## Saturday, October 27, 2012

### Student Work on the Quiz

At the end of this first run at algebra, I used student work on the actual quiz.
Here's what it looked like:

I'm not sure what this means about creating assessments though.

I think I'm looking to make it relevant for the students. But this seems like it's relevant in a very 'meta' way.

In the build up, I asked students to create patterns with toothpicks, and they did. They also discovered the numeric growing patterns and created the algebraic expressions accordingly. The work was displayed around the room.

And when it came to the assessment of their learning, they did seem somewhat excited to see their classmate's work on the quiz. I know I was excited because I've never done this before, so maybe my excitement rubbed off on them.

Some Initial Questions and a Comment:
1. Does this "somewhat excited" justify the use of actual student work on the quiz?
2. In the past I've used student work as an assessment AS learning, but is it just as valid as an assessment OF learning?
3. Put another way, is this just a novelty/semi-interesting use of technology or does it give me some return on investment?
4. It was a relatively small investment in time and effort to take the pictures and throw it on the quiz.

Now I'm wondering what you think.

## Friday, October 19, 2012

### Let the toothpicks fall where they may.

I wanted growing patterns.
Actually, I wanted patterns that grow by the same amount every time.
We might call these linear patterns. At least that's what I was hoping for.
And for the most part that's what I got.

See...
But then something like this happened:
And this:

Q: "Hey Mr. Rowinsky, we couldn't seem to find the algebraic expression for this one."

A: "Umm, ya, those aren't, ummm linear. They don't grow by the same amount. What you've discovered is a pattern that grows by an amount that grows. That's advanced."

Cheers

Q: "But do these have an algebra expression we can use?"

A: "Good question!" as I think to myself, 'not one that I know off-hand.'

Awkward smiles and blank stares

I rush to Wolfram Alpha to find the solution. And Wolfram Alpha came through.
See...
You can check it out here and here.

Some initial thoughts:
1) Sometimes students take it to the next level without any help
2) Sometimes I need help with the next level
3) We live in a world where we shouldn't fear 1 or 2.

## Monday, October 15, 2012

### What happens when you've never even heard of a square root?

On the board,
Using the chart paper, draw a square with an area of area 40 units2
I circulated around the class and recorded the conversations. I didn't answer any questions but I did repeat, 'it has to be a square' a few times.

I hadn't even asked a question yet and this is what I got in return.

What I heard and what I saw:

Class A
Wait, how big is that?
Can we use a calculator?
Does it have to be a square…can it be a rectangle?
I know what it is! ~ (…what does he mean by it?)
But 40 is not….
Is 40 a perfect square!
Points…we have to use points… point 5
6 x 6 = 36.
40 is not a perfect square.
It must be between 6 and 7
What's 40 divided by…
(and again)
What’s 40 divided by… (not sure how to complete this sentence)
Does it have to be a perfect square?
What can be divided by 40 so that it makes a perfect square?
What number by what number will get us 40…but it has to be the same number.

Class B
Does it have to be a square?
No rectangles?…is a rectangle a square?
It has to be times by the same number
(found written on a page)
2x20
6.6 x 6.6
4x10

7.5 x 7.5
6.7
6 x 7 =42

What’s 40 divided by…
Can we do point 5’s
HAS TO BE A SQUARE RIGHT?
Factors of 40
40 is NOT A PERFECT SQUARE!
6 is closest to 36.

Class C
Guys, it’s 40 divided by 15
2.66
Wait, 40 isn’t a square.
20 x 20 that’s it...no wait.
20 x 20 is not equal to 40...it’s like...400
Let's find something that = 40
Something that multiplies to give 40
Has to be lower than 6.5
Try multiplying 6.5, 6.8, 6.7
More than 6, because 6 x 6 is 36.
6.3 is too low.

Class D
A rectangle is a square
When you say a square what do you mean?
Area is 40 units (2)
Rectangle is a square or a square a rectanlge?
2 of the same...numbers
Something times something....= 40
It needs to be squared
Can we go to decimals
Definitely not 7, no wait, definitely not 6
6 x 6 = 36
6.5 x 6.5
What’s on these sides? (points to the sides of the square)
It’s a decimal point!
8 x 5 = 40
6.3 x 6.3

Some End Results

I also noticed a surprising amount of reluctance at not being exactly at 40 units2. So much so that some groups were paralyzed and did not show any numbers, like this:

And, to be honest, I even saw some of these (despite work being done somewhere on the side.)

And then, after all that, I mentioned square roots.
I feel like my return on investment is very high here.
Which of these student comments stands out for you?
Are there some comments worth noting more than others?

## Friday, October 5, 2012

### Teaching an Old Word Problem New Tricks: PART II

If you missed PART 1, I emphasized how impressed I was by the number and variety of different solutions provided by my students.

And then this happened!

Keep in mind, this student is just in grade 8 and is still learning the terms, multiples, factors, primes, composites and perfect squares.

By the way, that's the sound of my jaw dropping and my teaching life changing all at once. Who knew it would sound so good with a little Shayne Ward playing in the background.

Initial Conclusions:
1. The kids need more ways to express their solutions (and I might not even know what those ways are)
2. Always review student solutions in front of the whole class, it can change where you go with the lesson
3. Grade 8 students are stepping up their game
4. I hear ya kids...time to step up my game
6. And it got to me...gonna need a moment.

What initial conclusions can you add for me?
Help me out, I have to take advantage of this going forward.

## Wednesday, October 3, 2012

### Teaching an Old Word Problem New Tricks

Today, I used the fairly well known 'Locker Problem' with my students. If you haven't heard of it, here's my version:

Imagine 50 closed lockers in a hallway.
One student goes by and opens every one.
Another students goes to every second locker and closes it.
A third student goes to every third locker and if it is open, closes it, and if it is closed, opens it.
A fourth student then does the same but for every fourth locker.
A fifth student does the same but for every fifth.
And so on.
At the end, which lockers will be open and which will be closed? Why these numbers?

I think there is merit in analyzing the best way to deliver the problem. Case in point, the students did question the words 'And so on.' "Do you mean forever?", "How far should we go?", "When do we stop?" were fairly common.

But here, I want to emphasize something that totally surprised me. About how the students solved the problem, and what they chose as their method. The options were wide open. I'm defining wide open here as 7 different ways of solving this.

1) Pen and paper (immediately popular when they were reading the problem)
2) Chart paper (the response was, muted)
3) A long roll of mural paper (the rolling, intriguing, but not much commitment from the crowd)
4) The white board (notable enthusiasm)
6) 2-sided plastic counters (aha moments for most groups, interest was building)
7) Sticky Notes (game over...close the door...the kids are getting rowdy)

But I didn't close the door. Instead, I offered two groups of students (per class) to take the problem outside, into the hallways, with their sticky notes. Inside the classroom, 2-sided counters were popular, followed by some independent pen and paper enthusiasts.

Having given some time for the groups outside to get going while I watched students inside, I wandered out the door to check on the progress. And there was the surprise. I thought, Sticky Notes outside the class, was one option. My misconception was that I could imagine how students outside were using those Sticky Notes. It wasn't one option. And I had no idea the students were going to do all of this:

Lay numbered Stickys on the Floor

Using O's and C's

Only OPEN Stickys without using Close Stickys

Choosing to skip lockers instead of all being labelled.

Tally Marks on each Sticky as needed.

Moving Stickys up and down, for Open and Closed

How long will 50 remain OPEN?

24 has been through some changes:

Some inside highlights can be seen here:

Here come the counters,

Some planning with Paper and Pencil.

I thought I knew what I was going to see. Today was a pleasant surprise. This is good.

At the end of the day, I still have these questions:

How could I have widened their options (choices) before starting?
Logistics and timing were an issue. How can I manage this better?
What prompting questions, if any, are needed here?
How can I better word the question, "Which lockers will be open and which will be closed"?
An attempt at answering that last one: Can I just say, "What will happen?"