# Perspective and Understanding

Here is a conversation I had with my 2 year old recently:

2: “I really want a car of my own when I’m a little bit bigger.”

Me: Why do you want a car?

2: “I want to drive in the back seat.”

Me: In the back seat? Why do you want to drive in the back and not the front?

2: “I want to be just like you and dad, mom.”

My confusion was was cleared up when I realized that she is in a rear facing car seat, so to her I am in the back and she is in the front. I bring this up because I think it has a lot to do with why I love having mathematical discussions in class as well as really push students to justify any written work that is turned in. My first assumption when my daughter told me she wanted to drive in the back was different than what she actually meant. Its a matter of perspective.

Many of my students know a lot more of the math than they are able to communicate in accepted mathematical language. The discussions allow me to figure out if there is a concept gap, a language gap, or some other kind of gap. Of course in order to succeed in math, a standard vocabulary and frame of reference might be needed, but why we choose to write and do math the way we do is not obvious to all students.One of my main goals for this year has been around helping students figure out how to express the math they do know, even when they think they don’t know anything. The key to doing that is to figure out what they actually mean and then help them frame that in mathematically accepted language.

I feel like I’ve had a lot of success with the discussion side of this piece. Students are talking about math in much deeper ways than they started with. I’ve also seen them make connections with the material in problems leading to more correct answers. I still need to figure out how to make the spoken word to written word jump smoother. The amount of times they ask me to repeat back what they said so they can write it is astounding. But, we are getting there. Writing about and discussing math is vital. I hope my students come out of class knowing that math is not an answer on a piece of paper.

# Probability Day 2

My district has some common material for certain CCSS units, especially those that are not covered in the text book. I don’t use our book much, but at least for the probability unit, I have been using some of the district provided material. I usually take a piece of it to use for warm up and to frame the topic of the lesson.

Example of Scope of Probability:

1. Review of Probability Ideas from Earlier Grades/ Single Events and Complement Events

2. Independence (or not) / Conditional Probability with Compound Events

3. Area/ Tree/ Venn Diagrams/ Two Way Tables and Union/Intersection of Events

Mixed into these are the analysis type questions and discovery of rules, etc. I have been taking the review section and the final Follow Up Activity which asks the students to put all the concepts together to solve a bigger issue. (It is sometimes a problem, an error analysis, or an explain x type of question)

Instead of the middle section, I usually use my own lesson here. After our first lesson discussed here, the students had an understanding of creating a sample space. So after of Day 2 warm up, I paired them off to play some games with dice. There for 3 different versions of 2 player, 2 dice games. 2 were unfair, 1 was fair. After they played the games, they answered some questions about theoretical vs experimental probability. Then, they had to create a new dice game that would be theoretically fair. I provided them with a blank area diagram model to create a sample space. They had to write the rules for a game. Afterwards, they played each others versions of games as time allowed.

I had two people observing this class. After the lesson, I saw that one of them had decided to try and create a game. She commented “I got one, but I broke a rule.” Her game basically had 17 outcomes for each player to win and 2 outcomes that meant no one won. I immediately said “That doesn’t break a rule, the only rule is that the game is fair, ie. theoretically the players should tie.”

That game/sample space became our next day’s warm up. I put it up, explained the rules and asked “Is this fair?” The class discussion on it was fun. Some students were convinced that only 18/36 and 18/36 would be fair. Others were really excited by this ‘new’ approach. I stayed out of the discussion, only asking follow ups or looking for dissenters. Some had a light bulb moment and asked to rewrite their own rules. By the end of the conversation, the majority of the class had decided the game was fair and were okay will the fact that I wouldn’t say if that was right or not, instead saying “The assignment said a fair game, just make sure you can defend why your game is fair.”

Then we moved back into the regularly scheduled programming. I am loving having a 5-10 class debate as a warm up or a check out. The confidence has increased over the year and they are getting better at not having a neat answer to every question. Now to get those great thoughts on paper!

***No photos 😦 Apparently being observed makes me forget to take pictures.

# Probability Day 1

We are just starting a unit on probability in Geometry B class. It is probably my favorite unit to teach in the class. (I wish we offered a probability/statistics class so I could spend all year on it.) I don’t know if this is because I use these more in my own life, or if its because it is the farthest thing from ‘regular’ high school geometry and I’m still nursing a really bad memory of my own high school geometry class. Seriously. I almost turned down my current job when I was going to have to teach geometry. But now it is one of my favorite classes to teach. Go figure.

Most of the students come in with some idea of the general idea from elementary and middle school. So I wanted to see how much and what kind of knowledge stuck. Our warm up on day one was “Remove One.” I could give credit to a hundred different sites, but I think the template I used to edit was from here. I gave no directions other than to place the blocks on the numbers for the first round.  The second time I simply asked them to think about the game and adjust as they saw fit. We only played twice, then got down into constructing the sample space and starting to refine the ideas they brought to the table.

Across the board “equal likelihood” meant 50-50 to the students. I don’t know if this is out on convenience in language or a gap in math, but it was a good place to talk about mathematical language versus what we hear in our daily lives and why precision in language matters into most (all?) fields.

To end class, students worked alone to create sample spaces for a different scenarios and write possible events that could happen. Then they turned to their seat partner to share out and clear up any misconceptions.

# Flying Higher

While one class was racing cars yesterday, another was flying drones. I liked the gather data and predict aspect of the car lab, so I created a similar format for the drone lab.

Students were given a drone and the task: Determine the maximum height the drone can reach. Find out how distance from the controller affects this number. Use the data to make predictions about other distances and heights.

We spent about 15 minutes outside flying the drones and gathering data. Then the students came back in and Desmos to create a scatter plot and line of fit for the data. We then had a quick group discuss/recap to talk about the model and why or why not the line would make a good prediction for other trials.

Take Aways: Everyone wanted to fly, so most groups switched off controllers. Our class theory was that the controller may have actually had a bigger effect on drone height than distance. Also, we used “paces” to measure. Some groups were much better at doing these uniformly. Either way, both of these introductions of error were good talking points.

More Notes: The students are getting good at explaining their thinking orally, especially when prompted or questioned about flaws. When they write about their thinking, they still need a lot of work. Even when I say, write what you just said to me, the paper version never makes as much sense. We need to work on this!

Notes: Our drone’s battery life is very short (~6 minutes) and charge time is long. I keep one spare with me, but I also let the students know this. They have to be able to get the data they need in a limited amount of flight time.

# Racing Cars

Today we did the Vroom Vroom lab designed by Fawn. Basically, each student got a little pull back car and took data on the relationship between pulling back the car and how far the car would travel. The students entered the data into Desmos and played with the sliders on a linear equation to find a line of fit. Then they had to predict how far to pull it back to hit different distances. The winner was the person who got closest to that distance. We also got into a discussion on when the models do and do not work. At a certain pull back distance the little cars would do a circle and shoot off in weird ways, or not go at all.

I used a great template designed by Mr Orr and made a few adjustments to the distances based on the cars I had. I wanted one to be obtainable and another to cause more problems for at least some of the students. The biggest issue was the lack of Desmos enabled devices (we had 2 for the class to share), but that turned into a positive when early finishers turned into mentors helping their classmates enter and predict.

The students seemed to enjoy it and the math talk it produced was fantastic. They became attached to the cars, giving them names and makes. They also found reasons to love their car. One went the farthest, one was very predictable, etc.

# Introduction to Functions

Algebra 1B is just starting a new unit on functions. It comes at a weird place in the pacing guide. as it feels it might make more sense to use function notation from the start, but I am looking at it as a good way to go back and make sure the information from Algebra 1A is still accessible and sort of serves as end of the year test prep review without having to just review.

We do very little note taking in class as I prefer more hands on discovery lessons. This particular class has asked for more notes though, so I figured I should abide by that. At least a little bit.

We took class notes Frayer Model style for Relation and Function and then the students practiced some examples as I walked around the room making sure the notes turnedinto understanding. As they finished the examples, they were given a card sort as a quick recap. After they were comfortable with the different representations of the data they were asked to come up with their own examples of functions and not functions using at least 4 different representations. Tomorrow for warm up they’ll trade their creations and try to categorize them as function/ not a function. We’ll then discuss student thinking and error analysis as needed.

Frayer Model Idea from here.

Card Sort from here.

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# School is Back

I must be really bad at breaks because I always come back feeling more tired than before. Or really good at taking full advantage of exploring/adventure time. We had a really good turn out today and it was much easier to jump back into the fold than I was expecting.

Algebra 1A worked on x and y intercepts. They got equations in all forms had had to find the intercepts and graph.

Algebra 1B finished up the exponent unit test. Weird timing. I meant to have it done before break, but the students asked to take the test in shorter segments even if that meant doing one on the Monday after break. Surprisingly, it didn’t seem to have negative results.

Geometry A worked on dilation, with a quick recap of all the rigid transformations first.

Geometry B is on probability and determining independence/dependence. We also spent some time with setting up area or tree models of events. I have been surprised at the difficulty of this part of the topic. It is not a task I had budgeted much time for and have needed to revisit often. In the future I need to find a better introduction to them.

3rd year math is all of different topics depending on which class they’ve chosen.

Advisory is tackling racism. They paired up to discuss types of racism (individual/structural/institutional) and then we had a class discussion. We’ll continue the conversation tomorrow with a focus on our city and what is happening to combat the issue.

I didn’t taken any photos today so hopefully I do better with that tomorrow.

On a side note, we didn’t have clocks today. I have never paid much attention to them (I am unfortunately really good at keeping the students in class past the end without realizing it) but it was weird not having them there so I must look at them much more than I think.