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.

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.

And trimmed it.

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…

… 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.

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The Real Value of the #MTBoS

Our friends over at Explore The MathTwitterBlogosphere have prepared a variety of opportunities for us math teachers to become better acquainted with this community that exists with one goal in mind. Take math teachers in and make them better. What distinguishes the #MTBoS as it has come to be known from other grass roots movements is that this movement is completely inclusive. We WANT you to join us because we all have something to share and something to learn.

As part of the exploration, Sam Shah (@samjshah) has asked us to consider an interesting question: What happens in my classroom that makes it distinctly mine?

Within my district, my practices and classroom policies are a little off the mainstream. My students get boatloads of exploration and practice activities. That isn’t unusual, except that more than half of the work I assign I neither collect, nor assign points to. Oh, and I try to avoid homework at all costs. I try to put students in an ability to use mathematics in ways that seem authentic. As a geometry teacher, this means that I use proof and algebra probably more sparingly than most, opting instead for visual explorations when appropriate. Also, I have an aversion to textbooks.

I believe in authenticity above all things. There is a belief that most students won’t do work that doesn’t have points attached. I have found the opposite to be true. I have noticed that students are more willing to explore and practice when we are honest about what it’s for. The same with homework. Homework doesn’t tend to be a very meaningful learning activity. So, I don’t use it very much.

We follow the same idea with Algebra. We use algebra… when it makes sense in the context to do so. We graph… when it makes sense in the context to do so. But transformational geometry doesn’t require these things. In fact, there are many different ways to explore the strategic transforming of shapes and figures. From an understanding of transformations, you can move toward congruence and similarity discussions. Similarity leads to proportional reasoning and then trig naturally follows. In my opinion, this is  fairly simple way to look at geometry course, but it has lead me in directions that are, as I said before, a little off the mainstream. It forced us to ditch our textbook, which was heavily, heavily algebraic. We had to replace it with our own handouts which were homemade or found on the #MTBoS. That opened up a ton of conversations about what good math looks like and what it will take to lead our students to it.

This is where the real value of the #MTBoS has shown off. The #MTBoS means never having to be alone, regardless of your thoughts or ideas. There’s someone who has thought that thought and discussed the possible implications on a blog somewhere. There’s someone who has tried an activity and tweeted about how it went. There’s someone who has a link to a handout and an e-mail address so that you can ask questions about how it got delivered. If you try it and blog about it, there’s people who will read your post and leave comments. It’s a form of individualized PD that is difficult to find elsewhere.