Andrew Busch

We did a project day in Algebra yesterday.

Students chose between 3 options:
1) creating letter patterns,
2) exploring Lego staircases, and
3) learning to program a TI-84 graphing calculator.

More than half my class decided to work on making letter patterns. Each student worked on 3 different letters--figuring out how to make the letter out of squares (mostly) and then enlarge them. They did great!

Students who normally struggle with math got a chance to showcase their other strengths. When we presented our projects today in class, students who normally look at the floor and sink into their chairs when I ask for volunteers were excited to share what they did.

I can think of an interesting extension of this project. Instead of having students create equations for their own letters on the sheets, I would have students create equations on a separate piece of paper. Then have students trade papers and create equations for each others' patterns. Practice with creating equations from patterns and tables--done.

3 students went for playing with Legos. They worked together as a group exploring how the number of bricks relate to surface area and volume.

Surface area almost ended this group. When I walked by, they were staring blankly at the paper and the pile of Legos. I gave a little support and asked some leading questions to get them started. After that, they were off and running.

I had 4 freshman boys giddy with excitement that they were going to learn how to program something. I spent most of my time holding the hands of this group as we walked through the syntax of BASIC and the concept of if-then-else statements. Throughout the day today, I had conversations with each student at least once with their calculator in hand asking how they could either fix or extend their program.

Based on the recommendations of multiple people, I read 5 Practices for Orchestrating Productive Mathematics Duscussions. The title isn't all that creative, but it does speak for itself. In 13 years of teaching and a masters in math education, I don't think I've read anything nearly this helpful. Anyway, they parse apart a tiling task in one of the chapters that I thought worked brilliantly. It starts out concrete and moves towards abstraction with baby steps.

In Algebra 1, we've spent our last week looking at how to create rules and equations from patterns we see in numbers. Being a thoughtful caring teacher, I decided to let someone else do the pre-planning work for me--I used one of the lessons from the book. I decided to spend two days on the concept--modeling is a big deal later on in my class; it's worth losing a day to deepen the foundation. But that meant I needed another task dealing with the same ideas without being the same thing over again. Enter Dan Meyer's work on makeover Mondays this past summer. He remade a checker boarder tiling problem with many of the same problem-solving skills in the 5 Practices task. I spliced the two together in the following handout for students.

I have more of the files for this lesson located here.

Students jumped right in. Drawing the 4th and 5th patios ended up being incredibly helpful for my more concrete learners. It seems so trivial to me at times, I'm tempted to skip over such steps. In Algebra 1, especially at the beginning of the year, I remind myself to slow down and move from concrete to abstract. Bring in all the students and then start climbing up the ladder of abstraction together.

Students came up with several expected solutions to the patio problem:

And two separate groups came up with a way I had not thought of:

These groups said, "When I look at the patio, I see three rows of tiles that are each two longer than the patio number with the center cut out. So, on patio 4, there are 3 rows of 6 tiles each and then I can subtract the 4 out of the center". Brilliant.

I did add an extra step of complexity during our class discussion of the cafeteria tiling problem. After figuring out how many colored squares there are in the 100th iteration of the pattern, I ask what it would cost to purchase all of the tiles included in the floor design (white and colored)? I hopped on Home Depot's site and found tiles I liked: light colored tiles are $3.19 per square foot and the darker colored tiles were $3.45. It gives a little twist to the problem that isn't trivial after we've already found the number of colored squares.

I totally stole this from Fawn Nguyen last week. I'm doing a review of linearity with my Algebra 2 kids; we just went over point-slope form. At this point we normally jump into a study of the gender gap - pay inequality between men and women workers in the U.S. However, this gem was just too much fun to pass up. I even went to the store to buy more pull back toys for my kids (4, 2.5, 4 months) so I could use them in math class. (Good Daddy and good teacher at the same time--2 for 1!)

The data was all over the place. I had hoped the distances would be somewhat linear but that didn't quite happen. It will never be said of my children's toys that they were anywhere close to being consistent. Students did a great job of analyzing the data and trying to find something that works.

When the bell rang for the end of class we were still finishing our races. Not one student made a move for the door or even packed up their stuff. That's a good day in math class!

Taco Math - Algebra 1

Pixel Patterns - Algebra 1

Racing Cars with CBRs - Algebra 2

We began the experiment by first discussing what we wanted to do. We want to roll a ball off a cliff and predict where it's going to land. We discussed how we could set up an experiment that would do this and the class came up with the following.

Off to the physics lab! Students created ramps out of meter sticks, ring stands and masking tape. Students could test the ramps with a steel ball as long as the ball never actually hit the ground. Then, we took videos of the balls on our various mobile devices and transferred them to the laptops. We used the Tracker Video Analysis software to calculate horizontal speed of the ball as it moved along our track.

Students used quadratic equations to calculate the time in the air based on the vertical distance the ball will travel. Then, with that information, calculated the expected horizontal distance the ball will travel based on that amount of time and the speed of the ball on the ramp (basic Algebra 2 stuff or even Algebra 1 2nd semester).

Now the moment of truth. Can we predict where the ball will land with any level of accuracy? To up the stakes a bit, full credit is only earned if student groups can get the ball into a can on the first try.

They nailed it!

Making math matter is a tricky thing. Is there a way to "do math" from a Christian or even a generic moral perspective? Isn't 2+2 the same for everyone?

In undergrad I never really cared about whether math had applications; it was beautiful and that was incentive enough. I would teach my students to value aesthetics of proofs and for them, that too would be enough. And then came Pre-Algebra in a 95% free and reduced lunch school in rural Northern Michigan. My students hated math. There was no beauty, only previous failure. All of a sudden, I had to think of some way to motivate my students to engage with the subject. For several years I floundered; Jimmy shovels at a rate of 8 ft every minute, if the driveway is... You know what? It doesn't really matter how long Jimmy is out there, because he's going to stay out there until he finishes shoveling the driveway. Trite doesn't work. Does math matter or not? If it's just some intellectual exercise I'm forcing on my students because politicians and textbook makers think its a good idea then we should change the good idea.

There's a long road to my infatuation with problem-based learning and using experiments in my classroom, but developing lessons for the Kuyers Institute was pivotal. Can teaching a math lesson be about more than just the math? Can we talk about social inequality using numbers? What role does math play in convincing people of the rightness or wrongness of an opinion in our society?

One of the few times I've been successful with such a lesson is in studying pay inequality between men and women in the U.S.A. I've done this lesson in both Algebra 1 and Algebra 2. I think the older students are better able to delve into the underlying factors (not surprising).

We start by looking at the data from the census bureau: 1955 to 2003. Before beginning group work, we have a discussion about why we use data for median incomes of full-time year round workers. (Women are more likely than men to work part-time in the US and that would skew the data, hence 'full-time year round'. Also, men are more likely to be top earners in the US (think Bill Gates and the like), which would also skew the data, hence 'median'.

Then we look for trends in the tables. Almost always students say that men earn more than women and that as the years increase people make more. Then we see if the graphs can help us be more specific (page 2 of the doc). What do you see? Can you use math to describe that? Students suggest exponential curves. We try them and they don't work. Then we talk about cutting off parts of the data. If you cut the data at 1975-ish, it becomes approximately linear.

After laying the foundation for the lesson, it's group time. In groups of 3 or 4, students start to work through the questions in the lesson The lesson is divided up into 3 tasks to make for natural discussion breaks as a class for formalization of content. Here's a quick overview:

In Task 1 students find equations for both men and women incomes using lines of best fit and two points. Students must also explain what each aspect of the function stands for and whether it is appropriate to predict backwards using the functions.

Task 2 tackles linear regression, correlation coefficients and whether our equations have any merit when comparing with current data (I have 2010 incomes in the lesson).

Task 3 takes a brief foray into rational functions (cents on the dollar). I couldn't help bring up a conceptual problem by asking students to create a line of best fit for this function as well (it looks linear). However, this yields drastically different results than the findings from the previous tasks. Lovely discussion about why this doesn't work.

All in all, it's an interesting lesson that leads to great conversations about the structure of our society, concepts like 'justice' and how we use math to 'prove' our points.

I'd be interested in your thoughts and/or recommendations for improving it.

Well, this isn't the quickest way I've ever taught combined variation but it is the most fun. Instead of using prescribed examples, we investigated the amount of weight it takes to break a board made out of spaghetti based on its width, thickness, and length (distance of the supports).

The data didn't turn out to be completely direct or inverse variation (as I hoped). The basic relationships were what I expected them to be: width = linear, thickness = quadratic, length = inverse the distance squared. However, the students did a great job fitting equations to the data.

We spent two days making and breaking spaghetti boards (I only planned on one) and one day pulling it all together as a class.

Below is the sheet I handed out to help give some more direction to the experiment.

Next time I think I'll use something a little smaller than a washer for weights. There wasn't a whole lot of accuracy in student measurement--the data was kind of everywhere. I also think I'll strongly suggest students make a scatterplot for each of the relationships. Never underestimate the power of visuals when attempting to model data.

Continuing in my quest to make Algebra 2 more interesting, we're using flashlights to explore direct and inverse variation relationships. I'm using the lesson I posted over the summer that reworked one of my textbooks projects.

Day 1:
I introduced the lesson using one of our school's stage lights. I had three students come up to the whiteboard--one to hold the light, one to measure the distance from the whiteboard and one to measure the diameter of the light on the whiteboard. We recorded three measurements at 10cm, 20cm, and 30cm. At the end of the experiment we are going to try and predict the width the light will make 1 meter (100cm) from the board.

Here's some pictures from students working on figuring out the relationship between the distance of the light from a surface and the size of the diameter.

Day 2:
Some students had difficulty with the data they collected. When subtracting out the diameter of the flashlight lens (we're doing direct and inverse variation so we need to go through (0,0)) their data wasn't even close to linear. Bad news. For the most part we did pretty well figuring out whether it was y=kx^1 or y=kx^2 by looking at the table and the graph.

As I write this, I realize it's kind of absurd. Maybe I need to rethink how long the spaghetti bridges activity I have scheduled for tomorrow will take. I've planned for two days...

Oh. And why yes that is an overhead projector shining on my wall next to a flat screen TV. Don't judge me.

In my quest to make my math classes more engaging, I'm trying to do at least one exploratory activity every week in every class. Hopefully, my entire class will be a bunch of neat tasks and projects but that will take time. Here's a taste of what we've done in the last week or so.

Creating patterns with pennies and generalizing to equations - Algebra 1

Measuring and maximizing speed with dominos - Physics

Trying to find how many cars can fit in a parking garage - Algebra 1

Using pennies to talk model inverse variation - Algebra 2

Rolling soup cans and cars up and down ramps to talk about velocity and acceleration - Physics

I want to change the way I introduce problem solving and the creation of math models for situations to my Algebra 2 students. Let me give you some background to my class. The first two chapters are all about jogging students' memories on math functions they've seen in Algebra 1 and then introducing them to new models--mostly variation: direct, inverse, joint, combined.

The way I've always taught this is to give students a situation and the information that goes along with it then ask them to explore the information and figure out how to model it. Here are some of the situations:

1. The pressure of a gas and its relation to volume
2. The intensity of a sound based on the distance from a set of speakers (like at a concert)
3. The length of a wing (bird or insect) and the number of beats per second.

I could go on but you get the picture. The class conversations are great. The kids to a great job of diving into the data and figuring out patterns and shapes. But I'd like to get the kids more involved in creating their own data. I want every student vested in the discussion.

I found the following 2 options. Project #4 on the left and Activity 4 below. I think both are great ideas but leave something to be desired from an inquiry perspective. They do a little too much leading. I want the students to figure out on their own whether they want to use a table or a graph. I want them to think through the process of needing to find a variation constant. If the book tells them exactly what to do all the time, where is the problem-solving in that? If you look at Project #4 above, part e just gives away the extension. That's lame.

Directions - Flashlight
This could totally turn into an Act 3 lesson except that instead of showing a video, I would do a quick demonstration on the board by placing a student at set distances from the whiteboard and then draw circles around the illuminated portions of the board. At this point the students could guess at the relationship, figure out what information they need, and then hop to doing an experiment of their own to figure it out.

Use a small flashlight and a sheet of paper or poster board. Use a measuring tape or a ruler to shine the light from various heights on the paper. Measure the diameter of the circle created by the flashlight. Create a math model describing the relationship between the height of the flashlight and the diameter of the circle.

Make a prediction about how large the circle should be from a different height. Test your prediction.

(If there is time at the end and I can find a light meter, checking the relationship between the distance of the flashlight and the intensity of the light would be brilliant)


Directions - Ball Drop
Either use a CBR to measure or use an iPad to video a ball drop.

CBR: Place the CBR above the ball to get data relating the distance the ball drops in relation to time. Transfer the information from your calculators to something the whole class can see.

iPad: Video a ball being dropped next to a tape measure taped to the wall. Start the zero at the height the ball is dropping. Using the stop and forward options on the video bar, measure the distance the ball dropped over time. Transfer the information to something the whole class can see.

Both: Create a math model based on the data and make a prediction about how long it should take to drop from a different height. Test your prediction. Be ready to share how you arrived at your conclusions.

This summer I'm spending my time looking for rich contexts or problems to engage my Algebra 2 students. Generally, Algebra 1 is a piece of cake whereas Alg 2 gets a much more booky class. I intend to change that.

I'm taking a no-holds-barred approach: surfing the internet for good contexts, activities begged borrowed and stolen, and--this is a first for me--using the projects section out of the back of the textbook I use. Surprisingly, I like some of these. What a dolt I am for never having really looked before.

The project on the left is rich enough that I think I can use it to replace two sections 2-8, 2-9 I currently teach. (sorry about the picture--I used an iPad 2) Sections 2-7, 2-8, and 2-9 deal with data modeling and combined variation. The combined variation aspect is really neat because you can investigate each variable involved in a situation separately and then put them together for a combined variation function. For example, in this situation, the Weight (M) a board can hold varies directly as the width of the board (w), directly as the square of the thickness (t) and inversely as the distance of the supports underneath the board (d). The variation constant (k) changes with each particular situation. When you put them all together you get: M=(k*w*t^2)/d.

The difficulty with this project is finding the materials to make it work. Finding strips of Balsa wood was doable but I couldn’t find enough weights to make a multi-group experience of the project. Here’s how I plan to modify it: Instead of Balsa wood, each group uses spaghetti noodles. Suspended paper cups from the spaghetti using paper clips. Then place metal washers or marbles in the cups as weights. They’re not standard weights, but that could even lead to a discussion of whether or not we could just weigh one washer and scale up or whether we need to weigh the whole stack when predicting how much weight it will take to break a "board" of this many noodles.

Here’s a quick write up of what I’d give to students. If you have any suggestions, I'd love to hear from you.

Directions
We will do 3 different experiments to explore how different variables change the amount of weight a board can hold: width, thickness, and the distance between supports. Because wood is much harder to break and I don’t have lots of weights laying around, we’ll make pasta boards by taping together spaghetti noodles. Please do at least 5 trials in each of the 3 categories. As a group, you may either choose to use washers or marbles as your weights for the hanging cups.

Spaghetti Width
Investigate how the width of a board changes the amount of weight it can hold without breaking. Be sure to keep all other variables the same and only change the width of the spaghetti board.

Spaghetti Thickness
Investigate how the thickness of a board changes the amount of weight it can hold without breaking. This one is a bit more on the tricky side of things. I suggest using tape and to help force your spaghetti to stand vertical. I also would use a width of more than 1.

Support Distance
Keeping all other variables the same, change the support distances to gather information about how the distance between supports affects the amount of weight a board can hold without breaking. Spaghetti noodles are generally 25.5 cm please use metric units.

Questions to Ask:
After collecting your data, analyze it as a group. (I expect you to be able to share your analysis with the class using math.)
What relationships do you see?
Create models describing each of these experiments.
Make a prediction using your data and test it with your spaghetti and weights.
Make a mathematical model that takes into account all three variables and how they relate to the weight a board can hold.
How can we test our model?

This is where I’m currently at for Algebra 2 (outline below). All starred items indicate an activity or rich context. All “Ch” indications are for projects I thought had some merit at the end of the chapters. I haven’t done the work of writing up each lesson yet--just gathering the resources. Most of the “Activity” listings are from the UCSMP activities workbook. There were some real treasures in there but none usable for me right out of the box. They gave too much away by telling the students directly what to do and making connections for them. I think 6 years ago, I would have thought them perfect; now, I need to help make them more inquiry-based. I'll share what I have when I have it.

You’ll also see the concepts I’m cutting and those I’m adding in order to align properly with the Common Core. With the addition of more constructivist opportunities for kids, some stuff doesn’t have time to get revisited if we’ve already covered it in Algebra 1 (goodbye linear unit?). If you've got some ideas, I'd love to hear them.

I plan to write up lessons for each of the starred sections and post them eventually both here and on michedmath.weebly.com

Algebra 2 Course Outline
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