# Number Sense – Dots and Powers of Ten

Number Talk Today: How many dots in this image?

Huge variety of answers. Three observations:

• No student (or at least no student who wanted to share) saw it as a 10×10 box with 4×3 and 5×4 pieces cut out. Not a big deal, but I thought that was interesting.
• Many students saw it a a series of rows or columns, rather than overlapping/adjacent rectangles. I think this speaks to the lack of number sense I see — they don’t have fluency with rectangular representations of multiplication. Still plenty of students who did, but there was a pretty sharp divide between the two groups in all of my classes.
• My students love dot patterns. It’s awesome to see them all counting the heights of the rows and columns, and in some cases leaning up out of their chairs to make sure they count right. I would say classes averaged about 2/3 of students who wanted to share their approach. Also, classes ranged from 4 – 8 different answers at the start (I’ve been starting by taking every answer anyone has, with no judgment given on the quality of these answers). Again, speaks to their lack of fluency with rectangular representations of multiplication, and in 8th grade!

I’ve thought more about the idea of longitudinal structure to number talks. I’ve only been doing them for a few weeks, but they’re my favorite part of class. My students in general don’t love doing what I ask them to do, but engagement is high during number talks and almost everyone has volunteered to speak at one point or another, including the vast majority of my lowest-skilled students.

Anyway, I’ve been thinking a lot about next year. Between losing a few classes as students take finals and a 4-day trip to Washington D.C. I don’t have much time left with my current crew, but I’m excited to figure out how to make number talks even more awesome for next year.

Here’s my idea of the day, which I’m sure will change radically by the time I implement it.

1. Students will have a weekly sheet that they keep with them to track number talks.
2. Each week there will be a theme to number talks — multiplication, division, dot patterns, spatial sense, “does this answer make sense”, estimation, and more.
3. Students will still do the math mentally, share all answers, then share strategies, but while sharing strategies, students will have the chance to scribe strategies they like.
4. At the end of each number talk, students will write the strategy they liked best, or, if they liked their own best, why they preferred it to others.
5. At the end of the week, students will have an additional few minutes to write what they learned from the number talks that week, and note any strategies that were new to them that they will use in the future.

Finally, we we’ll be talking about exponents and scientific notation today. I opened with this oldie but goodie from 1977 on the powers of ten and the universe. I stopped it after it reach it’s outer limit, and tomorrow we will watch it zoom all the way into a proton. The questions I got were interesting — mostly around the speed of light and being awestruck at the size of the universe.