The Effects of Immediate Feedback

A real big shout out to Desmos, Dan and Christopher for their work on the Function Carnival.


 

Science: Completing the Square

Veritasium: 

"If you think that something is true, you should try as hard as you can to disprove it.  Only then, can you really get at the truth and not fool yourself."

This video could not have come at a better time.  In fact, Derek sums up in about 4:40 what's taken me a semester to convey to my students.

Last week's quiz asked students to investigate a parabola given these three points.

Naturally, many assumed that (-5, -3) was the vertex.  However, upon close examination, they should have realized that the rates of change between the points wouldn't allow that.  So, the next day, we set out on an exploration.

"Ok, open up Desmos, plot the points and find a quadratic that fits these points."

It didn't take long before a student came up with y = x² + 8x + 12. We all verified it and then I suggested we try something else.

"Enter the function y = a(x - h)² + k and make sliders for a, h and k.  Now find a function that fits."

Soon we had y = (x + 4)² - 4.  Students loved this form because of the obvious horizontal and vertical shifting that was going on.

Problem
Is y = x² + 8x + 12 the exact same function as y = (x + 4)² - 4?  If so, is there a way we can take a quadratic in standard form and re-write it in this magical form?

Research
We spent the better part of a period graphing functions in standard form and then matching them in vertex form.

"Now, look at all the different functions you have.  Write down what you think is going on here.

Hypotheses

Note: To this point, all quadratics have been a =1. 

Their job over the next day is to determine which hypotheses (if any) they'd like to accept.

I'll keep you posted.

I Like Triangles

Last night, I asked if anyone could point me back to this fantastic animated factorization visualization. (h/t @calcdave)

Now, I'm kicking myself for not thinking to use this in the first weeks of the school year.  Talk about some Fake World math doing a number on pseudo engagement strategies.

I started the animation at the end of each period and walked out to greet students as they walked in.  Once everyone got settled, I walked back in the room and each time the dots would circle up, I'd yell, "PRIME!"

"Alright, I'm up 1-0. PRIME!, man I'm smoking you guys."

Kids caught on really quick and started looking for the circled numbers.  In fact, it took many students a while to realize that the applet literally said "prime."

We started out by looking at the patterns and how each number was represented visually.

But, next I said, "You know what, I really like triangles. What is the smallest number that will give us a triangle?"

This one's easy. 

"Alright, what's the next number that will give us nothing but triangles?  Write your guesses on your easel."

Guesses were about 50-50 between 6 and 9. 

"Alright, what about the next one?"  

Still guesses were a little sporadic.  But by the time we got to 81, most students thought they figured out a pattern.  From 243 on, we were at about 100%. 

After about 10 minutes of doing this and discussing our results, I put up this slide.

There was a nice discussion on clarifying our question.  Three different student offerings illustrated the idea of First Idea; Best Idea. 

Student 1:  At what stage is each triangle?

Student 2:  How many triangles are in the green circle?

Student 3:  How many dots are there in each circle?  *Boom*

Now get to it and be prepared to justify your answer.  

The first student said there were 9, 27, 81 and 243 dots.

"Ok, great. So how did you do that?"

"Well, the green circle has 9 dots, then I multiplied by 3 to get the red. Multiplied by 3 again to get the blue and then by 3 again to get the black."

"Alright, so let's press on this idea a little."

I know what question I want to ask, but I just bit my tongue until a student speaks up.

"How can you be sure that there are 9 dots in the green circle?"  *There it is*

"I estimated."

Here's where it gets good. 

 From across the room, I hear, "It looks like there are 9 dots in the green circle, but we have to look past that."

Wait, what?

"Yeah, we can't trust the picture because the dots are too small.  We know there are 6,561 dots on the whole page and there are three black circles of dots.  We have to start there."

So, Dave, keep this link handy, I'm sure I'll be asking for it again next year.  

First Idea; Best Idea...

...and the Worst Idea

Creating a Culture of Questions was, by far, the most popular post on this blog until someone somewhere starting linking to the post on Exponent Rules.

I think a natural follow up to the Culture piece would be with regards to establishing a classroom culture where feedback is given and accepted.


The First Idea is the Best Idea and the Worst Idea

The first time students hear this, I usually get, "Gosh, that's mean."

But we discuss how the first person who puts forth an idea holds the best idea as there is nothing to which we can compare it.  But using the same logic, this idea should be the worst.  This assumes the flow of ideas that should follow.

I think this encourages two important things:

1.  "If I go first, it doesn't matter that my idea isn't fully formed."  This student has established a floor on which each other student can stand and/or build.

2.  "I can take someone's idea and help them make it better."  The real work is done by the first follower.  This student chips away at any imperfections and helps the first student refine her idea.  Subsequent students then follow suit.


What's this look like?

Yesterday, we trying to determine the equation between the points below and students wanted the y-intercept.

Students were using what they knew about slope to find other points and had to wrestle with the fact this particular line doesn't have a lattice point for a y-intercept.  Once we were finished, I asked students to write down any questions they had.

Student 1:  "I have a comment."

"Ok, what is it?"

Student 1: "No matter which points we choose, the slope simplifies to the same thing."

"Can you turn your observation into a question?"

Student 1: "Will that happen all the time?"

Now here is where it happens.

"I can misunderstand [Student 1]'s question, can we make this more precise?"

Student 2: "Will the slopes always simplify to the same thing?"

Student 3: "Will the slopes between two points always simplify to the same thing?"

"Are we only using two points?"

Student 4: "Will the slopes between three points always simplify to the same thing?"

Student 5:  "Will the slopes between any two pairs of points always simplify to the same thing?"

Student 6: "Are the slopes between any two pairs of points always equal?"

"Are we really talking about any 4 points here?"

Student 7: "Are the slopes between any two pairs of points on a line always equal?"

The Farming [Thing]

I called this a Project. It's not.  It's more of a problem-y kind of performance task learning opportunity assessment of  for of for? learning that hits close to home.  Literally.  We live in a huge agricultural area and kids don't know what an acre is.  Anything that gives students a chance to wrestle with the fact that a piece of land can't have dimensions of 20 acres x 20 acres, is a win.  Anything that allows me to answer the question "What's an acre-foot?" by doing this, is a win.

In this project  problem's first iteration, I was focused on the skills of equation writing, line graphing and solving mixture and work problems.

In the second iteration, I was less focused on the skills and more interested in having students explain what each component of an equation represented, why we'd want that equation and how graphing inequalities made sense.  We got to discuss why understanding the problem makes sense--kids tried to hire crews to prune cotton.  For you city-slickers out there--you don't prune cotton. It doesn't grow on trees.  Students had to sign up via Google form to interview with me as they finished a task.  I did something north of 175 interviews for one class that year.

This year, I've changed it a bit more.  They are no longer tasks, they're constraints.  There are fewer of them and they don't specifically tell kids what to do.  Before, I told them to create inequalities and graph them. Now, I'm removing some of the scaffold.  They get to decide what tools they want to use.  Before, I did this project after we had done systems, mixture and work problems.  This time, we have only done systems. They're going to have to work through the mixture/work stuff.

That's been the highlight--the mixture problems.  I have a few students who went straight for that constraint and have been on a mission to figure out how to make sense of it.

Today, one boy asked, "Mr. Cox, how accurate to I need to be?  I'm accurate to the trillionth, but I can't get it to be exactly 36%."

I said, "How accurate do you think you need to be?  We're killing weeds, not sending someone to space."

So, with all that, here's the updated version complete with dynamic answer key.

The Real Flip

If we can get students to flip their thinking from this:

If I know the rules, then I can do the math. 

to this:

If I do the math, I can know the rules.

Then we've won.

I'm Bringing Multiple Choice Back

So here's the idea:

One problem with multiple paths to solution.  Students connect as many skills as they can to the problem.  I listed eight possible skills two of which wouldn't necessarily apply to the problem. Students had to assess themselves on the skills they demonstrated. 

Question of the day:  "Mr. Cox, is it possible to use all of these skills?"

Answer to Question of the day:  "It's possible that some of the skills don't apply."

For this first iteration, I used the standard Ticket Problem. 

Below are samples of student work. 

As an exercise for the reader:

1) What are your thoughts on this process?  

2) How did each student do?

Let me know in the comments. 

Student A

Student B

Student C

Student D