How Many?

I teach an Algebra 1 class that meets every day (our block schedule means classes usually only meet every other day) so I needed twice as many openers for that class. The reason they meet every day is due to substantial learning disabilities, and the majority of the class has a language based learning disability on top of challenges with math. Based on all of this I decided to start each of their bonus classes with the simple question: "How many?" followed by an image. I started with the idea of doing number talks but was quickly captivated by Christopher's simple question (blog post, book coming soon?!) The best part of this activity was the blend of math and language. Wait, maybe the best part was the low entry. Or perhaps it was the opportunity to share all sorts of cool things that helped build foundational skills (ex: subitizing and multiplicative thinking) that these students struggled with? Okay there were too many best parts!

The first week kids declared this baby work. They were in high school, they didn't need to practice counting! They definitely said this because they didn't see the value, but also because it was hard. Many days I would see kids standing at the board counting individual items. My co-teacher and I encouraged students to come up with strategies so they didn't have to count every item. We shared strategies such as counting rows and columns, looking for repetition and grouping nice numbers. Here's one we did in September:


First kids counted each item (green numbers). Then kids recognized that two halves made a whole and so they counted the number of wholes (red numbers over the image). I remember discussing "what do we call this?" since it wasn't an object we recognized (lime? grape fruit?) but it was important to differentiate between the 8 and the 16. We decided it was definitely a fruit and so we were able to apply that word - language in math class! Then someone (probably me since it was the beginning of the year but not necessarily) pointed out that there were four rows and four columns. One of my students would consistently yell out "array!" when we were discussing rows and columns, I love that he had that word in his vocabulary. We connected counting pieces one by one to multiplying rows by columns. Importantly, I did not belittle anyone for counting one by one. I was happy to hear any and all quantities and strategies. Early in the year I did not get as many contributions, but as students felt more comfortable they stopped complaining and started getting creative.

Just last month we did another image of sliced fruit:

The list on the right is the initial brainstorm (the class is down to six students so one idea from each of them). Then we wanted to get more specific - how big are those two slices? It was an incredibly happy coincidence that we were practicing exponential functions that day, I couldn't have planned that better if I tried! If I hadn't been so focused on the fractions I'm sure they would have added "two seeds" and "one brown background" to their list. Naming the background became the default if someone had already said what they wanted to share. I love that this activity encouraged them to think creatively and pay attention to details.

One day in December we had a particularly vocabulary-rich discussion:

Not everyone agreed when we got the first round of ideas, but I wrote them all down anyway. After gathering ideas without comment, we began our discussion. In addition to a discussion of the differences between a square and a rhombus, someone also coined the term semi-octagon! I was so excited to see kids who generally lack confidence in math and language to be creating their own math vocabulary. Also? Their definition of diamond is the basic pentagon tile because it looks like a cut gemstone diamond. No one called the rotated square a diamond!

I really enjoyed doing this activity every other day. Students had three minutes from the time the bell rang to record everything they noticed. I asked each student to contribute one answer to the question "How many?" before I allowed any students to contribute additional quantities or comment on the other answers. I had fun choosing photos from this number talk site, things I'd seen on twitter (#howmany or #unitchat) and even photos I took myself. All the images I used this year are in a dropbox folder. I look forward to playing again next year. Perhaps I will encourage them to bring in photos or we will go on a photo scavenger hunt someday together!