Category Archives: Tutoring

Testing the “Open Middle” Waters

I have been itching to try an Open Middle math problem.

As a tutor, I am always looking for ways to enrich and deepen the understanding of the students I work with, as opposed to just band-aiding their current math struggles.  I finally got my chance last week.  I chose this 3rd grade question: Draw three rectangles with a perimeter of 20 units.

First, I asked my student to draw one rectangle with a perimeter of 20 units.  Here is his initial attempt.

G_OpenMiddle_Perim 1








Rectangle…check.  Perimeter-is-the-edge…check.  Perimeter = 20…not so much.

We add the side lengths and he sees the total perimeter is much more than 20.

Aha! #1  “Oh, I get it!”  Immediately, he draws a square and proudly labels each side 5 units long.  Yay!  

G_OpenMiddle_Perim 2






Wonderful!  Now:   Draw another rectangle with a perimeter of 20.

Complete puzzlement.

So, we talk about what perimeter means and I ask Is there any other way to have the sides add up to 20?  Aha! #2, and he draws the following “rectangle:”

G_OpenMiddle_Perim 2b






Hmm.  Okay.  Perimeter = 20…check.  Rectangle…not so much.  So I draw his “rectangle” to scale.

G_OpenMiddle_Perim 2c1





Hmmm.  What’s going on here?  We talk about the properties of rectangles and his next attempt is a rectangle, but again the perimeter is not 20.

G_OpenMiddle_Perim 3a






So, I ask him to try again.  He thinks for a bit and then draws:

G_OpenMiddle_Perim 3





Great!  We have two rectangles with a perimeter of 20.  Now draw another one.

Much less thinking time this time:

G_OpenMiddle_Perim 3b





And we get to Aha! #3, at which point he stops drawing, and starts talking:  “Or, the sides can be 2-8-2-8, or 1-9-1-9, or …!)”  Pretty soon he’s listed all of the different configurations/rotations of rectangles (with integer-length sides) and HE GETS IT!

This was more than an “aha!” moment; it was a groundswell of understanding – like a wave of comprehension crashing on the beach:  I get it!  I get it!  I get it!

And it didn’t happen after creating one rectangle, or even after two.  It required him to come up with three different rectangles before he understood.

SO fun to watch.  It literally made my day.  Can’t wait to try more Open Middle problems!

Desmos Graphing Exploration

Graphing Exploration 1

Graphing Exploration 1

I have been reading many blogs lately, some of which have lead me to Benezet’s experiment,  A Mathematician’s Lament, and reminded me that “Perplexity is the goal of engagement.”  As a result, I have adjusted how I am approaching the super-speedy-but-in-depth review of Algebra One I am doing this summer for one of my tutoring students.  I know, that sounds like an oxymoron and it definitely is.  I had planned a lovely, and still too short, 10 week course based on one of Geoff Krall’s Problem Based Curriculum Maps when I found out my student (I’ll call him Josh) was going to be gone much of the summer at various camps and on vacation.


I’m not sure how quickly Josh and I will cover the material this summer. He’s very bright, and with only one student I should be able to proceed much faster, while still achieving depth of understanding.  On the other hand, he also has many holes in his math fundamentals, which I deal with as they come up and which take time.  Still, I’m optimistic we’ll make it through enough of my planned curriculum that he’ll be reasonably well prepared for next year.  Also, I know Josh’s Geometry teacher reviewed selected algebra topics during the last month of class, and his Algebra II teacher will review concepts as well, so I’m not too panicked.  (Funny.  I can no longer think about Algebra II and Geometry without a wry smile.  If you aren’t also wryly smiling, read A Mathematician’s Lament – be sure to read all the way to page 25.  It’s worth it, and a little depressing at the same time.)

What to do?  My current approach to solving the speedy-but-in-depth challenge is to begin by helping Josh obtain a solid (and dare I hope, intuitive?) grasp of functions and relations, using the lessons outlined in the Algebra I curriculum map.  I also hope to finish the initial linear and quadratic functions sections and still have a couple of days to look over the Algebra II textbook so I can pre-teach some of the concepts.  I know, I know; summer is shorter than I think.

But back to the graphing exploration.  As part of my recent reading frenzy, I came across Fawn Nguyen’s cool Des-man lesson, in which she very creatively uses the Desmos online graphing calculator.  (So fun!)  I plan to work Fawn’s Des-man lesson in later, but for now I created a simple graphing exploration activity I will use to introduce Josh to the unit on functions.  Although my activity doesn’t start with a problem per se, I hope to get him intrigued (if not perplexed) about what makes linear, quadratic, and other functions change before we start a more in-depth conversation.

The activity starts with two graphing explorations, one for a series of linear equations and one for quadratics, and then has a blank template that can be used for as many student explorations as desirable.  In the first two explorations I plan to teach Josh how to use Desmos to enter equations, add a slider for variables, and record observations about the graph of each equation. (Okay, after about 3 minutes he will probably be teaching me how to use Desmos, but I have to start somewhere!)  At the bottom of each page are questions intended to provoke thoughtful reflection on his part (I can dream, can’t I?), or at least remind me of the questions I wanted to ask.

It’s easy to see when Josh is intrigued and engaged (or not), so I will get immediate feedback from a student perspective.  I’ll let you know how it goes.  Meanwhile, if you have feedback, comments, or questions, please leave a comment.  Thanks!