A long-awaited solution (featuring Geogebra)

So, a while back, I posted “Circles from Cedar Street” with an intriguing (at least to me) picture designed to kick off the conversation about rotations.

Then, I left it alone. Like, literally. I didn’t solve it. I should have solved it. I let you down.

I especially let guys like Dan down, who also found the problem intriguing. Sorry about that.

But I realized today that it’s never too late to make it right. So, using Geogebra, here’s is one possible solution to the Circles from Cedar Street problem from back in August of 2013.

Why I love this picture #2

A while back, I discussed why I loved this photo. It’s possible that I have a series starting here (sort of like I have with “Real or Fake”. Anyway…)


Today’s photo is a beautiful example of tech integration because the tech is actually integrated. Integrated with what? Well, in this case, with manipulatives.

See? The student has the problem presented online and will record her answer online, but there clearly is no expectation that the work will be fully digital. Which is a good thing because it can be difficult to ignore how effective manipulatives can be in helping student model and visualize mathematical topics (in this case, 3-D images. She’s building a rectangular prism given the front, stop, and side view.)

That’s why I love this picture.

A model of talking math with your kids

Christopher Danielson (@trianglemancsd) started a cool thing. It’s called “Talk Math with your Kids“. There’s a hashtag (#tmwyk) that is pretty cool to check out, too. According to Dr. Danielson “We know we need to read with our children every day, but what should we do for math? Answer: Talk about math with them as we and they encounter numbers and shapes in our everyday lives.”

I try to do this as much as I can. I have an 8-year-old, a 5-year-old, and a 2-year-old. And shapes, numbers, sorting, more, less, etc. are all things that I try to talk about with them when I can. Mostly because it is interesting to me, as a former math teacher.

Recently, I recorded one such conversation with my 5-year-old as we prepared breakfast (listen for the crackling of delicious bacon in the background.) I am submitting it as a model of how these types of conversations can look and feel.

What do you do to talk math with your kids?

A word about fractions

This story begins with a tweet that I read.

This tweet poses a nice engaging situation where addition of fractions would be a very useful tools. But, addition of fractions involves common denominators. And, then I began to remember my students attitudes toward fractions, which can be summed up by the following…


… clearly fractions are so difficult that it requires someone with the reputation of Chuck Norris to be able to deal with them effectively.

Except, they aren’t. Or maybe they are, but they certainly don’t need to be. The logic that says that 2 min + 31 sec doesn’t equal 33 of anything is perfectly understandable to most. It’s the exact same premise as requiring common denominators to complete a fraction addition problem. And THAT is confounding to many. It seems like an arbitrary rule that math teachers invented to trick students.

And the teaching of it carries with it some strong opinions, too. I remember during my undergrad, one of my professors asked this:


One of my classmates changed his major that day. He got so angry that it we would be discussing the possibility that a student could write that equation and could be thinking something mathematically accurate. Dude literally stormed out of class and I never saw him again.

It is possible, by the way:

About the same time I was reading the IES Practice Guide for teaching Fractions. Are you familiar with the IES Practice Guides for mathematics? The Institute for Educational Studies gathers high quality research studies on educational and catalogs them in the What Works Clearinghouse.

The Practice Guides are documents that synthesize the multiple research studies that exists on a certain subject and operationalize the findings. Recently, I explored the IES Practice Guide for Fraction Instruction K-8.

I’d encourage you to check it out. To summarize, making fractions and conversations about portioning and sharing things a common part of math conversation from the beginning can help take the natural understanding that kids have and build fractions into that context. That will give us a chance to use math talk as a tool for students to need more exact language. My preschool son right now uses “half” extremely loosely at the moment. (I’ve drank “half” my water could really mean anything quantitatively.) In order for him to effectively communicate, he’s going to need to develop a more precise definition of “half”. That will require him adding additional fraction vocab to his toolbox.

As teachers, this gives us a chance to build in some more effective language, clearly defining the fractions as numbers. As such, encouraging a lot of conceptual sense-making about the different operational quirks that are required to effectively compute when fractions are involved. (If fractions aren’t numbers, but instead are just made-up, goofy ways of writing numbers, then the rules for computing them are goofy and made-up, too.)

The practice guide provides some tangible steps to achieve this. I’d encourage you to check it out. Lots of steps forward to take in the area of student comfort and effectiveness with fractions.

Physics, a helicopter and cameras

First watch this (and be amazed… well, if you’re anything like I am.)

So, when the students can get past the idea that there is some foul play involved, then it becomes a wonderful opportunities to discuss the idea of frequency.

Frequency is an odd discussion because it’s got a strange unit. The “per time” can be a little challenging for students to wrap their heads around.

And the opportunity that this video provides is that here, we don’t need to immediately concern ourselves with the quantitative value of the frequency (maybe 300 RPM for the helicopter rotor, for example, or 5 frames per second on the camera), but we can begin with the qualitative value of the frequency (that the frequencies, whatever they are, are the same.)

And then it opens the door for them discussing some quantitative issues. For example, the fact that the standard unit of frequency (the “per unit time”), obviously isn’t constant. So, the helicopter rotor is RPM and the camera shudder is typically in frames per second (at least, I think. Not a photographer…) So, you’ve got some nice dimensional analysis opportunities.

Where could you take this next?

One thing’s for sure, I’d hate to waste a video like this. Fully captivating, and it only costs your 30 seconds of class time.

Source: I owe TwistedSifter credit for blogging about this video first.

Perplexity and how it appears…

Here’s a video (by Derek Alexander Muller) I think you should watch.


The critique of #flipclass aside, I’m intrigued by the way the narrator describes the value of “bringing up the misconception”. It’s almost like a thorn that creates some discomfort that only learning will relieve. This gets close to Dan Meyer’s use of the word “perplexity”.

From Dr. Meyer: “Perplexity comes along once in a while. What is it? It’s when a kid doesn’t know something, wants to know that thing, and believes that knowing that thing is within her power. That right there is some of the most powerful learning moments I’ve ever seen – so powerful that it’s really hard for me as a teacher to mess those up.”

There’s power in perplexity. I’ve seen this in my classroom on multiple occasions. It’s important to remember that there’s three distinct parts to creating what Dr. Meyer is describing. First, there needs to be something worth knowing. Second, you have to create the want. And finally, we need to empower the students so they feel enabled to know that thing. What Dr. Alexander suggests is that becoming aware of your misconception seems unsettling (leading to claims that videos were confusing), but also leads to more learning. The discomfort fed a drive to resolve the discomfort.

The tricky thing is that misconceptions are a tool you can use when they are available. Science provides a particularly fertile ground for misconceptions because so much of it is drawn from experiences many of us have regularly. Alexander uses the model of a ball flying through the air. This video uses the phases of the moon and the seasons.

The potential for misconceptions is necessarily lightened when there’s no misconceptions, so the quest for perplexity in math needs to take on a different look, proper planning and timing, and different strategies for when perplexity isn’t an available option. (Preconceived notions are just as good at times. After all, we ALL think we know something about squares!)

It’s like Dr. Meyer says, those wonderful perplexing moments only come along once in a while. We foster those moments when we have them, try to create as many as we can and we do our best every other time.