One Week in Mr. Haines’s Math Class – Wednesday

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

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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.

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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.

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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?

 

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One Week in Mr. Haines’s Math Class – Tuesday

Tuesday

Warm-up:

I give my students a worksheet with three problem types:

  • Find the area of a square
  • Find the side length of a square whose area is a perfect square such as 25, 36, 49, etc,
  • Estimate the side length of a square whose area is given but not a perfect square.

I like recapping every problem type from yesterday since it reminds students of our progression on Monday. It also provides an easier on-ramp for students who were absent on Monday.

The only difference today is that students are no longer allowed to use calculators. I want them developing their own strategies to estimate square roots.

Activity:

Once we’ve reviewed the questions from the warm-up, I ask students to get out their $1 Textbooks. This is my name for my class’s interactive notebooks. I was skeptical of the time commitment required to do interactive notebooks well, but this year I buckled down and got it started. I realized that I don’t use a textbook in my class, so it’s my responsibility as a teacher to provide a resource to students so they can review the vocabulary and concepts that they are responsible for in my class. We write in our $1 Textbooks about 2-3 days a week, and rarely more than a single slide or two of notes. But at the end of the semester, my students have a marvelous study guide for their final exam.

We write down some review material about squaring numbers, and then I give a short speech about square roots and what that term means to me. I say that trees come in all shapes and sizes, but the thing that fundamentally determines their size is their roots. Similarly, squares come in all sizes, but their size is determined by their “roots,” which is the size of the base. If you want to draw a square with an area of 25 square inches, you need a square with a root of 5 inches. This is a bit of a silly analogy, but it provides some hook for this new vocab.

Once we write down our notes on squares and roots, I ask students to turn to the back of their warm-up sheet. Along the top you’ll see that I ask students to calculate the squares of the numbers from 1-10 and then find or estimate a bunch of square roots. I provide the top line because I know that some students will not immediately come up with their own strategy for estimating square roots.

Squares and Roots Worksheet

Almost every student can find the square roots of perfect squares because the first row of the worksheet acts as an answer key. So if a student is having trouble with estimating roots, I ask them how to find the square root of 81. Then I ask why they are having trouble with the square root of, for example, 38. When they say something like “38 isn’t in this list of numbers,” I ask them to find the square number that is closest to 38 but a bit smaller. Then we find the closest square number that is close to 38 but a bit too big. We use those numbers to make our estimate, which should be between 6 and 7.

Once most students have completed the worksheet, we check the work as a class. I typically project my own worksheet using a document camera. I don’t love that I have to sit in the back of the room, as I can’t read my students’ facial expressions. But I like being able to manipulate the worksheet itself. That way my students can easily scan my sheet and their own to check their answers. Also, while I am writing estimates on the worksheet, I can find the square root using my calculator and show that on the document camera as well.

By this point, most students have noticed that all these square roots of numbers like 47 and 6 keep going to the end of the calculator. So I go into a quick math lesson about rational numbers, and how ancient Greeks thought that they were the only type of number that existed until they tried to find the area of a circle. I know this history lesson is rough (and not exactly accurate), but it taps into my students’ prior knowledge of pi and helps them to believe that the square roots of certain numbers are  also irrational.

Then it’s time to write down some notes about rational and irrational numbers in our $1 Textbooks before the bell! Not the best time to introduce new vocab, I know. I’ll have to remedy that tomorrow.

Homework

IXL work on finding the square roots of perfect squares. My goal is for everyone to say “Mr. Haines, I finished the homework in like 2 minutes!”

Tomorrow’s Goal:

Build fluency with estimates of square roots.

One Week in Mr. Haines’s Math Class – Monday

Sometimes I read about a great lesson or idea, but I am not sure how the teacher strings all these ideas together. So I’m going to blog about an entire week of teaching and explain how I tried to meld together several different types of lessons.

Weekly Goals:

My goals are for students to meet the Common Core standards shown below:

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Explain a proof of the Pythagorean Theorem and its converse.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

That’s a whole lot for one week! To be fair, there is a lot in these standards that I won’t be getting to. Here are my more specific learning goals:

  • Find the square root of perfect squares
  • Estimate the square root of other numbers
  • Understand the difference b/t rational and irrational numbers, and know that many square roots of whole numbers are irrational
  • Classify triangles as acute, right, or obtuse based solely on side lengths using the Pythagorean Theorem
  • Find the missing side of a right triangle using the Pythagorean Theorem

Still, it’s ambitious. And to be honest, I wasn’t sure on Monday if I would even get to the missing side of a triangle. Fortunately, I have 3 days next week to use and apply the Pythagorean theorem before exam prep starts. So if I only get through classifying triangles, I’m happy.

Monday

Warm-up: Number talk about the problem 14*8.

This is the only day this week that I did a warm-up that was not connected to the material. I wish I had done more warm-ups, but unfortunately as the semester exam approaches I get nervous about running out of time. My favorite method a student used was to multiply 7*8 to get 56 twice, then add 56+56=112.

Activity:

I handed out this worksheet. The front asks students to find the area of squares. I know that geometry is my students’ weakest topic in our standards, so I try to start at the very beginning. Some students didn’t know or remember how to find the area of a square. Is this shocking to me as an 8th grade teacher? Definitely. But I just walked around the room with blank graph paper. If a student couldn’t find the area of the first few squares, I asked the students to draw each square on graph paper and count the number of boxes inside each one. Every student pretty quickly realized that they could multiply the base of the square by its height.

Intro Worksheet

The bottom half of the worksheet I included non-integer side lengths to prime the students for the possibility that squares can have side lengths other than whole numbers. This will come in handy on the back of the worksheet when they have to estimate square roots. In my A period class, I didn’t allow calculators, but so many students were having trouble multiplying with decimals that I let them use their calculators. After all, fluency with the multiplication algorithm isn’t the point of this lesson. Noticing patterns in square numbers is the point of the lesson.

(My favorite part of this activity was the student who got 36 instead of 1/4 as his answer to problem 6, which had a side length of 1/2 ft. I asked him how he got 36 and he said “Well, I know that half of a foot is 6 inches, so I changed the measurement to inches and got 36 square inches. Am I allowed to do that?” High fives all around.)

Then we moved to the back of the worksheet. On the back, I provide the area of each square and  students have to find the side length. I was prepared to walk around with graph paper again so students could try to draw a square with 36 boxes in it, but nobody seemed to need it. Kids blew through the first four problems.

Intro Worksheet 2

The bottom half of the worksheet was harder. I had kids saying “I can’t find any number that multiplies to 19,” to which I replied “What’s a side length that’s just a little too short?” Kids would usually say that 4 was too short since it gets an area of 16. Then I said “What’s a side length that’s just a bit too long?” and kids would usually answer 5.

So I would say “Hmm. 4 is too short, but 5 is too long. Weird.” and walk away. By this point in the year, the kids are used to me doing stuff like this.

(Note: Initially I used A=20 instead of 19, but I had two students who thought the best estimate was 4.5 because 4*5 =20. Then they put down a side length of 6.7 for A=42 since 6*7=42. These are both decent estimates, but for the wrong reasons. I changed problem 5 to A=19 to avoid this misconception in the future.)

Some kids came up with decent estimates for all four problems on the bottom half, while others got stuck on problem 5. Kids would show me their answers and I would ask them if they got exactly 19. Invariably they would be reeeeally close but not quite. With about ten minutes left in class I said “Competition time. The table who gets the closest answer to problem 5 wins a jolly rancher.” With this added incentive kids were furiously plugging numbers into their calculators and refining their estimates.

I would always get one or two kids in each class who knew how to use the square root button, but they only shared their solution with their table since they didn’t want to jeopardize their chance of getting a Jolly Rancher. 8th graders are teenagers, but they are still kids.

With 5 minutes to go, I collected all their estimates on the board and we tried them out as a class. Usually one table had the “right” answer of 4.358898944, which gets exactly 19 on the calculator. But I told the kids that they were ever so slightly incorrect. In fact, if they had a more precise calculator, they would see that their answer was only correct when rounded. We finished the class talking about the fact that the answer seems to keep going and going into the decimals without stopping…

Homework

Students did IXL homework on exponents, reviewing a concept from last year. I am trying to do lagging homework or “refresh” homework when I can, but I am still not good at it.

Tomorrow’s Goal:

Introduce square root notation, work on estimating square roots.

Resources:

Squares and Roots Introductory Worksheet