Maker Geometry – What can blocks do for you?

So, today I got to play with blocks called Keva planks. A set of Keva planks are nothing more than a whole bunch of congruent wooden rectangular prisms. You can build towers, mazes, bridges, and… of course… geometric shapes.

So, a task came to my mind. Let’s suppose we were going to take the planks and make the smallest possible cube.

  1. How many planks would we need for the combined widths to match the length of a single plank?

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The answer’s five, by the way. So, each face is 5 planks wide (which is a single plank long)

Looks kinda like this…

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So, naturally… now we can calculate stuff. Like surface area and volume. But to do that, we’ll need some numbers.

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So, in my haste,  I clearly did a lousy job of lining up the blocks on the measuring tape. Sorry about that. Not exactly a deal breaker, but annoying.

But since maker materials are becoming more common, I figure you might prefer your students pull their own measurements.

There are a couple of ways that I could see variations of these: different shapes (different kinds of prisms, for example). I could also consider challenges like, given x-number of blocks, who can build a figure with the largest volume or a flat shape with the largest area?

Keva blocks seem like a low-risk, high reward manipulative simply because the start-up would be so quick. What could you do in your class with a set?

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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.

If haven’t met Desmos yet…

I would like to use this post as a shout-out… an atta way… an unpaid advertisement, if you will.

There are a lot of tech tool providers out there. Not all of them want to work with you. Not all of them want to hear your feedback. Not all of them provide their products openly for free on lots and lots of different platforms. (In fact, I sat in a focus group with an instructional tech developer once. The producct in question was being sold to districts for $10,000 per building. When we all picked our jaws up off the floor they said that they set the price high on purpose. If too many people bought it, they didn’t think they would be able to properly support everyone who was using it.)

Desmos is not any of those things. I’ve never gotten responses from instructional tech providers and developers quite like I’ve gotten from the folks at Desmos. I have two anecdotes that will help illustrate this.

1. I had an idea and made this. When I finish a post, I send out a tweet. The folks at Desmos responded to my tweet.

I want to make sure you clicked both my “made this” and their “few possible ideas.” If you didn’t do that, do it now. Do you see the difference? Do you see what happened there? They took the little bit of an idea that I had and added to it stuff that I didn’t even know how to make Desmos do. The tweets that followed were an exchange between them and me that helped me learn how to do that stuff.

That was a unique event. That had literally never happened to me before. The developer of the tool reached out to my feeble attempt to use their tool and personally improved it and instructed me all the while giving me the credit?

At least it was unique until it happened again. Last week I had an idea. This idea. And I tweeted out the post. And then…  

 

Once again, be sure that you check out their idea compared to mine. Mine was a nice start (I hope you see the progress I’m making learning how to use the tool). There’s was expert level. The folks at Desmos are eager to build on the ideas of the educators who are trying to put their free tool into play in the classroom.

If you haven’t gotten a chance to try some Desmos work? Maybe consider redesigning a lesson and then, just maybe, they’ll have a few ideas for you, too.

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.)

Update:

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.

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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.

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And trimmed it.

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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.

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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.

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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.

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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.