My Other Math Sites
Lessons From the Classroom
The Right Question Institute
From www.aish.com:
Isidore Rabi, winner of a Nobel Prize for physics, was once asked why he became a scientist. He replied: “My mother made me a scientist without ever knowing it. Every other child would come back from school and be asked, ‘What did you learn today?’ But my mother used to say, ‘Izzy, did you ask a good question today?’ That made the difference. Asking good questions made me into a scientist.”
A week ago my superintendent, principal, and 7 of us teachers attended a full day workshop The Right Question Institute in LA. Luz Santana and Dan Rothstein, authors of Make Just One Change, facilitated a worthwhile and engaging session, so I just want to share some highlights and my takeaways from it.
Highlights
(Some of these might be direct quotes. I’m just writing from my notes.)
Not knowing what to ask is the fundamental obstacle to participating and therefore to learning.
The skill of question formulation is the single most powerful renewable source of intellectual energy.
How can we easily develop students’ question formulation skills? It’s simple. But simple does not mean simplistic — it means doing it so everyone can access it.
Six components of the Question Formulation Technique (QFT):
A question focus — the teacher gives a prompt related to topic currently being covered in class, prompt can be visual. It should be a statement or phrase and not as a question. The simpler, the better.
Producing questions — in small groups, students share questions that they have related to the question focus (prompt), one person records on large poster paper or whiteboard. But before starting, everyone is reminded of these rules:
Ask as many questions as you can.
Do not stop to discuss, judge, or answer any question.
Write down every question exactly as it is stated.
Change any statements into questions.
Categorizing questions as “open” or “closed.”
Prioritizing questions — choose 3 most important questions from list, pay attention to the question focus.
Next steps — what’s one way you could use your priority questions?
Reflecting — what did you learn and how did you learn it?
Takeaways
Luz and Dan are really lovely people. Warm, hard-working, fun. Their book — and this Institute — mark the arrival of a twenty-year journey for them! (Arrived, yes. Settled, no.)
This task of having kids ask their own questions is not unlike Act 1 of a 3-Act math task that many of us are familiar with.
But the QFT process can be used — and is used — in a variety of academic disciplines and in communities outside of school. (Their journey actually began when they worked on a dropout prevention project and heard from the parents who were not coming to the school meetings because they “don’t even know what to ask.”)
This is another powerful structure that empowers students when asking questions becomes a natural tool for them. They think more critically because the QFT process helps them hone in on their questions. Dan and Luz categorize the learning of asking questions as going from divergent thinking to convergent thinking, then that last component of reflecting is metacognitive thinking.
Teachers and students will get better at implementing the QFT. It’s building classroom culture, so it takes time.
We’re doing this already with our students to some degree with varying expertise. Maybe we just need to be more intentional about it. Give it a name.
If not, perhaps Make Just One Change.
Deconstructing a Lesson Activity – Part 2
Previously: Deconstructing a Lesson Activity - Part 1
No matter what I write in this Part 2, I hope it's not a wrap-up of this topic. On the contrary, I hope it opens up and extends our conversations on improving the implementation of rich tasks in our classrooms.
Physical arrangements, whiteboards, groupings. The student desks are set up in rows and columns in my classroom. Not terribly exciting, but don't judge a teacher by his/her furniture arrangement. When I'm given 38 desks (I had 37 students in Math 6 last year) and x square units of floor space, my creativity is stifled.
Work with whatever space you have. Push the desks together, pull them apart. Kids don't mind sitting on the floor. (If they do, ask them to stand and see if they like that better.) Can they work in the hallway — or outside if you're in warmer climate — where you can see them from inside the room? Just make sure you are constantly roaming among the groups.
Big-ass whiteboards. Nathan just wrote a letter asking for these. Last summer I sent my principal the link to Frank Noschese's post — and the whiteboards were waiting for me when school started. I got ten 2' × 2.6' boards, each at $10.50 from here. Please get them. By far the single best school purchase, worth their weight in gold. Just how much do I love these? You touch my whiteboards, I'll kill you.
Randomly assign kids to groups. We do group activities often enough that eventually pretty much every kid ends up with somebody new in the group. If you try to group them "heterogeneously" with high-medium-low kids, then you accomplished just that — you just told them who's high, who's medium, and who's low without saying a single word. Kids aren't stupid. I use Instant Classroom to randomly assign. You can always use your discretion to change a few kids around after the computer picks them — but still let the kids think that the computer did all the choosing. I never heard any whining. Kids don't whine at what the computer says.
Group roles. What are these?? (No, I'm asking you!) Like "facilitator," "recorder," "reporter," "budget person," "dietitian," "hairdresser," etc... These roles wouldn't work with problem-solving tasks. I don't want a kid sitting there doing nothing because it's not time for his role to occur yet. Please, no assigned roles. Except the one about trying to solve the problem.
Grading this type of task. I don't see dead people, but I hear student voices all the time. While I enjoy grading as much as I enjoy poking needles in my eyes, I hold certain beliefs about grading problem-solving group tasks (and the student voices that guide mess with me). And my possible reasons/solutions for them?
It is wrong to give a lower grade because they socialized too much instead of focusing on the problem. (We're teenagers and you expect us not to socialize? OMG! Did she just put me and Joey in the same group? He's sooooo cute. How's my hair? This problem is just too hard anyway! We really tried but we got stuck and you were too busy with another group to help us. Laura is such a show-off. I wish Andrew would grow up.) If the whole group is off task, then I'd seriously reconsider the relevance/engagement level of the task and the social dynamics of the group. It's back to that Step 0 of picking the right task that's engaging and has low entries so everyone can get on board. It's my fault that the kids are not on task.
It is wrong to give a lower grade because they did not come up with the correct solution when the bell rang. (Sucks that we didn't win the game today, but we still had a good game, right? Didn't we work well together as a team, especially on defense? Nice block there, Mitch. I almost had a pick right there if my damn leg didn't cram up! Ha, now I know what Coach meant by the hook-and-lateral play!) Our goal of wanting kids to engage in problem solving is to honor the process that they go through — their thinking, their collaborating, their critiquing one another. We want to tap kids' two most abundant natural resources: their curiosity and their need to socialize. I simply cannot justify putting a grade on this.
But bottom line, you grade it if you want to. Don't grade it if you don't want to. I graded fewer than 50% of the tasks that were done in class last year. When I did "grade" them, I gave full credit. To worry about how to grade group tasks is really to sweat the small stuff. That said, if you had a handout for each student that went with the task, then it's fair to give the individual grades.
Establishing a classroom culture of problem solving and finding time to do so. Stephanie Reilly's question in Part 1 helps me shape what I'm trying to convey in this section.
I can't think of a better day to start doing problem solving with kids than Day 1 of school. Kids pick up on what we say we value and what we do to back that up. Set a goal to do one task every two weeks. Too ambitious? Then once a month. Just please don't give up. On Day 1, I might just start with Pyramid of Pennies (Ha! I nailed the spelling there) or the new Bracelet Craze problem. If I were a student and knew that all my teachers would go over "Rules & Procedures" on the first day of school, then I'd be tempted to feign high fever and induce vomit to stay home.
However, you need to come up with guidelines for group work that you will share with kids before they begin. Culture takes time. It takes a lot of reminders too. I'll share what I say [for guidelines] to the kids under "group time" in the last section of this post about implementation.
Post the strategies for problem solving in your classroom. I have these on just regular size paper, but laminated, and we refer to them all the time. You know, strategies like these ones.
Teacher concern: I'm afraid I don't have time to do this because there's still so much to cover in the textbook. You can't do this and feel guilty. (Remember how crazy in love you are supposed to be with the tasks you choose for them?) You have to be okay with not being able to go cover the textbook front to back. The person who tells you that you have to do so is delusional and mean. Common Core does have fewer domains and standards at each grade level. Spend this summer mapping out key concepts and lessons. I believe in having some sort of pacing guide, but I don't believe in having it dictate how we move through the year — the kids and your formative assessments of their learning should govern the flow. I haven't done research or have hard data of my own to give you, but I believe your kids will do better on year-end assessments if they have been exposed to problem solving throughout the year. Trust me? :)
Carve out time by re-examining and possibly eliminating things that you normally do.(For the last two years our students had two periods of math each day. I think this is going away next year, so I have to re-think this through too.) Besides just having better classroom management — meaning it's not taking you 10 minutes to get the kids to settle down and start class — how effective is your use of class time when you do these items?
Warm-ups
Games
Review games before a test
Pre- and post-surveys
Benchmark tests (beginning, mid-year, end-of-year)
Stuff that kids can do at home blindfolded (I think we know what these are.)
Class parties (What the hell are these? I like parties too, but let's have them at lunch time.)
Lastly, please don't forget that these are perfectly good SCHOOL days for doing mathematics: First day of school, last day of school, last day of the quarter, first day of the quarter, whatever day. The day before Christmas or spring break. The first day back from an extended vacation. Sub days. Your sub is perfectly capable of passing out a meaningful handout (it's meaningful because you made/selected it), and it will go well because you have already pre-taught the kids what's on that handout — give them a sneak peek at it! — and shared with them your expectations the day before you leave. If there's one assignment worth grading, then it's the work that they do while your sub is there.
Finally, implementing the task. Thus far I've covered the behind-the-scenes stuff that was missing from my lesson posts. The task itself is actually a lot more straightforward — pretty much what you read on my lesson posts is my best storytelling of what went on in the classroom.If you're doing an actual 3-Act lesson a la Dan Meyer, then you're good to go! These are some of my favorite 3-Acts that I'd written up:
File Cabinet
Taco Cart
Equilateral Triangles
Penny Pyramid
So, this is an outline of how I implement a non 3-Act problem-solving task.
Ask for a volunteer to read the problem aloud (5 minutes). Each kid gets a copy of the problem to follow along. After it's read aloud, everyone reads the problem again quietly to self. Then my questions begin for the whole class:
What are we trying to solve for in this problem?
What information do we know?
Is there information that you wished you knew? Why is it not given then?
What's the first strategy you have in mind that might help you attack this problem? And why did you say 'do a simpler problem'?
Depending on the task, I've also begun to ask — instead of the questions above — these two questions from Annie Fetter (YES! Please watch the 5-minute video if you haven't.) The kids write down their answers, and I randomly call on them to share.
What do you notice?
What do you wonder?
Quiet individual work time (10 minutes). You have to allow for some individual thinking time. I can't work on a problem when others around me are talking. So I set a timer for 10 minutes. Of course it doesn't have to be 10 minutes, it's up to you and depends on the problem, but this is NOT the amount of time in which I expect any kid to solve the problem. If a kid does solve it quickly, then hopefully you have an extension — you should always have an extension — ready for this kid. If a bunch of kids could solve it quickly, then you've chosen the wrong task, too easy. Back to Step 0.
I say something like, "I'm setting the timer for 10 minutes so you can think about and start the problem on your own. There's no talking and no sharing at this time. You'll get in groups to continue to work on the problem after the quiet time. You're welcome to get up without my permission to get any tools (protractor, compass, ruler, graph paper) that you need. Do you have any questions for me before you begin? Remember our rule of NEVER TELL AN ANSWER. Go!"
While they're working, I use Instant Classroom to form the groups and move some kids around if necessary.
Group time (30 minutes). Again, this is a very generic time allowance. You're the teacher, you'll know by how much to shorten or lengthen the time depending on the groups' progress or lack thereof.
I say something like, "You will now continue to work on this with your group mates. You will use the large whiteboards to show your work. Everyone has his or her own marker to use. But now I'll explain more by what I mean by 'never tell an answer.' If you think you have an answer already from working on it just now by yourself, then please don't share it with your group. Choose to be the last person in your group to speak because I actually need to speak with you first.[1]
... Also, every time we do a task and I hope we get to do lots of them, I ask the computer to randomly assign you in groups, so if you have a complaint, take it up with the computer. If your group would like more individual work time, like 5 more minutes, then that's great and fine by me....
I'm interested in your working together to solve this task. I'm not asking you to become best friends. One person speaks, everyone else listens. Argue about it, but be respectful. Ask questions of one another. Don't take so-and-so's word for it, ask him or her to explain it. Don't let others think for you. Help each other out. Maybe this whole structure is new to you, don't worry about it. I'll walk around and listen in and smack you in the back of the head when you don't quite have it right. Just kidding. Not. Yeah, I'm kidding. Go!"
I actually repeat much of this same spiel throughout the year.
[1] So I talk privately to the student who does have the correct solution and suggest a few things:
Can you solve the problem a different way? (It's important that you do not force a student to find another strategy especially when you can see that she has found the most elegant one already — this just seems counterproductive to me.)
I'd like you to try the extension to this problem. What if...?
I need you to go back to your group and practice really good listening skills. I just want you to listen to your teammates talk. Then see if you can help them by asking questions only. Kinda like what I normally do with the whole class. You may give them one hint if they're really stuck. You want to give that a try?
Teacher role during this group time. This is where the book 5 Practices comes in for me. I've been presenting its contents at workshops over the last two years. There's no way I can do it justice here, and I've already written a brief post on it just as a quick review.
The gist of it is that I go around and listen in and check on the groups' progress. I ask questions of specific individuals in the group.
Hey, Julia, can you explain to me what I'm seeing here on the whiteboard? Maybe you didn't write it, but whoever wrote it, did he/she explain it to you?
Jonathan, I'm not sure where this equation/number comes from. Please explain.
I saw this same strategy at Erika's group. Allie, did you come up with this strategy? If not, what is yours? Where is your understanding of the problem so far?
Joey, what has Cindy contributed to the group thus far? (If Joey says, "Nothing," then I'll ask Joey again, "What have you contributed?" I don't remember ever having two people in one group who have not shared anything. Remember they had 10 minutes of quiet time to work on this already. They have something to share!)
Cole, your group is over here. I don't want to tell you that again. (And I never have to.)
Now, when there's one group that has made a lot more progress than the other groups, I ask for the whole class attention and say, "Julia's group has made an important connection, so I'm going to ask someone from her group to share with the class one hint, one strategy, or one something that would help all the groups along. Listen carefully."
If none of the groups has made progress, then the teacher needs to jump in with a hint. But be patient too!! You have to watch the clock. How much time is reasonable? Are the kids mostly working and asking questions of one another? If they're exhibiting productive struggle, then let them be. Nobody is going to die if you extend this lesson another day.
The SHORT version can end here after the groups have figured out the solution. Maybe not all the groups finished, but remember, most of them did. Depending on the task, depending on the students, depending on time, depending on whatever you deem as important, you can end the lesson here and not feel guilty that there was no large-group sharing at the end, no connections made among the different strategies. Instead, focus on all the mathematics that you did allow the kids to be engaged in. I see enough teachers feel discouraged that they "didn't get to do everything that I wanted to do" — it's not about doing everything, it's about doing something to get started, to get better, to suck less each day, to remember why you went into teaching in the first place.
The FULL version includes the "connecting" piece that the 5 Practices refers to. It's about making connections between the different strategies, and you accomplish this by having the groups share their work on the whiteboards. (This step is moot if the task didn't have more than one strategy.) Kelly O'Shea is my whiteboarding goddess. And connecting is also about you the teacher making the connections of all their work back to the original intended learning goal of the task.
Who says you can't add the connecting piece to your short version 2 or 3 days from now (hell, even two weeks later) and make it a complete kick-ass full version? In real life we return to problems all the time. Snap a photo of each whiteboard if the kids need to refer back to their work at a later date but you have to use the whiteboards for another class in the next period. Problem solved.
You can do this. We can do this together.
Deconstructing a Lesson Activity - Part 1
[I've decided to break this post into two parts because I don't want to bore and/or discourage you, and I need to take a breather. These two posts in particular are truly my labor of love because if there's one thing that I find myself proficient at implementing in the classroom it is problem solving. I hope you'll find some parts useful.
I've done a fair share of posting actual lessons and pictures from my classroom in this space and on my 180 blog [that's no longer available, sorry]. But I'm afraid they appear polished and therefore unhelpful to teachers who are trying to implement problem solving, white-boarding, 3-Act lessons, or any task-oriented activity in their classrooms.
So here's my earnest attempt to deconstruct the structure of a lesson, get down to the nitty-gritty, take small bites (and spit out what you don't like), and make it real because if it ain't real to you, then it ain't gonna happen for your kids.
Some important prerequisites. I need you to have this mindset or else we're not going to accomplish anything.
You care that kids learn something meaningful in the 45-minute period that they are with you. You might be thinking, Of course I care or else I wouldn't be teaching. No, I don't mean that. I mean the "something meaningful" part. What did you intend for your kids to learn today?
You might very well fail at implementing a 3-Act in your first attempt. And fail again and again. But you can't give up. You can't give up because the kids need you to persevere, it's the same MP1 that you ask of them. Cry and bitch about it at home. Adopt a puppy if you live alone. Eat ice cream. Drink a beverage with higher alcohol content if you need to. Get a punching bag or go to the gym.
You need to be okay with leaving some children behind on some days. I'm not a miracle worker. Neither are you. One hundred percent student engagement 100% of the time is a myth sold by the snakes-oil salesman. You can't differentiate every lesson. You can't reach and motivate every child. But you will reach and motivate the ones that you can. You'll die trying because you love these kids, but you're going to realistic about it.
Surround yourself with helpful colleagues. They listen and are willing to observe your class and give critical feedback. They remind you to eat. They eat lunch with you. They don't badmouth kids when their lips are moving. They believe Happy Hour is invented for schoolteachers and feel it's sacrilegious if they went without you. The toxic people in your life can just piss off. If no one is around and you really need to vent, please email me at fawnpnguyen at gmail dot com. I am a much better listener than I am a writer.
Piece de resistance. I carry out the lessons through the lens and language of these bodies of work. It's okay if you don't have the 4 books, but I highly recommend them if your school or personal budget allows.
5 Practices For Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein
Thinking Mathematically by John Mason with Leone Burton and Kaye Stacey
Mindset: The New Psychology of Success by Carol S. Dweck, Ph.D.
12-minute video of Dan Meyer's TED Talk: Math class needs a makeover
Improving learning in mathematics: challenges and strategies (PDF file) by Malcolm Swan
The Art of Problem Posing by Stephen I. Brown and Marion I. Walter (I just borrowed this book from UCSB last week. It's wonderful. Nat Banting reviewed it here.)
Picking the right task. Think of this as Step 0 of the 5 Practices. If I picked the wrong task, then no sound pedagogy or fancy technology would be able to save me. I'm done. Lesson sucks. Game over. That's how important this step is. What is a "right" task?
It's age appropriate. I don't mean for you to go searching under tabs that read "6th grade" or "algebra 1" either. What's age appropriate for your 6th graders might be too high for my 6th graders. Know your kids. An inherently good task would cover a wider range of consumers. If it's early in the school year and you don't know your kids well enough yet, then choose a higher level. It's a crime to underestimate children's mathematical abilities. Dan Meyer speaks volumes about "low-entry high-exit" tasks and Ladder of Abstraction.
It has multiple strategies. At least 2 ways of solving. Single-strategy tasks are like culs-de-sac. There's nothing to do except to turn around. Think how fruitless and boring it would be if you asked kids to share their different strategies and there wasn't one to begin with. Also, just because you'd struggled with a task does not necessarily mean all your kids will. Maybe there was a more elegant solution that you did not see. Be humble, ask another colleague or throw it out on Twitter for others to give it a try. (Please tell me you have a Twitter account. Mine is @fawnpnguyen.) And if your gut thinks there's another way to solve a problem, then be honest with the kids and say exactly that. They'd be thrilled to death to learn that they'd helped you see something differently. My favorite moments for sure.
You are crazy about the task. I can't speak for what tasks/problems turn you on, but I know what my favorites are. (And every.single.time we do a new task, I half-jokingly say to my students, "This one is my absolute favorite!" The kids roll their eyes at me, but they know I'm passionate about it.) It's hard to get kids to like things that we ourselves do not care much for. That's kinda fraudulent. I'm be a big fat liar if I say that I like all the tasks Andrew and Nathan have on their sites. Your chosen task is your baby — you've personally nursed and nurtured it. Kids sense this and they'll handle it with care too.
Throw a curve ball. Meaning offer a task that does not line up with your current topic right now. I know this sounds strange. (I've never read or heard anyone else suggest what I'm suggesting). Normally teachers look for a task that lines up with what the kids are learning. Sure, I do these "tasks" too, but I really call them "exercises." Exercises help you practice the skills you're learning. The tasks that this post is referring to are problem-solving tasks, and true problem-solving has no prior diagnosis and certainly no given prescription. Don't do Taco Cart right when you're teaching Pythagorean Theorem. Unless you want to do it as an exercise, then sure, go for it! The beauty of this is you can reach into your folder of best tasks, close your eyes, and pick one! Be a rebel, break the rules. (Don't forget it still has to be an appropriate level task.)
Custom tailor the lesson. This is hard work. I don't care if His Holiness the Dalai Lama himself wrote the lesson, I still need to tweak it so it works for me and my kids.
Great lesson but terrible handout. I see this all the time. Take the time to re-type it. Is there wasted space or not enough? How's the font size, the heading, the outline? Can you improve on the graphics? Are students asked to work on page 2 but keep having to flip to page 1 to see the sketch or data?
Do you have to pass out the handout at all? How about starting from scratch as in having the kids take out their own paper to create data and meaning for themselves? Remember that any question that appears on a handout may potentially rob a kid's opportunity to ask that question for herself. There are certainly good handouts that have just the right amount of information and provide easier access points for kids. Please create them.
What level of technology is involved? Disaster abounds when an activity requires at least 15 computers and you only have 5. Adapt it or scrap it. No more than 2 kids should have to share a computer. Will the server crash if everyone got on the system? Did you review the YouTube video for all the potential peripheral garbage and comments that might be on there? To be safe, how about you projecting the video from your teacher computer instead? And I need to add here that I beg you not to incorporate technology into a lesson for technology's sake. Technology should enhance the student's learning. A shitty lesson on the interactive whiteboard is still a shitty lesson.
Venturing Into the Sequel of Penny Pyramid
I didn't get around to doing Penny Pyramid when I first saw it last year. But Dan's 3-post series and Nathan's recent mention of it were the reminders I needed to make it happen.
Act 1
how many pennies
how much money is that
how long did it take
who in their right mind would do this/who has that patience
how much does it weigh/is the table gonna collapse
what is the volume/surface area/height
what is the ratio of pennies from one level to the level above it
(Student who gave the highest high guess did correctly say her written number as "one hundred quadrillion." It made me happy that she knew this.)
Acts 2 and 3
Lauren F: Is there a way to multiply consecutive numbers quickly? You showed us the addition one...
Maddie: Isn't that the exclamation point operation?
Gabe: But we're not multiplying consecutive numbers!
Mia: Doing 40 by 40 then by 13 gives all huge numbers, so we're doing a simpler problem, then find an equation.
Lauren P: Our group is finding a pattern and making a table.
Gwen: We're doing layer by layer. There are more of us (4 instead of 3), so it's pretty quick to divide up the work.
Gabe: I already have the answer because I was too eager to do the math, but I didn't say anything to the group. (He got the answer about 2 minutes after we formed groups.)
Julia: And I got it 3 or 4 minutes after Gabe.
Angela: And I got the answer after Julia. Without her help.
Me [to Gabe, Julia, and Angela who were in same group]: Aren't you guys special. You seriously just sat there and did nothing then while I walked around?
Julia: Well, yeah, we're kinda admiring our work.
Me: Geez Louise. What do you think I'd have asked you if I knew you'd found the answer to this pyramid?
Gabe: If it was 100 high?
Me: No. A million high. A billion high.
Gabe: Hehe. That's why we didn't want to say that we're done.
Two students figured out why each stack had 13 pennies.
Their other questions were answered to their satisfaction, except we didn't know exactly how long it took Mr. Bezos to build it, but we talked about how we might be able to estimate this.
Kids remembered from last week's lesson that a square pyramid has 1/3 the volume of a cube with same dimensions, but that our penny pyramid had jagged lateral edges.
While everything up to this point had gone as well as I'd expected. Kids immediately responded to the video with WOAHs and WOWs. They asked solid questions in both Acts 1 and 2. They worked well in groups.
However, the kids and I knew that no one really struggled with the task of just finding the number of pennies. The math was pretty basic and with a calculator, 40 layers of pennies didn't make anyone break a sweat.
What was meant as an "extension" or "sequel" really needed to now become the focus of our lesson — at least for this group of students who valued a good struggle. We needed to try to figure out the equation for this penny pyramid.
But I also realized that it would be very unlikely for my 8th graders to come up with the equation because it involved summation of a sequence. (You're right, Nathan, it is unlikely, even for Gabe.) But the process of getting there might be worth it. I wouldn't be their teacher if I didn't ask them to explore the patterns that they might see along the way.
I gave them small interlocking cubes and colored chips so they could build smaller models of the pyramid.
Their collective frustration arose from how "simple" the pyramid was built — nothing more than a sum of layers whose square dimensions were consecutive.
Incomplete Cube
We started with a smaller problem. We did a 5 x 5 square pyramid with a height of 5. We didn't like the "jagged" lateral edges of the pyramid either, hence we pushed the cubes into one corner like this so at least the cubes stacked squarely.
One way would be to imagine that we had a whole 5 x 5 x 5 cube, then subtract from this the small cubes that were missing. We noticed the missing pieces were these L-shapes.
We see a pattern in these missing L-shapes:
4 pieces of (2n-1) or (n-1)(2n-1) or (2n^2)-3n+1
3 pieces of [2(n-1)-1] or (n-2)(2n-3) or (2n^2)-7n+6
2 pieces of [2(n-2)-1] or (n-3)(2n-5) or (2n^2)-11n+15
1 piece of [2(n-3)-1] or (n-4)(2n-7) or (2n^2)-15n+28
Incomplete Rectangle
How else can we see this pyramid? Because my mind has a tendency to reshape things into rectangles, I flattened the pyramid into an incomplete rectangle like this:
The dimensions of the rectangle were straightforward enough, and unlike the missing L-shapes of the incomplete cube, the missing pieces here were rectangular and came in pairs. For example, in the above right sketch, the missing pieces were two 1 x 4 and two 2 x 3 rectangles. But if n were even, then the number of missing pieces would be pairs of rectangles plus 1 lone square piece.
I talked with them about the sigma notation, and since they knew how to add {1 + 2 +... + n} quickly — we refer to this as "Gauss addition" in class — they thought it was fun to learn the new symbol.
Then we went into WolframAlpha and typed in what we wanted. The equation came up with the "newly" learned summation notation.
The kids saw patterns. They learned a fancy new sign. They knew that the right math could help solve for any penny pyramid. But I really think they look forward to learning more math in high school.
I'd like to feature this comment from the old blog:
May 19, 2013 2:21 PM
l hodge wrote:
If you draw two copies of the rectangle sketch mirroring each other, with a 1 unit space between them, you have a nice sum of squares proof. The space between the two copies is easily seen as a re-arranged sum of squares. Divide the area of expanded rectangle by 3 and you have your formula.
May 20, 2013 2:41 PMfawnnguyen wrote:
Thank you, l hodge! Mind blown. So happy to know that we were on the right track of flattening out the pyramid into an incomplete triangle. We did make another copy of the flattened pyramid but turned it around (1800 rotation) to look at that double-pyramid-with-extra-spaces rectangle, but time ran out. So, we drew this together in class today. So #nguyening!!
Stack 'em High
My algebra kiddos are doing Dan Meyer's Stacking Cups because Andrew Stadel did the lesson and wrote glowingly about it. But Andrew used only one-size cups, I used three different sizes.
Just eyeballing
I ask students to look at me, look at this 12-oz Styrofoam cup that I'm holding, and estimate how many cups they would need to stack to reach my height. I tell them I will not answer any clarifying questions regarding this, "just make your estimate in whatever way you think I mean by this." (Their hands shoot up anyway, but I remind them I won't answer any questions right now. It's clear that they want me to define "stack.")
They write their answers on a quarter sheet of paper. Here are their 29 estimates, the median at 24 cups.
Given the heights
Then I tell them my height is 163 cm with the flat shoes that I have on. I carefully measure the cup's height in front of them, we get 11.25 cm.
Equipped with this knowledge, I ask the same question as above. Their 28 answers yield a median of 14 cups. Of course Dan had already anticipated this — most students just divided my height by the height of the cup.
Stack 'em like this
I now ask them what it was that they'd wanted to ask me earlier. Sure enough Eddie says, "By stacking do you mean bottom-to-bottom, then top-to-top, or... one inside another?" I give him some cups to show me. At this time at least 2/3 of the kids admit that they answered the previous questions believing that I meant to stack the cups the way Eddie just described formerly. (This is consistent with the low guesses we see in the top image.)
I ask the next question, "Okay, I need you to answer the same question again of how many cups it'll take to reach the top of my head, but you now know exactly what I mean by stack, and you also know my height and the cup's height. Here, I'll even stand on this table with 6 cups stacked at my feet so you can see. Go, give me a number.
Here are their 27 estimates, and 113 is the median.
We're just getting started
I randomly pair kids up. (Normally they are in groups of 3, but I think it's better to be in pairs for this activity.) I give each pair 6 cups. They have the last 25 minutes of class to figure out:
The equation for this problem
The number of stacked cups for my height
Some groups need help with finding the y-intercept. A few groups don't know where to begin. I ask them some questions and walk away. They plan to nominate me in June as their most non-helpful teacher. Whatever.
I like this group's drawing, even though the lip and body of cup do not add up correctly.
Using their equations
The majority of the groups figure that the lip of the cup is the slope, but many groups also think that the full height of the cup is the y-intercept. Using their equations to figure out the number of cups, they give me these numbers.
Not too shabby. My equation yields 102 stacked cups to reach top of my head, and that happens to be the mode and median.
And the actual number of cups is...
This is the moment they've been waiting for! I have 81 cups stacked already when the kids come in the next day, then they count out loud as each additional cup is added to reach my height. Our principal is paying us a visit today, and he keeps telling me how impressed he is with how engaged the kids are and how hands-on the lesson is. He helps with the countdown too and officially announces that it takes 100 cups to stack up to my height! (Three groups whose estimates of 99, 101, and 102 are having pizza with me next week!)
Getting our principal in on the fun
Working backwards. I tell the kids that it takes 116 cups to reach the top of our principal's head. How tall is he? Their answers give a median and a strong mode of 184 cm. His actual height is 183 cm!
Twice the volume and half the volume
Just when they think they're done, I pull out the 24-oz and the 6-oz cups. It's not often that I hear them shriek in delight to do more math! The pair of students now gets only 3 cups of each new size. They go to work.
My height is 40 24-oz cups and 126 6-oz cups. Their calculations are great for the big cups, not so much for the little ones.
This ranks up there as one of my favorite lessons. Thanks much to Dan for another fab activity. But due to us having already passed linear equations, I honestly would not have done this lesson now without Andrew's push.
Updated 02/06/13
Eight more kids will join me for a pizza lunch for getting the equations to the 24-oz cups correctly. And here's a pic of me and the 3 stacks. The boys are 6th graders helping me hold up the stacks.
Equilateral Triangles
Thank you to Dan Meyer for this great task idea on equilateral triangles.
Act One is this video which asks for a ranking of how well each teacher had drawn his equilateral triangle.
But as soon as I saw the video, two thoughts came up:
We're in a 0:1 classroom right now, and having the whole class come up to the big screen just isn't efficient.
I want to be in the competition! The kids will all want to do this.
So I knew this would have to be a pencil-to-paper activity in my class.
Eyeballing
My instructions to twenty 8th-grade geometry students:
Here's a blank sheet of paper for you to do this task. Be sure your pencil is sharpened. Put your name in upper-right corner.
Using only your eyeballing skill and your pencil, mark three dots in the shape of an equilateral triangle.
Gabe normally asks amazing questions that make my heart sing. Today he asked, "So you want us to draw the best equilateral triangle?" I replied, "No, Gabe, I want the crappiest one you can draw."
Now, connect the dots with a straightedge.
Pass your papers forward, I'm going to make a photocopy of your drawing so I can have a clean copy of it just in case.
While I'm making the copies, I want you to think about how you are going to decide which triangle is most equilateral.
I dashed quickly to the copy room a few doors down. I wish you wouldn't tell anyone that I'd left the children unattended for 3 minutes.
I'm now going to randomly pass the papers back, meaning you should have someone else's drawing to work with. Write your name on their paper as the "tester."
Okay, I want to know which one of you drew the bestest equilateral triangle. To do that, we need to come up with some kind of criteria, a way to test it, a way to score it. Talk to me.
They said that an equilateral triangle had to have three congruent sides or three 60-degree angles. So, I said:
I guess you'll be measuring the sides and angles. Then you get these six numbers. Do you need all six? What are you going to do with the numbers?
We know what perfection looks like: 3 congruent segments, 3 congruent angles. Let us safely assume that no one drew a perfect triangle. So how far from perfect is it? What score would it get? How fair is your test?
I need you to work quietly by yourself for now. Get your tools: ruler, protractor, compass, calculator, whatever. Then figure out a way to test for equilateralness.
Working alone
They worked diligently and carefully. I appreciated seeing this student use her ruler to extend the side length to help her spot the angle measure more accurately. Julia asked, "Measure in centimeters, right? To the tenth?"
Working in small groups
Now I'm going to randomly put you in groups of three. In your small group, share your scoring strategy. Fight about it. Defend your methods. Eventually I will ask you to choose the best method from your group to present to the class.
By the way, just because someone in your group is in possession of a drawing that you know is far from perfect doesn't mean that his/her method for testing it should be dismissed.
Oh, hey, should a larger triangle deserve more points in your scoring system? I mean, is it harder to throw down three dots that are spaced farther apart?
I moved from group to group, listening to their discussions and observing their calculations.
Watching them, listening to them, asking them questions — I didn't want to be anywhere else.
Presenting to the whole class
One by one, the groups were eager to share. They questioned each other: Why 100? Why 30? Why divide by 3
Summary of what they shared on the board. You can see that 4 of 7 groups used either side lengths or angle measures and not both.
Voting on the best one
The scoring method from Gianna's group got the most votes with 9. Gabe volunteered, "None of these is spot on. But I don't know what the best way is either." I said, "Thanks for saying that, Gabe. Me neither. But I love what you guys are all coming up with!"
Over two days, no one mentioned using perimeter or area. And I vowed not to say anything — I wanted the kids to drive this entire lesson to wherever it needed to go.
Okay, Gianna's famous now, we'll refer to her group's method as "Gianna's formula" from now on. I need everyone to go back to the triangle you have and use this formula to find a score for it.
Testing another triangle
I made another set of copies from the originals (during my prep this time) and randomly passed these out. This very diligent work was still human, and I just felt each triangle deserved another pair of fresh eyes on it.
Now, you're going to apply Gianna's formula to another triangle. This way each person's triangle gets tested by two different classmates. Record your numbers on the board.
About 7 of the 20 sets of numbers had enough variance that I had to ask both scorers of each set to re-do their calculations and/or measurements. I then took the average of the two scores.
The results
I went back to the kids' original drawings and measured all the sides (with a ruler, thanks), applied Dan's formula using this calculator and here are the results. The names highlighted in yellow share the same rankings via Dan's and Gianna's formulas. The greens are off by just one.
We thought this lesson was pretty great. Maia said, "Our way was not too bad at all.”
Mr. Meyer's Taco Cart
Won't be long before I have to change the post title to Dr. Meyer's Taco Cart. This lesson went really well today.
Act 1: We watched the video clip. Their guesses:
Me: That was fun. Kinda split in the middle there with your guesses. But that question of who gets there first only gave you two choices, Dan or Ben. What other fun questions could we ask?
Student 1: The length of the road.
Me: I did say "fun."
Student 2: What their walking rates are?
Me: That's funner than "length of the road"?
Nathan: What is the fastest route?
Me: What do you mean by that? Can you come up and show the class what you meant?
Nathan came up and traced out the blue path with his finger, "They can walk to the cart like this..."
Me: Oh, okay, what is the fastest route? Yeah, right, because what if neither of the guys' routes was the fastest? This will give us a bunch of different guesses. Or, a similar question, where on the road should Dan or Ben enter to reach the taco in the shortest amount of time?
I gave each student this working placemat (Dan's Act 2 slide) for work space and the road strips to throw down a guess. They needed to line up road-on-paper perfectly with road-on-strip before marking their guess so that both marks indicated the same position.
Their guesses:
Act 2: The questions began.
Me: Okay, so now figure out how much time it takes to walk the route that YOU chose.
Student: Can we have the dimensions?
Me: Which one?
Student: All of them.
Me: Wrong answer. Try again.
They yelled out for the legs, the sides, the road, the hypotenuse.
Me: I'm just gonna give you one of the sides. Just one. Ask wisely.
For whatever reason, they agreed it should be the hypotenuse. (Dan did not give the length of the hypotenuse on his slide. I purposely put a white box there pretending like maybe they could ask for this length too.)
I gave them the hypotenuse as 650.0 feet. They stared back at me, faces scrunched up as if they were begging. They knew they needed another length to use their trusty little equation of a^2 +b^2 = c^2. I also gave them the walking rates on sand and on road.I walked around the room, peeking to see what they were doing on their papers. One student plugged the rates into the equation. He wrote this:
The 2 and 5 in the bottom equation came from the walking rates. The 105625 seemed to have come from multiplying the distance (650 feet) by the time to walk that distance on sand (325 seconds), then dividing this number by 2.
Half of them were just quiet — daydreaming, thinking.
Almost 10 minutes had passed. I said, "I gave you the hypotenuse, correct? But before I gave you the hypotenuse, don't you have the hypotenuse on your paper?"
Five seconds went by, then...
Gabe: Proportions!
Lauren: O my God, I hate that when I think too hard!
Nathan: I thought about measuring!
Janelly: Me too! But I was afraid to.
Maia: Do you measure with a ruler?
Janelly: No. Measure with your toe.
Ha!! Measure with your toe!! How can I not love these kids. So, they got busy with their rulers.
Maia: Should we measure in inches or in centimeters?
Someone: Centimeters! We never measure in inches in here.
Me: Not never. But why might we want to measure in centimeters for this problem?
Someone else: It'll be more accurate.
They measured carefully and checked each other's proportions. And because I only gave them the hypotenuse, their calculations for the legs weren't all the same, but close enough. We respected the margin of error when using a ruler.
It was another day that I didn't want this class to end. But with only five minutes left, I had to wrap up the lesson.
Me: What's the chance that your guess happened to be the fastest route?
Various students: Not likely. Way off. I think mine is perfect — I picked exactly half-way on the road. More to the left, I think.
Me: So, after you figure out the time for your route tonight, I want you to pick another point on the road. Do two sets of calculations.
Nicole: Like do it again for a different location?
Me: Yes. And if you chose your second point to be on the left of your original guess, then why? Or to the right, why?
Thomas: Will you tell us who got closest tomorrow?
Me: Sure. I have your original guesses already [on the strips]. And guess what? The student with the closest answer gets a Del Taco lunch!
Students: Really? What about Taco Bell? I love tacos! I hope mine is closest! Are we taking a field trip there?
Me: Fine. Taco Bell or Del Taco, your choice.
Act 3: We'll have to wait until tomorrow to see everyone's work and answers for their chosen paths. I'll reveal this again with the green line drawn in to indicate the best spot to enter the road. But it would be so cool to get some numbers right near the minimum time. Maybe I'll bring up differentiation. Maybe we'll plug something into Wolfram Alpha.
What made this lesson work for my kids:
Nathan's question of "what is the fastest route?" allowed for a lot of entry points than just "who would get there first, Dan or Ben?"
Giving them just one side of the triangle was really the best thing I did for this group of kids. Because once they figured out the other legs, they just tapped away on their calculators — already comfortable with the equations they've worked with before.
The strips provided me with an easy way to display student guesses.
Printing out Dan's Act-2 slide as work placemats for students allowed uniform access to the beach scene and made measuring the triangle sides possible.
I only thought to ask the kids to pick another point to figure out for homework just seconds before. Glad I did because we just doubled our data points!
The promise of Del Taco. No, I can't just give to the kid with the closest answer. I'll make sure I bring in enough for the class. This fun lesson deserves a fun closing.
Thank you, Dan.
Updated 11/07/12
