Monthly Archives: January 2016

[2016 Blogging Initiative] Week Three: Understanding Questions


After making the transition to Standards Based Grading a few years ago (which is awesome!), over time I realized that my class had become too skill-oriented. To fix this I first tried to create standards that were “understanding standards”, but this overwhelmed my students with too many grades.

It took me an entire semester, but I realized that what I should be doing is asking “understanding questions” on assessments instead of only skill-oriented questions.

I try to limit my assessments to three questions (sometimes a question might have multiple parts), but I now always try to include an “understanding question” as one of those three questions. I never grew up answering these on math assessments, and they’re harder to grade because there’s usually not “one right answer”, but it has helped me get a better grasp of what my students understand (or don’t).

What do these look like? Here are some examples.


Exponents: Explain why b^x \cdot b^y = b^{x+y} is true.[1]

Polynomials: What does multiplying polynomials have to do with the distributive property?

Polynomials: Why can you combine some terms of a polynomial but not others? (3x^2 + 4x^2 can be added but 3x^2 + 4x^3 cannot)

Rational Expressions: Before factoring was the opposite of simplifying. What has changed and why do we factor first to simplify rational expressions?

Functions: Give an example of a function and a non-function outside of math class.

Transformations: Why does (x+3) move a graph left and (x-3) move a graph right? Isn’t that that the opposite of what you would expect?

Logarithms: Explain why \log_b{M} + \log_b{N} = \log_b{(M \cdot N)} is true.[1]

Reflection/My Own Questions

(1) Are these “understanding questions” enough to check for understanding? Probably not by themselves. So I need to get better at assessing repeatedly over time to check for retention of understanding.

(2) Should I give students the questions beforehand? Right now I do because if they want to figure out the answers on their own, great! As long as I have enough possible questions so they’re not simply memorizing and spitting back what I say, but really understanding it. (Or should they be able to get these questions even without me providing them ahead of time?)

(3) Is there a place to get these types of questions? I primarily look in the textbook or come up with my own questions, but surely there’s a bank of these somewhere online that I haven’t found yet.


These questions are ones that get at understanding, though harder to grade (at least they take longer), are worth it. When I started these questions, I was sorely disappointed how little my Precalculus students understood (even though they have seen some things, like exponents, in Algebra II and probably even in Algebra I!). I’m really curious what people think about the three reflection questions above.

Thanks for reading!


[1] I’ve flip-flopped between using the vocabulary “prove” and “explain”. The former suggests there’s one right answer to students whereas the latter allows for various explanations. “Explain” also is harder to grade, but I’m very excited if I see students start to write out examples in their explanations. No, it’s not as rigorous as professional mathematicians, but it shows me that they’re starting to understand.



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[2016 Blogging Initiative] Week Two: My Favorite = Desmos


Desmos. If you’re a math teacher and you don’t know about it, stop reading this and go to their website now.

It took me a long time to even think of Desmos as one of “my favorites” because it’s so integral and every-day in my math classes that I guess I’ve started taking it for granted. I forget the days when TI button mashing ruled the day.

Here are a few things I appreciate about Desmos.

Browser Based & Free

The fact that students can access these from any computer with internet, or install the app and have them without internet access is incredible.

I have a class set of iPads, but increasingly students are asking to use their own phones. I point out to them that their eyes are going to go bad, but many of them have become proficient at using the tiny screens.

Sharing Graphs

I am increasingly using the “share this graph” feature. A student will create an awesome graph and instead of just presenting with the projector, all the students in the class are interacting with their own version of the graph on their device. Or I’ll create a graph (usually a table that I don’t want to waste class time having students type in the data points) and share it quickly with the whole class. It’s great!


Nothing beats building intuition with function transformations like having students move the sliders to manipulate the values. Before Desmos, I used Geogebra, but I spent a lot of class time showing students how to make sliders–it was not nearly as streamlined and intuitive as in Desmos.

Activities & Activity Builder

I was a huge fan of Function Carnival, Penny Circle, and Central Park when they first came out. I even had my students do these, even though we weren’t exactly on those topics when each activity came out (it was review, okay?). Now Desmos has Polygraph[1] and, the latest that I’ve yet to try with my class, Marbleslides, each excellent activities.

But I think the best thing that Desmos has done in this area is the Activity Builder. I haven’t had enough time to dig in and create activities, but the possibilities are endless. And no need to reinvent the wheel–activities that other teachers have created are available for you to see, too! Holy smokes!

To access all these cool things, go to I haven’t even started talking about the awesome teacher-view for when all your students are working on these activities.

And So Much More

It would take me way too long to mention all the incredible things that Desmos can do and is doing[2]. And they’re constantly improving things. They respond quickly to feedback, both in communication and by doing the thing you asked for within Desmos.


Thank you, Desmos!


[1] There’s a huge bank of polygraphs since people can make their own!

[2] To list a few: recognizing function notation, derivatives, inserting images,intuitive click-on-the-point to find the intersection or x & y-intercept, domain & range restrictions, lists, click & drag points, regressions for any equation, implicit function, colors!, labeling axes, easy animation, and converting equations to tables. I’m sure I forgot ~90% of the features that I like.


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[2016 Blogging Initiative] Week one: A Day in the Life

A day in the life: 1/13/16

6:30 am My watch buzzes and wakes me up. I have an alarm clock across the room that buzzes at 6:31 just in case I don’t wake up, but this way I have 1 minute before I have to wake my wife up. Of course on Wednesday she’s already out the door at the gym until 7:30 I shower, eat, and get dressed, all hoping not to wake our 1 1/2 year old son. If he wakes up, I have to change & feed him, which means I can’t get to school 15-30 minutes early and make sure all my plans are ready for the day.

7:30 Benji (my son) wake up just before Anna (my wife) arrives home and I spend 10 minutes with them before heading out. Benji still gets really upset as I’m walking out the door, which makes me feel loved and sad for him at the same time.

7:40 I arrive at school. Since I live on campus, it takes about 90 seconds for me to walk and get to school. If we ever leave this place, I’m going to miss the commute. A few students have already arrived and one comes in to ask me how she did on yesterday’s quiz. I get ready for the lesson today in Precalculus, which involves making boxes out of 8.5 x 11 paper and comparing the volume to the size of the square in the corner, so I need paper (easy), rulers (little harder), and scissors (I knew I had a box somewhere… 5 minutes later I find it).

8:10 First bell rings, student pour into class as I give them high fives @gwaddellnvhs style. Some students still grin, even though I’ve been pretty good about doing it every day.

8:15 Class starts. I pray and then I demonstrate, silently, how to make a box from 8.5 x 11 paper without wasting any paper. Here’s a picture.



We didn’t get as far as I’d like, mostly because students really struggled to come up with creative math questions after I showed them how to make a box. Even after I made a second one with different dimensions, they still took nearly 15 minutes to say anything remotely close to “Which box has the most volume?” I need to work on improving their creative thinking. We’ve done a 3Act lesson nearly every week and yet they’re still quiet after the think-pair-share, which often gets off-topic very quickly in the “pair” part of that. At least we made a table on one whiteboard and a graph on another (though only half the students actually put their “point” on the graph. Ignore the student names, ha.


One new twist this year is I had students figure out how many cubes go in the box, which helps many of the students to get a feel for volume. It shocked me how many students forgot how to find the volume of a box (rectangular prism) in Precalculus.

9:50 Class ends. I have 10 minutes to get 12 iPads across the street (yes) to my 2nd classroom before Physics.

10:00 First bell rings. Second bell rings at 10:05, but I don’t wait until then to start class since you can’t hear the bell from the classroom anyway. I just wait until all 12 students are there. Today there are only the 9 boys–all 3 girls are absent at the start of class. We start the mistake game and this is the first year that I feel the activity has been a success (I’ve tried it 3 or 4 years in a row now!). The primary thing that helps, which I can’t remember whether Kelly puts this on her blog post or not, but I would highly recommend, is to make everyone’s situation different. We did this today (this is the 2nd time in this class I’ve done it) by having students go outside and shoot a tennis ball into a basketball hoop. They recorded each other using Vernier Video Physics and because each partner pair shot from a different location and threw it different heights, they each started with different values, so it wasn’t as simple as “oh, that’s what was different from our board”. Some students had really neat mistakes, stumping the entire class, even though (a) it’s only the second time we’ve played the mistake game and (b) many students make these mistakes in their own work!

The second part of physics involved setting up the momentum transfer model and intro experiment, but we didn’t get to actually do any of the experiment since the mistake game took so long (which is OK!). [1]

11:35 The bell rings and I have 35 minutes to get to the lunch hall (we get a free lunch once a week) and get home (90 second walk, remember?) to eat with Anna & Benji and get back to class, but a student stops me and asks to see his Physics exam from last semester, which is in my first room of the day. I get it for him and do all the above and am walking back to school just as…

12:10 The first bell rings. I now have a “class” of 2 remedial students who both failed my Chemistry last year. Jealous? Well, this is supposed to be my planning period [2], so don’t get too jealous. Oh, and we meet in the computer lab because there isn’t a classroom free for me to have them in, so I’m kinda in my 3rd classroom of the day. These two students really, really failed Chemistry, so I’m taking a different approach in the form of 3 steps: (1) get them excited about science, (2) jump into chemistry via a modelling-like curriculum [3], (3) make sure we’ve covered all the parts of chemistry and that I feel comfortable passing them. Today they were working on this PhET simulation and playing the game. They both got 10/10 on all 4 games. I told them “let’s do a quiz on this on Friday” (their smiles faded), “where the quiz is you have to get 10/10 on all 4 games again” (they smiled a little more), “and you get as many chances as you want to make it” (they broke into huge grins again). These smiles coming from students who hated chemistry last year when I taught it. It’s amazing how successful students can feel when they’re not comparing themselves with others who just get stuff before they do. Here’s a list of “atom rules” that they came up with just a little prompting from me.


1:50 The first bell rings for the last class of the day and I move across the hall (again) to teach in my 4th classroom of the day. It’s Precalculus again, but since this mostly group is juniors (the other was mostly seniors), we’re a little ahead of the other class and we’re working on exponential growth. I showed them, and we have a good discussion about, this graph of the Ebola outbreak from 1.5 years ago [4]. We talk about why ~4,000 (a small number compared to the number of people in the world) people having such a contagious disease is a very frightening thing. They take a quiz on exponential functions and all work on an old practice AMC test for “fun” while waiting for others to finish. When everyone is done, they divide into groups and talk about the “homework”. I’m trying a flipped classroom, so it’s really about the guided outline they had to fill out and answer questions on as they watched a video I created. The topic was an intro to logarithms.

3:20 The bell rings to go home but I hang out and answer a few students’ questions. We’re supposed to stay until 4:00 (or 3:45 if you arrived at 7:45 in the morning) but our boss is away on a trip and I think the secretary and myself are the only ones in the building by 3:50.

4:00 I get home and play with Benji in a “cave” we’ve made out of our comforter and one of his pack-n-play cribs. See photo.


5:25 We go to Wednesday night dinner, which happens to be in the same building as where the students eat lunch earlier in the day.

6:30 I finish eating and help wash the dishes. There’s a lot of help this week since it’s the first one of the semester so we get done early and I go home at…

7:00 This is the time that I have to prepare for tomorrow, but I foolishly hop onto, among other websites and follow Wake Forest basketball. We lose to VT :/

8:00 We go across the street to visit our elderly neighbor as we do every night. We do that because he doesn’t have family that lives nearby, but he loves seeing Benji every night.

8:30 I go play pick-up basketball with several of the teachers and locals which include some alumni, whom I’ve taught.

10:15 I get home from basketball, take a shower, ignore the grades I was planning on doing cause I’m tired, and go to bed.

6:30 am My watch buzzes…


[1] Yes, I steal almost my entire Physics curriculum from Kelly O’Shea.

[2] No, this semester I don’t technically have a planning period, but one class of mine is 1 student and another is 2 students, so I plan around them while they’re working.

[3] I’ve never done a modelling curriculum for chemistry, though I’ve done some modelling-like lessons in the class.

[4] I did not create that graph and I’m sorry that I’ve lost where I got it from. Please let me know so I can credit them if you know!


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