) and what does it even mean to talk about higher dimensions; (2) Thinking carefully about what assumptions we’re making, even when those assumptions seem obvious, can allow us to think about what would happen if those “obvious” assumptions didn’t hold. For example, it seems obvious that if I head off in a fixed direction I’ll never return to where I started, but If I keep traveling due West around the globe I will return.

        Tanton’s books do a nice job in playing with (2). As for (1) I like the idea of using the idea of thinking about higher dimensions, as a hook and a thread to run throughout a geometry course. Much of our time is spent on one and two-dimensional geometry, but we’re careful about our thoughts there because we want to understand how we might generalize those ideas to higher dimensions. Thomas Banchoff has great book on this topic, Beyond the Third Dimension. My understanding is that he is working on an online/interactive edition of the book.

  2. I love the old Discovering Geometry book. It would be worth finding one for good inquiry lessons. Also patty paper geometry is nice for investigations.

    I love love love geometry. You will have a blast.

      1. Agree. I taught out of it twice, and I didn’t like the idea of waiting to the end of the course to make things deductive. But I do think that it is a great resource for activities that give kids insights into geometry. If I were to teach geometry again, I would love to use those activities and introduce formal proof alongside it. Truthfully, I ended up doing it anyway, because my students would develop enough insights into geometric properties that they would say, “Well, wouldn’t it make sense that this is true? If you think about it…” Since that was ultimately the goal, I would take up those moments and link them to the idea of proof.

    1. You should teach online then. My daughter is struggling miserably with geometry and I have no idea how to do any of it.

  3. Sam – I have been teaching Geometry for 6 years, and I love it. So many things to do, try, manipulate. Discovering Geometry by Key Curriculum Press is a textbook based entirely on discovery, and over the years I have pulled a lot of ideas from it. I love to include some Patty Paper explorations (book full of them by Michael Serra), but I would also [given freedom from the NYS Regents exam] love to teach the course based on constructions and building from there (just like Euclid). You could start doing them by hand and then transition into GeoGebra or Geometer’s Sketchpad. I’ve also moved away from emphasis on 2-column proof towards paragraph/verbal explanation; after reading The Mathematician’s Lament, I took a hard look at what I was doing and how it was driving students away from the subject I loved.

    Definitely, definitely work on the algebra skills; if the students move on to Algebra 2 after this course, they will need to have kept them current, and kept them growing while they are learning geometry. Some regular practice on equations, factoring, and relating those topics to geometry is very helpful (and next year’s teachers will thank you).

    Just today I received a cool activity in the Mathematics Teachers’ Journal from AMTNYS in which plain old envelopes could be used to illustrate parallelogram and triangle properties just by drawing lines and cutting them apart, as well as an exploration of DaVinci Sketches and curvilinear shapes. The resources for geometric coolness are endless!! I’d be happy to share resources/brainstorm with you anytime (can you tell I like geometry?).

  4. YAY! Welcome to the Geometry club :) I’m still pretty new to teaching so I’m super excited to see what you do/hear your reflections about your couse…

    1. Still figuring those out… My school has these things called Gizmos (www.explorelearning.com), which are pretty cool. I mostly like the free versions at Illuminations better, though. I also really love the tasks/activities at the Math Assessment Project (http://map.mathshell.org/materials/index.php).
    2. Andrew Shauver is awesome: http://thegeometryteacher.wordpress.com
    Also, Tina and Fawn and Dan and Andrew Stadel and all the other MTBoS regulars
    3. I haven’t really gotten there yet since I’ve only been teaching for 2 years and this past year I pretty much scrapped everything from 1st year. I tried INBs, but I want to emphasize them more next year.
    4. I try to teach Geometry thinking about dimensions/units. My students often label angles as sides and vice versa, so trying to get them to see the differences between those shapes. I’ve also noticed my kids will get tripped up especially on volume and surface area so I try to get them to build intuition on what they’re calculating based on how many dimensions they need.
    5. My school does a fluency section of each day- it’s about 10 minutes for a “review” skill. This is where I typically bring in algebraic skills that can be useful in Geometry (solving equations, simplifying square roots, etc) as well as skills that we learned earlier in Geometry that I can now make a little more algebraic (write equations to help find angles in a transversal diagram, for instance).
    6. I’m definitely still compiling my must have list (it was my second year), but next year I’m going to push more intuition tasks first. Mathshell and Shauver are excellent resources for this. I also want to do more story telling- geometry has some really rich history!

    So excited to hear your thoughts and how you see some connections in Geometry that we can help students think about to lay foundations for Calculus :)

  5. Here are some things that I love:

    “Fostering Geometric Thinking,” Mark Driscoll
    “CME Project: Geometry,” by Bowen F. Kerins
    Christopher Danielson’s Hierarchy of Hexagons and Measurement, Explored
    All of the Things from the Shell Center (especially check out Rolling Cups, Pythagorean Proofs, Floodlights, Transformations of 2D Figures)

    You’ll need to decide what proof means in your class. This is, hopefully, going to be our focus during the geometry morning sessions at TMC this year. I tried to think through some of these ideas in a post this year. Especially check out that Michael Serra piece that I linked to. I think that Lockhart’s polemics against overly rigid Geometry proofs are helpful to think about as well. Like for so many things, CME Geometry has was strikes me as a very sensible approach.

    If you can find a copy of “Proofs and Refutations” to browse through, I think that was helpful for pushing me to see definition giving as a creative, mathematically challenging act.

    As far as the connections to algebra go, I’m not a huge fan of embedding non-trivial algebra in trivial geometry. (Like, you give supplementary angles and label one of them “2x + 30” and the other “40 – x” degrees and ask them to find the measure of both angles.) That’s a thing people do, though. I think it’s more important to develop Algebra 2, Trigonometry and Calculus ideas than it is to review Algebra 1 notions, and there are a lot of chances to do that via transformations (especially rotations), area, volume, scaling and limits.

    Ummm…my brain’s stalling out right now but I teach a bunch of Geometry so I’m sure I’ll think of more things later.

  6. I’ve not taught geometry, but I like the Art of Problem Solving “Introduction to Geometry” book by Rusczyk http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry
    It uses a somewhat informal proof style (but mathematically sound) and a problem-solving approach that feels more like real math than the rather antiquated Euclidean approaches that often feel more like history classes. The problems are probably too difficult for most students, but they are great for mathy kids who are bored with the usual stuff.

    1. _the art of problem solving_ is a great book. our honors geometry class uses this text with much success. coxeter’s _introduction to geometry_ has some great HARD problems that i use for final project. i also start the course with a reading from dunham’s _journey through genius_ (chapter 2). it’s definitely worth checking out.

      i haven’t had a chance to read all of the comments here, so forgive me if i’m reiterating, but i like throwing a little bit of non-euclidean geometry in my classes as well. i actually did a lot more in past years, but after a while, the students were so into it that we were able to offer an advanced geometry course that was entirely non-euclidean + more. now, in our intro geo courses, i only have a day or so on non-euclidean stuff, just to wet their appetites. :) for our advanced geo, we use greenberg’s _euclidean and non-euclidean geometry, rucker’s _the fourth dimension_ and weeks’ _shape of space_. jeffrey weeks is incredible mathematician and worth reaching out to. he came to speak at our school for a math symposium and ended up skyping with my advanced geo class after we read portions of his book. really amazing guy.

      happy to talk more if you’re interested in the non-euclidean / shape of space stuff!

  7. I love the dimensional-ness of geometry! Points and lines: 1D, Euclidean planes: 2D, volume: 3D

    I start my year with logic, simple to complex. I tell them about the wonderful historical figures as sidebars to whatever we are doing, gradually building a timeline of history and discovery on a nearby wall. I teach by discovery: turning a circle into a triangle by folding, carrying this idea into parallelograms, trapezoids, and congruence. A circle (using cutouts from foam plates serves as a lesson for similarity (same shape, different size) congruence (depth of field) and proportion. I used a simple YouTube video of drawing to give them the idea of one and two point dimensionality and scale drawing. From this, they designed and built a scale model of our school.
    I love the books mentioned by other commenters, I also use the text called Geometry: Seeing, Doing, Understanding; Third Edition, Harold R. Jacobs; WHFreeman&Co, NY 2003

    I connect geometric images to algebraic expressions when appropriate, and I informally assess the student skill levels in this area as we work through lessons, especially in the area of building expressions from patterns. I also love teaching the quadratic and conic sections (just the circle and parabolic w/focus & directrix) and connecting the algebra to the geometry.
    I find number talks to be incredibly powerful- not just by themselves, but for everything we explore. I encourage the students to share their paths to solutions, and while I want them to successfully solve for an answer, I really want the learning to be about the process, the problem solving, the right tools.
    Constructions is another favorite part if mine! Straightedge (not rulers- no markings!) and compasses should become extensions of their thoughts. Everything can be drawn this way!
    I do let them construct their own measurements: using whatever arbitrary length they want, they set up and solve distance, area and volume problems in their scale; compare their scale to others in the class and then convert from their scale to others.
    I also cover probability – an extension of logic – both geometric (area) and basic (the coin and dice variety) with real coins, dice, spinners, bean bag throws.
    For coordinate geometry lessons, we gridded the floor with blue painters tape, they would plot things out by standing in the correct spot on the grid, using rough green rope to make the lines (it was thick and didn’t sag- great textile experiences!) Each student made a rope “triangle” for Pythagorean’s theorem (a Pythagorean triple with knots). We went around the school and walked through our woods to explore for right triangles. We also took wood stakes and string and laid out a building on our school lawn, using nothing but our rope triangles for the 90 degree angles. The flagpole is 30 feet high- we know because we measured our and it’s shadows and calculated the proportions.
    I also love teaching tesselations and transformations: the art world is filled with classic works: I let the students create their own with construction paper and template. We start with a triangle, and add bumps and curls, turning the triangle to add the same image to each side, then observing how the pieces still interlock and combine (old file folders are great for this!) They start the triangles with a compass and a straight-edge, of course! And then there is curve stitching! You’ve seen it: straight lines make georgeous curves at their tangents. Precision in drawing (or string and pins) make it beautiful, and there are many 3D possibilities with this.
    Circles are a whole world unto themselves. Begin by folding a unit circle with coffee filters (or construct from party paper (yep- the compass again!) and proceed from there!
    Origami is a classic discovery method for angles, especially bisection and congruence. My room took flight from all the cranes! Kites (real ones) and geodesic domes made from rolled newspapers and tape (lots if tape!!!) created the shape knowledge of all gons poly, and laid the groundwork for prisms and compound shapes. (My colleagues said my room looked more like the art class than a math classroom!)
    I know you will enjoy the blend of beauty and art with math in your new discipline. Keep it light and airy, and connect the algebra wherever appropriate, not because you think you should. And don’t be surprised if students struggle with this way of “seeing” math. The students who do well in algebra have a tough time with geometry because it is pictures and words, not algorithms. I’ve even had students tell me they want “real” math, not all the shapes! Students who are tactile and visual learners will blossom. Students who like expressions will need extra comfort- don’t let them turn the explorations into a formula! Pair these types together so one will pull the other into the beauty, and the other will help with the calculations! Enjoy, and welcome to the world of geometry! I am going to be teaching Alg II next year, so I will miss it!
    Contact me if you want details on any of these, or other, lesson ideas.

  8. Background:
    1) One of the things I love about mathematics is proof. Therefore, it should be no surprise that geometry was my favorite high school course.
    2) Our school uses tracking (4 levels for each course) and I have taught all the levels of geometry we offer except for the highest track. All together that probably adds up to about 15 years of geometry experience as a teacher (this is my 11th year teaching – some years I taught multiple levels of geometry).
    3) With that said, I am planning a major overhaul of the way I teach this course next school year. I’ll be watching with interest to see what people post on your blog, especially with regard to maintaining algebra skills (a primary goal for my course next year).

    I thought I would offer a comment regarding themes of the course. The text that our district uses (which I wouldn’t recommend) focuses heavily on proof in the first semester, and slowly shifts it’s focus to the more algebraic aspects of geometry in the second semester. As such, I have a theme for each. For first semester I focus on mathematical argument. One hurdle you’ll have to get over with regard to 2-column proof is the “when will we ever use this” question. I take that away from them day one, I say, “you won’t”. But then I spend the whole rest of the semester emphasizing that it doesn’t matter that you’ll never use a 2-column proof. We are simply using it as a tool to teach mathematical arguing (and what kid doesn’t love to argue???). In fact, I typically can get the kids to buy into the idea to the extent that they appreciate the 2-column proof because it is easier to write/format/structure than the more often applied “paragraph proof”. Second semester we talk about how we can take the facts that we have proven in first semester and couple them with algebra and analytic geometry to solve problems about spacial relationships. We go into perimeter, area, volume, loci, and other problems with the frame of mind that we will use theorems to set up equations or make graphs that can be solved to find an answer to a specific problem. I suppose my second semester emphasis is more on problem solving, but I don’t really call it that (maybe I’ll change that next year too).

    I know not everyone will agree with my approach – even within my district my technique is in the minority, but it works for me in regards to inspiring mathematical thought and not just jumping through a bunch of hoops that look mathy.

    1. Two-column is a great form for *presenting* a proof but much less so for coming up with the proof in the first place. I really like using flowcharting tools along with the typical other tools (mark a diagram, redraw overlapping triangles) for constructing a proof, then a two-column or paragraph for presenting the concise argument. What you don’t want to see is a kid writing, in two-column, an exhaustive list of all the things they can figure out, they get lost in the list and won’t find their way through to a conclusion.

      1. I agree. I try to get kids to by into the idea of using flowcharts and other techniques as a way of doing a “rough draft” proof. I don’t always get buy-in on this one.

  9. I’m not sure what “Advanced Geometry” means at your school, but the go-to text on Advanced Euclidean Geometry is Coxeter’s “Geometry Revisited“. it’s not necessarily a good student text, but it’s a great teacher text.

    In my advanced geometry courses, a few units that I always make sure to teach are:

    Non Euclidean Geometry (Great chapter in Jacobs)
    Solid Geometry (Great chapter/problems in AoPS book mentioned above)
    Plane Geometry with Non-Euclidean metrics (e.g., Taxi-cab geometry)
    Vectors (your experience with multivariable calculus should come in handy)

    To counter Bowen’s remarks above, I wouldn’t shy away from algebra in a geomtery course. To me, analytic geometry is a fundamental thread in mathematics. But it is important to teach it in a way that highlights its shortcomings as well as its strengths.

    1. I agree with Patrick on this. Definitely do not ignore analytic geometry! And definitely highlight its strengths and weaknesses.

      There are some Geom textbooks that use analytic all the way, unfortunately; kids never get a sense of what is or isn’t part of Euclidean geometry, and it becomes difficult or impossible to carry the concepts into higher dimensions or alternate rules. Analytic also becomes a liability in most geometric optimization problems, when coordinatizing becomes cumbersome while cleaner options are available. In high school I was absolutely a “coordinatize everything” kid, and it works until it doesn’t!

      My personal preference is to do some but not much analytic/coordinate geometry in a Geom course, then return to hit it again a few times in Alg2 and Precal.

  10. I’d consider looking at the Math Design Collaborative for a potential scope and sequence? This is their course outline.

    Click to access Geometry_Outline2014.pdf

    I don’t know if it is “advanced geometry” or not.

    One thought I had was that it would be interesting to give students tools to explore various aspects of geometry (one example of a tool: http://www.sciencevsmagic.net/geo/ another: http://geogebra.org), and then have the entire class revolve around making observations with those tools, making conjectures based on those observations, and proving those conjectures are true in a variety of different ways. Everyone’s work in this classroom would be shared, perhaps visible on the wall, or in some online space, and people could either offer feedback on each other’s work or spend their time refining and collaborating with other students. In this classroom, your role would be to help support students designing proofs and to steadily push the definition of “what constitutes as proof.”

    Another option is to use Paul Lockhart’s book, Measurement, as a source of curriculum and/or problems. He starts off fairly basic, and proves some ideas, but also includes many examples of problems or areas of focus students could work on.

  11. I’m so excited to see the beautiful things that come out of your Geometry classroom!

    1) Everyone already mentioned all the books I like. As far as interactive things, I tend to make my own Geogebra explorations. Exeter math has some great problems that bring together locus and analytical geometry. TI has some great free resources – the ideas behind them can be adapted to your environment. Jen Silverman has posted a ton of Geogebra stuff.

    3) My favorite geometry lessons are open-middle-ish problems that can be approached with a variety of tools. One group sketches it out in geogebra, one group grabs compasses and straight edges and scissors, one group sketches a diagram and tries to equate things, one group puts it on a coordinate grid. That’s the dream. Of course they have to get comfortable with all those tools which is a challenge. I tended to start the year with lots of constructions (analog and digital) and try to keep coming back and back to them.

    4) The biggest one I always come back to is “how do we know for sure that something is true?” (what is a valid proof?)

    5) Agree with Bowen to not start with coordinate geometry. But I’d say definitely highlight it as a problem-solving method whenever it makes sense.

    6) Depends on your kids, but I found it important to keep things as concrete as possible. The real meanings of length and angle measures and area are slippery. What i mean is, unless kids are actually measuring things, and drawing things with some precision, and building 3D objects, on a regular basis, they start to see labeled distances as devoid of meaning and turn to memorizing rules that don’t mean anything to them.

  12. Sam –

    Coming late to the party, but glad to hear you’re getting a geometry class! I really enjoy teaching geometry, and I think there are plenty of resources and references other people have given that are great. Serra’s book fascinates me, as does the Lockhart “Measurement” book. A couple of other possibilities:

    1) If you’re going to use GeoGebra (highly recommended as a great sandbox for exploration), “Exploring Advanced Euclidean Geometry with GeoGebra” is a good book: http://www.amazon.com/Exploring-Advanced-Euclidean-Classroom-Materials/dp/0883857847

    2) I have heard good things about the new Apostol & Mnatsakanian book, “New Horizons in Geometry” – have not read it yet, though. http://www.amazon.com/Horizons-Geometry-Dolciani-Mathematical-Expositions/dp/088385354X/ref=sr_1_1?s=books&ie=UTF8&qid=1400163960&sr=1-1&keywords=geometry+apostol

    I tend to fall on the “less concerned” side of the whole “STUDENTS WILL FORGET ALGEBRA!” debate, frankly. If students can’t write the equation of a line after a year of a typical geometry course, they NEVER KNEW IT TO BEGIN WITH. And the other stuff that goes on in algebra 1 that may/may not come up in geometry – factoring, the quadratic formula – can be retaught in algebra 2 if necessary.

    The hardest thing in geometry is the construction of original proofs. I am agnostic about the importance of doing so – if you’re dealing with a bunch of future math majors, then sure, do it – if not, you might want to focus more on the underlying logic (as applicable beyond geometry).

    Have a great time! Oh – and before I forget – HAVE THEM MEASURE STUFF. LOTS OF STUFF. Distances, areas, angles….


  13. I’ve taught geometry for over 12 years and am planning on changing things next year to be more hands-on and investigative.

    I’d say using Geogebra / Sketchpad to investigate & explore ideas is a good place to start. Key Curriculum Press has some very good books tied to Sketchpad & discovery.

    NCTM’s Mathematics Teacher had a couple of cool articles that I’ve worked into my classroom over the years. Ayana Touval’s (May 2009) article on “Walking a Radian” is excellent.
    Michael Todd Edwards, James Quinlan, Suzanne R. Harper, Dana C. Cox, and Steve Phelps (Feb 2014) article on angle chasing was a good one too.

    Proofblocks (http://www.proofblocks.com/) are a different way to attack triangle congruence proofs.

    I’ve seen teachers do some cool things with the book Flatland.

    If you’re class is truly an “advanced” class of students, there is a lot of things you can integrate with logic. Truth tables, knight / knave problems, logic puzzles all can provide extensions from a typical geometry text.

  14. 1) I think you would really enjoy Chekarian’s book, Geometry: A Guided Inquiry.

    In this book, each chapter has an opening problem. Chapter #1 starts right away with a brilliant “shortest path” problem. The exploration is awesome for the students and the proof is elegant (hits on triangle inequality). This book, written by Chekarian called Geometry: A Guided Inquiry, is all about inquiry. I highly recommend it for an advanced geometry class.

    I found a link that shows you the first problem about Shortest Path.

    As it says in the link, “This single problem leads to the development of several important concepts: distance from a point to a line, perpendicular lines, and reflection of a point across a line. ”

    2) I’ve always wanted to teach from Euclid’s Elements but have not explored that. I bet you would enjoy trying to impement the material to fit your curriculum.

    3) I love old school constructions with compass and straightedge (and also supplement with Geometer’s Sketchpad or Geogebra to help students make conjectures –easier to make conjectures when you can drag and click rather than have to redraw all the time).

    4) I think “what is the locus” type questions given certain criteria are really thought provoking..

    Good luck and I look forward to hearing about what you discover about teaching geometry next year.

    1. Ah, we call that the “Burning Tent” problem. It’s a terrific problem, pretty tough as an opener for an entire course though (CME uses it as an opener in a later chapter on geometric optimization). Chekarian’s book gets very formal very quickly, overall it’s solid and leading with experience is a great way to go.

  15. Goodness! I’m glad I wrote down some thoughts when I first got your post last night and typed them up just now before I saw the numerous other replies, because now that they’re here I probably wouldn’t have felt my comments added marginal value. (You may not be as glad.)

    Sorry I’m not really following the structure of your questions, but I hope this helps. As a bit of background, I had a similar experience as a geometry student, and I think it profoundly affected the way I look at problems (for example, I am not very inclined to use geometric representations, even when the alternative is cumbersome and precludes insight; this really hit me at PCMI when I worked with Henri a few times). Later in my own student years, I had some good experiences with problem-solving that have helped me clear this up a bit. Still, I haven’t taught geometry much and haven’t really enjoyed it. My more recent work with teachers has given me some new perspectives that may be helpful.

    (A question: What is an “advanced” geometry course? A different track? A subsequent course to an initial geometry course? Different expectations or emphasis? Just wondering.)

    Some big ideas that I think need to be emphasized (some still at the thinking stage):

    • Thinking about the flow of an argument in the big picture – chains of reasoning – and distinguishing among premises, conclusions, and reasons at various “scales.” Concretely, I would strongly recommend ProofBlocks and three-column proofs as tools (I can send you more). Michelle Cirillo has also done some good work (still at early stages) on helping students understand proof in the geometric context.

    • As a related but more “tactical” habit, helping students think about “reasoning moves” (which are often also problem-solving or modeling moves) like working forward and backward, asking “wouldn’t it be nice”, “salvaging” statements, and pulling out and re-assembling sub-parts (whether geometric objects or arguments).

    • Working with triangles and angle-chasing. The basic books from the Art of Problem Solving crew have a few good chapters on this. Also Michael Todd Edwards has done some nice talks about this that he’s posted online.

    • On this note, I think the use of dynamic geometry tools needs to drive a fundamental overhaul of how students make sense of some geometric objects (especially angle relationships) from an early stage. I got to be pretty fluent with the Geometer’s Sketchpad but never really crossed over into getting students to use it productively. A lot of folks are using GeoGebra in impressive ways now, and sharing and writing about it (Todd also helps organize a free annual GeoGebra conference in Ohio).

    • Finally, I think the dynamic geometry could connect to looking at transformations as a central organizing principle. (From my passing experience with topology the big idea of invariants is closely related to this.) If “advanced” geometry means “after a lot of algebra” then this could be connected with matrices as well (and would give a lot more meaning to ideas like determinants, matrix multiplication, and inverses).

    More comprehensive resources I’d recommend (second-hand; I didn’t get to use them):
    • EDC’s Connected Geometry and later CME Project Geometry (as mentioned, including by Bowen himself). The former is older and a more “pure” incarnation of the approach but may be harder to use as a textbook (though I get the impression that’s not what you want anyway).
    • Michael Serra’s Discovering Geometry (Key Curriculum Press) is another good book (again, as mentioned, likely by people with more and better personal experience than I have).
    • Anything by Henri Picciotto (I can put you in touch with him if you like, and you’re probably connected with him already through math bloggers like Justin and Avery as well as others who have previously commented whom I haven’t had the pleasure of meeting myself).

  16. YES! Discovering Geometry is teaching GOLD. tbh, you and Dan Meyer were my go-tos! This is my last year teaching geometry (next year, we move to integrated math, and I move to pre-calculus), but I think I learned more from you two about teaching this class well than any other blog. Except maybe Fawn.

    My year has been pretty successful, and I teach a class that is very… non-advanced. My focus has been wavering between “real-world” (art and nature) and preview of advanced mathematics. Ex: my class topped my district in Trig performance – much to everyone’s surprise – and I approached trig with an eye for calculus & the unit circle. For application, we found the distance between planets. We estimated the distance along the equator. The kids thought that was amazing. (Students really liked learning how people “discovered” new concepts. If I did this again, I’d use more Euclid.) More than anything, this got my students to consider higher-level mathematics like they hadn’t before.

    Were I teaching advanced students, I would teach it the way I was taught — through proofs and construction. 17 years after taking Geometry, I still remember practically everything I learned because we discovered every new geometric concept. Construct parallel lines by copying angles? Then you will remember that corresponding angles are congruent. Show how and why that is always true? Then you will always think about explaining each step of your thought process as you work toward a conclusion. That was the best mathematical training I had and served me well in Trig & Calculus classes later (because I formally learned logic & process rather than memorization of concepts).

    Best of luck!

  17. So excited by this news- geometry is why I love math and it is what I loved teaching math – we will talk!

  18. You said a lot of good things. I have been in Geometry for 6 years now and I came to it with the same apprehension you described from earlier in your career. You are also right when you describe the need for sense-making and connection-building. I have found that this requires the appropriate balance of algebra and almost ELA-style writing. My students get used to essay questions and applied vocabulary exercises. The tricky thing is that the sense-making extends beyond the content into the methods. To help the students understand why we are writing about this topic and crunching-numbers about this other topic.

    I am looking forward to following this development. I would like to see what content you choose, how you organize it, and how you have the students explore it.

    1) I rely pretty heavily on three already mentioned: Dan M., Andrew S. and James Tanton.

    2) None that haven’t already been mentioned.

    3) I’ll get back to you on this.

    4) We base much of our year on transformational geometry. We use rigid motions to connect to congruency and non-rigid motions (dilations) to set-up similarity which carries us into circles, Trig and Special Right Triangles. Then we are onto 3-D.

    5) We don’t focus on specific algebra skills as much as the strategic use of skills they’ve already learned. Plus the distinction between solving equations (which is what Algebra tends to discuss heavily) and strategically writing equations (which often gets a lot less practice). These are important algebraic ideas that we push.

    6) I use Kolams to discuss rotational symmetry. That tends to be a nice blend of crowd pleaser with good content exploration. http://wp.me/p1AxHx-62 Also, I am going to try blogging next school year to try to establish a more holistic dialogue between the students and the math, but that idea is still in beta form.

  19. I designed a project-based geometry unit with a couple of people a few years ago. It focuses on properties of circles and exploring the “livability” of communities. If you’re interested you can download the unit here: http://solvingworldproblems.wordpress.com/lesson-plans/

    If you end up adapting it, I’d love to hear how it goes. Teachers I know in Texas, California and Massachusetts have used a version of the unit. But we’re always looking for ideas to improve upon it.

  20. I hope to be teaching quite a bit of Geometry next year, too! (Our curriculum is being adjusted as we speak.) I love Geometry, but it looks like you already have lots of great resources here. I think of Geometry as an opportunity to develop a class that is as tactile as it is mentally rigorous. I work in algebra whenever it seems appropriate, but I don’t force it because when the geometry gets really rich, the algebra arises naturally. I’ll keep reading on what you’re doing and add/recommend stuff where I can!

  21. I took a great Geometry course during my first year of teaching – incredible class! At the time, I was teaching Geometry, and rather badly – I haven’t taught it since, but have always wanted to try the approach used in that grad level course taught by Dr. John Mayer, UAB – he’s seriously amazing.

    I’m linking to a presentation he gave on how the course is structured – an inquiry approach with huge collaboration component. Really great structure with proofs done at home, discussed in small groups, then presented and discussed as a class – incentives for presenting and/or questioning the presenter(s). Quizzes based on proofs. Very meticulous work recorded in notebooks and assessed.
    Also linking to the book he used – we were fortunate to get a free copy, since it was a draft at the time. It’s available on Amazon:

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