teaching math in middle school
learning from three teenagers
picking up dog poo
A Lesson Salvaged
The intended activity did not go well last week.
I gave these instructions:
- Pair up. One person is the Describer, the other the Drawer.
- The Describer is going to look at a Roman mosaic and describe it to the Drawer to draw it.
- Both people may talk to each other, but no hands or any other bodily gestures.
- The Describer may not look at the Drawer's paper as he/she works on the drawing.
I gave each Drawer a blank sheet of paper. I gave each student a ruler and a compass. And I gave each Describer a picture of this.
I also told the Describers to get their Drawers to get the size just right — thus the ruler. I encouraged the Describers to use words like rotate, reflect, symmetry. After a while, I walked around looking at the papers and thought, Holy Cow, if-only-you-could-see-what-your-drawer-is-drawing.
I called for the Drawers' papers after about 30 minutes. The results:
What the...? Oh, let's not forget No. 7. Sketches 4 and 7 took up most of the 21.5 cm x 28 cm paper; the actual mosaic was 8 centimeters in diameter.
Then, I thought, Let me try being the Describer!
The results:
Sketch D was very close, considering we were all hurrying through as the period was ending. Still, this was not my assignment and not a fair comparison when I had a lot more time to look at the shape. I took the stack home and let it mingle with the other piles of papers on my desk.
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Late last night I looked again at the sketches on my desk and thought, I should try to construct this shape in Geometer's Sketchpad (GSP).
It was not an easy task!! I had to really think about each step and kept wondering during my construction if there was a better or more efficient way to do this. I felt great when I was done because it was challenging. No reason to keep this experience to myself, so I asked my geometry kids this morning to construct the Roman mosaic in GSP!
I reminded them that their constructions must pass the "drag test," meaning their Roman mosaic must not collapse when any vertex got dragged about.
Five minutes into the construction, Bobby said, "This makes me crazy!" Yes!
Some works in progress:
Karie was one of the first ones to finish; she talked about her construction:
Alex talked about his construction. You could hear the dismissal bell while he was talking, but no one got up to leave or put away the laptops. I had to tell them!
A lesson salvaged by GSP. A teacher schooled by her 8th graders.
Designing Buildings
I purposely did not give the kids interlocking cubes to complete question 1 — wanted to see if they could count the number of windows and rooms from just the diagrams. A few students had some trouble counting the number of "windows" in Buildings D and F.
In question 2 below, students needed to create three buildings, each with a specified number of rooms and windows. I passed out interlocking cubes and isometric dot papers at this time.
While some kids could create a building, they have a tougher time drawing it on dot paper. Rapha had a suggestion for us.
Matt built this and wanted to know if it could be considered a "building." The class redefined it as "It's a building if it can stand without tipping over."
But before they attempted to draw the buildings, I asked them to give their building to at least one classmate to see if he/she would get the correct number of rooms and windows. They were very engaged...
... to construct the buildings and draw them to complete question 2.
We were into our second day on this. At least half of the kids were now working on questions 3 and 4.
And these were some of the answers for questions 3 and 4.
They were very much into the building and drawing, so I went around to ask them what they thought of this lesson thus far:
- It lets me be creative by drawing and making, see how it comes out.
- It's difficult to count the windows. I check with my classmate and it's wrong!
- This is awesome fun, question 4 is challenging, I'm still trying to figure it out.
- I'm stuck on creating Building 3, it's difficult, but I'm pretty close.
- I like the different ways that I can build.
- I never thought question 3 would make me think so hard like this!
- Drawing is easy for me, it's more difficult to build.
- The 3-D drawing got easier after Rapha helped me.
- It's difficult. It's not working out (referring to creating Building 2), but it's fun to try to find it.
- Very fun, I like it because I can help people with it, show them how to draw perspective.
My conversation with Sam about his paper:
Me: Help me read your answer to question 4.
Sam read it and added: Imagine a gigantic cube building... Well, all those rooms on the inside don't have windows. So, no passing the code!
Me: How big is gigantic? What size?
Sam: Hmmm... 25 cubes on each side?
Me: Okay, like 25 by 25 by 25.
Sam: Let me get a calculator.
Class was ending. I asked the students to finish this for homework. I let a few students take some cubes home because they asked. I look forward to Sam's calculations of rooms and windows for his "gigantic" cube. I look forward to our class discussion tomorrow —
I've never heard both words "fun" and "difficult" to describe a lesson as often as I'd heard it here in this lesson. Yeah, maybe I don't suck at teaching any more.
Why Wait for Calculus?
(I just had my very best lesson yesterday, on a Friday, thank you. I feel almost brilliant right now. And I only feel like this once every 47 years, so please stay and read this post!)
My own kids tell me they will stock up their dorm rooms and apartments with junk food and soda when they move out to make up for these years of deprivation. (And this is supposed to make me feel bad.) So when I intentionally bring home a snack, like this bag of kettle corn, I usually find it empty within 24 hours. But seeing the empty bag made me think of a volume activity that I could do with my 6th graders with all these other bags of Orville Redenbacher's popcorn.
But the activity I had in mind — maximizing the volume of a box — is commonly done in a calculus class. These are my 6th grade babies. But didn't we do okay with approximating the volume of a torus via my donut lesson? So, why can't we do this too? I have to get rid of the popcorn.
I randomly assigned the kids in pairs, gave each pair two sheets of white copy paper. I told them to use one paper at a time to make a box — the goal is to make the box as big as possible so it'll hold the most popcorn. But the box must be made simply like this: cut off 4 corners from the the paper, then fold up the sides and tape them together. I used a half-sheet (so they couldn't duplicate mine) to demonstrate what I meant.
They quickly went to work. A few students were NOT cutting off square corners, so the top edges of the sides didn't line up. Two groups folded in their papers, in addition to cutting off corners, so they had to re-do.
Ryan and Annamaria wanted to make a shallow box. Ryan said, "... it doesn't matter how high it is."
Rapha and Cristian made the biggest corner cuts that I saw in the first round.
Mike's and Roy's first box was the shallowest in the class, but they changed their mind for their second box.
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With 10 minutes left in our first hour together, I asked the kids to measure the box and find the volume. They had no trouble with this since we did the donut. They recorded the volume inside each box, and I tacked them on the board. (The butter seeped through in few of the boxes.)
Well, that was fun. I pointed out that two of the bigger boxes were over 1,000 cubic centimeters. The bell rang. I said, "We'll wrap up this afternoon."
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I didn't know what I was going to do to "wrap up" the lesson. The microwave actually overheated — my room stank of greasy popcorn.
There was a confidence in me, however, that the kids would help me figure out how/where this lesson could go next.
I began the afternoon hour by going over what they'd learned in the morning. They said:
- The four corners must be of the same size. (I never told them this in my instruction.)
- Each corner must be a square. (I didn't tell them this either. Not everyone was convinced of this, so I cut non-square corners to show them.)
- There was a limit to how big the square could be. (I loved this! And this made me ask, "Is there a minimum to the size of the square?" Their eyes squinted, almost as if they were trying to "see" how small these corner squares could get. Or it was just my imagination. One kid said, "No. Technically, no.")
- The volume numbers that people wrote down could be wrong.
By then they understood the different boxes and their volumes depended on the size of the corner squares that would get cut off. We focused on this. I asked them to draw a 10 x 12 rectangle in their math journal. We removed 1 x 1 corners from this rectangle and found the volume. I guided them through the next 2 x 2 corners. They continued on their own.
Then I gave each kid another white piece of copy paper. We measured the length and width of the paper and agreed that the paper was 28 cm x 21.5 cm. I asked them to build a systematic table like the one they just did in their journal. I said something like this, "Because you now know how to figure out the volume without actually cutting and making the boxes, see if you could figure out what size square the corners should be to maximize volume."
I saw kids high-fiving each other, "The corner has to be 4 by 4!" Rapha and Cristian beamed after congratulating each other, "That was one of the boxes we made!"
We ended class with that. I swore I felt myself tearing up.
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On Monday we'll play around with this applet.
And we'll ask Wolfram Alpha to take the first derivative of the volume for us. (I'm pretty sure the class could write this equation V = (28-2x)(21.5-2x)(x) for me to enter into WA.) Well, I actually just did it, and WA gives the side of our corner square as approximately 4.01965. My kids got 4 — pretty damn good for 6th grade brute-force math.
Now that I'm writing this, however, I am really most proud of how well the kids had worked together. I randomly paired them up — a handful of the pairs were like the odd couples: high/low, shy/outgoing, squirrelly/quiet, jock/nerd, princess/cowboy. There was not a whisper of whine when their names were called to pair up. How did I get so lucky?
Just For Fun
Even though these two tasks lack any rigorous math, the kids tell me they enjoy doing these after two long weeks of reviewing and taking the State tests. Many kids love to draw, work on the computer, take pictures, talk to each other — and they appreciated the chance to do so.
I stole A School Year Is... from Ms. Roitz's blog. She did hers on the full calendar year, I changed ours to reflect the school year of 180 days. It was fun to see the kids wanting to make their numbers "bigger" by converting them into milliseconds and micrometers (I didn't bring up this unit, a 6th grader did). Some students wrote theirs in scientific notation. Like Ms. Roitz said, it was also wonderful to learn a little more about the students by what they chose to work on.
These are from my 6th graders:
I placed the 8th graders' work in the showcases outside our classroom, and they get a lot of attention! From my 8th grade algebra kids:
From my 8th grade geometry kids:
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I suggested Math Comic Strip when Kristen @Fouss and I were discussing a possible review project. Kristen is doing this with her precal and algebra kids. She tweeted an insanely cute pic from her precal senior who'd turned it in early. I hope she'll share more on her blog soon!
I did this with my algebra kids. I wrote the main algebra topics/concepts onto slips of paper and let the students work in pairs or alone to choose a topic to create a comic strip on. I just requested that each strip be comprised of a minimum of five squares, each square can be 4"x4", 5"x5", or 6"x6".
Next year, if we do this again, I'd do what Kristen did — have the kids make their comic on ONE piece of paper! (What the heck was I thinking with all the squares and then having to glue them together?!!)
What is a Function?
Mixture Problems
Rational Equations
Work Problems
Parallel and Perpendicular Lines
A Famous Bridge Problem
So I looked in the back of our geometry textbook, curious to see what was there — hadn't ventured this far back before and because we'd just finished with the State tests — and I saw the Königsberg Bridge problem!
I was all excited because what little I know about graph theory I find very interesting and useful. I found three lessons under the heading Discrete Mathematics:
- A Famous Bridge Problem, pages 676-678
- A Traveler's Puzzle, pages 678-681
- Minimizing the Cost of a Network, pages 681-683
I thought, Yippee, this could be my geometry lesson for the entire week!!
But half way through reading the first page of A Famous Bridge Problem, I came to this paragraph:
Euler reasoned that in order for a person to travel every bridge once and return to the starting point, every vertex must have an even valence. This is because a person traveling into a vertex must also leave it. So the edges must be paired, one "in" with one "out."
It got better. Or a lot worse, actually.
In the 7-bridges problem, none of the vertices has an even valence, so a circuit over all 7 bridges is impossible. However, if two more bridges were added, giving the 9 bridges as shown at the right, then every vertex has an even valence, and a circuit over all 9 bridges is possible.
Then, finally, right before the Exercises section on the next page, a question is posed:
Can you find an Euler circuit for the graph of the 9-bridges problem?
The answer to this question, printed in my teacher's edition, is Yes.
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What?!! Really?!! This textbook just gave away the answer [... so a circuit over all 7 bridges is impossible] to what I thought would be a natural first question that I would pose to my kids: Can you walk in the city of Königsberg so that you will cross each bridge exactly once?
Just like that, in one swift sentence [... every vertex must have an even valence], McDougal Littell not only gave the answer but also decided not to give students an opportunity to reason it out for themselves.
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I searched Google and picked this image, copied and pasted onto a Word document with nothing else on it, not even space for student name. I was that lazy and miffed that I couldn't tell the kids to turn to page 676 in their textbook.
I gave the students a brief history on the Königsberg Bridge problem and reintroduced them to Euler. Question #1: Can you walk in the city of Königsberg so that you will cross each bridge exactly once?
Bobby: Yes!
Josh: This is too hard!
Jacob: Don't say that. Mrs. Nguyen loves it when we say that!
Zach: The answer is no.
Taj: No.
I walked around the room. All but two students answered NO to the first question. What they said and what their papers looked like:
(Many): I tried a few times, can't get it to work. I'll try a couple more times.
Michael: You'd have to have equal number of bridges evenly.
Bobby: I know I said yes, but I can't replicate, repeat it.
I had students number the bridges, then I posed question #2: What if bridge 2 got destroyed, no longer there, can you walk in this city so that you will cross the remaining six bridges exactly once?
Again, they tested more routes and wrote NO for their answer.
Question #3: Suppose a new bridge gets built to replace the collapsed bridge 2, where should it be built so a person can walk through the city crossing each bridge exactly once?
Michael: I perfected my theory!
Me: What was your theory?
Someone said something to him that I couldn't hear.
Michael: Okay, maybe not. But I discovered something else.
This is a clip of Michael explaining his thinking.
Slater explained why he'd build a new bridge next to (parallel to) bridge 4.
It made me happy that Austin called me over so he could explain his answer. Normally he's shy about sharing his work.
Colin made more sketches than anyone else. He tried a lot of different paths, and this is him explaining his placement of the new bridge.
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I recorded all the possible placements of the new bridge (to replace bridge 2) on my sketch on the projector — there were 5 different locations that were shared.
Bobby normally is slow to leave class, today was no exception. He came up to the board and started talking.
Tomorrow we'll talk about vertices and edges and valence and convert our Königsberg Bridge into a network diagram. We'll practice drawing and exploring more of these diagrams. And maybe my kids will generalize that "every vertex must have an even valence." And maybe they won't. But I think today I gave my kids a chance to think about math in their own language.
Barbie Bungee
This was last night:
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.
Crazy coincidence. Just hours earlier I didn't know who Mr. Vaudrey was but became an instant fan after reading his post. And today we're doing the same lesson! Well, same "Barbie Bungee" anyway, but I just have rubber bands and a ladder, and I stole this lesson from someone years ago. He has real platforms, a mullet, and who knows what else.
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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:
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!
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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.
Circles Galore
I don't have the original source for this lesson, so if someone knows or when I find it, I'll be sure to give due credit. Using graphics from the original source, I created this recording sheet — in pdf and in docx.
But instead of passing out the worksheet and starting right in like I had intended, I placed the stack in my desk drawer and decided to ask the kids to follow my instructions to create the diagrams. (I'm getting better at asking kids to do more, including following a set of directions.)
I gave each kid one sheet of unlined paper and a compass. These were my instructions to them, and I followed along also:
- Put a dot in middle of paper.
- Draw a large circle using the dot as its center.
- Draw in the diameter.
- On this diameter, mark half the radius. (I could ask them to bisect the radius, but to save time, we just estimated "half" using a ruler.)
- Now draw a circle, centered at this half-radius mark, using a radius half that of the original circle, draw two adjacent circles along the diameter.
- Shade in the outside of the two small circles — their drawings should now look like Stage 1 on worksheet.
We stopped here, and I asked Question 1) What fraction of the large circle is shaded? They quietly did their thing.
After seeing about half of the kids finished answering this question. I said, "I know not everyone is done with #1, which is perfectly okay, but I'm going on with my instructions because some people are done, and I need you to follow along. Then if you need to, you can go back to #1."
I repeated directions 4 and 5 above, and before I asked Question 2, Josh said, "What fraction is shaded now?" They went to work again.
I then repeated the instructions to get their drawings to Stage 3. However, our compass can't make a circle smaller than 0.5-inch radius, so we freehanded them in. Everyone knew next came Question 3) Now, what fraction is shaded?
And Josh immediately asked, "Are we doing this again, make even smaller circles?!" We drew in the next 16 shaded circles for Stage 4.
You guessed it. Question 4) What fraction of the large circle is shaded? I asked them to work alone on the questions, but that they would have a chance to discuss with other classmates during the last 15 minutes of class. They plugged away.
Then the talking began; and there was a lot of it, so I started recording their conversations. I was more interested in those who did not get 1/2 for Question 1. I happened upon Zach telling his two classmates that the answer was 2/pi:
Did you catch what Zach said? He reasoned that (16*pi) + (16*pi) = 2*pi*32 by saying "You can't get rid of stuff because we haven't done any mathematical operation to get rid of pi." I was so glad this came up! Sometimes I feel a wrong answer is just as important as the correct one.
(But I thought it was funny too how Slater reacted, "That's wrong." Then there was peer pressure, "He made me change my answer." And there was alleged copying. Then there was, "Taj showed me the way." I will miss this class.)
Daniel started the discussion with this group.
The "formula" that Daniel had trouble articulating was never written on his paper, it was all in his head. Then I noticed that he'd carried his calculations to stage 8! I was so pleasantly surprised by this that I was speechless like a complete dork in this next clip.
Class ended and everyone agreed that the answer to question 1 was 1/2.
I passed out the worksheet above for homework, asking them to try to figure out the answers for stages 2 through 4 if they hadn't already, then to give stage n a try if they had time. Daniel never saw the worksheet to know that I'd wanted him to keep thinking of the pattern, but he was already heading there on his own.
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Tomorrow we'll finish out this lesson, but I wanted to capture what we've had so far before my amnesia sets in.
Some thoughts:
- When I search for lessons/activities for my kids to do, I always search for lessons "recommended" for kids at least one grade level above mine. Meaning, I look under 9th and 10th grade activities for my 8th graders. Why not? I could always work back from there.
- I remember adding "stage n" to the worksheet just because there was some left over space at the bottom. This made Daniel happy.
- I'm glad I had them draw the circles. This was a last minute decision that paid off — they made pretty sketches!
- I learn that my kids are smarter than I am. (But let's not tell them this.) Daniel's "equation" was so elegant that he could do all of it in his head. I had to work hard to get to his answers!
- Stage n, if we get it or not tomorrow, is icing on the cake. I feel pretty full.
Speaking of stage n. What do you think? Is there a limit that we are approaching?
Staircases and Steepness, Continued
Here is the first part that we did on 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.
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I was on cloud nine — the same way I felt after doing Mr. Stadel's File Cabinet lesson with my geometry kids. Mr. Stadel wrote a wonderful post of how the lesson went in his classroom.
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.
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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.
Staircases 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 wanted to do this because we had a great lesson couple months ago on "squareness," and I wrote a little bit about it in this post. Kate Nowak wrote two wonderful posts on a similar squareness lesson, here and here.
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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.
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I made this worksheet during testing. I'd be flattered if you wanted to download the lesson — in pdf or docx. I purposely wrote the lesson 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 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.
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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?
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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
Summer Plans
But this summer's agenda is filling up fast too. I'm committed to two big-ticket items:
Professor Nathan Carlson, my colleague Erin, and I just got word last week that our team had been selected to participate in the workshop "How to run a Math Teachers' Circle." I'm excited to spend a week learning and doing math at the American Institute of Mathematics (AIM) in Palo Alto, California. I was only looking up Math Circles to be a participant in, but there isn't one in my area, so when this opportunity came up, it was a no-brainer to apply.
Under a grant from the UC-Santa Barbara Mathematics Project, I will continue to co-lead a one-week workshop "Common Core Institute for Mathematics Educators" in July. I immersed myself in this project to learn more about the CCSS, but specifically I wanted to learn how we can connect the Practices to the Standards — I've seen enough in print, now I want to see it walk off the page, march into classrooms, and act up!
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I'll be spending many summer evenings planning lessons for next year. I'm happy with Geometry (two summers ago I spent a solid month working on Geometer's Sketchpad projects), but I need to create better lessons for Algebra and Math 6. This assumes that I have the same preps for next year. I'm really looking forward to perusing the many great math blogs, reading math books (I read these like other people read novels), checking out all the math links/lessons/videos/tweets that I'd tagged for summer fun.
I also hope to spend a week this summer in the Pacific Northwest, my home, where beer and oysters roam freely. Sabrina and Nicolai will be with their father for five weeks, Gabriel will only go for two weeks as he needs to be here for football. (I cannot cannot wait for his season to start!!)
We bought the Hollywood Bowl five-concert package again this summer: Glen Campbell, Smokey Robinson, Neville Bros, Nora Jones, and Diana Krall. But I'm really looking forward to Roger Waters (Pink Floyd) and Glenn Frey later this month!
Maybe this summer I could learn to ride the new Ducati that we got last month. (Nicolai and my niece are having a blast riding it.) I tell people that I can ride just fine, I just can't stop. I dropped a bike 10 years ago and have been gun shy ever since.
