Andrew Busch

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.

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Math In Action Session -- GVSU

2/23/2014

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).

math_in_action_presentation-_gvsu.pdf
File Size:	10226 kb
File Type:	pdf

Teacher Cred
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) ______________________

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Spaghetti Bridges - Linearity in Algebra 1

2/21/2014

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).
-or-
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.

7.7_spaghetti_bridges.doc
File Size:	34 kb
File Type:	doc

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Measuring Water Height - Algebra 1

2/9/2014

This activity went way better than I thought it would. This summer I paged through my textbooks accompanying resources and found an activity suggesting using soup cans, centimeter cubes, water, and rulers to introduce slope-intercept form. My initial reaction was "you're kidding right?". It seemed like a good activity but giving groups of freshman open containers of water sounded like an invitation to several very wet students and multiple detentions. I filed it away, just in case, with no real intention of coming back to it.

After a little over a semester of intentionally including multiple experiments every chapter, I wasn't nearly as intimidated when this file stared back at me from my chapter outline last week. I printed out the book's worksheet, gathered some resources for the next day and went home. The morning of, I decided to rework the task a bit (a have both the original and my file at the end of this post).

The students did swimmingly. Every student was on task for most of the period. The class did well seeing the relationships between the initial height of the water and the y-intercept as well as between the number of cubes added and the rate of change.

When I do this again next year, I'm going to gather some different containers beforehand. Glass worked better than metal containers because the water level was much easier to read. So, I think I'll use some salsa jars, big fat canning jars, and some of the baby food jars from my house (maybe even the plastic Gerber ones will work). Goodness knows I've got enough of those laying around.

This is my counter after class. The science teacher was so proud of me.

My version (also shown below)

7.4_water_height_activity.docx
File Size:	50 kb
File Type:	docx

The original.

7.4_activity_16_alg_1_ucsmp.pdf
File Size:	83 kb
File Type:	pdf

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Bungee Apparatus - Modeling Linearity in Algebra 1

2/8/2014

The setup:
How many of you have ever been bungee jumping? (No hands) How many of you want to go bungee jumping? (6 or 7 hands and a whole lot of excited responses of "that would be so cool" or "I don't want to die before I can drive")

Today, we're going to do some research for a bungee jump. What do you think are some important questions to ask before you step off the cliff? (Students almost always mention the strength of the bungee cord, how long it is, how much you weigh, and whether or not you have a death wish) Do you think there is a relationship between those things? Does it matter how much you weigh? How much does it matter?

Here's our bungee jumper. (Show students the paper clamp with some washers attached) We don't want to send a real person down while we're experimenting because it'd be a shame to kill someone just to gather data. Your groups will use our state-of-the-art bungee apparatus to figure out the relationship between how much the bungee cord stretches based on the weight of the person.

Demo:
Hand out student worksheets. Demonstrate how to and how not to drop the bungee apparatus and gather data.

Let 'em loose:

Reconvene:
I often start by having student groups present their graphs and lines of best fit. We compare and contrast group work. We talk about what we like and might do differently. Then we shift the conversation to a compare and contrast of what we like about the table, the graph, and the word descriptions. What do we like best? What information does each one present well? What information do we loose in each of them?

Then I have them use their graphs to estimate the amount of stretch for 18 and 30 washers. 18 isn't much of an issue. For 30 washers every group need to extend their graph and line of best fit. Plus, we get to have the conversation about the domain of our math model. Will the linear relationship between weight and stretch continue forever? Why or why not? It doesn't take a lot (if any) leading from me for students to see that the stretch will begin to level off at a certain point, or the rubber bands will simply break.

I picked up this activity from the 1st edition of Core Plus Course 1 back in 2004. It's similar to the infamous Barbie bungee jump but instead of varying the number of rubber bands, students vary the weight. I like it for a couple of reasons:
    -it helps students make connections between tables, graphs and words (I don't get into equations on this one those that would be completely doable).
    -it allows for some great conversations surrounding variables. What does it mean when we say something varies?
    -once you have the materials, you can do this experiment with virtually no setup from year to year.
    -it only takes one class period.

Materials for each group: 1 Large paper clamp (which will never ever hold papers again), 2 rubber bands, 15 washers (I have a box of old washers I picked up somewhere that I use as weights in multiple experiments throughout the year), 1 centimeter measuring tape (or meter stick).

7.1bungeeapparatus.doc
File Size:	44 kb
File Type:	doc

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Too Tall To Measure - Proportions Activity

1/28/2014

When teaching proportions or similar figures I generally throw in an example or two of measuring the height of objects you wouldn't normally be able to measure using shadows or various other known distances. Since I've been on my activity/experiment kick, I wanted to go measure shadows to find the heights of trees and telephone poles. However, it's January in Michigan. Nobody wants to go outside and measure shadows, which assumes there will be a sun to cast a shadow. There are sometimes weeks that go by without seeing the sun up here.

Enter a brilliant adaptation by Jonathan Claydon for the cannot-go-outside-for-5-months-because-we-live-in-the-midwest group of people. I have my adaptation of it at the bottom of this post.

Students start be measuring each member of their group (see below). This measurement is known as the group member's 'real height'.

Then groups drift off to the gym to find the height of objects that are too tall to measure easily. I picked three of them: the gym ceiling, the top of the backboard, and the height of the railing on the weight room. Using a ruler, group members find the 'fake height' of a each group member standing next to the too-tall object and the 'fake height' of the object. We found both measurements need to be recorded each time because of differences in how students hold the rulers.

After completing our measurements, we went back to our classroom to finish the calculations. Each group member had to find the heights of the objects using their own fake height measurements. Then groups posted a group height for each too-tall object. Lots of good conversations happened when calculated heights didn't agree. Most groups just averaged the calculated heights for each too-tall object.

Next, we figured out how we would go about agreeing on the actual height since I don't have the actual measurements myself. With our ceiling calculations, we decided to remove the group's height which was very different before averaging the heights.

The class's backboard heights weren't nearly as close. Here we thought the median might be a better descriptor of the 'real height'.

Aside: I am horrified at the rounding in this next picture of my blackboard. I wrote the numbers students told me to write without correcting their rounding errors. In hindsight I should have forced them to be correct (or attend to precision) but we were hurrying to get done before the bell. ...Plus it makes me look like an idiot.

Here again, the 'real heights' were all over the map. They thought the median worked out well last time and wanted to use it again.

When we talked about difficulties we encountered, several students said they had difficulty eye-baling the measurements because they couldn't hold their hands steady. When I do this again next year, I plan to hand out a jumbo paper clip to put on each student ruler to serve as a marker for the top measurement.

Thanks to http://infinitesums.squarespace.com/commentary/2011/9/30/proportions.html for the idea!

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M&M Catapults! - Quadratic Transformations in Algebra 2

1/9/2014

For an activity to do before Christmas vacation, I borrowed this from 8ismyluckynumber.blogspot.com (who borrowed it from sweeneymath.blogspot.com). With a little revision to fit my style and students, we were ready to go--the file is at the bottom of this post.

I handed out the project description 2 weeks before attempting it in class together to allow for catapult construction at home. That may have been my 1st mistake (more on that later). A week before the project, students brought in their catapults for a show-and-tell of sorts. The idea was to test the catapults and make any necessary revisions. I had at least one group break their catapult on testing day.

The contenders:

When I do this project again, I'm going to suggest a couple of designs; none of the plastic spoon catapults lasted through the entirety of the project nor were they accurate enough. Along those same lines, almost all of the catapults broke at some point or other during the project; the question was whether or not they could be fixed. Mr. Sweeny gives step-by-step instructions for creating one. I'll suggest his method next time.

Gathering Data:

We spent Day 1 in class gathering data about our catapults and calculating equations modeling the flight path of the M&Ms. As with all student created and measured data, the results were less than consistent. I'll definitely have to stress this more next time. Although, because of the errors, we did have some excellent conversations regarding what different aspects of the equations meant and whether or not we thought they were accurate. I had to forcibly move several groups on to the next steps because they were attending to too much precision.

Letting the M&Ms fly!

On Day 2 we took our catapults and our calculations to the proving grounds. Originally I hoped to use my classroom but some of the catapults had more lift than I anticipated. When raised the 100 some odd centimeters to the table they hit the 10 ft ceiling.

Only one of the catapults was able to hit the paper plate target. Next time, I plan to go for a target distance rather than an actual target. I think there would be a lot more student success and a lot less frustration from all of us.

Lab Report

This is the first time I've ever had students do a write up of what they did, how they did it and what they learned. The science teacher down the hall was so proud of me. All in all, I'm impressed at the engagement with the material and how well it helps me understand what students understood.

6_m_and_m_catapult_project.doc
File Size:	120 kb
File Type:	doc

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A Day in the Life of Math Class

12/23/2013

A quick peek at December in math and physics.

Mechanical advantage and efficiency - Physics

Everyday percent problems from my breakfast table - Algebra 1

Inverse square variation relationships - Physics

M&M Catapults - Algebra 2

Building Hotels - Algebra 1

We also did some stuff with social justice and quadratic transformations. Since I've already written up posts on them, I'll just link to them if you're interested.

I'm missing a couple of the projects for the month because I don't always think to have a camera handy until after all the action is finished. Think of this as the sampler plate at Ruby Tuesdays.

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Quadratic Transformations - Geogebra Rocks It Out

12/10/2013

2 Comments

I love using real data in my classes but it's usually easier said than done. I don't have any fancy pants TI-NSpire calculators with little WiFi receivers that transmit data instantaneously to all my students (if anyone wants to buy me a classroom set I would send you a thank you card and a drawing from each of my toddlers). I've tried with regular graphing calculators but it takes almost an entire class period to get the data to everyone. This means I generally don't use student created data as much as I'd like.

This year I'm doing something different. I bit the bullet and signed up for the computer cart for a couple days. On day 1, we introduced transformations using Desmos. After playing around with it for 5 mintues we explored different parent functions and their generalized forms: y=x and y=a(x-h)+k, y=x^2 and y=a(x-h)^2+k, and y=2^x and y=2^(x-h)+k. On day 2 we looked at data using Geogebra files I created beforehand (see below). Each of the files is based off the video of a basic event.
At this point you need to know I am a complete Geogebra newbie. I had to look up how to get the spreadsheet numbers to show up as points. If I can do it, you can do it.

Based on the student questions over these last several days, this is probably the best students have understood the concept of function translations and transformations after the initial introduction.

free_throw_quadratic_-_slow.mp4
File Size:	378 kb
File Type:	mp4

6.6_basketball_free_throw_data.ggb
File Size:	4 kb
File Type:	ggb

rolling_a_can-main.mp4
File Size:	1143 kb
File Type:	mp4

6.6_rolling_a_can_data.ggb
File Size:	7 kb
File Type:	ggb

projectile_quadratic_video.mp4
File Size:	100 kb
File Type:	mp4

projectile_data.ggb
File Size:	4 kb
File Type:	ggb

2 Comments

Gender Gap - Social Justice in Math

12/8/2013

4 Comments

I wrote this lesson several years ago for the Kuyers Institute. The question we were trying answer was whether teaching a math lesson could 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.

MS Word file of the lesson

gender_gap_student_lesson-alg_2.doc
File Size:	1800 kb
File Type:	doc

MS Excel file with data for student calculators

gender_data_through_2010.xls
File Size:	44 kb
File Type:	xls

4 Comments

Virtual Filing Cabinet - MTBOS Sharing is Caring

11/25/2013

2 Comments

I've been inspired you online math community. Over the past two years, I've borrowed/stolen more ideas than I care to think of. People like Geoff Krall (@emergentmath), Dan Meyer (@dydan) and Fawn Nguyen (@fawnpnguyen) spend more time than I care to think of putting lessons and ideas online to benefit the rest of us. I've taken these ideas, shared them in my department and to whoever else will listen and I've grown as a teacher. As I move towards richer and more engaging problems and tasks, I think it time to share the wealth.

I dig the idea of a virtual filing cabinet--a place where people file away e-resources for personal use but available to all. Instead of filing away every resource I can possibly find on a topic I want to post what I actually use (or plan to use) in my courses.

I started this past summer with my Algebra 1 and Algebra 2 courses. Slow and steady the resources come online. I've already gotten quite a bit of my linear stuff up. Next week we break into quadratics and then to exponential functions, polynomials, rational functions, etc.

http://www.andrewbusch.us/math-tasks.html

Take a gander. Check back in a month or so for new stuff. If you feel like adding some suggestions, please give a holler. I'd love to use them.

2 Comments

Lifetime Cost of Light Bulbs - Algebra 1

11/18/2013

I recently went to the store to purchase light bulbs. The options were a bit overwhelming. Do I buy standard bulbs (incandescent) or energy efficient ones? If I buy energy efficient ones, which kind: LED or Compact Florescent (CFL)?

After fruitlessly trying to find 100 watt bulbs with two toddlers in tow for several minutes, I gave up bought some cheap 60 watt incandescent bulbs and decided to figure it our later. When later came I learned the old 100 watt bulbs are no longer sold--hence my difficulty finding them. So, before I have to purchase bulbs again, I decided to figure this thing out. And then I thought to myself, there's a great math lesson in here somewhere!

Here's my 1st take at creating a lifetime cost lesson for light bulbs. The MS Word document is 3 pages: incandescent, LED, Compact Fluorescent (CFL). I'm probably going to include it in our system of equations unit.

5_alg1_-_light_bulbs.docx
File Size:	446 kb
File Type:	docx

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Lifetime Appliance Costs - Algebra 1

11/11/2013