I like to start off my two-way tables unit by polling the class about something, like whether they think people in our class who play sports are more likely to also play a musical instrument. The kids love it! They understand right away how two-way tables are set up and everyone wants to vote! But then when I want to give them some different sets of data to analyze I can never come up with anything that’s very motivating. The examples in the textbook include a variety of boring contexts that range from how many arbitrary 7th graders and 8th graders chose English as their favorite class to what sizes and colors of t-shirts a made-up store sold.
This year I was thrilled to come across a lesson plan created by Chris Shore that gives students two-way tables with Titanic passenger data to analyze. This lesson plan particularly piqued my interest because I’m an occasional docent at the house museum of the Unsinkable Molly Brown and possess an abundance of Titanic facts that I don’t get to share very often. The Titanic two-way tables were interesting to talk about and led the kids to ask lots of questions. I also liked using them because they presented a motivation for wanting to calculate relative frequencies. Since the number of men on the ship was so much greater than the number of women it made sense to find percentages in order to compare how much more likely it was that a woman survived than a man. Any other examples of intriguing two-way tables that you’ve used?
We did this today to help kids review at the end of our 8th grade linear relationships unit. I feel like I’ve had a little bit of a hard time motivating kids to graph linear relationships, so it was nice for them to have such a strong reason for wanting to graph them. As I walked around the room I heard so many kids helping other kids out by explaining slope and y-intercept to them. I helped one student get the right equation by asking him what the slope was between two of the stars and then I heard that student asking his group member the same question when he got stuck. Yay!
The other day I asked my 8th graders who could recall the formula for slope. The first answer was y = mx + b. “Which part of that equation is the slope?,” I asked. Another student gave us the right formula: m = (change in y)/(change in x). Then another student said, “Wait! I saw a meme on Instagram the other day that was like ‘y = mx + b is the equation of the slope of my life going downhill.’ Does that mean the meme is wrong because y = mx + b isn’t actually the slope formula?” I just answered her with a beaming smile because I LOVE that kids are making math connections to memes.
A few days ago I was trying the think about WHY students would ever need to know whether two relationships are proportional. The first thing that occurred to me was mixing paint colors together, so I put together this problem for today’s lesson:
Greg and Wanda are mixing paint together to paint a room in their house.
Greg combines 4 parts blue and 6 parts yellow.
Wanda combines 6 parts blue and 9 parts yellow.
Will they both make the same shade of green?
I wasn’t sure the children would really be that excited to talk about paint, but they LOVED it. I think it helped that it was such a familiar context to them from art class and they could visualize what shades of green would look like with more or less yellow and blue amounts. I gave the kids some time to think about it on their own and almost everybody wrote: “No, they’re not the same color because Greg has two more parts of yellow than blue, but Wanda has three more parts of yellow than blue.”
Luckily there were at least a few kids in each class who started thinking about ratios and scale factors. The two most common ways to prove that Greg and Wanda have the same shade of green were:
Both of them have 1.5 times as much yellow as blue. (This idea turned out to be hard for a lot of kids to understand. Many of them were really confused about where 1.5 came from because they’re not super fluent with decimals.)
The ratio of blue to yellow for both Greg and Wanda reduces to 2/3.
Some kids also found the total amount of paint they would have all together (10 to 15) and then noticed that this also reduced to 2/3. After seeing these ideas everyone was convinced that the colors would turn out the same.
I had asked my kids at the beginning of class if they knew what the word proportional meant. They said a few things about portions and spreading things out evenly, but didn’t have a good sense of what it meant. After we kept seeing the fraction 2/3 pop up in the paint problem one of my kids asked if things is proportional when they reduce to the same fraction. WHY, YES!!! YES, THEY ARE!!!!
Another girl had approached the paint problem differently than everyone else and had been trying to scale up the paint amounts in a rate table earlier. When she heard the comment about reducing all the fractions she asked, “Can you show that things are proportional by making the numbers in the fractions bigger instead of smaller?” OMG SUCH A GREAT QUESTION!! I yelled out, “Oooooo!” and ran to the board. She responded with, “Um, you just made a weird sound and didn’t answer my question. Am I right or not?” We talked about how much blue paint each person would need if they had 18 parts of yellow paint. Sure enough, we got more equivalent fractions!
After today our working definition of proportional is: when quantities can be written as equivalent fractions. Tomorrow we’ll start investigating cross products.
I know a lot of middle school math teachers have probably heard of the Barbie Bungee project. If you’re not familiar, here’s a summary: Kids build a bungee cord for a doll out of rubber bands. They collect data in the classroom and then have to calculate how many rubber bands will be needed for the doll to safely jump from the roof of our school (a height of 631 cm). It’s a great application of linear relationships and making a line of best fit. In this post I’m going to share how we’ve implemented it at our middle school!
Here is the packet we give kids in math class to complete for this project.
The kids can bring in a doll if they want, but we mostly just reuse the same dolls each year. We’ve been doing this project at our school for over ten years so we have a lot of dolls! Below are some pictures of what my classroom looked like today as students were working on the project. 100% engagement and every student knowing what they’re supposed to be doing: a teacher’s dream come true.
The project is really, really helpful for kids to have a context for applying linear equations. It’s also a great opportunity to talk about the difference between proportional and nonproportional relationships. Does the doll drop twice as far if you double the number of rubber bands??
A few other things we’ve implemented at my school: Two years ago we added a science component to the project to make it a little more cross-curricular. Science class focuses on calculating the force acting on the doll, applying Newton’s second law and drawing free body diagrams. We also added presentations that are made to an authentic audience of community members. Kids have to figure out how to explain the math they did to a group of adults who might not remember middle school math and convince them that they’ve created a fun and safe bungee experience. Their end goal is to be awarded a permit for their bungee jumping company.
My talented friend Ben wrote a theme song to play at the start of math class every day. The children are pretty excited about it.
A running list of student commentary about the theme song:
This song sounds Egyptian!
Ms. Peterson can you PLEASE play it again? I need to learn all the words!
My little sister can sing better in the shower.
Did you just find this song on the internet and change the words to Ms. Peterson?
When is your friend going to make this into an actual song? Is he working on an album?
Your friend should make the song longer, like into a real song. *I thought you hated the math theme song, why should it be longer?* Well, I mean your friend just sounds really depressed and sad when he’s singing. He should be, like, happier.
*upon hearing me listening to the new Dirty Projectors album before class* Oh, this person sings way better than your friend! Your friend should take singing lessons from this person.