Seeing through the eyes of 5th and 6th graders

The assignment was simple enough. Take a photo that reflects energy changing from one form to another. It could be a photo that you find funny or interesting. Or it could be something that you’re curious about or have questions about. That part was up to them. The “why” behind the photo was their business.

Here’s a few of the highlights. Enjoy. It’s not very often you get to see physics through the eyes of 10-12-year-olds.


Which one is your favorite?

Gallons and Gallons of Pennies

Sometimes, “real-world” problems just go ahead and write themselves. And I say take advantage. Why be creative with the actual world can do the heavy lifting for you, right?

This floated across my Facebook feed. Pretty sure you’ll see where I’ve made some edits to the original texts.

Pennies Problem

Sequels could potentially include:

If Ortha wanted to exchange them for quarters, how many 5-gallon jugs would she need? You could do the same with nickels or dimes.

What would be the mass of each penny-filled jug?

What do you think? What other questions could come off this wonderful set up?

Exploring Reflections with Desmos Activity Builder

Geometric transformations take up a good chunk of the first quarter of Geometry (at least the way that I had it sequenced). The tricky part of teaching transformations in Geometry is the delicate balance between the non-algebraic techniques and understandings and their algebraic counterparts.

For example, consider the two following statements.

Two images are reflections if they are congruent, equidistant from a single line of reflection and oriented perpendicularly from said line of reflection.

The reflection of A(x, y) is A'(-x, y) if the line of reflection is the y-axis and A'(x, y’) if the line of reflection is the x-axis.

As a geometry teacher, am I to prefer one over the other? In my experience, they both present challenges.Students are often a little more enthusiastic about the first (which is why I start there), but can often be more precise with the second. The second requires the figure to have vertices with coordinates and the line of reflection be an axis. The first requires the figure be drawn (and fairly accurately, at that.)

So, in my attempt to learn how to use Desmos Activity Builder, I wanted to produce something useful. So, I made a Desmos Activity to bridge the transition from a visual, non-algebraic understanding of reflections to an algebraic one.

Full disclosure: This activity presupposes that students are familiar with reflections in a visual sense. It isn’t intended to be an introduction to reflections for students who are brand new to the topic.

Feedback, questions… all welcome.

The Math Reading Classroom

In my previous post, I make a case for reading becoming a necessary component of the math classroom.

It’s interesting to consider what would it look like to integrate “the reading of mathematics” into a secondary math course as an essential learning target. “Essential” in the sense that we explicitly teach it, assess it and report out student status on it.

You’d start by creating a learning objective (or borrowing one that’s already written). Then you create some success conditions. Then you create an assessment (or series of assessments) so that you have a tangible experience in your mind when you are designing the learning activities.

In my mind, reading would need to be treated like one of the Common Core’s Standards of Mathematical Practice. It isn’t math content. The reading experience would be designed to a certain grade level, but in order to properly assess the reading, you have might need to back off the intensity of the math.

Obviously, word problems are nothing new. But this would be a different kind of word problem. Using the word problems as a READING assessment instead of a math assessment is not something I’ve seen before… or done before. Reading assessments look like a reading passage with some strategic follow-up questions designed to examine a student’s reading comprehension. Seems like a word-problem-esque scenario could take on that feel.

I imagine something like this:

“Danny and Sandy both collect bottle caps. Whenever they get together, they bring all their bottle caps with them. Danny has four Coke caps, three Sprite caps, five Mountain Dew Caps, and a Faygo cap. He’ll have more Mountain Dew caps when the 12-pack that his mom bought is gone. Sandy has nine Coke caps, five Pepsi Caps, and nine Mountain Dew Caps. Sandy wants more Pepsi caps, but that will have to wait because her Dad came home with a Pepsi 12-pack of cans yesterday.”

Okay, this is not perfect and people who write test questions for a living would probably come up with something much better. But, this a fairly standard word problem set-up. So, what questions would we ask if what we’re really trying to do is assess a student’s ability to read instead of assessing his/her ability to do compute? Maybe questions like these:

From the evidence, which of the two do you suspect drinks more pop? Why?

How many more bottles of Mountain Dew do you think are left to drink at Danny’s house? Why?

After Sandy finishes the 12-pack of Pepsi that her dad just bought, how many Pepsi bottle caps will she have? Explain how you know that?

The reading passage is written at about a 6th grade level, depending on which index you use. The math in the questions is probably first or second grade. So, giving that passage and set of questions to a seventh grade class would only be valuable as a reading assessment. How well are the students comprehending the details of the situation? Details like Danny’s 12-pack of Mountain is eventually going to yield 12 caps and he’s already 5 caps into it. Sandy’s 12-pack of Pepsi will yield zero caps because the pops are all cans.

Asking those questions gives you a window into the ability that each student has to comprehend the text. But considering the possibility of assessing our students in this way leads to a couple of confrontation points.

First, I don’t know of any math teachers who have learning standards written for mathematical reading. Those would have to be developed. That’s not a small or insignificant step. We don’t want to get in the habit of assessing without clearly defined learning targets.

Second, our students usually skip word problems in their practice sets. We would have to build in structures that change that. Whether it’s taking some pages from the #FlipClass playbook, or using some cooperative learning structures, somehow the attitude around word problems would have to change. We don’t want to get in the habit of assessing things that we know the students aren’t practicing.

Third, if a student begins to fall behind, or regularly is assessing at a low level on the math reading assessments, most math teachers are not well-equipped to provide appropriate curriculum-based interventions in the area of reading. These exist, but math teachers are typically not trained in their use. We don’t want to get in the habit of asking teachers to do things they aren’t trained or equipped to do.

Confrontations aside, there’s a lot of potential here. Potential for student growth. Potential for interdepartmental collaboration. Potential for more holistic math classes. But as with all updates, redesigns and revisions, it needs to be done strategically, thoughtfully, and with the best interests of the teachers and students in mind.

Desmos-Enhanced: The remodeled Pencil Sharpener Problem

Lately, I’ve found it tremendously enjoyable to revisit some of my favorite homemade problems and use Desmos to model them.

I decided to remodel the Pencil Sharpener Problem this time. If you’re not familiar, go check it out. Here’s how it goes.

Three boys are held after class for detention. I told they have to stay for a half-hour, but they can leave earlier if they can grind down 100 pencils by hand in less than a half-hour.

So, the three of them decide to take the me up on my offer and begin cranking the pencil sharpeners as fast as they can. Each of their top speeds is recorded on video. If we assume that they keep their top speed up the whole time and don’t slow down, then how long with their detention last?

This problem has created some fantastic student work. Enough so that it is almost tempting to force pencil and paper work.

However, I couldn’t resist the temptation to create a Desmos worksheet for it.
Besides, by now, Pencil Sharpener Problem is ready for an extension. How’s this:

It seems safe to assume that the boys will tire as they crank the pencil sharpeners over and over and over. How about we say that the each subsequent pencil takes 5% longer than the previous pencil. So, if first pencil took 60 seconds, the second one took 63 seconds, and the third one would take 66.15 seconds, and so on.

Would it still make sense for the boys to grind away pencils? Or should they just sit quietly for 30 minutes?

Redesigned Wedding Cake Problem with Desmos

It all started with this tweet:

I got a taste that Desmos was capable of conditional images and text and stuff. So, after I did a little of the ol’ one-man PD tinkering around with their link, I decided I wanted to try to model something. So, I figured why not Wedding Cake Problem? It’s gotten a few redesigns already, so what’s one more?

And so I created this on Desmos.

What I like about this one is that, after all this time, I believe that I have finally figured out a way to deal with the issue of the variable “frosting thickness”.

Also, I like that I was able to include the reality checking piece that the top had to be the smallest section, the middle had to be the second smallest, and the base had to be the biggest.

I encourage any feedback.

Also, I have a question for anyone who is able to answer it: Is there a way that I could make the function round up to the next whole value? I’d like the line to only snap to whole number values since the question is “how many jars of frosting will you need”? (That is, if you need 2.3 jars, you need to buy 3 jars.)


The last question received an answer by John Golden (@mathhombre) who taught me how to use the ceil() function. You can now check out the final Desmos worksheet.

Algebra, Geometry, Desmos, and the number line…

Number lines are wonderful tools. Simple, elegant, and useful to everything from beginning to count, to categorizing number groups, to helping students make sense of irrational numbers.

So, naturally my mind started racing when my daughter handed me this.

2015-01-26 10.06.29-1

I think she intended for me to throw it away, but, sticking to the old adage “One man’s trash is another man opportunity to integrate several pieces of instructional tech to create a delicious opportunity for young people to learn mathematics” (Paraphrased), I decided that this was just to powerful a tool to chuck.

So, I thought 3-Act. The problem is, unless I’m extremely resourceful, I don’t have enough for an Act III here, so I switched to a different number line.

2015-01-26 11.46.19

And trimmed it.

2015-01-26 11.46.21 (2)

There we go!

Okay, so let’s start estimating the locations of some missing values. To give it a bit of context, I placed the number line scrap on a grid on Desmos.


Then, to collect the data, I created a Google Form for the students to enter their estimated coordinates at points 100 and at 55 in the missing portions of the number line.

Once you get all of the student data entered, you can simply copy the x and y columns in the Google Sheet…

Desmos Forms

… and paste it into a new line in your Desmos graph.


Then, the second part of the lesson becomes the teacher facilitating a conversation about the different ways we could either hone down our estimates, or calculate EXACTLY where the 100 point would be. (The two ideas I had were modeling the number line as a linear function or as a hypotenuse of a right triangle. Both scream for proportional reasoning, which my experience suggests is a useful activity anytime it can be fit into the curriculum.)

Then, once they feel like they’ve come up with answer they like a bunch, you can reveal to them the answer.

You may also choose to re-paste their estimate points so they can see how well they did as a group.


I like activities like this because it provides ample opportunity for focused guessing, collaboration, and a variety of solution processes. It also asks a pretty simple question at the beginning, which helps to include everyone, regardless of level.

It just seems the longer I’m involved in math education, the more and more functional uses I’m seeing for the number line.