My Other Math Sites
Lessons From the Classroom
Always Sometimes Never
I did Always-Sometimes-Never (ASN) questions with my 6th graders. The kids were randomly paired up to work on placing 18 mathematical statements into appropriate columns: always true, sometimes true, and never true.
The students were involved in the discussion and coming up with examples. They had to translate some statements into equations or inequalities and defend their answers. They learned to give counterexamples.
ASN works for any math level (and could be used in other subjects). The kids are comfortable working and having that math conversation with just one other person initially; then, they build up the confidence to share their reasoning with the whole group later.
I got this particular set of 18 from Swan and Ridgway. Sixth Sense has a set also.
Max gets a pay raise of 30%. Jim gets a pay raise of 25%. So Max gets the bigger pay raise.
When you cut a piece off a shape, you reduce its area and perimeter.
If you add the same number to the top and bottom of a fraction, the fraction gets bigger in value.
In a sale, every price was reduced by 25%. After the sale, every price was increased by 25%. So the prices went back to where they started.
(a+b)/2 ≥ (ab)1/2
If you divide the top and bottom of a fraction by the same number, the fraction gets smaller in value.
It doesn't matter which way you multiply; you get the same answer, like a × b = b × a.
If you add a number to 12, you get a number greater than 12.
The square root of a number is less than the number.
It doesn't matter which way you divide; you get the same answer, like a ÷ b = b ÷ a.
If you divide 12 by a number, the answer will be less than 12.
The square of a number is greater than the number.
p + 12 = s + 12
(n + 5) is less than 20
2(x + 3) = 2x + 3
3 + 2y = 5y
4p is greater than 9 + p
2(3 + s) = 6 + 2s
We Won First Place!
Today, the Naval Sea Systems Command (NAVSEA) held its 12th annual engineering competition during National Engineers Week. We were one of 14 junior high schools invited to participate. Each school brings a team of 5 students and one teacher, and NAVSEA assigns one of its engineers to assist us in building a rocket launcher.
Our task was to design the launch itself. Each school was given a box with materials to build from and a one-million-dollar budget to purchase additional materials. But we'd get bonus points if we stayed within this budget. Our "rocket" was a ping pong ball, so we had to build something to launch this ball at and into a target set 10 feet away.
We first built a sling-shot contraption using rubber bands and popsicle sticks but couldn't figure out an excellent way to stabilize the unit. We then thought of making a catapult—figured if it was good in medieval times, it ought to work here. It turned out our catapult worked great. We were given two one-minute test launches at the target, and we were delighted with our design because we came close to the target on most trials.
The kids did an excellent job of collaborating, designing, and constructing our model. Daniel calibrated each launch to ensure the ball's projectile motion landed on its target. Michael was our architect; he drew pictures of both prototypes before we built them. Slater gave us the catapult idea and made a cone to hold the ball; he also had the most significant role as our main presenter before a panel of seven judges. Jacob and Josh constantly helped improve our model launcher.
I couldn't be more proud of these five kids. They worked hard on the project—but I was most impressed with how well they worked together. Daniel leaned over and said to me before they announced the results, "I hope we place." I hoped so, too, but I knew we had a shot at first place, too, because no other team made all three of their competition launches as close to the target as we did, and the boys did a great job answering the judges' questions. So when our school was not named for 3rd or 2nd place, I could see the boys quietly squealing in their seats (I guess I was too). We celebrated at a nearby Costco snack bar before returning to school.
Each student received a $30 gift card from Barnes & Noble and a "first place" certificate. They will also be invited to visit a real naval ship later this year. Our school got a framed certificate. The local newspaper cameraman had been at the event the whole day, and a news reporter came to interview us.
This was our second time winning first place at the NAVSEA Engineering Challenge. We won back in 2008. That year, the team of five students would never forget that winning day, which was the day Mrs. Nguyen got a speeding ticket. On our way to the event—with five students in my car—I got pulled over by a policeman on a motorcycle for doing 50 in a 35-mile zone. Quite miffed by this, I turned to my students and said, "You guys better win this competition!" They did.
Now we have two of these. Go Tigers!!
Playing with Barbies
I got most of my 20-plus Barbies from eBay and use them well. I use the Barbies with my 6th graders when we study proportions. I use them for Barbie Bungee with my algebra students during linear equations.
The students work in groups of three, one Barbie per group. They take various measurements of Barbie from head to toe; then, they take the exact measurements of someone in their group. (I did this activity with math teachers, and some teachers were concerned that students may not wish to be measured. I've done this activity for the last seven years, and if anything, the kids want to volunteer to have their measurements taken.)
I then assigned each group one measured body part—Group 1 has "feet," Group 2 has "waist to ankle length," Group 3 has "head height," Group 4 has "legs," etc. Students now take the ratio of their group's assigned human body part to that of Barbie's. Students then multiply all of Barbie's measurements by this ratio, and they sketch their "humanized" Barbie based on these scaled-up numbers.
Below is one of the ten posters that will go up in our school's cafeteria.
What my kids wrote in their reflections on this activity:
The Barbie was weird. Ms. Nguyen was right, Barbie does look like a freak in human form.
We did a life-sized Barbie and my group got "feet," so that meant she was super tall.
Also that Barbie project is very fun but creepy at the same time. Barbie was freakishly disproportionate.
Our Barbie's feet were REALLY small.
We did a really fun project with Barbie. I learned a lot about proportions because of it.
People were really creative with their Barbie; our Barbie's name is LOLA! She is cute.
We work on Barbie and it was weird because Barbie is so weird. My group messed up and now Barbie is messed up, it doesn't even look like Barbie.
We had an insane task! We had to see what Barbie looked like in human form and she's a FREAK! It was pretty fun to do though.
Barbie was a hard, hard, hard challenge. She is so tall if we did her legs for our body part it will be so long.
Updated 02/09/12
Math and Literature
I read If You Hopped Like a Frog by David M. Schwartz to my 6th graders. They are learning about ratios and rates and how to solve proportions. It's a beautiful book and beautifully illustrated by James Warhola. At the back of the book, the author has questions you can ask students to work on. I made this worksheet based on that.
If you were as strong as an ANT... you could lift a car!
Even junior high kids still like to be read to. They squeal with delight when I pull out a book— and it does not have to be an illustrated book. Another book I'd read to my 6th graders earlier this year was How Much is a Million? It's also written by Mr. Schwartz and illustrated by Steven Kellogg. We were working with number sense then, and the kids said oohs and aahs as I read to them, "How big is a billion? If a billion kids made a human tower... they would stand up past the moon... If you wanted to count from one to one trillion... it would take you almost 200,000 years."
A great resource that I faithfully use is Math and Literature, Grades 6-8, by Jennifer M. Bay-Williams and Sherri L. Martinie. One suggested reading is A Drop of Water: A Book of Science and Wonder by Walter Wick. I read this to my students, and afterward, they blew bubbles on their desks and measured the diameter of each burst bubble to find the circumference. We had a lot of fun!
5 Practices for Orchestrating Productive Mathematics Discussion
The post title is a book by Margaret S. Smith and Mary Kay Stein. It's a wonderful little book that can make a big impact in a math classroom.
I'm lucky to be part of a team of presenters—there are four of us: Chris is a math professor at UCSB, Maria is a high school math teacher, Jeff is a district math coach, and then there's me.
We started the UCSB Mathematics Project this past summer. About 30 participants, most of whom are school teachers, attended a one-week workshop in late July and will continue to gather for three days during the school year.
While the Project's focus is on the new Common Core Standards in Mathematics, we also present and discuss leadership and equity issues, engage teachers in math activities that model the new standards, and examine the 5 Practices for Orchestrating Productive Mathematics Discussions chapter by chapter. I was assigned to cover the Introduction, Chapter 1, Chapter 5, and Chapter 7.
The book persuades and guides teachers to lead more thoughtful and productive discussions during a math activity. It outlines the necessary steps:
Find a math problem that is high-level with multiple strategies.
Launch the task by telling students what tools are available and what type of task products they can expect.
Discuss and summarize by using these "5 Practices":
Anticipate — the teacher must do the problem ahead of time and anticipate the different strategies and solutions students may develop.
Monitor—The teacher needs to pay close attention to students working in groups, listen to their mathematical thinking, and observe their strategies.
Select — the teacher needs to select which groups or which member(s) of a group will share with the whole class.
Sequence—The teacher must arrange the order in which the selected people will share (It's no fun if the group with the "best" strategy shares first!)
Connecting—The teacher is responsible for asking students to connect the solutions presented by the different groups and identify the key mathematical ideas in the problem.
I like to practice what I preach, so I've been using The 5 Practices whenever appropriate. My algebra kids were working on a problem involving systems of inequalities. I monitored closely to learn the following:
Luis' group was talking about so and so dude on YouTube,
Bella's group was on task, but they were too focused on one strategy that may not take them to the solution they needed,
Miranda's group was drawing graphs, but everyone focused on the same graph instead of branching out to get more done,
Martin's group smiled at me and covered up their empty papers,
Eliana's group had the first part completed correctly but had a hard time going further,
And Dean's group was... "Dean, what are you doing over there?... Did you ask me if you could get out of your seat?... Nobody cares that your back is hurting right now, Dean!"
What step was after monitoring?