Last week I posted an overview of what (for me) is the fundamental remediation activity to do with students K-10: race games. This week’s subject is Diffies, an extension of race games, whether they involve big or small whole numbers, fractions (common fractions or mixed numbers), decimals, every number to the right of zero on the number line. The link here is a fresh revision of my how-to instructions for Diffies, the paper-and-pencil extension to hands-on race games. Diffies will remind you of the 2nd half of a race game, the race back down to 0, where students have to take away the amount shown on the dice they roll, working their way down to zero. The graphic below shows a blank diffy (click here for a pdf). Once you get a number in each of the four corners, you put the differences in between, working your way down to zero as in a race game. You keep putting the differences in between the new corners until you work your way down to zero. Game over!
Diffies are typically done by a single student, although a second student could join or help. I like rolling dice to get the numbers in the corners, but you could use Excel to get a number at random from a range you specify.
Composition and decomposition (breaking a number into pieces or taking putting numbers together) is a key skill, whether the numbers are whole numbers or fractions, the bread and butter of addition and subtraction in grades K-5, and the prerequisites of 5-8 and beyond. It really helps students and teachers if the new skills and understandings we’re supposed to teach and learn are a lot like old ones, stuff we already know. (Math is often like that—or can be, when properly taught.) I’ve been playing race games with my students and with pre-service teachers for the last four decades or so, and I try to capture what it is we do that is so engaging and instructive for everybody.
Usually I introduce students (and their teachers, if I’m doing professional development) to race games in-person, by playing a game or two but I’ve realized that I need some kind of write-up that explains how to play a race game and what I thinking and saying to my students as the game progresses. Here’s a link to the latest version of that write-up in pdf format.
Adapted from Richard Lesh’s translation model and elaborated over the years to reflect what I think makes for effective teaching of K-8 mathematics and for thinking about and anchoring math instruction at any level.
Counting, naturally, is the foundation for the whole edifice, conceptually and chronologically. Teaching and learning, we build our way up—and keep in mind where we’ve been.
To find the difference between two numbers, you can add up from the little one or subtract down from the big one: your result will be the same. When kids do diffies, they get lots of practice in additive differences—exactly what students do when they play race games—especially in the 2nd half of a race game: the race from your target number back down to zero.
This morning I finally got around to sorting through, spiffing up, and posting these PowerPoint animations that show how to do diffies. I’ve got a brief overview of diffies here. I’ll add more details later.
This is what a blank diffy looks like. Or you can do one from scratch on a sheet of paper. Here’s an example of one done from scratch on the back of an oversize envelope:
(Click below to download these how-to diffy animations in PowerPoint pptx)
Lazy me! I haven’t posted anything since last year. Instead of boring you with the details, I’ll just crow about my bikes4fish channel passing 300,000 total views, with the Root Spiral of Theodorus still in the lead. Here’s what I just now grabbed from youtube.
Spurred by springtime (or something!)—maybe invigorated by my long layoff—I’ll be adding more pages to this website in the coming days and weeks, especially (freely) downloadable pdfs and slideshows of grids and activities, that teachers, parents, or whoever is interested can use with their students, children, or whomever.
Also, I’m happy to say, my Root Spiral of Theodorus pops up right at the top of Google searches for “root spiral”, “root spiral of theodorus”, and similar phrases:
Is there an animation you would especially like to see? Let me know on the Contact page.
Last week I saw that my video “Root Spiral of Theodorus” had passed 200K total views. I posted it to YouTube a little more than 4 years ago after putting it together in PowerPoint and then saving it as a video. Most of the views have been from India (over 50%), followed by the USA (at a little less than 30%), then the United Arab Emirates. If you google “root spiral”, my video comes up right away, just below Wikipedia’s explanation. I’m glad it’s so popular, but I’m not sure why. If you have what you think is a good explanation for its popularity, please let me know.
Now I’m going to try to come up with something new. If you have any ideas for what I should work on, you can let me know by using the Contact form. If you’d like to download my original PowerPoint slideshow so you can modify it to suit your own needs or whimsy, or see how I put it together, just hit the Animations tab.
My bikes4fish youtube channel has 339 subscribers (as of this afternoon). I haven’t submitted any new content in awhile, but I’m working on a few more animations that I hope you will like when I “release them into the wild” sometime soon.
This morning when I checked the youtube stats for my youtube channel, I saw that the Root Spiral of Theodorus had hit 100K views.
Here’s a screenshot that shows the most popular videos:
This morning I checked in on my youtube videos at www.youtube.com/bikes4fish and was happy to see that the total views had cleared 145,000. Over 50% of the views have been from India—this year over 70% of the views have been from India. The most popular video is still the “Root Spiral of Theodorus“; in 2nd place is what I meant to be its companion, “Constructing Square Roots on the Number Line“. Formerly, the big months were October and November; now the big months are April and May. I’m wondering about the reasons driving all this, and about the fact that most of my more than 250 subscribers are from India. My best guess is that it all has to do with who’s trying to learn—or teach—what and when.
Ryan claims there are 5 essential questions that we all should keep in mind—plus a bonus question at the end. I’ll list them here (except for the bonus question), and include a transcript so you can read through the text and pore over it; but for starters, you should really watch and listen to his whole speech—the excerpt, actually—it’s not even 7 minutes long. Anyway, here are Dean Ryan’s 5 essential questions and his claims about them:
“Wait, what?” is at the root of all understanding.
“I wonder” is at the heart of all curiosity.
“Couldn’t we at least…?” is at the beginning of all progress.
“How can I help?” is at the base of all good relationships. And,
“What really matters?” gets you to the heart of life.
Back to wondering about my videos’ popularity in India, I’m stuck on #2. If you’ve got something with better explanatory power than my rather obvious “who wants to learn/teach what/when curricular explanation, I’d love to hear it. I suppose #3, “Couldn’t we at least…?” was my motivation for making the videos in the first place: I thought it would be good to have a slide presentation (that later became a video) guiding students (i.e., James-Ryan-HGSE-5-questionspeople) slowly and surely through Theodorus’ square root spiral and another one that showed step-by-step how you could construct/locate some of the most commonly encountered real numbers—the square roots of 1…9 on the real line, all in the comfort and privacy of your own home/library/study space.
My geometry animations cleared 100K views and are closing in on 110K! Half of those views are from the insanely popular Root Spiral of Theodorus:
And about a sixth of the 100,000 + views are from its companion video, Constructing Square Roots on the Number Line:
As always, you can download the PowerPoint slideshows these animations were made from by clicking on the Animations tab. That way, you can modify them and pace them to suit your students and your style.
