## 4.28.2011

### April Showers

If April showers bring May flowers then my yard will be a freakin garden shrine.

Before Sandbagging

After Sandbagging Day 1

After Sandbagging Day 2

We've literally lost entire towns.

And the rivers aren't supposed to crest until Sunday.

And there's a 60% chance of rain.

### Multiplying Binomials

Last year I taught the Box method and quickly explained the Foil method. My students really took to the Box method and just went at it. I think they actually enjoyed it.

I've recently talked to my students about flipping the classroom next year and they really love the idea of it. I wanted to try it out but couldn't figure out how to convert Jing .swf files to a file that I could burn to a dvd or even play on any type of player. I can almost always count on the Internet to view the .swf file but because of all the tornadic flood weather, you just never know. So instead of a cool screencast and voiceover, I just made PowerPoints. I figured the students would have to think harder without me explaining and there's nothing wrong with that. Besides, I love a good PowerPoint.

Here's the setup. I split my classes up into 3 teams. Each team was sent to view a PowerPoint about multiplying monomials but every team would be viewing a different method.

 Box Method
 Break Up Out of Jail
 FOIL Face

Each student was given the same handout to act as guided notes. They would be taking notes on two examples and trying a third example on their own. Every team was completing the same three examples. After everyone in their group was finished taking notes and could fully explain their method, we reconvened as a whole. Each group came to the front and presented their method. They wrote it out on the SMARTboard and explained each step of the process. The rest of the class were watching at this time and not taking notes. Once they were done and students had no further questions, the entire class, even the presenting team, completed a new example on the back of their sheet. This way every student was presented with 3 different methods and practiced each one. From here on out, students can pick whichever method they prefer.

My students really liked the way this lesson was set up. They rated the methods from easiest to hardest as: Box, Break Up Out of Jail. and FOIL Face.

For those of you who are wondering about my title, my students were taught in middle school to remember the distributive property as 'breaking out of jail'. The number on the outside of the parentheses is the jailer and he has to go in to each individual member to break them out of jail. Building on that theme, I called it 'breaking up' out of jail to help symbolize breaking up the first binomial before breaking out using the distributive property.

Here are the resources:

Box Method PowerPoint
Break Up Out of Jail PowerPoint
FOIL Face PowerPoint
Multiplying Binomials Guided Notes

Please download the PowerPoints to see the full effects of the color coordination and animation. No cheezy transitions, I promise!

## 4.25.2011

### SBG Guilt

For those of you involved with my twitter conversations, you know my students were really struggling with subtracting polynomials. Out of 36 students I had to pull out about 10 of them to work with individually.

It took me a while to figure out their misconception.

Say the problem looks like this (3x - 8 + 2x) - (4x - 3 + 6x).

I taught them to change the sign in the middle and flip every sign behind it.

Here's what they would do: (3x - 8 + 2x) + (-4x + -3 + 6x).

In middle school they were taught to change every minus sign to plus a negative. When I told them to flip the sign, they would, but they added in the negative which doesn't flip the operation. And even though I stressed every sign about a million times, they would always politely skip over the plus signs. There were other errors too but this was the most common misunderstanding.

With those 10 kids, I pulled them out individually and tried to explain and re-explain what to do and why (I just now realized I should have said to flip the operation instead of flip the sign). Later I realized it was even more confusing because it looks like they are changing the minus sign in the middle to plus a negative. Egad!

Too late I asked the other math teacher what I should have done and he suggested just practicing lots of problems like -(7x - 2 + 4x) so students can really see over and over how the negative sign is distributed.

Anyway, to get to the point of this post...I pulled students out and we went over their mistakes on the quiz. Then I gave them new problems to practice. Then I give them a new quiz. Quite a few students got a 100 on their retest. Some needed to do it three times instead of two but still got the 100. Now I feel guilty. I feel like it's cheating. I feel like I should have pulled out every single students who didn't get a 100 instead of just the ones who did terrible. Shouldn't every student have the opportunity to make a 100? I don't know. It's kind of like forced sbg. I took away their initiative to improve and forced them to 'want' to do better. I practiced with them and we conversed and we identified misconceptions. All the perks of sbg, right?

So why do I feel so guilty?

On my assessments for each method of systems of equations my classes did really well. But maybe we had just practiced enough times that they had the process memorized. How do I know that they truly learned? I guess that just means I need better assessments. How do you assess understanding other than having them show by doing or show by explanation?

I am afraid now that my administration might look at the grades and assume that I am not doing my job or not doing anything for my students to all have high grades. Does that mean I need to make my assessments harder so that more students will fail? Will I feel less guilty if my grade distribution looks more like the 'normal' bell curve?

## 4.23.2011

### Depth of Knowledge

One of my real stumbling blocks this year throughout all of our school improvement, leadership team, instructional team, student support team, and graduate class meetings has been "What do I teach?" Anytime I look through textbooks or online for resources, I find different ways of presenting the same concept. But hey that's differentiation and multiple intelligence styles and so on and so forth, right? What is messing me up is wondering if my students could solve any problem thrown at them pertaining to a specific concept I've taught.

For example, I felt like a did a good job of teaching systems of equations. We did graphing on the calculator, we did word problems, the whole kit and kaboodle. As I was creating the test, I found a problem where one of the equations had a variable on both sides, some like 2x = 45-5y. We had done many examples but not one with variables on both sides. Oh, but we solve regular equations like that all the time, right? So I put it on there. I had a ton of questions on that one. Some said it was infinite and some said no solution and some canceled out the variables altogether. Not everyone got it wrong but fewer knew how to handle it than I would have liked.

I know I've posted before on increasing students' critical thinking skills and I guess I can attribute some of this to that issue but my real question is "How do I teach to mastery so that students can apply the concept anywhere?" This is probably an obvious question to most or even the definition of teaching to some. But for me, it's my latest realization. Of course I'm always thinking about test scores but truly, how do we ever know what depth to teach to? Our district has a chronic problem of repeating material over and over without ever going any deeper. We're currently trying to align things to Common Core as well as K-12. But no matter what list of standards we look at, I could probably say I teach all those things. The problem is, have I taught them to the same depth of understanding that the ACT is assessing? The obvious thing is to look at the ACT but even then, I'm not sure how much that can help. If a bunch of students get a question right, they throw it out. If a bunch of students get a question wrong, they throw it out. It's impossible to know exactly what will be on the test and what it will look like.

Does the answer lie in assessment, critical thinking, more practice, or better teaching? Do my students need harder problems and more time to think things through? How can I ever be sure that they can apply the concept in new situations that we have not practiced together? Is that an okay expectation to have? Isn't that the definition of math or of learning- using what you know to figure out something you didn't previously know?

I feel frustrated sometimes because even if we had the perfectly aligned harmonious pacing guide, how would we be sure that each teacher is teaching to mastery so that students can reapply the concept in new situations throughout their mathematical career?

For example, addition. Students learn addition and it carries them through many new situations. We throw in variables, exponents, square roots, absolute value, and even imaginary numbers. But for the most part, do students ever not know how to add, even in new situations? Is it due to the fact that they've been adding for years and years? Is it the fact that no matter the situation, adding itself never changes? How can that work for other concepts?

How can my students become so comfortable with a tool that they can figure out how to use it in any situation?

## 4.22.2011

### Card Sort

We are in the middle of our polynomial unit and I decided now would be an appropriate time to teach exponent rules. I had prepared a concept attainment for the exponent rules before I talked to my coach. She suggested that we stray away from teaching 'rules' and rather teach the concept behind it. So basically, I only taught the concept of multiplying the coefficients and adding the exponents. When it came to the power to a power rule, we literally expanded it out and then multiplied coefficients and added the exponent. The more that we practiced the concept and writing them out, the more students figured out the shortcut of taking the coefficient to the power and then just multiplying the exponents.

I was skeptical of this method at first until I realized that it was a better way to scaffold the lesson. Present one concept at a time, that will build on prior knowledge. Practice it until it becomes second nature and they will naturally look for a shortcut. We humans are mighty efficient like that.

I taught one concept but we were really practicing three rules:  power to a power, product to a power, and the distributive property. My coach is a big fan of sorting. I made a set of 18 cards, 3 groups of 6 that represented each rule.

 Product to a Power

 Distributive Property

 Power to a Power
These are index cards (my favorite!) cut in half by the way. So each group had the same cards and they were all mixed up of course. The students were instructed to spread them all out and then begin sorting into piles.

I did not give any parameters to sort by. In every class, without fail, my top students' team quickly sorted them by color. I then charged them with the question, "If I am asking you to sort them, do you think I would have color coded them for you?"

If teams were really struggling, I told them they should have 3 piles. A lot of teams sorted them by sets of parentheses which works for the power to a power group but not the other two. Every single group mistook two of the distributive property cards because they were written backwards compared to every other one in the group.

See above. (4x + 7)(-2x) and (2x - 5)(-4x) look different and so students put them into the product to a power group. I guided them to look inside the parentheses and see what's happening. From there they realize the distributive property means there will be addition and subtraction.

Once students had them correctly sorted, they distributed the cards evenly among themselves (some from each group) and then solved them. Depending on quickly the group got done, students could rotate cards and work more problems.

Each card had a letter on the back. Students wrote the letter next to their solution so that we could check the answers.

Students tried to sort by alphabetical order but that only created one big group. :) They also tried to sort by what 'looked' easy, medium, hard or problems that looked short, medium, or long. Also some tried to  group based on how many negatives or positives in the problem and even how many exponents existed.