My Other Math Sites
Lessons From the Classroom
A First Grade Lesson Using Vertical Whiteboards
As we wrapped up our book study of Peter Liljedahl’s Building Thinking Classrooms, and in my role as TOSA (teacher on special assignment) I do not have a classroom of my own, I asked one of our first-grade teachers if I may facilitate a thinking task using vertical whiteboards with her students.
To supplement the teachers’ classrooms with more whiteboard space, I reached out to Wipebook for durable dry-erase chart paper surfaces (with grids on one side!). The reusable Flipcharts they sent were the perfect tool to use over her glass doors and windows! They can easily be removed and reused as many times as you want — an eco-friendly solution to reduce paper waste. Our district has also purchased whiteboard easels with adjustable heights.
The teacher already had her groups paired up with their “thinking partners.” Each whiteboard space was labeled such as “Blue 1 and 2” so the students knew where they would be working.
We had the students sitting on the carpet area for me to launch the task. I taped this flower image on the board and asked how many circles they saw. They said seven.
Then, I counted out along with them seven sticky notes, marked from 1 to 7. This great Flower Petal Puzzles thinking task is from Dan Finkel, founder of Math for Love. The intent of the task was for students to write the numbers inside the circles/petals and to keep track of numbers they have used by crossing them off at the bottom of the page. Dan also suggested using counters. However, I wanted the students to do these vertically, so counters would not work, and writing numbers in and erasing them is challenging and cumbersome, especially for young students. The use of sticky notes was really key in helping students stay focused on the task because it was easy to swap out numbers and keep their brains moving.
I called up one student at a time to place a numbered sticky note into an empty petal without further instruction or stating what the goal was. When there was only one space left, I said it was my turn and placed the number in. I wish I had taken a picture of the board at this point. But since I didn’t, below is a picture of the flower with randomly placed numbers. I then told them these three petals are connected and asked them for the sum and wrote the equation.
We found the sum of the other two groups.
It was not until then that I told them the challenge of the task was to have all three sums be the same. And that the sum is 10. So off they went with their thinking partners to their whiteboards!
Sure enough one pair got the solution within 5 minutes, and we asked them to keep their equations (not erase them), and to do the next challenge which was to get a sum of 12 for all three equations. Some time later, all of the groups except for two got the 10 puzzle. We came over and applauded them for their perseverance and asked if they would like a clue. They nodded and we let them know to place the number 1 at the center of the flower. When this group got the solution with the hint, the girl excitedly said to me, “This is a lot of fun!”
I tweeted this out later as truly it was the best hour with first graders for me. So much perseverance, so much thinking, so much fun. The groups were in different stages (finding sums of 10, 12, and 14), and two groups ended the hour working on the “ultimate challenge” of creating their own puzzle using a 9-petaled flower.
Updated 04/11/2022
I just thought of a way to take this 1st grade task to next level. https://t.co/7vQzKk4kuP
— Fawn Nguyen (@fawnpnguyen) April 1, 2022
7 circles: possible sums r 10, 12, 14 (what #s go in middle circle of these sums, respectively?)
What r possible sums for n (odd) circles? How many equations [of equal sum] for each n?
Euclid's Algorithm
I show my 6th graders this image, pointing out that this picture represents the two numbers 1 and 1 that I'd entered at the top.
I then ask them to give me two new numbers — any two positive integers [that are 10 or less, for now] — and the computer will draw a new picture. As each set of new numbers is entered and the corresponding picture is generated on the screen, I ask students to jot down their "I notice, I wonder" in Google Form and to draw a rough sketch of it in their journal. After a few sets of numbers, I ask students to imagine and/or draw a rough sketch of what they think the picture will look like before I hit the update button.
These are the pairs of numbers they'd asked for and their corresponding pictures, listed in the order that was asked.
I love that the kids are asking for...
6, 6 after the initial 1, 1
3, 8 after 3, 1
8, 3 after 3, 8
But, when sets 9, 9, and 7, 1 are asked at the end there, I say to the class, “Hey, what if figuring out this puzzle — which is how the computer draws the picture given two numbers — gets you a million dollars. And you get to ask for sample sketches like you've been asking, except that each sample costs you some money! So, make each request worth it. Let it prove or disprove your conjecture. Ask carefully."
I love the OHHHs and AHHHs after each picture is revealed. But no one is claiming that he/she had drawn the same diagram. I pause longer for them to write down their noticing and wondering.
I now say, "You may only ask for four more sets of numbers. Remember, make a request that would test your conjecture."
I ask a normally quiet student. She says, "10, 3."
Another student wants to know what "100, 5" looks like.
"What about 8, 5?" I reply, "Sure, but draw it in your journal first." They are fully engaged. Then I say, "Now, share your drawing with a neighbor."
I ask, "Did anyone sketch the same thing as their neighbor?" They're shaking their heads, and I say, "That's pretty crazy! Do you think yours is more 'correct' than your neighbor's?"
I reveal 8, 5.
The last request is 23, 75.
What some of them have written [with minor edits from me]:
When we did the same two numbers the shape didn't change but when we did different numbers it changed. Why does it divide into little parts within a square when we put 3,8? When we did 8,3 the number switched around. I wonder if the two numbers are dividing to make the shape. How can you figure out the number when it can't divide easily. My drawing for 8,5 was one whole and 5 little squares. The 23,75 was a little confusing to me.
They're different, they are the length and width, and when the two numbers are the same it's just one cube. I notice that if it can be simplified, it is. Example: 6,2 = 3,1. I don't understand 8,3, 5,9, 23,75 or 8,5. But I did notice that the smaller the parts of the shape are, the lighter shade of blue they are.
I observe that when the same numbers are entered it equals to a blue square. If the first number is bigger than 2 then it will add one more square. I wonder if you double the number for each number will it be the same shape. I wonder why for 3,8 it has one square with three parts. I observed that if you divided the first number by the second it will equal to the number of squares. For 8,5 I didn't get the right sketch. The sketch was one square with half of a square cut in half, then in one half is has a strip that is cut in half. I wonder why it has half of a square. I think that my answer for how it figures it out is right, but I don't know how it comes up with that picture for 8,5.
When you do 1 and 1 it doesn't change because we tried 6 and 6 it didn't change and if we put 3 and 1 it did change. I saw that when we did any number like 3 and 1 is 3 ones. So I think that all you have to do is divide something by something = the first number that you put in but if you can't divide by 2 then I'm wrong. I'm not sure that I got this right but this is what I think.
For the first one 1 and 1 I thought it would be a small one by one cube. What threw my off was the 6 by 6 because the size did not change. For the 8 and 5 I drew a big block and and 5 little ones, but my image was wrong. I also wondered if the first number was the amount of shapes that would appear, but I was wrong again. I don't understand yet. I tried looking for a pattern, but couldn't find one.
When I tried 8,5, my answer was almost right. I had the one big square right, the half square right, but then I got the little squares wrong. I think that the way the computer does it is dividing the first number by the second number. I am confident that if you put the numbers 10 and 5 in, it will show 2 squares. When using the diagram, the second number will represent the vertical side.
What I've been noticing was that if you put the bigger # in the front and the small # last then it would be like a rectangle. I've also been noticing that if you put the same #'s it would like keep on drawing a square. So someone said what could (8,5) look like and Ms.Nguyen showed us the drawing and the I notice that nobody got it right. I was expecting something like smaller because the #'s were small they weren't as big, but at least I tried to get it correct but I drew something a little bit smaller than that. I also wondered why when we put the same #'s together why do they all become a square that's what I wonder.
For 5, 12 I notice that it is two big squares, two smaller squares, and two tiny squares, I thought it was going to show 1 big block and another big block but that one would be cut off at the bottom or not a whole block. I also notice that the pattern is the first number multiplied by what equals the second number and the number that is missing is the amount of blocks that is created. I thought I knew it but I don't really get the ones with a bigger number first and the smaller one last. I thought I knew what 9, 23 was going to be but it the result was surprising. It didn't look at all how I thought it was going to. The website is pretty cool, but one thing I didn't understand was the placement of the small blocks and what they stand for. Like some were really tiny and some were small but I don't know what they stand for. But I bet if someone explains it to me I probably will understand perfectly and feel dumb.
The only thing that I am sure that I know is that when the first number is larger than the second, the shape is wide and when the second number is larger, the shape is tall. Other than that I am very confused.
Then, together as a whole class, they agree on the following;
When both numbers are the same, then the picture is one square.
The computer simplifies the two numbers, such that a picture for 6, 3 is the same for 2, 1.
When the second number is a 1, then the picture shows the first number of squares. For example, 7 and 1 would form a picture of 7 squares, and 100 and 1 would form 100 squares.
The first number is the horizontal dimension, while the second is the vertical.
They are all squares.
The dismissal bell is about to ring, and I want to teach forever.
Tomorrow, we'll spend some time with one set of numbers, like 10, 3 or 8, 5. We'll dissect the diagram. Play around with a few more. Practice sketching a few. We'll write out the equations that go with each diagram. I'll guide them into noticing the size of the smallest square in relation to the two numbers.
I found this investigation at underground mathematics. The site describes itself as having "rich resources for teaching A level mathematics." From what I understand, "A level" means advanced level mathematics consisting of core modules ranging from quadratic, logarithms, geometric/arithmetic series, differentiation/integration, etc.
Perfect for my 6th graders who continue to torment me with their arithmetic atrocities, such as, 3² = 6 and 5 ÷ 10 = 2.
While the original task is scripted for older and more advanced students, I found in it what I needed to make it rich and appropriately complex for my 6th graders.
Hail, Euclid's algorithm!
Good-Enough-for-Now Curriculum
I did my first webinar last week as a precursor to my talk at NCTM's Innov8 Conference next month. I thought it went okay — or horribly — just tough to be the only person with the mic and not being able to actually see the attendees. It was weird.
There are a few slides from the webinar that I'd like to share here mainly because I'm still thinking about them and writing anything down helps me set the wobbly gelatin.
Two weeks ago I presented at an independent school that's Preschool through Grade 8. Afterward, I was given a quick tour of the school — the 33-acre campus gleamed with pride in its thoughtful architecture, manicured grounds, state-of-the-art this and that, and a smorgasbord of elective offerings, including Mandarin and photography.
My school is Kindergarten through Grade 8, and the similarity between my school and this independent school pretty much ends there. I teach four classes, my smallest class has 23 8th graders, the other three, all 6th graders, have 32, 35, and 36 students. We're a Title 1 public school.
I bring up the private school and my public school because, like apples and pomegranates, they are quite different. So, when we do PD and share whatever it is that we share about education and serving children, we need to be mindful about the space that each teacher occupies in her building and be mindful about the children who come into that space.
When someone shares something with me, one or more of these thoughts cross my mind: 1) I can see how that would work with my students, 2) I can see how I might adapt this to fit my kids, 3) This person is afraid of children or unaware that children are people, 4) Nobody cares.
Likewise, when I have the stage to share, I'm assuming you have similar thoughts of my work. But I beg you to think about the space that I share with my students.
Below is a quasi rating scale of "critical thinking demand" that I'd created to place the types of tasks that I regularly give to my students. And this scale is only possible because I'm mindful of the tasks' contents and my own pedagogical content knowledge to facilitate these tasks.
What are these six things? The resources for these are on this spreadsheet.
1 & 2. Assessment and Textbook: We're using CPM. [04/07/2022: We now use Open Up Resources and Desmos.]
3. Warm-up: Due to our new block schedule, we've only been doing number talks and visual patterns.
4. Problem-of-the-Week
5. Task
6. PS (Problem Solving)
Do these 6 things align to the curriculum?
The slide below shows the 4 types of tasks that are aligned to the curriculum, or that when I pick a PoW or Task, I make sure it correlates to the concepts and skills that we're working on in the textbook. Therefore, it's entirely intentional that the warm-up and PS are not aligned because critical thinking and creative thinking are not objects that we can place in a box or things that I can string along some prescribed continuum.
All 6 types of tasks are of course important to me. I try to implement them consistently with equal commitment and rigor to support and foster the 8 math practices.
Which ones get graded?
I don't grade textbook exercises, i.e., homework, because I can't think of a bigger waste of my time. I post the answers [in Google Classroom] the day after I assign them. I don't grade PS because that's when I ask students to take a risk, persevere, appreciate the struggle. I don't grade warm-up because I don't like cats.
I'm finally comfortable with this, something I've been fine-tuning each year (more like each grading period) for the last 5 years. I could be a passive aggressive perfectionist — or just an asshole when it comes to getting something right — so it's no small admission to say that I'm comfortable with anything.
It's about finding a balance, an ongoing juggling act between building concepts and practicing skills, between problem-posing and answer-getting, between teacher talk and student talk, between group work and individual work, between shredding the evidence and preserving it. Then ice cream wins everything.
Here's the thing. We want to build a math curriculum that makes kids look forward to coming to class everyday. I trust that that's true for more than half of my students — this could mean anywhere between 51% and 80%. I think we're doing something wrong when kids look forward to just Measurement Monday, Tetrahedron Tuesday, or Function Friday. Math should not be fun only when students get to play math "games"!
Greenies
Our black lab Mandy is 3.5 years old, weighs a ton, and her breath used to smell like death. Until we started giving her one Greenies a day. My husband orders them from Amazon, he also gets them for our neighbor's small dog Bailey.
Although both boxes weigh the same 36 ounces, Bailey gets 130 treats in the Teenie size and Mandy gets only 24 in the Large size. This caught my attention which led to this task with my 8th graders who happen to be working with similar shapes. (Like I had planned this all along.)
Greenies are sold in various size packages. I'm interested in the 27-oz and 36-oz.
To launch the task, I hold up the 2 treats: 1 Large and 1 Teenie. I tell them that there are 24 Large ones in a 36-oz package, and I want them I guess how many Teenie ones are in a 36-oz package.
Then I give the students these:
The photo of the 2 packages (so they know only the highlighted information in the table above).
Each group of 3 students get two real treats: 1 Large and 1 Teenie.
In return, the students need to give me these:
A 2-dimensional outline (with dimensions labeled) of what a Petite, a Regular, and a Jumbo may look like. For example, these are the actual outlines of the Large and Teenie. They may write down the thickness also.
A completed table with the missing counts filled in.
Highlights of this task:
Kids use some known information to construct new information. They use modeling to figure out what the other sizes may look like and how many of them would fit in a 27-oz or 36-oz package.
It's kinda messy and weird. While the kids can measure whatever lengths of a treat, how do these numbers translate into the mass of each treat?
It's good to work with solid objects instead of just flat polygons when learning similar shapes.
The reveal (Act 3) of something like this is always a lot of fun. Not only the reveal in the count per package, but also how close their outline sketches are to the actual treats when I bring in the Petite, Regular, and Jumbo.
How much does a 36-oz package cost?
Is the Jumbo a shot in the dark? Would kids think to ask me for the size of the dogs? How does this help, if at all?
Driving Them Nuts
I'm proud of my students. I'm proud of what we do in room 15. My classroom. My home away from home for the last 11 years. These bopping teenagers, sullen one minute bubbly the next, hormonal but invincible. They don't all love math like I do. (Not everyone loves college football, but we get along.)
What is happening in room 15 is the loud and proud math culture that we have set in place. We build it from day one — then we continue to do, say, and write stuff to sustain and strengthen the culture because we know our behaviors are our best evidence that this culture exists.
Here's one piece of that evidence:
Her frustration resulted in her loving the problem. Her last sentence is an enormous celebration of how much we honor the process of problem solving. Her classmates had their own reflections — short snippets of how they engaged in the problem.
They worked hard on the problem because it was driving them nuts. It's not unusual to hear kids blurt out, "This problem is making me crazy!" Or, "I won't be able to think about anything else until I get this!" Now, they've owned it. This isn't about a letter grade any more; and it certainly isn't about me.
This was the problem they'd worked on.
The Missing Area
A 10 by 16 rectangle is attached to a triangle as shown below. If the purple section is 24 square units, then what is the area of the yellow section of the rectangle?
John Golden, GeoGebra extraordinaire, created an animated gif of this problem.
Mike Lawler solved this problem using similar triangles.
A Simpler Solution
I'm guessing this was about 5 years ago. I was at an all-day workshop when a high school math teacher, sitting next to me, asked about the PoW (from mathforum.org) that I had assigned to my students. I happened to have an extra copy in my backpack and gave it to her.
Dad's Cookies [Problem #2959]
Dad bakes some cookies. He eats one hot out of the oven and leaves the rest on the counter to cool. He goes outside to read.
Dave comes into the kitchen and finds the cookies. Since he is hungry, he eats half a dozen of them.
Then Kate wanders by, feeling rather hungry as well. She eats half as many as Dave did.
Jim and Eileen walk through next, each of them eats one third of the remaining cookies.
Hollis comes into the kitchen and eats half of the cookies that are left on the counter.
Last of all, Mom eats just one cookie.
Dad comes back inside, ready to pig out. "Hey!" he exclaims, "There is only one cookie left!"
How many cookies did Dad bake in all?
Maybe you'd like to work on this problem before reading on.
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The teacher started solving the problem. She was really into it, so much so that I felt she'd ignored much of what our presenter was presenting at the time. She ran out of paper and grabbed some more. She looked up from her papers at one point and said something that I interpreted as I-know-this-problem-is-not-that-hard-but-what-the-fuck.
It was now morning break.
She worked on it some more.
By lunch time, she asked, "Okay, how do you solve this?" I read the problem again and drew some boxes on top of the paper that she'd written on. (Inside the green.)
She knew I'd solved the problem with a few simple sketches because she understood the drawings and what they represented. I just really appreciated her perseverance.
I share this with you because a few nights ago I was at our local Math Teachers' Circle where Joshua Zucker led us through some fantastic activities with Zome models. We were asked for the volume of various polyhedrons relative to one another. Our group really struggled on one of the shapes. We used formulas and equations only to get completely befuddled, and our work ended up looking like one of the papers above.
Over the years I've heard a few students tell me, "Mrs. Nguyen, my uncle is an engineer, and he can't help me with the PoW." Substitute uncle with another grown-up family member. Substitute engineer with another profession, including math teacher. I remember getting a note from one of my student's tutor letting me know that I shouldn't be giving 6th graders problems that he himself cannot solve. (The student's parent fired him upon learning this.)
I like to think that my love of problem solving will rub off on my kids. I hope they will love the power of drawing rectangles as much I do.
Hotel Snap
[NCTM Illuminations has my blessings and thanks to re-write and feature this lesson on their site.]
Andrew Stadel and I recently presented this task at the 2013 CMC's North and South Conferences.
The Challenge
As a team, build a hotel that yields the highest profit [score].
Rules and guidelines for building the hotel
Each cube represents a hotel room. All 50 cubes must be used.
Hotel must stand freely on at least one side of cube. Here’s a non-example because it’s tilted.
Entire hotel is one piece. A non-example due to yellow cube not attached:
All rooms must have at least one window, a window is any exposed vertical side of cube. The white cube below has no window.
Building costs and tax (daily rate)
Land costs $400 per square unit.
Land refers to outline of top view of building.
All enclosed land is charged, for example, squares marked 8, 9, 10, and 11 in the left outline are open space, but because there's no access to the outside, you are charged for these 4 square units.
A roof costs $10 each, roof is any exposed top side of cube.
A window costs $5 each.
Tax on height of building is calculated by multiplying the tax rate for the highest floor by the total land cost.
Floors 1-10 —> 50%
Floors 11-20 —> 1000%
Floors 21-30 —> 2000%
Floors 31-40 —> 3000%
Floors 41-50 —> 5000%
Income from each type of room (daily rate)
The more windows, the more income.
4 windows, 1 roof = $600
4 windows, 0 roof = $500
3 windows, 1 roof = $300
3 windows, 0 roof = $250
2 windows, 1 roof = $200
2 windows, 0 roof = $175
1 window, 1 roof = $150
1 window, 0 roof = $125
Scoring
Your net profit/loss income will be checked for accuracy. A deduction of 50% of your error will be applied to the actual number. For example, your building nets a profit of $13,500, but your group submits a profit of $15,000, therefore you're off by $1,500. Then 50% of this error ($750) will be deducted from the $13,500 to give your team a score of $12,750.
If your calculations are right on, then your team's score will be awarded an extra $1,000.
Adapting this lesson
Change the number of cubes, as few as 10-15 cubes for younger kids, and maybe up to 100 cubes for high school students.
Also for younger kids, have the Excel file (more on this later) readily available on computers so kids can go back and forth between checking their profit margin and tweaking their hotel rooms — so no calculations needed on their part, they just need to be able to know how to count the different types of rooms.
Older students can create the spreadsheet, it's great practice for understanding how cells work and formulating equations.
Adjust the time for individual and group work based on your expectations.
Modify, take away, or add to the rules and guidelines.
Change any of the costs/income/tax numbers.
Change how you reward accuracy or penalize mistakes.
Ask each group to estimate and rank the profit margins of other teams' hotels just by looking at them (like on a -5 to +5 scale, -5 for biggest loss and +5 for biggest profit).
Ask, "What if all costs and tax stay the same, but now the incomes for the rooms are all reversed so that 4-window-1-roof earns only $125 while 1-window-0-roof earns $600? How would you build your hotel using the same rules?"
If I were to do this with my 6th graders, I'd first have everyone build the same 10-cube hotel with me, then we'd use this hotel to familiarize ourselves with the different types of rooms and tally them up. We could calculate the costs and income together for practice.
How I ran the lesson with 8th graders
(I did this lesson with two classes of 8th graders, one geometry and one algebra. We have 57-minute periods. This lesson took 2.5 periods. I'm so scripted here because the one thing that kids wished they had more of was time. Your teacher instruction needs to be tight and supplies distribution needs to be efficient to allow for all the student work time needed.)
State the challenge of task.
State the rules/guidelines.
Give each student a zip bag with 50 cubes — but I first took time to show them that how they see the cubes now is how they need to be put away when we clean up.
Give each student this cost_income_sheet:
Set timer for 15 minutes for individual work, reminding students that later they will be randomly assigned into groups of 3 to work on one hotel.
While students are working, I use Instant Classroom to put kids into groups of 3.
When the timer goes off, I give the following instructions:
I now need you to listen to directions for working with your teammates, and when I'm done giving these instructions, I will set the timer for 25 minutes for you to work. During this time, you'll need to do the following:
Share and discuss the best model to represent your team's hotel.
You may modify this chosen hotel, or you can even start from scratch, but watch your time.
Use this tally_sheet to record your room counts, land, roofs, etc. — remember this is just a tally sheet, so you want to do all your calculations on whiteboard.
On the large whiteboard, divide it into 4 quadrants, and your group will need to fill in 3 of the 4 quadrants answering these questions. Watch your time carefully because when the timer goes off, the hotel you have in front of you is the one you must keep as is.
Allow a couple of minutes for groups to get supplies and settle down together with their individual hotel models. One tally sheet needs to be passed out to each group.
Timer is set for 25 minutes. Monitor the groups, check for understanding, and listen in for building strategies. Keep counting down the time every 5 minutes.
When timer goes off, ask groups to break down the hotels that were not selected and put the cubes neatly back into the zip bags and return to the front.
Next day...
Kids get back into their groups with their whiteboard and hotel. Some groups may still need to finish filling in their tally sheets and whiteboards.
The question for the 4th quadrant is now asked: "If you could relocate just 5 cubes on your hotel, where would you place them?" Remember that you are no longer allowed to change your hotel, this question is just a what-if scenario.
Allow 10 minutes for groups to finish filling in their tally sheets and answer all 4 questions on whiteboard.
As indicated on bottom of tally sheet, groups are asked to bring up the tally sheets to teacher to check their calculations and arrive at the final score.
The Excel sheet
I have two files: one locked and one unlocked. The "locked" one so of course no one can inadvertently change the cells, especially wise if you give kids access to the file to use on their own - only the blue cells to enter data are open. The "unlocked" file is so you can edit as you please.
Notice that all the cells appear in the same order as the tally sheet, makes for quick entries with instant results. My husband gets most of the credit for this, I still claim credit for making it simpler.
Math Practices
I found this form 8_MP_questions_to_ask online and can't find it again to cite the source.
I'm making good use of this form after almost every task now. Students do this for homework. I ask them to highlight the questions that came up during the activity - from teacher, peers, and self.
We discuss this as a whole class the next day and generally come to a consensus of which math practices our task had fostered.
Reflections
We presented this lesson because it went really well in the classroom. The kids were completely engaged. They collaborated, talked a lot among themselves about what they were building and thinking, but they were pretty much dead silent during the initial individual work time. I love seeing the structural varieties — this was true too among the hotels built by math teachers.
To not spoil the fun, I won't post pictures of the high-profit hotels here, but I'd love to learn what some of your profit numbers are from your class.
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.
Technology and Construction Papers are Cool
Big-time struggles for my 8th graders on this problem from Five Triangles mathematics.
The one student who got it also struggled, but he was good about our rule of "never tell an answer." What I gathered from seeing their boards and listening to them explain:
They wanted to find the area of the parallelogram first.
So they needed to find the height.
To find the height, they reasoned that triangle ABC was isosceles, making the perpendicular bisector AY also be the height of the parallelogram. Then they could use the Pythagorean theorem to find AY. (I can't get AB and DC to be exactly 6.00 cm.)
So I showed them the parallelogram below just to be less helpful and to remind them about assuming something is isosceles because it looks it.
Without a clear way to find the height, frustration mounted.
But major props to them for persevering as you can tell from their boards that they tried to dissect the parallelogram into even more pieces in hoping they'd find what the shaded piece would be equal to.
The next day... I let them struggle some more. They weren't seeing the key pieces (at least to me they were key), so I told them to look at the "relationships" among the pieces. They were pretty sure the diagonals of a parallelogram cut it into 4 triangles of equal area. Then they were stuck again. I then asked them to construct this parallelogram using Geometer's Sketchpad (GSP) and find the answer to the question posed. Soon one student asked, "What should the angles be?" I replied, "It doesn't say in the problem, so I don't know. But it is a parallelogram, so make sure you have the same side lengths." Without much trouble, they found the answer. They also realized that even though they may not have created congruent parallelograms among themselves, they all came up with the answer of 0.17 or 1/6.
Me: So why did I ask you to use GSP to do this?
Student: Because we couldn't do it by hand?
M: Maybe. Or maybe you didn't have enough time. But why do we use GSP in general? What's the difference between doing a construction by hand and doing it using GSP?
S: So we can move things around.
M: Right! It's dynamic! Then this is what I want you all to do. Move things around. Drag the side lengths. Change the side lengths. But while you're doing all this messing around with the parallelogram, I want you to pay attention to the numbers, the measurements.
Not long before many of them chimed in excitedly: The ratio doesn't change! I came by each kid's screen to check their work. (These are my reconstructions of some that I saw.)
To make the most of their construction (and a good problem), I then asked them to find the ratios among all the pieces that they saw. I wish you could hear them. A bunch of them kept making the same exclamations: So cool! Mind blown. While only one student found the solution by hand, the rest of them felt pretty good about their construction of parallelograms, and they discovered important relationships among the pieces. Technology came to the rescue. Here's how I used colored construction papers to help students see the relationships better.
Advantages:
Instead of having to write down or say "triangle ABC," "triangle BEC," or "quadrilateral FDGE," I can just refer to each one by color.
By not having to follow the letters of ABC or GCE, kids can focus on the visuals and see the relationships more quickly.
I can show congruence by placing one piece on top of another.
Something like this:
The diagonals cut the parallelogram into two pairs of congruent triangles. Therefore, each blue is 1/4 of whole. (I reflected each pair so they can see the congruence.)
Below left: green is 1/2 of whole.
Below right: red + yellow = 1/2 of green, or 1/4 of whole, because they share the same height and half the base.
Below right: red + pink is 1/4 of whole (same size as a blue). Since left and right images below are equal in area, and red is red, therefore yellow equals pink.
Blue + yellow (combined) is a similar triangle with pink due to AA postulate. And because their side lengths are 6 to 3, or 2 to 1, then their areas are 4 to 1. Let's call the area of pink as 1, and we learned from above that yellow equals pink, so yellow is also 1. This leaves blue to be 3.
Putting all the pieces together, we have a total area of 12, so the shaded part (red) is 2/12 or 1/6.
Kate
Add to what Robert said, your kids have enough GSP skills to use it to investigate a problem. Kudos! That's a tough trick to pull off. I did it a different way! I found constructing a supporting parallel line dissected the quadrilateral in a helpful way. http://www.geogebratube.org/student/m83241
February 10, 2014 12:10 PMfawnnguyen wrote:
Hi Kate. Yay, beautiful work!! We did this activity just last week and one of the kids asked at the start, "How do you find the area of a non-parallelogram or trapezoid [FDEG] anyway?" I replied, "Well, can you divide that shape into other shapes that you can find the areas for?" So, I saw the groups draw in that parallel line that you have. They were able to find the area of pink triangle as 1/8 of whole but struggled with finding area of blue triangle during their paper/pencil work.Thank you, Kate.