Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Friday, March 18, 2011

Getting More Out of Dynamic Geometry Technology

My school is ramping up use of the Ti-Nspire. This year is the first year we are using them in Geometry. Next year they will make their way to Algebra 2/Trig1. But this post applies to the use of any dynamic geometry software (DGS) including Nspire, Geometer's Sketchpad or Geogebra. (Disclaimer: This is not a post about the pros and cons of different technology. All platforms have their benefits and drawbacks, this is what we decided to go with, and now it's my job to use them as effectively as possible. So razz me all you like, but I'll most likely ignore you.)

DGS is awesome but poses unique challenges. Kids can look at a million different examples of something (yay!), but then they think that constitutes a proof. Sometimes they don't bother looking at the million different examples - they just look at the screen as it first appears, and guess. Kids can measure anything (yay!), but they can use that to circumvent your goal of teaching them to use their understanding of a property to solve a new problem. You can set out an activity with lots for them to do, and kids like pushing buttons and it keeps them busy, but some of it is just busy work. The translating of the button pushing to the learning is where the work lies for a teacher.

So let's look at an example of a typical lesson from my Geometry curriculum: G.G.40: "Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals." Surely someone somewhere has posted a decent lesson about trapezoids using DGS. I head over to TI's repository of Nspire lessons, and look under Geometry/Quadrilaterals. It has a lesson called "Rhombi, Kites, and Trapezoids." Oh, goody. This is what I find about trapezoids in the student document:
...and that's it. Hm. Thanks, TI.

So we head over to NCTM Illuminations, search 9-12 Geometry for "trapezoid," and the only result is a lesson making an origami pinwheel:

Which is made out of trapezoids and looks kind of fun, but is not going to help us with "Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals."

How about Geogebra Wiki? Maybe there's something I can adapt there. But a search on the High School Geometry page for "trapezoid" returns nothing, as does the same search on the Quadrilaterals page.

So, yeah. I guess I'm stuck making a new thing. Awesome.

My file ends up being three pages, with questions to explore about each, with the goal of discovering the properties of trapezoids and isosceles trapezoids.

First a plain old trap:

Then an isosceles:

And then some "use the properties to find the missing thing" problems. I am not super into demonstrating examples and then making them do the same exact thing. Then they're exercises, not problems.

Then an isosceles with the diagonals constructed, which they are supposed to measure and change the shape of and conjecture about:

Then they're asked to prove the conjecture about the diagonals:

So, this lesson gets the job done, but it's not terribly satisfying. The kids completed the tasks gamely enough, and they'll remember the properties better for having seen a bunch of examples using the DGS. But it was pretty boring. (Although not as boring as listening to me wah wah wah my way through telling them the properties and doing example problems for them to copy down.) You got a better idea? Let me hear it. And I swear if anybody says do the same thing except put a picture of a bridge in the background, I'm going to throw up.

Here are the files for the trapezoid investigation, and also a similar lesson for kites.

1With reciprocal trig functions and logs to any base, features which will be disabled in Test Mode for state exams. Deity help us.


  1. Damn, I've been through the same steps for the past 5 years in Geometry.
    Before the (insert Geometry Topic here) unit starts, search online for DGS lessons on that topic.
    Spend hours being disappointed at what is available, then make the stupid thing myself.

    When I first created the investigation, I walked the students through each and every step, and took most of the critical thinking out of the process. I've improved the activities each year, taking more and more of the steps out, and (I'd like to think) adding to their knowledge each time.

    One difference: I've always had the students construct the quadrilateral by themselves, but you are giving them a fully constructed one with some of the measurements already there. Do you think there is any value to having them construct the quad? It takes more time, and it may also be a PITA with the Nspire. I've been lucky enough to have Geometers Sketchpad available. What do you think?

  2. How about the following theorem:

    In trapezoid TRAP where RT || AP:
    AT^2 + PR^2 = 2(RT)(AP) + RA^2 + PT^2

  3. @dandersod Starting from scratch and improving the activities each year is often better (IMO) than downloading an activity. You've learned a lot about the software and what you want to use it for.

    @k8novak I probably can't offer any great ideas, but I can tell you my general approach with this kind of software. I prefer to use it as a demonstration tool. That is, the students watch me build and manipulate it rather than do it themselves. About half the time I'll use a series of volunteers to actually do the building and manipulation. When I _really_ think there's a benefit to be gained, I get all the students on a computer and they do it themselves. As far as I'm concerned, that last approach takes a lot of time and cannot be done for every lesson/activity.

  4. How about not starting with the trapezoid but starting with a more familiar shape -- ie. the square -- and morphing it into the desired shapes to see gradual changes?

    If you start with letting the kids construct a simple square ABCD inside a digitized coordinate plane, they can immediately construct its diagonals and note that the diagonals are the same length AND all angles are 90 degrees. Then, have them drag C and D away from A and B, in order to modify the shape into a rectangle. The resulting diagonals would still be congruent and interior angles still = 90. At this point, nothing is surprising yet. Then, have them drag C further away, but keeping A, B, D at the same locations (inside the previous rectangle). Tell them that this new shape is called the "trapezoid" and have them jot down some observed properties of the trapezoid, as well as some observed non-properties that had been true previously. Then, ask them to move only point A or point B to create a new trapezoid whose diagonals ARE congruent. Have them jot down properties about this special trapezoid and to try to come up with a name to describe the observed relationship between the newly formed sides. (Hopefully they'll come up with "isosceles" on their own.) See if they can come up with some more examples of this type of trapezoid, with parallel lines running up-down, or running horizontally, or running *diagonally* in the coordinate plane!

    I haven't quite thought through how the angles relationships would work yet. Maybe have them drag one point around (keeping within the constraints of a trapezoid) to see the effects on resulting angles, and asking them to draw a conjecture and to justify it based upon their knowledge of angles inside parallel lines. I'm also a fan of giving kids four points, such as (1, -4), (2, -2), (7, -2), and (5, -6) and asking them to graph and decide whether it's a trapezoid. (This one is diagonal, so it requires some working knowledge of slopes. Nice spiral review.) They can then verify their conclusion by digitally measuring the angles inside the trapezoid formed by the 4 points, and "applying the trapezoid adjacent angles theorem" to verify that this indeed is a trapezoid.

    Personally, I think it's a bit ambitious to jump directly into the proofs on Day 1. I'd start by having them copy down one nice isosceles trapezoid of their creation onto actual graph paper, and to fully label all sub-sides and all sub-angles with tick marks to show which ones are congruent. Ask them to generalize what they notice using words, and then challenge them to come up with a counterexample that does NOT show the same relationships amongst sides and angles...

  5. Oh, and Kate: I always let the kids do their own investigations, and I give them worksheets with very detailed tech instructions. I only circulate to facilitate but don't lecture or demo myself. Sometimes Geometry software isn't necessarily helpful for all topics (as you well know), but when I think it is helpful, I think it's worth the extra time spent on it to make sure every kid arrives at the concept on their own. So, no demo. When I say "tell the kids that this is a trapezoid," I mean it should be in the worksheet, embedded under Step #n.

  6. I think what helped most this year with my Geo students (using GSP and Geogebra) was always trying to modify the figure to something we had already studied. Area of rectangles led to modifying (cutting and re-arranging) parallelograms to become rectangles. Same formula for area.

    Then, trapezoids at first really threw them, until they found out they could duplicate it, rotate 180 degrees, and form a parallelogram. From there it was easy to derive the formula for area.

    Of course extreme cases required more work, but at least they had some confidence looking into them.

    I really can't imagine NOT using DGS. I think the main problem right now is that the curriculum usually is not made to best utilize the software.

  7. @dandersod I do prefer it if they construct the thing. It's just a time tradeoff. As it was, this lesson was pretty perfect for my 43-minute period. If I use the laptops, we have even less time for the lesson because of the time lost to distributing and logging into laptops. And the Nspire is indeed more of a pain than the computer-based apps. And, unless there's an easier way to construct an isosceles trapezoid than I came up with, that would be too hard for them to accomplish.

    @Hao I'm not sure what you're suggesting be done with that theorem.

    @GS I haven't had much luck with the demo model of using DGS in lessons. I've certainly done it to save time, but they get way less out of it.

    @Mimi I like your idea, but I don't understand how, in the part where you "drag C further away" you'd ensure they were still looking at a trapezoid. Unless you gave them an existing sketch where the bases were constructed to be parallel. But I do like the idea of starting with a square and morphing it into less constrained quadrilaterals. Something akin to this. Maybe like 1st page is a square and no matter what you drag it remains a square. Next page you can drag one side to make it a rectangle. Next page you can drag a vertex to make it a trapezoid. Then challenge them to make a trapezoid with congruent diagonals.

  8. @Kate I guess you'd have to give them more specific instructions. Like for example, "Drag C horizontally farther away from the other 3 points." And then you'd circulate to make sure that they're actually reading and following the directions as written.

    Personally, I don't really prefer those exported demo things because I feel like they're too restricting; when the kids are actually dragging their points around on a totally free grid, it's easy for me to walk around, catch their misconceptions, and fix it, versus if they can only drag the points in a certain way, any misconceptions they might have are invisible to me. You can also guide them a bit by providing some pictures of some already-created trapezoids and ask them to compare their created shape to those to make sure they see what's in common and that their shape "makes sense" when compared with the other examples.

    But it sounds like the Ti-Nspire might be more time-consuming to manipulate, so your suggestion of having different tabs could be better for your amount of class time. (I was basing my suggestions off of playing around with GeoGebra last night. I'm not trying to sway you one way or the other, but I do like GeoGebra for what it is; but our classes are also a bit longer, usually 50+ minutes 4 times a week.)

  9. @Mimi I'd be afraid I wouldn't catch some kids who didn't read carefully and started looking at non-trapezoids. Then I'd have to be like "Wait a minute wait a minute" and show the whole class what the directions meant and kind of blow the punchline. Of course I'm only afraid of this because it's happened many times before. :)

  10. @Kate My "fix" for that is that I have notes next to those steps (in ALL CAP GIANT FONTS, BOLD AND UNDERLINED!)

    "Ask your teacher to check your shape before you move on!!!"

    You still have to run around to make sure kids are even reading that note, but not so bad if they're working in pairs. (Cuts your monitoring work in half.)

  11. I'm not sure either. I just figured there had to be some interesting property of the diagonals, so I played around with it until I got that. (and checked it using mathworld -- closed-form equations for the length of the diagonals)

    If you wanted some demonstration of conjectures vs. theorems, it might be nice to have 2 expressions, one which is provably true, and one which is true under some limited conditions (e.g. opposite angles on the trapezoid are both acute or both obtuse) and then have students check the expressions using DGS software (e.g. play around with examples and see which ones hold up).

  12. I don't have anything productive to say. I'm just surprised there isn't a lesson out there called "It's a trap!"

  13. I don't have an NSpire to check out your construction of an isosceles trapezoid, but this is what I did in geogebra:
    create a segment AB
    construct the perpendicular bisector m of AB
    construct a point C not anything yet constructed
    reflect the new point C around the perpendicular bisector m to get point D
    ABCD is an isosceles trapezoid (unless, of course, you drag C to the wrong side and the legs intersect)
    I find that transformations cut down on construction steps rather a lot.

  14. @Lsquared you're so right... I often think while I'm making sketched "I bet there's an easier way to do this with transformations" but I'm not as confident with using them. Thanks for explaining this one.

  15. @Kate,
    I have been using TI-Nspire for about 3 years and I believe I am finally figuring out how to create appropriate lessons for my students. I have found the same problem you mentioned, the lessons at the Exchange site do not quite fit. So I end up creating an activity from scratch.

    I created a similar activity and include some short answer questions, multiple choice questions, and have students construct an isosceles trapezoid. I believe even though it may take students longer to work through a discovery activity, it is time well spent. Students tend to remember the properties they discover, but quickly forget those told to them!

    It has been my experience that students adjust to using the TI-Nspire well when I introduce them to using the device during the first week of school. They soon become comfortable with using the device and actually like the ability to "see" things change. To be sure students are learning they are required to transfer their file to my computer and I check for completion.

    Although these lessons may not be considered very "exciting". Most students, I believe, would rather use the TI-Nspire than hear a lecture from me.

    I also use GSP to help students discover the properties of geometric figures. To keep students from being "bored" I try to change up the discovery lessons between GSP and the Nspire.

    I believe the ability to drag vertices and segments to observe relationships is important to help students learn.

    I am new to this forum and am not sure how I could add my TI-Nspire file to share. Let me know and I would be happy to do so.


  16. Hi Joanne - Welcome! It's interesting that you use both Nspire and GSP. My normal way of thinking is that it would be better to stick with one or the other so the kids can get comfortable and really master one thing, but it's occurring to me that that's an erroneous attitude. The different platforms lend themselves to different applications, and they're very similar, so I'm just going to stop feeling bad about using both.

    It would be awesome if you were willing to share your files! The quickest way would probably be to make a shared folder on or dropbox, and post the link in a new comment here. If you have a ton of stuff to share and feel that commentary and reflection would help, consider starting up your own blog!

    Thanks for sharing your perspective. I'm looking forward to pawing through your stuff. :)

  17. Hi Kate,
    I am relatively new to blogging and am spending a lot of time this summer reading everything I can about blogs and twitter. I have been using twitter as a PLN for about six months. There is so much information about blogs and google apps that I find I am spending all day just reading and going from one link to another.

    My point is I will look into the or dropbox. But I would like to reflect and get advice from others so I believe I have the topic for my next post!

    Thank you!


Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.