Last week I introduced my students to Desmos. After doing a short lesson to show off the power and “what-if” capabilities of desmos, I pretty much let them loose on creating equation-driven art. My main objective here was for students to get a deeper understanding of linear equations, and specifically more experience with restricting the domain and/or range of a linear equation. So I showed the students how to restrict the domain and range, and my only requirement of the art is that they include at least one instance of a restricted domain/range. Very soon the kids wanted to know how to create and manipulate parabolas, circles, sine waves, etc. The students absolutely loved this activity; I was hearing comments like “Desmos is the best app I ever”, and “oh, now I see how that works”. I gave them two class periods to work on their art, and told them they can complete it at home and save it to the google folder they have shared with me. Here’s some of the artwork I received back. This first one, the Hunger Games’ Mockingjay uses 40 equations and the student worked on it for over 3 hours.
To see Mockingjay in all its glory in Desmos: https://www.desmos.com/calculator/ihxor3kijv
To see butterfly in Desmos: https://www.desmos.com/calculator/zwmkq6gxx5
To see Birds by the Sea in Desmos: https://www.desmos.com/calculator/poq9oyptce
To see star in circles in Desmos: https://www.desmos.com/calculator/tweud3vjdw
I’ve been doing a lot of professional development lately in preparation for common core. Everything from monthly meetings at the district office to a week with Silicon Valley Math Initiative this past summer. Also I am taking a Stanford MOOC called: Constructive Classroom Conversations, where I am waking up to the fact that my students need to learn how to talk to each other, and this class is providing me tools to teach them and ways to measure success (I’ll do a post on this later).
A common thread to all these PD sessions is for students to articulate their thinking and to listen to other students’ thinking in order to expand their own. I’ve incorporated Number Talks into the start of my classes (especially on Mondays since I’m not checking homework on Mondays). My students and I like these because most feel comfortable participating because there’s no one right answer, and there’s lots of aha moments.
Number Talks is when you pose a math problem, math dilemma, or other math visual to the students, and they are challenged to think about a strategy for solving the problem mentally or to form an analysis on the problem. Next you have them share their thought-process with the class, often after they have shared it with their partner. Also you get feedback on students’ progress by having them do a thumbs-up close to their chest (way better than raising their hand since students aren’t pressured).
These are the topics/sources I’ve been using for my Number Talks:
If Number Talks are new to you, I highly recommend learning about them and introducing them into your classroom. There’s lots of resources about them on the internet, and even youtube videos.
This was a hoot! A couple of weeks my algebra students and I were going over graphing relationships. I had a story about Bob driving a car to demonstrate a speed .vs. time continuous graph. Later we used the example of a squirrel gathering acorns to demonstrate a discrete graph. The students started mashing the stories together to create a story and graph of Bob chasing the squirrel….it didn’t end well.
Anyway, scary graphing stories were born. So today for Halloween, I created this Scary Graphing Stories worksheet. After the students created the stories, we shared out. Here’s a small sampling of student work:
What makes my classroom uniquely mine? Well, my students would probably say my occasional rapping and our many choral response ditties.
But when I think about what I do that I am most proud of, I think about the questions I pose to them and how I listen to their ideas… (btw, these are all a work in progress):
– Really listening to student ideas and learning from them! I try to follow their train of thought, and I try to be open to allowing them to adjust my thinking. This reminds me of something interesting that happened a couple of weeks ago. I gave the students a MARS performance task where they had to create an equation for a situation. The graph of the situation was a horizontal line (constant > 0) that at a given point increased linearly. The rubric expected the students to create an equation for the linearly increasing piece only. So when one of my students said he had created an equation for the entire graph, I listened although I was, inside, skeptical, that it would work. As a class we tested his equation, and amazingly it works. The equation included an expression in absolute value, and it was really brilliant. I am so glad I listened.
– Asking open-ended questions for the students to think about. When they respond to a question, I try to ask clarifying follow-up questions. It’s exhausting sometimes, but worth it.
– Lately I’ve been trying to get the class to answer each others questions more. I’m trying to follow the advice: A teacher should never answer a student question that another student in the class can answer.
– Each week, as part of their homework for the week, my students write about a reflection topic that I give them. These are topics where they either synthesize information we’ve gone over recently or think more critically or creatively about a topic. Some examples are:
- What’s the difference between 0/x and x/0? Is there a way to prove why they evaluate to differently?
- Create a math story for the equation 3(x+2) + 5 = 23 . What is x in your story? Solve equation. Does the solution make sense for your story?
- Summarize the conditions for when you would get a compound inequality, a single inequality, no solution , or all real numbers as the solution for an absolute value inequalities.
I love these. I think it gives students a chance to use the math vocabulary in their own words, to organize their thoughts, and to respond creatively in order to make more sense of a topic. A little time consuming to grade, but many of these response put a smile on my face. Math teachers don’t get to see original written work like this enough.
My 7th grade math intervention recently sharpened their fraction skills using this jigsaw puzzle from the wonderful nrich website; a website that I just learned about from the webinar on Problem-based Lessons and Tasks. My students loved solving the puzzle while working out the problems on their whiteboards. They didn’t finish the day they started, so unfortunately they had to start over the next day, but they were very eager to do so. They learned that the same task is accomplished much faster the second time around . I am sold on these types of puzzles; my 8th grade algebra kids loved their simplifying algebraic expression puzzle so much also.
A couple of years ago when I was teaching the Pythagorean Theorem to my Advanced 7th grade class, I gave the students a “formula” for creating Pythagorean Triplets (a,b, c values that form right triangles). They went mad with it. I had students creating lists of triplets into the the hundreds, just for fun!
Here’s how you do it:
If a ≥ 3 and a is odd, then b = (a²-1)/2, and c = b + 1
a=3 b= (3²-1)/2=4 c=4+1=5
a=5 b=(5²-1)/2=12 c=12+1=13
If a ≥ 6 and a is even, then b=(a/2)² – 1 and c = b + 2
a=6 b= (6/2)² -1=8 c = 8 + 2 = 10
a=8 b=(8/2)² – 1=15 c = 15 + 2 = 17
There are other variations on these formulas to finding the triplets, but I like how these work. Also note, that if a Pythagorean triplet has an odd number in it, there has to be two odd numbers. The triplets always have either all even numbers, or 2 odd numbers and an even number.
This week my Algebra 1 classes studied and practiced simplifying algebraic expressions. To cap off the learning, we did the puzzle that I lifted from Middle School Math Madness Blog.
However, instead of using Tarsia, which looks like great software but not available on the Mac yet, I just created the puzzle the old fashion way. Anyway, the results were amazing. Total engagement by the students. The puzzle was hard enough to be really challenging, but achievable at the same time. About a quarter of the students finished it, and many more were very close. They were eager to see a picture of the finished puzzle the next day. Here’s some pics:
BTW, I love taking pictures of the kids in action. And they really enjoy seeing pictures of themselves the next day as we summarize the activity. Such hams!