Here's a little showcase of what we've been doing:
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I wrote a small Impact on Education grant this year to buy our department math toys.
Here's a little showcase of what we've been doing:
We didn't have enough pentagons to make a dodecahedron. Students decided to fill in the gaps using triangles.
Lots of great discussions between students. Lots of arguing. Lots of off-topic designs. Lots of on-topic designs.
Here's a lesson my brilliant colleague Tom Seibel did with his pre-algebra class:
Here's some of the things I've been doing.
A couple of months ago I complained about a math conference I attended. My wife asked me what I was going to do about it. I didn't want to be part of the solution woman, I wanted to vent and complain. Then I made a commitment that I intended to present whenever organizations I'm affiliated with asked me to.
So, with all that as background, my local professional teaching orgainzation, CCTM (Colorado Council for Teachers of Mathematics), put out a call for presenters. My spiel is a bit longer than the 2-5 sentences they asked for. Gladly, I only made a commitment to attempt to present.
Session Title: Hacking Science into Math Class
"Creating interesting math tasks doesn’t have to be hard--just look to the sciences. Physics is chocked full of relationships begging for math teachers to steal. Linearity? Done. Quadratics? Done. Exponential, trigonometric and rational functions? Done, done and done. Shifting towards these types of rich tasks can also make implementing the CCSS math practices and the NCTM teaching practices feel a bit more natural. Let’s talk about using experiments and modeling to help our students engage with the content and have more fun in math class."
During the session, I plan to have the participants do an experiment (not sure which one yet) and then have a whole-session discussion where groups present how they went about running the experiment and modeling the data (a la 5 practices). Then we will make some connections between how using experiments and rich tasks makes it a bit more natural to incorporate the CCSS math practices (below) and the NCTM Principles-to-Actions math teaching practices (also below). It will be a fun-filled 50 minutes. Or at least I'll have a good time with it.
CCSS Math Practices I can easily make connections to:
-CCSS MP2: Reason Abstractly and Quantitatively
-CCSS MP3: Construct viable arguments and critique the reasoning of others.
-CCSS MP4: Model with mathematics
-CCSS MP5: Use appropriate tools strategically.
-CCSS MP7: Look for and make use of structure.
NCTM Principles to Actions Teaching Practices I can easily make connections to:
-MTp2: Implement tasks that promote reasoning and problem solving.
-MTp3: Use and connect mathematical representations.
-MTp4: Facilitate meaningful mathematical discourse.
-MTp6: Build procedural fluency from conceptual understanding.
-MTp7: Support productive struggle in learning mathematics.
Today, I got all my ducks in a row and headed over to the proposal's page. At the end of the speaker's registration page, there were two check boxes indicating you understood and agreed to the statements.
Then there is the issue of tech. CCTM is severely limiting the number of people who are willing to present by forcing presenters to get their hands on a projector. Things get even more complicated when you think about logistics in a presentation room. There are 30-50 people exiting the room and/or waiting to talk to the presenter in the 10 minutes between sessions. The presenter is kind and gracious and is going to chat people up rather than tearing down his or her tech. I'm then left feeling frustrated and guilty for being frustrated that I can't set up my own tech and get everything just so in the room before all the attendees arrive. Can't we just rent a school?
After some reflection, I think my anger and frustration stem from my wanting to make my local professional organization's PD offerings better but feeling that through their policies they are putting up barriers for teacher presenters to participate. I want to see what excellent teachers around the state are doing in their classrooms. I want people who don't think they have anything to offer--which is the way I feel--to put themselves out there and share. People won't share if they have to: spend days making a presentation, still pay money to attend, and find, bring, and set up their own tech. They'll give up, sit quietly in the seats, and suffer through some corporate song and dance trying to convince us for an hour that we should buy their materials. I hope I'm wrong; I hope other teachers will jump through the hoops and share.
Do other conferences do it like this?
I checked both boxes but I have no clue where I'm going to find an LCD projector.
It's not very often something I teach turns out better than expected--but this is it! This year, for the 1st time ever, I successfully taught transformations to Algebra 1 students. Yes, that's right--Algebra 1.
I bit the bullet and spent more time than normal designing a unit which forces me to take my classes to the computer lab. I hate the computer lab--mostly because of the sheer amount of time lost to travel and transitions. Maybe someday my students will have computers with them at all times, or I won't be out in the portable trailer park where it takes me just short of forever to go get a chromebook cart.
I designed the task for a two-day intensive. On day 1, students use Desmos to match graphs to the paths of skiers and snowboarders.
By the end of the period, students mostly understand the vertex form of a quadratic equation: y=a(x-h)^2+k. They are able to describe what the a-value does and how to find 'h' and 'k' without sliders. I made a handout for the students to use during the class period that also includes some practice problems.
Day 2 is about modeling a situation using the vertex form of the quadratic without using sliders. This means students look at real data, estimate the vertex, then calculate the a-value. They do this in two different situations.
Rather than using another projectile example, I used the simple situation of rolling a can up and down a ramp. Below left is a student response on how they fit an equation to the data. Below right is an example of how well the curve fits the data.
I use a basketball example here for two reasons: 1) the students dig it, and 2) I didn't do the greatest job tracking the data. Messy data forces student discussions about why the graph of their equation only works for the middle part of the graph but not the edges.
Here's an example of some student work on the practice problems.
A huge thanks to John Stevens and Matt Vaudrey for responding to my cry for help on Twitter with insightful comments. Once again the #MTBoS math community demonstrates the benefits of online collaboration.
If you decide to give it a whirl I would love to know how it goes. If you change it up, I'd love to know how so I can tweak it with my students as well.
I moved to Colorado this Summer and started a teaching math at a public school (it's been 6 years since I taught in the public schools). One of the new requirements for me was putting all of my assignments online. Because of the Christian resources on this site, I started a new webpage for use at my new school: andrewbusch-bvsd.weebly.com.
Things have been crazy busy--moving, starting a new job, 3 small kids under 5. That means I've been neglecting my blog and not consistently updating my resources on this site for the past several months. I plan to do such things... just when I can catch my breath. However, my work website gets updated regularly... because I'm paid to do it. Funny how things work like that. Anyway, we've only been in school for a couple of months, but the website is fleshing out
So, if you're interested in what we're doing for Algebra 1 homework, feel free to take a look around. In previous schools most of my homework was from the book. Here, it depends on the chapter; my books are so old I regularly beg/borrow/steal stuff from other people and sometimes even create my own assignments. Like when covering exponents, the only application problems the book came up with were ones dealing with scientific notation. Really? Next week we're talking about cleaning up oil spills and modeling the process using exponential decay functions. Then we're going to talk about the Fukushima nuclear catastrophe that continues to unfold in Japan and the half-lives of Caesium-137 and Strontium-90. Next up, the population implosion in the city of Detroit from 1970 to 2010. All this book can hack up is scientific notation? No wonder we have a generation of adults that hate math and can't think of how basic algebra relates to "real life".
I will now step away from the soap box.
Hey, if you're interested, I'm also teaching an intro to programming class using the language Python. If your looking for ideas on how to structure an introduction to Python class in middle school, this my 1st attempt. It's not perfect but it's a start.
This year I find myself teaching a class called Algebra 1B. It's basically the second half of Algebra 1--except that I have almost the whole year to teach it. So pretty much, almost every activity I've ever collected I get to do with these kiddos. Here's our last two weeks together.
Next week: Barbie Bungee Jump
Here's my presentation for tomorrow's Boyne Technology Conference. I had a really hard time paring down what I wanted to say. I decided to make this more of a hands-on workshop learning how to use the free Tracker video software. I'm not making this an apple-only event. This isn't about all the neat techy things you can do in your classroom. This isn't about how to create or incorporate problem based-learning (though it does touch on it). This is about how to expand the types of fun things students get to do in math class: throwing stuff, dropping stuff, racing stuff, breaking stuff, etc. Without expensive sensors, getting reliable data from the experiments is next to impossible--having to create functions based student collected data is why "Attend to Precision" is one of the CCSS math practices. I'm sure of it.
With this tool, anyone with a way to capture video and access to a computer can do most of the experiments associated with 1st semester high school Physics courses without all the fancy instruments found in well-funded physics labs. Which means math teachers can now get in on the fun without the funding.
As always, people are interested in more detailed instructions for the DIY iPad stand, I edited the original post. I hope it's helpful.
My 126 MB PowerPoint slides stripped of the videos as a 2 MB pdf:
And here's my speaker notes for anyone interested on the topic:
Right off the bat, let me say that I like trig. Or at least I like all the triangles and the easy applications for finding all sorts of lengths. Students in my classes generally like that too. What they find difficult, however, is translation of trigonometric functions. I mean, who looks at y = a*sin([p/(2pi][x-h])+k and says "ooooh I want to do that"? My students initially look at that and panic. You think I'm kidding.
This year I hope to change that--I'm stressing transformations for every family of functions we look at rather than limiting it to the chapter dealing with transformations of functions. Those of you with access to Desmos think that's easy. Without student devices I can tell you it's not. I'm also introducing trigonometry by embedding it in the unit circle. If all of the triangles we look at are always brought back to the circle, then all of a sudden the Law of Cosines isn't such a big deal. Shifting functions up, down, left, and right is less scary (I hope).
Anyway, that's all to say we starting looking at trigonometry today by playing with hula hoops.
We started the class talking about what students remembered about trigonometry--just to get the proverbial wheels turning. They started off slow but then started picking up steam. Then I threw a stick into the spokes: we're going to start talking about trigonometry by looking at circles. We watched the following video:
Me: If you were the person recording this video, what would the graph of your experience look like? Guess. Try to sketch it.
Student1: How do we graph that?
Student2: What are the variables?
Me: So you have to identify your independent and dependent variables? (I probably gave too much away there).
Student2: Are we assuming constant speed?
Me: Sure, I think we can make that assumption.
Students gave a lot of triangle-looking mountains as graphs. Brilliant.
Me: We're going to refine our ideas by making our own Ferris Wheel out of a hula hoop. *cue student rejoicing*
We ran out of time to formalize what I wanted to talk about with them. Finishing the questions on the lab/worksheet was homework. The next day we started class by having students show their graphs. Some student had graphs looking kind of like a bell curve and others had graphs looking like a sine curve. They quickly came to the conclusion that this difference was due to where people started measuring from--either halfway up or from the bottom.
Me: So if trigonometry is about triangles, where are the triangles in our circle?
Enter Geogebra and interactive websites:
These led to some great conversations about the geometric nature of sine, cosine, and tangent (not to mention cosecant, secant, and cotangent). Sine is the vertical distance of a point on the circle above the x-axis. Cosine is the horizontal distance of a point on the circle from the y-axis. Tangent is the length of the tangent through the point on the circle to the x-axis.
Then we talk about ratios, SohCahToa, and solving triangles using neat stuff like:
Introducing trig this way adds extra day to the crazy end of the year I-can't-believe-we-had-more-than-10-snow-days blitz to try to cover all the standards. What I hope is that it deepens student understanding and cuts down on the number of days we need to review these concepts.
We explored exponential functions in Algebra 2 recently. I had planned on using the NCTM Illuminations lesson: Too Hot to Handle Too Cold to Enjoy since this past summer. However, as I've become more comfortable doing experiments in my classroom and gotten better at creating my own investigations I decided to steal the idea and make it my own.
I introduced the experiment talking about the McDonald's lawsuit back in the 1990's over a elderly lady getting burned by a cup of coffee. We told some stories and ended up discussing that once coffee is 120 degrees, it's cold enough to not scald you and hot enough to still enjoy. I ladled out water to the groups and they set to work collecting temperature data based on the amount of time that had passed since they first started measuring.
Things didn't go well--or at least as well as I had hoped. In the picture below, you can see the tools the students were working with. The thermometer was hard for them to read accurately. As a result, several groups got data that was pretty much linear.
Once the students started in on the process of modeling the data, I had to make a quick decision. I hadn't planned on the data showing up linear and students were asking me what kind of model I wanted. I chose to have them make the model they though best fit their data. Here's some samples of student work.
Here we had a great discussion about how well the model fits the data AND the situation. All but one of my groups went for an exponential decay model. However, all but one of those groups got the model wrong. They did well enough finding growth factors and starting values but they didn't take into account the horizontal asymptote for an exponential decay function; it's always at y=0. This means for all of the above functions, when we made a scatterplot on our calculators and graphed our equation at the same time, the equation went down much faster than the data--even when using exponential regression.
Then the group below presented. They realized the horizontal asymptote needed to be at room temperature so they looked at the thermostat in my room. Then they proceeded to subtract 70 degrees from each of their data points before creating an equation. After finding an equation, they translated the equation back up by 70 degrees. Brilliant!
After students get somewhat used to exponential functions and nice neat numbers, I like using this activity. It helps students use what they know about a situation to come up with an ideal equation but then to fiddle with it in order to find a better fit.
The process begins by pairing up and playing with pennies (my handouts are at the bottom of this post). I need to be up front with you--you need a lot of pennies. I have a large yogurt container full of pennies and I normally run out. Because I don't have an infinite supply of pennies I have groups alternate whether they start with the growth or decay portion of the activity.
To model growth, each group starts with 10 pennies and then drops them on the desk. Students count the number of pennies heads up and then add that many pennies to the next drop. This process happens a total of 7 times and by the end groups have a desk full of pennies. Think of a lot of pennies, then think bigger. My highest group had something like 220 pennies by the end.
To model decay, each group starts with 100 pennies and then drops them on the desk. Students count the number of pennies heads up and then takes away that many pennies for the next drop. Again, the process happens 7 times. This one is a whole lot more manageable. By the end it's not unusual for groups to have a 0 as the last value in the table.
After collecting the data, students come up with an ideal equation to model the situation. We have to talk through this one. Students should not use the information in their tables to find this equation. I want them to think of the situation and tell me what the equation should look like: growth, y=10(1.5)^x and decay, y=100(.5)^x. Then I have students use their calculators to create a scatterplot of the data and graph the equation at the same time (step-by-step instructions on the handout). This is where we talk about needing to find a better equation. Some students use their tables to find and average growth factor. Some students just tweak the equation on the graphing calculator. Either way leads to a good discussion of the process of modeling and the messiness of real data.
The kids have fun. I have fun. And it all happens in 42 minutes.
We just finished looking at wave motion. To start, we looked at pendulums and the relationship between the period, the amount of weight, and string length. A couple of stop watches, ring stands from the chemistry lab and weights borrowed from physics and we were good to go. It adds an extra day on the front end but I think it helps my students get a better understanding of what's going on, plus it's fun/interesting. Fun/interesting is an core objective in my class.
While I was shopping for supplies in the physics lab I stumbled upon a spring weight set. I couldn't help myself. I set up some demo videos of pendulums and springs for the students to model. Since my students aren't equipped with laptops, tablets, or the like, I needed some other way to get them the data. I ended up using and Physics Video Tracker to get a data file. Then I imported the data into a Geogebra spreadsheet in order to plot the points and create a data file for the students. I only have 42 minute classes, so any time wasted means multiple days on the same concept. I only have two days budget for the modeling aspect of this exploration of trigonometric transformations.
The first two files have sliders for all the variables. The last several do not. I required each student to come up with 4 equations. Two from the easy column and two from the not-so-easy column. We were able to complete the modeling portion lesson and a summary classroom discussion within two class periods. Not bad. Not bad at all.
Trig Transformations--Pendulums and Spring Weights
I absolutely love it when kids are out of their seats in my class... and on task. There was a point in class today where almost all my students were laying on the floor trying to figure out the height of the ball bounce. How cool is that?
The lesson is pretty basic. I wrote up the lesson requirements on the board (see picture below). I'm not proud of it but it's all I had today.
Note: even though we've been doing experiments all year, Algebra 1 still needs the suggestion to make a table; thankfully Algebra 2 does not.
For the record, I would like to say that Freshman are awful at measurement. No matter how much we talk about precision, these kids coudn't get a correct height if their grade depended on it (it doesn't by the way). For all of our experiments/activities we measure in centimeters. Some of their measurements were at least 15 cm off. Really?
When analyzing the data, groups were pretty quick to realize that picking two random points to find the multiplier wasn't all that accurate. Several students suggested finding the multiplier between each successive bounce and averaging them; I felt like a proud parent. Then they proceeded to make linear equations using the multiplier like it was the slope; I no longer felt proud. We've still got work to do but at least we're having fun doing it.
Things have been more than a bit busy up in my neck of the woods. I feel like I'm doing good just to keep getting the kids out of their seats. Deep in my heart I know there must be a good question I can ask at the beginning which will elicit curiosity; I just didn't have the time to think of one. I'm all ears to those of you who have done this before. Luckily for me, playing with balls during math class was enough of a motivator for the students.
"We all want students to see relationships between the world around them and the math they learn in school. We want them to become curious—seeing math as a tool to aid in their explorations. As my classroom incorporates rich contexts and experiments, I’ve found even a single iPad to be an indispensable tool both in mathematical modeling and classroom discussions."
I presented at MACUL 2014 on how I use technology in my math class. For those of you who don't have access to the conference app, I thought I'd post my materials online as well.
Presentation as pdf
Along with incorporating technology into my classroom there has been a process of working out the kinks. You can see the process pretty well in the evolution of my classroom iPad stand.
Here are more detailed instructions for the DIY iPad stand.
L: Ring stand, M: D.I.Y. PVC version, R: MaxCases Handstand DX
I had enough requests for pictures that I made our 1st Barbie Bungee extravaganza into a video.
This last week we ended our focus on linearity in Algebra 1 (finally!). The Barbie Bungee has been around for as long as I've been a teacher (I had conversations with other teachers about it back in 2001). I've never done it before because generally because of my deficit in Barbie dolls. Well, this year I decided I wasn't going to let a little thing like having no Barbies stop me from doing a fun math lesson. I put out an all-call for Barbie dolls, preferably with their clothes (I teach in a Christian school after all). The response? Nothing. Not one doll. It took some doing but I finally convinced my Algebra kids to bring in some kind of figurine to drop. Here's what they came up with.
I had a couple of guys not wanting anything to do with Barbie--hello batman and dude from Halo.
Day 1: Data collection
Gathering data and making inferences is slowly getting better as the year goes on. I guess it really comes as no surprise that the more they do it the better they get. When I do this again next year, I need to make sure they have more time to gather data and make connections. The 20 minutes we had in class after questions on homework, the intro, and gathering supplies wasn't quite enough. I had to break my vow of silence and give suggestions on data collection to a group. Maybe it would be a good idea if your collected more information than simply one drop with all of the rubber bands. No, really. I felt bad, but I did it anyway. I'm on a timeline people: can you say 7 snow days?.
Many of the groups looked at the table and found the average rate of change and the y-intercept (both the winners and the runners up did this). One group graphed the points and used the line of best-fit (I love it when they see those connections!). One group doubled their data from 7 rubber bands and saw that it wasn't enough and then tripled it. They saw that it was too much so they found the average stretch per rubber band and then took away as many rubber bands as it took to get under the height from the floor to the hook in the ceiling (282.5 cm). Novel but unfruitful. They didn't take into consideration Barbie's initial height so they ended up being pretty high off the ground when all was said and done.
Day 2: The moment of truth.
I used a hook attached to the ceiling as the starting point for out plastic daredevils.
Then we used the iPad and the Apple TV to video and show the happenings up on the screen. The video came in handy several times for the "instant replay" feature (see below).
We had a hair's difference between first and second place. Our winning group had the photo finish above right. Their Barbie's hair just touched the ground but not her little noggin. We all agreed if you got twigs in your hair on a bungee jump and didn't die, that might just be the best bungee jump ever.
All in all, this was a great project to end our study of linearity in Algebra 1.
Embracing rich contexts, complex tasks and experiments in Algebra 1 and 2
One teacher's story of moving towards inquiry-based learning and peer collaboration in the math classroom.
Yesterday I presented at the Math In Action conference at Grand Valley State University. Below are my notes on the presentation and a pdf version of my PowerPoint (which means the videos don't show up).
I want to start out with some pictures of my classroom from this past year. My goal in showing you these isn’t so that you think I’m some wonderful teacher; it’s to give weight to what I’m saying. If I can do it, you can do it.
Several years back, I went to Western Michigan to get my Master’s degree in math education. When confronted with the research, it’s not hard to see the value in inquiry-based learning coupled with rich contexts in the math classroom...
___________________ (read more) ______________________
I keep trying to push my kids towards modeling. It's not near as easy as it sounds. Maybe my students are different; they need to do experiments over, and over, and over again. This time we made spaghetti bridges and found how many washers it took to break them. We wondered about the relationship was between the number of spaghetti pieces and the amount of weight. (Sorry, I didn't get any pictures of student work for this one).
It turns out the relationship is really close to linear.
I was proud of how well my students interpreted the data. Several groups had data like:
x| 1 | 2 | 3 | 4 | 5 |
y| 3 | 6 | 8 | 8 | 14 |
Some realized their problem right when looking at the data. Some needed to graph it first. It did this teacher's heart proud to hear students saying to each other: "I think we need to redo this one." or "This doesn't look right, it should keep going up but it doesn't. Let's try it again."
At this point in the year, we've gotten really good at interpreting and extending graphs. But I still have a couple of kids struggling with the jump in abstraction when I ask them to make an equation for our data. It's been almost 6 months and I still have kids wanting me to just tell them the answer. I keep reminding myself: ask a question back.
I don't show the kids regression until Algebra 2, so they generally attack it one of two ways:
1) They make a line of best fit, pick two points and create an equation (which is my objective for this lesson).
2) They find the average rate of change from the table and a) surmise that no weight will be required to break 0 pieces of spaghetti, putting the y-intercept at 0 or b) they go backwards in the table with their average rate of change and find y when x=0.
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
I teach Math and Programming at Summit Middle School in Boulder, CO.