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Lessons From the Classroom
Des-man
Not only was this so great to reinforce slope, y-intercept, and all the coefficients, it also allowed us to talk about domain and range.
What I heard around the room (that I can remember):
Oh, I get this now! I see what changing this number does!
Oops, I made his face too wide!
His smile is crooked. But I think I'll leave it because he looks cool that way.
Ha!! I see my mistake, I said x had to be greater than 4 but less than 2. Silly me.
I want the eyes to be oval shaped though. My plan is to make 2 parabolas opening into each other.
Can we work on this in 6th period too?
When I saw two students whose graphs were circles for faces, I knew they'd copied these from Desmos gallery as we haven't — and won't — learn circle equations in Algebra 1.
I reminded them of the minimum requirements, but I told the class that they may copy equations and tinker with them to add other features, such as hair and whiskers. (I actually said "whiskers," and Lexi had to tell me, "Whiskers? On a man? You mean beard or mustache?") People don't have whiskers? Good to know.
I made my guys' eyes elliptical and tweeted it, the good folks at Desmos responded.
So cool!
I think we got a lot of mileage from this activity. It's a good sign when teacher instruction is minimal and student engagement and discussion are high.
Just in case you missed the Grand Opening of Daily Desmos about 3 weeks ago, brought to us by Michael Fenton, inspired by Dan Anderson.
Parallel and Perpendicular Lines
Bleh. I don't know of a clever way to teach parallel and perpendicular lines. I'll try asking kids questions and maybe their answers will guide me and make me look smart until 2:50 p.m.
Me: Using your arms, show me parallel lines.
They do. One boy clasps his hands together like he's praying.
M: What do you know about parallel lines?
Students: They don't touch. They don't intersect. They don't meet.
M: Okay. Please draw this line in your journal. We're all drawing the same line, so make sure yours also passes through the points (0,-2) and (4,0)... Now, draw another line that runs parallel to the first.
I walk around the room and see that most of them have drawn pretty good "parallel" lines, not perfect.
M: How are you going to convince me that you drew parallel lines?
No one is talking. One kid has his hand up, but he's repeating algebra this year, so I ignore him. Poor kid is used to me ignoring him. His mother is adorable — and a friend of mine — and she ignores him sometimes too. Finally...
S: Check the lines?
M: What do you mean? How?
S: Extend the lines to see if they'll meet.
M: Extend until...? China? Where are we going?
S: The lines never meet!
M: So how do I check that? I need a way to check for parallelity. You say they never meet. Never, like infinitely-never-kind-of-never? I don't know how to check these infinitely long lines.
I'm stalling here. Say something smart, kids, so your clueless teacher can learn. I put my arms out, moving them together across my head like I'm doing a rain dance.
S: Measure the angle. Find the slope. Use a ruler.
M: Okay, let's find the slope because we know how to do this! So, find the slope of both lines. Don't talk to me again until you're done.
Jonathan: You can't die, Mrs. Nguyen. The Ducks would have one less fan.
M: I said don't talk to me. What are you talking about? Who's gonna die?
J: Remember you said that at the beginning of the year that if the Ducks lost a game, you'd die?
M: I do love you, Jonathan. Now, be quiet. Find slopes.
Jonathan draws the Duck mascot for me often, on his homework, corner of his test, on the big whiteboard. He also interjects random comments regularly.
The kids start talking, their heads nodding one by one, and they make this bold claim that parallel lines have the same slope. They say, "Let's do it again!" So we do it again. And again. After drawing three sets of parallel lines and finding their slopes, they tell me that parallel lines have the same slope. Hmmm.
They seem happy and gullible at this point. Best time to take advantage of them.
M: Show me your arms again. Wave them. Twist your hands like this. Do this... then this...
These kids kill me. So damn funny. They are like monkeys, doing everything I'm asking.
M: Okay, using your arms, show me perpendicular lines.
Yikes!!!!! Maybe, just maybe a fourth them show me arms crossed at right angles. The rest of them have no arm muscles.
M: What are perpendicular lines?
S: They cross. They intersect. They cross at right angles. Like an X.
So I put my arms up, crossing them like an X, but intentionally nowhere near ninety degrees.
M: Like this?
S: More straight. Like this. Yeah, that's it!
M: I heard right angles. Is that true?
S: Yes.
M: Draw me another line, please. This one going through (-2,0) and (2,1). Then, I need you to draw, as best as you can, a line running perpendicular to it.
Holy cow. Even the kids who say the words "right angles" and "ninety degrees" draw nothing near that. But one student, after I look at five others', does have a good sketch, so I show it to the class.
M: C'mon now. Please try again. Then what should you do after that?
S: Find their slopes!
Here's one student's attempt at drawing perpendicular lines. But you can see her second attempt is better.
(The first line has a slope of 1/4.) Unfortunately the kids who do have a decent looking "perpendicular" line find the slope to be 4 when it should be -4. Again, we do a total of three sets of these. They describe the two slopes: opposites, inverses, flip around, reciprocal, upside down. I don't really care what they call the relationship right now because when I give them a line having a slope of 3/7, they know the slope of a line perpendicular to it is -7/3. They know when one is -8, the other is 1/8.
Everyone draws the same original line together. But instead of trying to draw the perpendicular line to this (like we did on that first round) — a difficult task to perfect via eyeballing — and then find the slopes of both, I ask them to find the slope of the original line first, then find the perpendicular slope, then use this fact to draw in the second line. They think it's cool that they can start the second line using any point on the coordinate plane.
I'm saved by the bell. And I think know this is still better than my just telling them.
Barbie Bungee
I bought a 1-lb bag of rubber bands for my Barbie Bungee activity today.
We should have done Barbie Bungee earlier in the year while learning linear equations and lines of best fit, but better late than never. My collection of Barbies could use some refurbishing work — they get used so much each year for this activity and for our lesson on proportions.
OBJECTIVE: Create a bungee line for Barbie to allow her the most thrilling, yet SAFE, fall from a height of 3 meters.
I randomly assigned students to groups of three. Each group got their own Barbie and 7 new same-size rubber bands. My instructions:
Measure Barbie's height. Record this as rubber band length of 0.
Connect 2 rubber bands with a slip knot.
Wrap one of the two rubber bands tightly around Barbie's ankles.
Drop Barbie, holding the rubber band level with the meter stick.
Record Barbie's fall using the lowest point her head reaches in centimeters. This number is your rubber band length of 1. (The rubber band around her ankle is not counted in the length of the line.)
Add another rubber band, drop Barbie, and record. Do this for a total of six rubber band lengths.
Measuring: They quickly went to work. (We're lucky to have good weather here pretty much year-round because I need some students to be outside whenever we do projects like this — they need to spread out to do the work.)
Graphing: The groups graphed rubber band lengths vs. distance of fall. Then they drew in the line of best fit. From this line, students predicted the number of rubber bands for Barbie's bungee line that would be thrilling enough for her 3-meter jump without cracking her head open!
Once groups made their prediction — written on their papers and on the board — they may not change it. I had taped a small ruler to that rod to mark the 3-meter height. I can't have students on the ladder, so that's me getting ready to drop a Barbie. (The numbers on the left were their initial guesses before doing anything else.)
This was a blast!! I had two kids lying on the ground with meter sticks to watch and measure Barbie's initial plunge; they were our judges. We clearly had a winning jump when one group's Barbie came within 2 cm of the floor. They asked if they could get a second chance, so all 10 dolls had another jump after adjusting the number of rubber bands on the bungee line.
What I heard around the room
I noticed the centimeters went up by 10 on average.
Her height is the y-intercept.
Nine rubber bands is approximately 100 cm, so we need...
Stop stretching the rubber bands, you're gonna ruin our estimate!
Each meter stick is 98 cm. (His two teammates did not say anything when they heard this!)
I have to re-do our graph. I stuck it too close to the top, and the line of best fit has nowhere to go.
You're not supposed to connect the dots!
This was so much fun!
Oh, I didn't realize how stretchy the rubber bands got. (To which another student said, "Hello, it's rubber."
Ken is heavier [than Barbie]. We forgot this.
Hair centimeters! She was that close!
Notes
Barbies are Barbies. My kids are used to handling the dolls, a few of them have no clothes on (I got many of them on eBay). But I start every lesson with my usual warning that if I see anything remotely suggestive or derogatory done to the dolls, the student would be sent to the office and not be able to return to my class until I spoke with his/her parent. There has never been a problem in the last 8 years. The same applies to the rubber bands.
Staircases and Steepness, Continued
This happened Thursday.
By Friday morning the kids who did "base times height" learned that these numbers didn't match up with the steepness ranking. They said, "That just gives you area."
So I made these sketches, and hopefully the kids understood why finding area wasn't the same as finding steepness.
Those who did not do "base times height" shared what they'd calculated for steepness:
Because almost everyone got the correct steepness ranking the day before, they knew their homework calculations had to match the order, with staircases B and E yielding the same number.
Rapha: My dad helped me. I learn that the steeper it is, the number gets closer to zero.
Jocelyn: I measured one of them wrong. Did I just get lucky then?
Ryan: I tried base divided by height first, then I changed my mind to height divided by base because it made more sense for the numbers to go up instead of down... if it's gonna get steeper.
Matt: My base minus height did not work!
Arthur: Rapha's kinda the same as Ryan's, except backwards. And if you add her step widths, you get the base.
I was on cloud nine. Then I finally said the word slope, but I never said "rise over run." We ended class with this video "Tutorial — Measure Slope Steepness" by Bruce Tremper, Director of the Utah Avalanche Center.
We had a busy but fun Friday as it was also our 3rd quarter PFO-funded Einstein pizza party. Luckily my prep period is right before lunch to allow me to leave campus to get the 8 pizzas at Costco. The weather was gorgeous and we had 10 minutes left of geometry, so I took the kids outside to play Fizz-Buzz. If you've never played Fizz-Buzz with your kids, then don't start unless you want them to constantly pester you to play the game, even when there's only a minute left of class.
Staircase and Steepness
AHHHH, I love how this lesson has turned out so far!!
Today was our 3rd day of State testing, and because I have 8th graders for homeroom, we still have three more days of testing next week. Ick.
I didn't want to cram/review again with my 6th graders today, so I thought of doing a lesson on "steepness" (adapted from Swan and Ridgway).
I began the lesson by showing a bunch of images, like these, and asked,
What do you see?
Kids' responses: going down, going up, all using their legs, exercise, at an angle.
Here’s the lesson. I purposely wrote it as if it were unfinished because I didn't want it to end on my terms. I wanted the kids' conversations and discoveries to guide me to closure, if any.
Question 1: Without measuring the staircases, put them in order of "steepness," starting with the shape with the least "steepness."
Question 2: Explain how you came up with your ranking in #1. Because you were asked NOT to measure, what "tools" or strategies did you use to make your decision?
You can see from these photos how they thought of "steepness."
Question 3: Now discuss your ranking in #1 with another classmate. Are you going to change your ranking? If so, please indicate your new ranking.
Question 4: Now discuss your ranking in #3 with a different classmate. Are you going to change your ranking? If so, please indicate your new ranking.
Six students made no change to their original ranking after talking with two other classmates, 16 made one change, and 9 made two changes.
I was already so happy to see how the activity was coming along. I didn't realize how much more I was about to learn from Question 5 and beyond.
Question 5: You may now measure the staircases with whatever tool(s) you need. Use the space below to keep track of your measurements, calculations, and notes.
A few kids asked for a protractor, most used a ruler. Most of those who used a ruler measured this length.
Me: Now that you have these measured, what do the numbers mean
Matthew: (Silent, mumbling...)
Me, pointing to staircase F: I see this has the longest length. Was this your steepest shape?
Matthew: Oh no. D was the steepest.
Me: Okay. Shape C has the shortest length. What does that mean in terms of steepness?
Matthew: I don't think these numbers are right.
I went over to Troi's desk, she too had measured the same lengths as Matthew did.
Troi: These numbers didn't do anything for me.
Me: What makes you say that?
Troi: Well, the staircases are all different sizes, you'd have to make them all the same to compare them.
She then measured the rise of each step. I left her to do so.
Rapha: I'm measuring the height, but it depends on the width too.
She didn't do any more with the two sets of numbers.
Zoe: Can you show me how to use a protractor?
Me: Sure. Which angle do you want to measure?
She pointed to the middle of the staircase. I worked with her for a little while.
I checked on a small group of boys who seemed to be using the protractors correctly.
Mike: We agree on the rest of the ranking. We're just not sure about B and E. They're like one degree apart.
Ryan, holding up two different brands of protractors: These aren't even measuring the same.
Me: Hmmm. Tools aren't perfect, are they?
I was watching the time; we had 20 minutes left of our 2-period class. I asked them for a "final answer" after whatever measurements they'd done, "Put this ranking at the bottom of your worksheet."
By then the kids who used a ruler had abandoned the tool for the protractor. Make no mistake, these kids were pretty confident that if Ryan and Mike were measuring the angles, then they ought to be doing the same. More importantly, they noticed that the angle measurements correlated with their steepness rankings.
I asked Miles first — just because he sat front row, right side — for his ranking: D A B E C F. I asked if anyone else had the exact same as Miles'. Twenty hands went up. I asked for Moses' ranking: D A E B C F. Ten hands went up for this ranking. That only left Sierra with a different ranking. (Sierra was one of six who never changed her original decision.) Because B and E do have the same steepness, all 30 of 31 kids were correct.
Now what? Somehow ending the lesson here seemed weird, even though we had a lot of good conversations. They used angles to figure out steepness. I hadn't planned this! What about the ruler?!
So I said, What if you didn't have a protractor? What if you only had a ruler? What would you measure instead?
I then defined lengths on the staircase that could be measured with a ruler so we could all speak the same language about what was measured.
I said, It was great that you figured out steepness using a protractor. But now I want you to figure out how to find steepness using a ruler. Which of these lengths would you measure? Do you need to measure more than one? And if you measured more than one, what would you do with the two/three numbers you have?
They began measuring ferociously, calculators in hands. (Yes, we use calculators all the time!) I didn't see anyone measuring the slant. I got this much from them by walking around and asking:
Base minus height.
Base divided by height.
Base times height.
Height divided by base.
I ended with, For homework, please finish measuring and calculating for all six staircases. Do your calculations support the ranking?
Troi walked up to my desk when most students had already left, Can I change? I already knew... I did base times height. That didn't work.
#peedinmypantshappy