Before we get into today's lesson, which was my favorite lesson all week, can I rant about something for a minute?
Why in the world would someone try to teach about square roots without talking about squares? I'm not referring to "squares" as in raising a number to the second power. I'm talking about "squares" as in those pointy shapes with all the sides that match each other.
You will notice that I didn't even introduce the notation or the term "square root" until my lesson on Tuesday. This was intentional. I don't want my students to get hung up on this new vocab term or this symbol that kind of looks like a long division sign. No, I just want them thinking about how to find the length of one side of a square that has an area of 73.
So already, on day 1 of this unit, I have students who are accurately estimating square roots. They just don't know that they're doing it yet. They think they're finding the missing side of a square. Once they know that concept and build a strategy to solve that sort of problem, it's not a major shift to tell students "Ok, that thing you've been doing, where you un-square a number? That's called a square root. And it looks like this check-mark-with-a-bar-next-to-it symbol you've noticed on your calculator."
Conversely, if you start this unit by projecting a slide entitled "Square Roots" and introduce all the formal vocab and symbols from the start, then students don't have anything to ground their understanding of the operation. You are asking students to use a new symbol to enact a new operation that they've never tried before. And forget about asking them to find the missing side of a square - that would be a seemingly-impossible task to those students.
Introduce the challenge first. Once students understand the challenge, then provide the notation and the vocabulary. Don't dump them both into students' laps at the same time.
Ok, rant over.
Warm-Up:
None. We are diving into the main part of the lesson, something I have been dying to try since I saw Andrew Stadel post about it back in August.
Activity: Movable Number Line
My number line is huuuuge. Great for class, hard to capture in a picture.
Ok, so this might have been the most awesome thing I've done all year. I made a loooooong number line out of a string that stretches almost all the way from my window to my door. I put the numbers 0 and 10 on either side of the number line.
I told kids that I would be showing the whole class a number and then calling on one person to place that number on the number line. Nobody else could talk, but we would take a poll after the student sat down:
- Thumbs Up: Perfect!
- Thumbs Sideways: Your answer is in the correct order but needs to slide either right or left
- Thumbs Down: Your answer is out of order
Then I held up the number 4. I chose 4 on purpose because it's incredibly familiar to students and yet a bit tricky to place. It's closer to 0 than to 10, but how much closer? Not to mention, with such a long number line, it's going to be hard to get the placement exactly right. I was expecting a lot of "Thumbs Sideways" on this first number, and I wasn't disappointed.
By the way, I use popsicle sticks with names, often called equity sticks, to choose my participants in class for this activity. I know that some teachers feel that equity sticks cause students anxiety, but I think they are worth it for this sort of activity. First of all, they strongly improve engagement. Everyone knows that they could be called up to place the next number, so they are paying attention to each number I present. Secondly, this activity does not have a clear right-or-wrong answer. In fact, I usually end up polling the class and sliding each answer ever-so-slightly in one direction or the other. Since every answer gets improved or amended, the pressure to be exactly right is lowered. Everyone is just making their best guess.
But back to the game. The first student has just placed the number 4, and we have to decide - is it perfect, or should it slide right or left? In my first class I had a student place 4 verrry close to 0. This is a great opportunity to ask students to critique the reasoning of others in a respectful way. I had lots of great comments from students, such as "If 4 was that close to 0, you wouldn't have room for 1, 2, and 3, and you'd have way too much room for the numbers bigger than 4." As the lesson went on, the justifications became more precise.
My next choice, root(49), I chose because 7 is exactly halfway between 4 and 10, and I want to see fi my students will pick up on that.
My third choice, root(20), is where the real fun starts. Now students have to use yesterday's skill of estimating square roots without any benchmarks. Where should root(20) go? Where is 5 on this number line? Where is 6? These are all questions that my students are silently asking themselves as I hold up the card with root(20) on it. At least, it sure seems that way. My students are rapt. They can't wait to find out if their popsicle stick will get pulled.
From here, my sequence was 6, root(93), root(40), root 4, root(14), root(-4). I'm sure I could have sequenced them better, but I'm not sure how. How would you sequence this activity? Let me know in the comments.
Are these placed appropriately? If you could slide one card, which one would you slide, and where?
Are these placed appropriately? If you could slide one card, which one would you slide, and where?
Anyway, I like this lesson for a few reasons:
- Student engagement was through the roof. I felt like everyone was with me in a way that almost never happens. Kids were having fun! I even had a group of students ask me in study hall the next day if they could play "the number line game" again
- Because the number line is huge, almost nobody placed their card in exactly the right spot on the first try. We always had to shift someone's card a little to the right or the left. Conversely, almost nobody put their card in the wrong order. So all the students were participating in an activity where nobody got an answer totally wrong, but nobody got an answer totally right. The pressure that students feel when coming to the front of the class was lessened in this case. Sometimes, students had a legitimate difference of opinion and we had to agree to disagree since this activity has an inherent amount of imprecision. But that's great! I'd rather my students be disagreeing and debating as long as they back up their ideas with some evidence
- By placing radicals on a number line, students are beginning to interact with radicals as objects that have a specific value. It's not just a problem to be solved. Root(14) is a number that is somewhere between 3 and 4. That approach to radicals will be useful in Algebra 1.
- I threw root(-4) in as a challenge because I thought it would spark a great debate. I also wanted to add a little bit of new information into the day's material. Yes, it's a trick question. Maybe it will feel more memorable to students because they spent 3 minutes arguing over the location of root(-4) before I admitted that there is no place for this answer on the number line. At least, not this number line.
This took about 30 minutes, which sounds crazy, but it was my first time trying a number line, and we had great discussions between every number.
Here’s a good extension. Start with numbers other than 0 and 10. Here I’ve started with 2 and 8. Where would you place root(36)? Root(90)? Root(3)?
Once we had placed all the numbers on the number line, I pulled more popsicle sticks and got students to sort the numbers on the number line into the categories "Rational" and "Irrational." This is my attempt to cement the idea of irrational numbers within our existing activity. Kids could sort the numbers perfectly, but I still don't know if they truly understand what irrational numbers are, and how they actually differ from rational numbers. Something to think about before next year.
Lastly, I got students to create a foldable for their $1 Textbooks to help them classify triangles by their sides and angles. This is vocab that they should know, but it's going to be vital for tomorrow's activity, so it's worth the time investment to make a good foldable.
Note to self: draw triangles in the foldable so kids don't draw their own "obtuse" triangles that are clearly acute.
Homework
IXL Activity on estimating square roots. I am lagging in my implementation of lagging homework. I'll get better at this.
Tomorrow's Goal
Classify triangles by their sides and angles, maybe even discover the Pythagorean Theorem?