Not ODEs – The Variation of Parameters Method

VersatilityWhen one attempts to define versatile, something like “able to be adapted to many functions or activities” may come to mind. As a classroom teacher (and now working K12) this idea comes even more naturally as I know have the opportunity to see how successful and adaptable an idea or lesson may be with learners of various levels and ages.

Recently, I had such an opportunity to try marry a couple of ideas I have brewing in the back of my mind with four classes, two 6th grade classes and two 8th grade classes. The original idea stems from two separate sources a random search and from Mark Driscoll’s amazing book Fostering Algebraic Thinking in which the heart of the lesson follows. The other idea is my love of working a 3 Act lesson into anything that I can teach. So I will discuss both lessons and what I learned in what follows, but I’ll first set the stage for the lessons.

This year, I have had the pleasure of working a lot in the sixth grade, a grade I felt comfortable with my background (as I have described before) doesn’t lend itself naturally to the younger learners. I am working on getting some practice with many folks favorite age group, the infamous eighth graders. So I had this opportunity lineup, within a day of each other, I had the time to teach the 6th graders on an introduction on Equations and Expressions, and an introduction for the 8th graders on slope.

Enter my normal teacher brain, whose thoughts in no particular order went something like this: 1. I don’t have much time to prepare two very different lessons, and make them meaningful, 2. Students need hands on connection, an anchor, that we will be able to refer to throughout the unit,Teacher Brain

3. Students like to build things and I like to facilitate through questions and setup versus talking to myself in a lecture format, 4. Using 3 Act format gets a lot of buy in early, prompts students’ thinking in the direction I need, and makes for a natural direction to move forward, 5. What do I have around me that would allow me to use the same format, but change the questions (or purpose) of the lesson with the same manipulatives?….

Enter the work setup by Driscoll, he posed a Matchstick problem that extends, this would work perfectly, and the dollar store has popsicles by the hundreds for, well….

My manipulatives clearly in hand, my thinking went to the idea that I wanted the sixth graders to build the triangles, as shown in the video, counting the number of triangles formed and relating that to the number of sticks required to make the triangles. There is a natural differentiation here too, advanced learners may recognize that there are more triangles depending on how they build their triangles together. The relationships that start to form here, are well beyond the scope of this lesson, but what a great learning experience for those kids ready to go beyond. As the students build the triangles, they record the data, and then the extension question(s) go like: 1. How many sticks would we need if we were building 100, 1000, or 1,000,000 triangles? 2. How many triangles could we build if we have 100, 1000, or 1,000,000 sticks?

Good ThingsThe math nerd in me love these questions because it extends to the idea and in light of actually building it, this is a natural way for me to introduce the idea of generalizing, enter expressions and equations. In addition, this also shows the doing and undoing process of the relationship between the number of triangles and the number of sticks, an important piece that we sometime overlook.

Using a similar idea in eighth grade, the idea there was to tackle the misconception that these learners had in regards to scaling, the idea that the ratio of two similar triangles is the same though the lengths of the respective sides may be much different. The learning that I wanted the students to see was that as they build the right triangles, make measurements (by counting sticks) and comparing the ratios of “rise” to “run” they would discover that the ratio was the same. I was expecting some errors, like how the students build and counted the right triangles, which I was hoping would be a great “teachable moment.” Students would get a lot of collaboration and would work with a small data set of 9 other ratios calculated from other student groups, reinforcing their understanding.

What was used is shared is here.

What I learned through this experience is the importance of modeling and being very clear with what was required of the learners. After reflecting on these lessons for a while, and thanks to invaluable input from #MTBoS and #mathchat folks, the lessons need some tweaks here and there. Specifically, the amount of scaffolding and teacher direction are not calibrated for optimum performance. Having giving them a test run, I like the core ideas and I think the experience was very valid, but I know some major revamping I will be doing for next year.

What did go well was the conversation and engagement when students were actually building and working with the manipulatives. I am forever amazed that something as simple as small, wooden sticks could get even the most ardent stalwart working through the lesson. The classroom was buzzing, that’s the moment I love to hear, kids actively involved with their learning. I will update this and other thoughts as I move forward, getting back to blogging and catching up has been a hard road to toll.

Happy Mathing!

A Hard Act to Follow

After attending one of the most powerful mathematics professional developments with Matt Vaudrey and John Stevens as the chefs in La Cucina Matematica they shared a wonderful site that fits right up my teaching alley, Estimation 180. The site offers a day-to-day estimation scheme, which helps students increase their mathematical thinking and sharpens their estimating skills. I love the idea getting students to use their instinct and to reason through their thinking on figuring out these problems. I always find it amazing how even the most reluctant learner will throw their hat in the ring with a compelling prompt, taking a risk when there is no wrong or right answer. The learners are always so passionate about their reasoning and it sparks some great on topic discussions.

That aside, Estimation 180 offers another nugget of gold, which are pre-made lessons, based on the 3 Act strategy implemented by Dan Meyer (to the best of my knowledge he is the first person I know to coin this term). On the teaching front, I often adopt this model, I find it is a natural way to increase students buy in and a fun way to facilitate learning. My background is based heavily in secondary and college curriculum, so I have spent this year getting my feet wet with the elementary grades. Of the number of lessons I have done this year in lower grades, I was extremely excited to see an awesome lesson prepared at Estimation 180, ready to go, which I saw as the second graders I was planning the lesson for are just entering addends with three digit numbers. The lesson, Bester Egg Tower, was a perfect starting point, I built a presentation around it, and went to try out the lesson. A preview of the materials I downloaded and my presentation are here.

The lesson went off very well, the kids were so excited, we had them sharing in groups of three, building the trays of eggs with ice cube trays, and Styrofoam colored eggs (colored for each group). The pace was pretty quick, as I tend to be very high energy in the classroom (a behavior management and buy in tactic as much as my personality), and I like to keep things moving according to my quick formative assessments, I adjust as necessary.

Although I felt great about the lesson, there were many things I learned that I will be more aware of with more practice. First of all, when students wrote what they noticed and what they wondered, I was surprised that about half the class didn’t know what to write. As a teacher, I am concerned because the 6 year-olds in my life are suffering from the same condition, they are not as excited about school, and don’t ask nearly as many questions as I would imagine they should be at this age. If kids don’t wonder about anything, then learning doesn’t follow as readily. So front loading what it is to wonder, and having question stems (and sentence frames) will be great additions to this lesson.

That being said, the next thing I was surprised to have students share out as their questions (which are recorded in the notepad document here) were either about why the chickens moved so fast or why the teacher was stacking the eggs. I was anticipating the latter, but reflecting on the lesson the former makes a lot of sense, as kids wouldn’t be used to seeing chickens moving in high speeds….leave it to kids to point out the obvious and make the teacher wonder.

The biggest moment for me came, when I asked the kids about the ten frame the carton of eggs, I asked the kids to compare and contrast the two. One student shared he contrasted that one was even and one was odd.

Ten Frame vs Egg Carton

As you can imagine, my teacher brain was thinking but they are both even, and I was thinking I needed to ask for clarification. My normal inclination may have been to ‘jump in’ to ‘save this poor kid’, but I didn’t and I asked for clarification. The student said that the ten frame as five in a column and the egg carton had six, so one is odd and one is even (granted those weren’t his exact words, but that’s what he meant). Do I really need to say more here? Wow, I know.

One adjustment I would make for next time is to have a picture of just one row of the stack of eggs, I think having the kids get to the point where they are understanding just one row (3 dozen eggs, so 36 eggs in a row) is a great starting point to move forward. We finished on getting kids to the “doubles” plus 1 number of 12, and we talked about the strategies the kids used to get their result.

All in all, it was a tremendous amount of fun, and I can’t wait to do this lesson, in another second grade class. I can also tell the students loved the time we spent, because it took literally 3 full class hugs and several high fives for me to be able to get out the door. Two weeks later, I see a group of those kids on campus, they run over to me, hug me with unison cries of “When are you coming back….” Needless to say, second graders may be my new favorite grade level, at least until I teach Kindergarten again.

Here are some students’ work from the day, I tried to capture the range of responses that the kids produced, the gradient of abilities.

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A huge thanks to the chefs at  La Cucina Matematica and for the folks at  Estimation 180 for making these great resources available and for inspiring some alternative fun. I have loved the 3 Acts theme for a while, but seeing that it works with second grade was amazing. The last thing I forgot to mention is that the kids had a very hard time estimating how many eggs they thought would be in the tower, this shows that the kids may not have a strong sense of the size and quantity of these numbers. As they are just now learning about 3 digit addends, I think this reflects in their difficulties with estimation, which is another reason why the estimation practice may be helpful – especially in terms of thinking about the reasonableness of their answers in their own work (some SMPs) in the mix as well.