Advisory Student Reflections

On May 21, 2015 By altmathLeave a comment

Although we still have just over 3 weeks of school left, I had my advisory class do a bit of self reflection of the year. The next weeks are filled with testing and racing the clock to finish, so I wanted to capture their thoughts now as well give them a little boost to show them how far they’ve come.

I typed out a few prompts and had them finish the sentence, then I typed the response side into Wordle to come up with the main ideas. Reading the individual responses was more powerful though, so I wanted to throw up a couple of those ideas as well:

We are an alternative public high school, so some students choose to come and others are required to attend. Some are here for a few weeks, others years. They are all in such different spaces, but they are all really cool people. I’ve had a great year and hope we can finish strong together.

Puzzling Minds

On May 20, 2015 By altmathLeave a comment

Keeping with our problem solving short Wednesdays, we did math station puzzle rotation today. Certain students really dislike when I assign one problem that takes the whole 40 minute block, so I thought I’d give out a few shorter activities today.

First station: Arrange all five pieces into a square. All the students found the 4 piece version and more than half tried to get away with sticking the little left over square on top. Most students were eventually successful with the five piece square although I know a little hint giving was done. I like this puzzle since every student walked in and said “This is easy!” and then quite quickly, “This is impossible!” After finishing, they’d sketch it in a notebook for me to check off.

<— All the pieces

A tricky Student —>

Second Station: Logic Puzzle Fun. Students could choose any of the varieties of logic puzzles I put out and solve it. They have seen each one of the choices at least once during the year.

Third Station: Four 4s. Basically, using exactly four 4s and any mathematical operation, create expressions to represent all the integers 1-100. This was the biggest hit. Students were whining and complaining at the beginning, but as soon as they got into it, they really got into it. I had multiple students try to skip their next class and stay in to do four 4s. I told them they could take them with them, but most replied “But I want you to know I did it myself.” I told them I trusted them, but still something about doing the problem while I’m in the room was important to them, so I told them it could be something they pull out as pre-class work or end of class work if they ever finish early. I feel so corny saying it, but it was so fun to watch them work in this one. They’d light up when they found a ‘hard’ number or used a funny operation. I got to introduce a whole lot of math vocabulary that we never get around it using. They loved this so much, I should have spent the whole hour on it. I had figured it would be the least liked activity. I was very wrong.

Absolute Values

On May 18, 2015May 18, 2015 By altmathLeave a comment

Algebra B is working on absolute value equations. We started off using Dan Meyer’s “How Old” game where students see a picture of a celebrity and have to guess the age. I made sure to throw in a picture of my daughter at the end. It is amazing how excited the students get when I bring a picture or talk about her. Plus, it was a gimme answer. Who doesn’t like to be exactly right at least once. After about 10 people, ages are revealed and the best guesser is announced. It was a small class, but we had a good split of students who calculated their guess score by using absolute values and those that kept signs and even a few that explained the idea of the calculation correctly but still had negatives in the answer. We looked at two student’s calculations on the projector to pick a winner. The lower number actually was farther off (used negatives) so we talked about ow we wanted to determine how far off each was without caring if they were low or higher. It was quick, about 5-10 minutes for the whole thing and a good introduction to the idea of absolute value.

I introduced the mathematical notation and then asked students to come up with examples of when the sign didn’t matter (use absolute values) and when the sign would matter (not absolute values.) Then I threw up a quick number line on the board and had students place sticky notes where they should belong.

We then moved on to graphing. I had all the students predict what they thought the graph would look like, graph the parent function, and then discuss why they looked how they did. As time allowed, student then started to graph transformations of the parent graph, mostly by using tables and plugging in values. A few caught on that the transformations work like all the others we’ve done so far and skipped some of the plugging in steps. (All this work is still in the student’s notebook, and I didn’t take pictures as I was walking around working with students, but I’ll get it up soon.)

Lastly, we started looking at the graphs to find solutions to absolute value equations. I like starting graphically for two reasons. It reinforces the skill of find solutions from a graph (especially the multiple solution part) and it highlights the need for careful thought when solving algebraically since there are two solutions.

Our last few minutes together, I put up a couple absolute value equations which could easily be solved by sight and had students think of both solutions. We’ll follow up tomorrow with more involved equations.

**Last semester’s Algebra B class did a more involved graphic introduction which I really liked. We graphed the parent function and then in different sets 3-4 transformations of the equations on the same graph in different colors. I liked that graphing activity better, but it was also the first major transformation example. I did radical graphs and transformation first this time, so I skipped the longer discovery this go round. I think I might bring it back next time. Having 4 graphs together gives a better comparison. and allows for finding solutions to lots of equations at once.

Introduction to Systems of Equations

On May 7, 2015May 7, 2015 By altmath2 Comments

My Algebra 1A students are just starting the systems of equations unit, and what better way than a TrashketBall competition. I’ve seen a few people use Trashketball as a fun review game, but Mr. Orr shared an awesome three day lesson on his blog. I borrowed Day 2 and 3, adjusted them a bit and made it an introduction to systems of equations.

Students played 4 rounds of 1 minute each to find their average make rate per minute, converted that to makes per second and came up with an equation using that rate. Then they graphed their lines. With a partner, students discussed what their graphs meant and compared and contrasted the lines. (All start at 0, but all different rates etc). A few pairs shared out insights.

We then graphed all the lines on a projected Desmos graph. I asked who would win a basketball competition and how they knew. Then I asked how to make the game more fair. Eventually they settled on giving the slower rate a head start. They jumped back with their partner and overlaid their original graphs using patty paper on one graph. They had to decide who should get the advantage and how many balls that advantage should be. One group wanted to give the ‘better’ player a late start (moving the x-intercept instead of the y) which I said was fine. They graphed the new line on the same graph as the line that wasn’t changing.

Up to this point we had just thought about how to make the game more fair. I asked them to make observations about the new set of lines. The first observation was that they intersected. Another student pointed out the the originals intersected too, but that the intersection was no longer at (0,0). We ran with that and talked about what the intersection would mean and why did we want to move it away from the origin. That intersection point became new time length for each partner to compete against each other. I had them write down the intersection point and decide what each value meant (time of game, # of makes expected)

Almost all of these games were quicker than the original one minute, but one generous competitor gave a bigger advantage had an almost 2 minute game. New partnerships were formed and they played one more time.

Closing as a whole group discussion about the game. We had a few ties and most were fairly close to the predicted make values. A few were outliers so we talked about those as well. I then introduced the term “systems of equations” and “solution to a system of equations” and the students contented them to the context of the trashketball game.

Tomorrow, we’ll move into solving systems graphically more formally, with lots of in class practice time. Then we’ll move into pictorial puzzles to lead into the substitution method. Elimination/Combination will round out the introduction to systems – still looking for the best hook for that one.

I really encourage you to try out Trashketball. Every single student was engaged and doing lots of great math. I will definitely be refining it for next year, maybe even extending as a unit long project. I did also promise a trashketball performance task on the unit final….

Perspective and Understanding

On April 30, 2015 By altmathLeave a comment

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

On April 28, 2015April 28, 2015 By altmathLeave a comment

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

On April 27, 2015April 27, 2015 By altmath1 Comment

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

On April 23, 2015April 23, 2015 By altmath2 Comments

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

On April 22, 2015April 22, 2015 By altmath2 Comments

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

On April 21, 2015April 21, 2015 By altmathLeave a comment

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.