Category: Actual Math

Peas in a Pod
This year, teachers and principals on every elementary site in my district have asked me about 3Act Math. After a dozen demo lessons, many of them have delivered lessons in their classes.

In all the demos, I’ve pointed people to three places:
- Graham Fletcher’s database of K-5 tasks for 3Act Math Tasks
- This worksheet I cobbled together from several #MTBOS sources (script on page 3 for teachers who need hand-holding).
- Dan Meyer’s description of 3Act Math and how it works.
- A YouTube playlist with all the videos is linked at the bottom of this post.
Cheryl Demus of Allen Avenue Elementary School teaches Kindergarten, bravely charges into new ideas and learning, is confident in her ability to run a class well, and regularly takes on student teachers from nearby universities.

Of course, as the Instructional Coach, I don’t have favorite teachers and serve all my staff equally.
So I’ll just say that Cheryl’s class is one I’d be thrilled to enroll my kids into.

For those unfamiliar with the 3Act Math task, I’ll give brief rundowns as we go (videos at the end).

Act One – the Hook

Peas in a Pod (Act 1) from Graham Fletcher on Vimeo.

Start with the media and get students asking questions. This is the time to make sure all students can get on board, which means we don’t zip to the math too quickly.

For me, that means starting with,

“Tell your neighbor what you’re thinking right now,”

before asking,

“What do you notice and what do you wonder?”
[youtube https://www.youtube.com/watch?v=XzWbHXNq1Pk]

As you’ll notice in the video, I treat the student comment, “I notice there are hands in the video,” with the same degree of interest as “I wonder how many peas in all of the pods.” It’s important for Act One to celebrate all student responses, and this is most clear at the K-2 level, where most of the responses are hot nonsense.

It’s highly unusual for a student to suggest a question so close to the one that I want. Most of the time, the teacher has to fake it with something like,

“Wanna know what I heard walking around? I heard people asking, [the central question I want them to ask]. Show me a thumbs-up if you were wondering that, too.”

Of course, the whole room raises a thumbs-up, because sure now they’re wondering that, too.

Graham Fletcher (author of this task and many others) calls this, “co-creating the question.” I call it “nudging students when they need a nudge, but letting them wander for a bit first.”

You know your class; you know how much wandering is helpful.

Number Line

The addition of the barbell on the worksheet (which I totally copied from Andrew Stadel, author of Estimation180) provides two important scaffolds that include all students.

1.) Little students have number sense that’s all over the place. A guess that’s too high and a guess that’s too low help them to fine-tune their guesses. I’ve seen many Kindergarten teachers ask, “Is 400 a reasonable guess?”

To a six-year-old? Of course it is.

Tracy Zager notes in her fantastic book that intuition, like any skill or muscle, can be strengthened by practicing. So if we want better Number Sense and more reasonable answers, we must give students a chance to practice reasonableness.

Note the questions around 3:30.
[youtube https://www.youtube.com/watch?v=wKseaFya0zs]

Did you catch the spot where 6-year-olds think that “between” and “middle” are synonyms? Adorable. My inner high-school teacher took a seething breath and muttered, “SMP6.”

Act Two – the Journey

Once we’ve established a central question, now we need more information to answer it.

“What are some clues that would help answer that question?”

or

“What information might be helpful to figure out ______”

This was a great snapshot for how messy Act Two can be with little ones. In the playlist below, videos 5 through 7 are all dealing with Act Two.
[youtube https://www.youtube.com/watch?v=Apqp8dLWWqM&list=PLuXQQj0Y_GB7J63yPYsVte__6KCHKX3Jo&index=5]

I just realized that Graham (the author) made Act Two media from a Word Document download so that anybody can scaffold for their class. Here’s how it looks:

Thankfully, Cheryl had advised me to ask, “What math can we do here to answer our question?” As soon as we got too deep in the weeds looking for bugs, I caved and pulled that question out.
[youtube https://www.youtube.com/watch?v=Apqp8dLWWqM]
Now, Math Teachers can testify that I’m not subtle using confusion as my default mood for classroom moves with little ones:
- What was that number for? I got lost.
- Where did this number come from?
- Wait, so to do [operation], we should [casual language]?
- I think I get it, someone explain that a different way.
[youtube https://www.youtube.com/watch?v=Apqp8dLWWqM]
There’s a spot in there when students start yelling, “six” out of nowhere. I asked Cheryl what that was about.
“You do your twos wrong,” she smiled and pointed to the correct two in foot-high cardboard.

Apparently, kids this age still flip (reflect) their letters and numbers, and my curly 2 looks like a flipped 6.

Andbafter probably 15 minutes of peas and pods and additionbwe arrived at 17 peas.

Time to check our work.

Act Three – the Reveal

Peas in a Pod (Act 3) from Graham Fletcher on Vimeo.

After Act Three, many students didn’t quite get that we were correct with our math, so Mr. Vaudrey choral counted and injected surprise and excitement and I’m kinda glad there’s no footage.

I love surprise and excitement, but I’m not sure how I could have better tied the answer to the process.

Possibly because the process is way more important to me.

Stuff I’ll Change For Next Time

In Part 6, I asked Kindergarteners to count silently in their mind. That might be the record for “Greenest Move the Middle School Teacher Tried With Kindergarteners.”

Big picture, the pace was blisteringly fast when I rewatched the footage. The teacher had no issue, but I’ll bet a coding of student responses would reveal my attention going to about 1/3 of the students.

I didn’t check for EL students or students with processing delays, but it’s safe to say that the “good math” students in my lesson where the fastest and the loudest. Ouch.

Related to that idea; I’m being reminded how difficult it can be for me to implement equitable practices without some kind of a system. If I just trust myself with, “I know the research and I know bias is a problem,” there’s a good chance I’ll just do the same kind of teaching y’all saw in Video 6.

UPDATE: After reading this post, Cheryl weighed in with another sentence frame she likes to use:

I’m hearing a lot from the same people, and I would love to hear what someone else thinks.

What Else?

Primary grade teachers, I know y’all have some feedback for me.

Let’s have it. I wouldn’t post all this if I wasn’t looking to get better.

~Matt “Curly Two” Vaudrey

UPDATE: 22 April 2019 – Later that day, a screw popped out of the teacher chair. Students were convinced it was because the “big man” sat in it.

And here’s a playlist with all the videos:

[youtube https://www.youtube.com/watch?v=XzWbHXNq1Pk&list=PLuXQQj0Y_GB7J63yPYsVte__6KCHKX3Jo]
April 20, 2019
3Act Math for Kinder

Yesterday, Mary and I co-taught a lesson with 3-Act Math to her class of 19 Kindergarteners at Oak Mesa Elementary.

Mrs. Fabela: And itbs okay if you have different ideas, right?
Kindergarteners: Right!
F: Because if we all think the same, we are…?
K: Boooooriiiiiing!

And then we did a @gfletchy task together.
Blog post coming.#elemmathchat #Bonitians

b Matt Vaudrey (@MrVaudrey) April 19, 2018

Last month, she sat next to me at CUE as Amanda Haughs showcased a fistful of accessible ways to change math instruction. Mary was quietly scribbling notes the whole time and… I’m not sure who nudged whom, but we picked yesterday for co-teaching a lesson that Mary found on Graham Fletcher’s K-5 repository of 3-act math lessons.

During our chat, we opted not to use the handout I built with some other K-5 teachers, and you’ll see why.

Act One – The Hook

Fabela: Friends, this is Mr. Matt* and he’s going to do math time with us today!
Kids: [stare vacantly]
Vaudrey: Boys and girls, I have a video to show you, and here it is.

Vaudrey: [whispering] Whisper to your neighbor what you’re thinking right now.
Kids: There were eggs and they fell out and she opened the door and it’s probably too full in there do you think that’s his house how old is she I bet that’s his daughter.
Vaudrey: Fold your hands and look at me. Now I’m going to show you the video again, and this time, I want you to think of one thing you notice and one thing you wonder.
Fabela: Wonder meansB a question that you have, like “What about…?” or “How many…?”

[Show video again]
Vaudrey: Tell your neighbor; what do you notice and what do you wonder?
Fabela: “Notice” is something that you see.
[Kids chatter]
Vaudrey: Raise your hand, please; what do you notice and what do you wonder?

Mary wrote down the Notices and Wonders as students shared them, sometimes helping them to craft their sentences.

Damon: Um.. I think that the eggs felled out of the fridge because they were on the edge and there was too much stuff in there and when she opened they felled out.
Fabela: Thank you, Damon. It sounds like you wonder if the fridge was full, so I’ll write that up here.
Vaudrey: [after a while] Boys and girls, I notice that a lot of us are wondering how many eggs broke, show me a thumbs-up if you’re wondering that, too. [All thumbs up, of course] Okay, how many eggs could fit in that carton if it were totally full?

Kids immediately began to guess huger numbers than their neighbors. “Ten! Fifteen! Thirty! A hundwed!”

Vaudrey: Let’s count together [pause the video and point to the screen] One, two, … twelve! Ohhhh, so the most eggs that could have broken would be?
Kids: [silence]
Vaudrey: Twelve, right? If all the eggs broke, then all twelve would be broken!
Kids: [silence]
Vaudrey: Okay, what if … um…
Fabela: Boys and girls, what number shows us when we have nothing?
Kids: Zero!
Fabela: Right! So if no eggs broke, we can show that with a zero!

Fabela: Okay, boys and girls, I’m going to close my eyes and count to twenty. When I open them, I want everyone at their desk with their whiteboard out and a marker and… what else?
Kids: A smile!
Fabela: Ready? One… two….
[Kids scurry to their desks]
Fabela: …eleven… I think we’re ready… twelve… thirteenfourteenfifteensixteenseventeeneighteennineteentwenty! [uncovers her eyes] Oh, look at everyone ready to learn, I love it!

During this time, I rolled the chart paper easel to the front of the room and stood ready to point at my “barbell” number line. This is where my secondary-focused-brain forgot for a moment that we were in a Kinder class. We spent the next four minutes getting the mini-boards to be horizontal, not vertical (“Hold up your hand and show me vertical… horizontal… vertical… horizontal… make your paper horizontal”) and still 20% of the class struggled. Then three more minutes to get the “barbell” written with 0 and 12 in the boxes.

This stuff takes time and we secondary teachers don’t give K-5 enough credit for how much “decoding directions and following them” is a skill, taught like anything else.

Thank you, Elementary teachers; I didn’t realize my middle schoolers had to learn which way is horizontal. It was also a struggle to get students to realize what the midline represents (between 0 and 12), but Mary came to my rescue.

Okay, so several minutes later, all kids are in their seats with the barbell on their mini-boards and a star on the number line, labeled with their guess for How Many Eggs Broke?

Claudia: [distraught] What if I can’t draw a star?
Vaudrey: That’s fine, honey. Draw a circle, then.

Vaudrey: Now, boys and girls, I have something to show you.

Kids: [silence]
Fabela: Let’s read this sentence together. Ready, go.
Kids: THERE. WERE. NINE. EGGS. IN. THE. crrrtahhhn TO. BEGIN.
Fabela: Wow, I’m impressed so many of you knew this word. Everybody sayB carton.
Kids: CARTON
Vaudrey: Wow, guess what? My guess was 10 and that’s too big now, right? Because there were nine eggs in the carton, so there’s no way that 10 could have broken. If theyB all broke, how many would that be?
Kids: [silence]

I forget exactly how Mary saved me here. It happened so often, I started to lose track. There was almost a language barrier between the way I asked questions and the way Mary did. Eventually, we got to this image:

This led to a great discussion about whether four eggs had broken or five.

Vaudrey: Paul, can you come show me where you see five broken eggs?

Dori: No! I think those are the same egg! [Dori draws a big oval around the two pieces]

One of my favorite parts of messy math is wrangling a common answer. With six-year-olds, Mary and I had to give the final word, which felt weird, sinceB arriving together at an answer with consensus is one of the fun parts of three-act math.

Anyway, we drew another line/star/circle on our barbells to represent five broken eggs, and then we had a snack of Goldfish on the carpet.

We’re open to ideas here: what’s a good way to wrap a lesson for Kinder? What could we have done next?

~Matt “I prefer Mr. Vaudrey but forgot to tell Mary that and it’s fine” Vaudrey

April 21, 2018
Pattern Play Pedagogy

A month ago, I wrote about my comfort with my daughter doing something imperfectly. An update happened this week that I want to share with you:

Link to Amazon

Pickle’s birthday was last month, and she was gifted (along with lots of glitter and pink crap) the box of wooden blocks* seen above from my friend Patricia. This week, she asked to play.

Math Teacher Daddy immediately asked, “What do you want to do first?” and hid all the cards. Pickle opted to dump all the blocks out and reassemble them into shapes on her own. I did the dishes while she did that, becauseB unsupervised play is a great way to explore. Then she asked to try a card.

Do you see what I saw immediately?

I sat next to her as she began to stack blocks in the frame. Being an adult, I have no issue translating abstractions into their concrete selves. Pickle is 5.1 years old, and didn’t notice that the slanty part was supposed to be viewable from the top, not the side. Note the yellow blocks below.

This took forever. This was our first time playing, and I wanted to touch the blocks, to arrange them into patterns, and to try and find a new way to pattern them. Sitting next to a child andB not interfering as they explore is really hard, and as soon as I start moving them around, she’ll follow my lead. So far, she’s directing the play, and that’s important to note, because of what happened next.

My wife (as I’ve mentioned before) had plenty of success in the traditional math class, but is slowly coming around to my philosophy of the modern classroom with multiple and varied means to mastery. She joined us in the kitchen when Pickle was 1/3 of the way done with her first card.

“Honey, use the green one next.”
“Turn it over.”
“You should put the yellow ones together first.”

My … lovely and helpful bride was taking Pickle’s proverbial hand and leading her down a paved path. I want Pickle bushwhacking in the brush and finding her own way.

After Mommy implicitlyB suggested that Pickle should shoot for perfection on the first try, Pickle began to get frustrated and didn’t want to place a block if it didn’t match the card, which she couldn’t really read anyway. She pushed the frame away and said, “It’s too hard.”

I glared at my wife, who immediately realized what she’d done. Confidence and fun were lost, so I had to restore both, and quickly. “Pickle, I have an idea. How about we do some together? Hand me that pink one.”

Some in the audience will note that our board doesn’t lookB exactly like the card, to which I say, “Yep! Shush.” Because look how proud that kid is.

I don’t need perfection yet, because enjoyment of the task is more important. We immediately dove into another card.

We built the bottom half first, but when I attempted to spin the box so we couldB keep the bottom half as our focus, Pickle resisted. Okay. Fine; you’re the boss. Then on to a third card.

Our third card prompted this phrase form Pickle, which comforted me.

I know how these ones go together because I saw how to do the other one.

Then I asked, “Do you want to stop for now?” and Pickle said yes. So we congratulated ourselves as we packed up the pieces. “Wow, honey; you did three cards on your first try? Do you feel proud?”

Of course she said, “Yeah!” whether or not she actually felt it. That’s a path I’m happy to pave: she knows the correct answer to my leading question is “yes!”

And b as expected b someone on Twitter says it better than I:

We impede kids doing of mathematics by introducing vocabulary too soon and imposing structures. From @Trianglemancsd #NCTMRegionals #MTBoS pic.twitter.com/RfTyfXIGSg

b Andrew Gael (@bkdidact) October 19, 2017

Think about your own classroom. How quick are you to jump in and pave a path for students to quickly reach perfection? Sure, they’ll reach the answer quickly, but that’s mostly because they followed you. How will they do on an assessment once you’re not leading them? Or when there is no path at all?

Tasks like Pattern Play and the Appetizers found here allow for students to build Number Sense by bushwhacking and using whatever means necessary to reach the goal, and along the way, they’ll build skills and strategies that make the going easier.

Some might say they’re stomping their own paths.

~Matt “The Green Slanties” Vaudrey
UPDATE 14 November 2017: We played again last night and found an interesting situation:

Both designs filled the board, one design had four extra green pieces.#tmwyk #mtbos @FractionTalks pic.twitter.com/DA5gK63G8R

b Matt Vaudrey (@MrVaudrey) November 15, 2017

*I called the blocks “right trapezoids” when I first opened the box. Neither my wife nor Pickle were amused, and both insisted they were just blocks.

October 21, 2017
Krispy Kreme Me – Third Grade
Today, we mushed together four of my favorite things from #mtbos:
- Notice and Wonder (from Annie and Max)
- Krispy Kreme Me (3-Act lesson from Graham)
- The problem-solving framework (remixed from Robert)
- The Estimation barbell (from Andrew)
Since there are loads of adorable student talk in this post, Ibll take just two lines for background here: Liz and Monica (third-grade teachers and #bonitians) approached me and said, bWe want to do math teaching better, is there some kind of database or model for ways to teach outside of the book?b

Herebs how it went down in Lizbs class. In the dialogue, shebs the T, as in Teacher.

Notice/Wonder

T: Boys and girls what do you notice and what do you wonder?

Immediately, the murmur of 8-year-old chatter bubbles throughout the room.

I wonder how many donuts that is.
Itbs a lot of donuts.
It doesnbt stop at the edge, like it keeps going underneath.
Itbs like some cereal because the donuts are round like this in a box like cereal.
We already know how to solve the problem!

T: Boys and girls, I hear a lot of interesting noticings and wonderings. Letbs put some together. (moves to poster paper)
Jen: I wonder if itbs perimeter or area.
Elise: I wonder how many donuts are in the box?
T: Who else was wondering that? Thumbs up.
[Lots of thumbs]
Marie: I notice that there are more donuts inside than we can see.
Jane: There are two pairs of right angles.
T: We have a lot of notices and wonders today! Keep bem coming!
Tim: If you see the whole box, you can just count the donuts how high and across.
Bella: I notice that it takes more people to hold it than usual, because like on a normal box it only takes one person to carry it.
Diaz: I wonder if theybre actually fresh.
Jolene: I notice that the box is square.
T: So youbre noticing the shape of it.
Student: Itbs a quadrilateral!
Jeremy: There are lots of people and not much donuts.
Christine: I wonder if therebs more than 100 donuts in there.
Isabella: Why is the box so big?
T: Thank you everyone adding, I know you have more to say than I can recordb
Student: Is the box real?

T: I notice your groups had one main question: How many donuts [dramatic pause] are in the box? Did you all have that question?
Students: Yes!
T: Now I could have had you write these down on your paper, but we did it all together. Go ahead and write down at least one Notice and at least one Wonder on your papers. Theybre in the middle of the table.

[PE Teacher bursts in to announce the results of the Track Meet]

Teacher: Boys and girls, if you wrote several Noticings and Wonderings, please circle this one.

Estimation

Teacher: Boys and girls, look down to the barbell. Let’s try to answer the question, “How many donuts?”
What is a number that is too low? Put that number b without saying it b into this box right here on the left.
Now that you have that number, think of a number that would be too high, and write it in here on the right. This is just your thinking.

T: Somewhere on the line between them, put a hashtag and write in your guess.
Student: Hashtag school
Student: Hashtag math

Act Two

T: Is there any information that you need?
Tim: We need to open the box.
T: Why is that?
Tim: We need to see if there are more on the edges.
Marco: I agree with Tim because there might be some hiding below.
Tyler: There might be some hiding on topb
T: Can you go show us on the board?
Tyler: There might be some hiding on top, like out of the green line.

T: I hear some of you saying, “Ohhh,” like you noticed what Tyler noticed. Well, other people were curious about those things, too. So they emailed Krispy Kreme and were sent this email.

T: (pauses for 20-30 seconds as students read) Oh, some of this information is missing. In this part, theybre showing blank times blank for each layer, so theybre multiplying. Is there anything here that is important that you took away? Jen?
Jen: The times signal
Tyler: 3000 millimeters
Dan: 2300 millimeters
Jolene: Three times layers

After writing their information on the board, Liz says, bLet me show you some more info.b

Jim: What?!
Tyler: That cannot be true!
T: I see youbre thinking. Letbs attack this problem; flip over your paper and let me see your genius.

Ibve been sitting in the back, pecking away at the keyboard this whole time. Monica joined me shortly after she dismissed her class to P.E.
Monica leans over and whispers to me, bThat boy there is just staring. I would nudge him and ask what hebs thinking, or maybe say, bTalk to your partner.b to keep him moving.b
bMaybe,b I respond. bOr maybe he needs more thinking time and is formulating his idea in his head.b
Monica giggles, bBut I want talk-time right away, so I assume that all my students want that.b
bMe, too!b I whisper.

T: Tyler, why are you counting?
Tyler: b& because these two sides are blank (points to the arrow going down and right)
Dave: I wanted to see if there was 25 donuts there or if only at the top there was 25 donuts.¹
T: Ah, okay! Carry on.

T: Boys and girls, let’s look at some of what our classmates are doing. Michael, what did you do here?

Michael: On the box, I saw 25, so I added 25 for the top and the bottom. Then I added 50 + 32 and got 82

Cairo: I built a box and I put 32 + 25 and got 57 and I tried 32-25 and then you took it away.
T: Any questions for Cairo? [pause] Okay, I have more. This is Christinebs.

Christine: The 25 I counted on the other side and I added them up and it says 3 layers so I added 3 to it.
Doug: Why did you write …umb& whatever that is by 114?
Christine: I wrote bnot answerb to remind myself that Ibm not done yet.
T: Look at Dave’s work.
Class: Whoa!

T: Dave, Ibd love to know what youbre thinking, Dave.

[bPlease excuse the interruption. Teachers: we are on a Rainy Day Schedule for lunch.b]]

T: Now, look at Davebs math. Dave what are you up to here?
Dave: That may look like more than ten 3s, but itbs ten 3s. …Actually, itbs more than ten 3s. Ibm writing out twenty-five 32s, then Ibm gonna add them all up.
Charlie: Why are you adding them all up?
Dave: Because I heard that there was 3 layers, there could be a lot lot lot more donuts around here and I figured there could be a lot more 32s around here. Then Ibm gonna add them all up and see what my answer is and see if they put more donuts here.
T: So youbre thinking twenty-five 32s.
Doug: Why did you do twenty-five 32s instead of thirty-two 25s?
T: What do we know about math? Will that work?
Doug: Itbll be done faster and you can figure out the other sides and that will bring you to the answer.
Dave: At the end, Ibm gonna put the answer times three.
T: Whybs that?
Dave: Because therebs three layers.
T: What are they telling us with those numbers here?
Dave: There are 32 like this and 25 like this. (points to rows and columns) Opposite sides are equal and then times it by three.
T: Letbs take another 5 minutes of quiet thinking

Vaudrey: I notice that you were counting the dots, what made you choose to count the dots?
Doug: I thought that each dot could be a donut and if they go in this way, they could make an array.
Vaudrey: [Stunned that the third-grader knows the word barrayb] Did you find that the dots were 25 just like the column of donuts here?
Doug: Mm-hmm.

At this point, Liz had been deep in Act Two for about 20 minutes. Maybe a third of the class has a method that will lead them to a helpful answer. Liz crosses to me in the back of the room.
T: This might be a little high for them. We have a few more minutes, I think webll get there.
Vaudrey: This is the wild-and-wooly part, where theybre working in different directions. You gave them 5 more minutes, that should be enough for them to get deep into their method or to find some limitations.
Lee: Okay, good. Itbs so hard to not steer them toward my method.
Vaudrey: Totally, youbre doing an excellent job of affirming all their responses and not giving any indication which one you want to see.
Lee: Itbs so hard!
Vaudrey: (laughs) I know! And you even put some information on the board thatbs not important and some that is. That was a good call.

T: Tim, Leadora, Jen, hand me your papers, Ibm gonna show your work up here. Everyone turn-and-learn.
(Students all put down their pencils and face the board.)
T: So, walking around, everyonebs doing great math work. Herebs one, by your dear friend Kaylon. 32×25 and all this math. Do you mind telling us about your thinking?

Kaylon: I was adding up 32.
T: How many times?
Kaylon: Twenty-five.
T: Any questions for Kaylon?
Michael: Where did you get the numbers from?
Kaylon: I was doing 32+32 and got 64, thenb& um…
T: …added another 32 and got his answer, and another 32b& slow and steady wins the race.
Jen: I…umb& I had all that then three times.
T: Why three times?
Jen: Because therebs three layers.

Tim: I added 32 + 32 and got 64, then I added 64+25 and got 89 and 12 and got 101.
T: What questions do you have for Tim?
Vaudrey: (Waits five-Mississippi) Why 12?
Time: Umb& I donbt know. I just added a 12.

T: Ibm gonna give these back to you to put under your doggies. Webll have some more think time on this later this afternoon or tomorrow.
Class: Thatbs dangerous; doggies love donuts.
Teacher: What questions do you still have?
Bella: How many donuts are in it?
Doug: The truck was quite wide and the box filled the whole bottom. The box was this wide and the truck was this wide and it fit perfectly.
Michael: There was three times layers, from above it kinda looks like more than one layer.

Ethan: In the beginning, I wondered if they were delivering the donuts. I wanna order that.
T: It does look like that! I have a little tidbit for the end I canbt wait to show you.
Tim: Ibm wondering how much the box costs.
T: I bet we could figure it out once we know how many donuts are in there.
Tim: If we know how much one donut is worth, we could figure that out.

Debrief

Monica: Those students that were sharing, are those the students who score the highest usually?
Liz: No, theybre just out-of-the-box thinkers.
Vaudrey: Liz, did you feel like the task gave all your students equal access to the material?
Liz: Oh, definitely! Like Ethan hasnbt done very well on his fluency checks this month, but he dove right in. I think he enjoyed talking about donuts!
Vaudrey: Totally. And thatbs the point of class; the entree was donuts, and the math helped them answer a question they had about donuts.
Liz: So now what? We didnbt quite finish; should we come back this afternoon or tomorrow?
Vaudrey: Either one, but some kind of resolve should happen Theybre gonna want to answer the question bHow many?b but they might also want to get answers to some of their other bwonderings.b
Liz: (deep breath) This collaboration is so powerful. I feel like I have so many more things I wanna try already.
Monica: I wonder how a 3-act would look with both our classes together. Like a huge, co-teaching lesson.
Vaudrey: Ibm down. Howbs next week?
Liz: We could to it on Friday? Pull both classes into one room and do a big lesson?
Monica: Great!

Also in that conversation, Liz asked me to make a graphic so kids could describe the various types of arrays they used and . I made this.

Liz is planning to project it onto the board and let students draw their rectangular groupings onto the board around the donuts. If needed, a different student couldB decompose the array into

Resources

We reformatted the Problem-Solving Framework from Robert so it fit our classroom. Click here for your own copy of our version (and please share if you make any improvements).

Grahambs full 3-Act lesson is here: Krispy Kreme Me. I highly recommend his database of Elementary 3-Act lessons here; Graham is a thoughtful and intentional teacher who cares deeply about kids and learning math.

Also, here’sB more on Notice and Wonder from Tracy Zager‘s companion site to her paradigm-shifting book.

Finally, if you’ve never seen this lesson, here’s the grand reveal.

~Matt bPut this under your doggiesb Vaudrey

^{1. This was my favorite piece of student talk. I assumed that students would read the Act Two image with the red arrows as “25 rows of 32 columns.” It never occurred to me that students would seek more specificity, that they might assume it’s only 25 from the middle up or 32 from the middle to the right. As always, students find new and exciting ways to interact with content. Teaching is just the funnest job.b)}
May 22, 2017
Channel Your Inner Jeremiah

Today was day 1 of CUE Rockstar Math, where 120 of my fellow nerds descended on Dana Middle School in Arcadia to discuss math education with me and some Twitter friends.

When you wake up and realize it is #CUErockstar math day!!! Let’s DO this!! @edcampOSjr @fawnpnguyen @Jstevens009 @MrVaudrey #MTBoS pic.twitter.com/KFDC8GQe5X

b Lindsey Blass (@LindseyBlass1) April 29, 2017

Thankfully, CUE starts these events at 9:00, which gave me time toB rinse diarrhea crumbs off my baby (not an idiom; actual baby, actual poo) and make it there in time to give hugs and high-fives to a room full of people who are just like me: math teachers who want to become better at their job.

Class Culture of Critical Questions

As with all Rockstar events, I gave two 2-hour workshops on the same topic, separated by lunch. The second round ran pretty well off of my phone, because I tripped on my cord and…

Just spilled my entire water bottle into my MacBook.
Anybody at #cuerockstar math have a loaner mac or chromebook?#NotAJoke @CUERockStar

b Matt Vaudrey (@MrVaudrey) April 29, 2017

That’s not the point of this post, but it was one of the moreB eventful parts of the day.

The workshop starts with a demo lesson, modeling the 3-Act lesson from Graham Fletcher called Krispy Kreme Me. After the lesson, we make some notes about what phrases and procedures got everyone interested in sharing.

It was pleasant, fun, and not a great representation of an actual classroom.
Let’s turn it up a notch.

It’s a good day when a roomful of Math Ts can channel their students in an @Estimation180 task.#cuerockstar #mtbos pic.twitter.com/XlDO3AHT4o

b Matt Vaudrey (@MrVaudrey) April 29, 2017

“In my hand are a stack of yellow cards,” I say to the room. “Half of the cards say General Ed Student, and the rest have some kind of instructional challenge for the teacher. I’m going to pass them out to each of you. Keep them to yourselves.”

As I walked around b grinning like my daughter before Gramma comes over b I went a step further.
“In order to make this more like a real class, I want you to channel a student that you have currently or had in the past. When you saw the card, you thought of a kid. Be that kid during this next lesson.”

Invariably, teachers begin to tilt their heads, smirk, and ask, “Really? You want Jeremiah in this room?”

And I grin right back and say, “Yep! Let’s go!”

Then a room full of adults get to make silly jokes about cheese, ask to go to the bathroom, bring up YouTube videos on their devices.

They also hesitate with big words they don’t know (EL Student), get distracted easily (Quick Finisher), or cry out, “I can’t see!” (Vision Disability). It’s one of my favorite things to do; flex my teaching muscles and be vulnerable.

BecauseB afterB that…

Walked away from @MrVaudrey session excited about all these math resources. Thanks for sharing! #cuerockstar pic.twitter.com/Zfz4dkCJql

b Jan Dickson (@jdickson_ausd) April 30, 2017

We make a list of culture-building stuff they saw me do with the “real class” and things they would add, subtract, or change.

It’s powerful to hear a grown adult say, “I was the English Learner and you went way too fast for me.”

“That sounds awful.”

Yeah. Before the “channel your inner Jeremiah” part of the workshop, I’m honest with everybody: “This might go horribly wrong, but I’m gonna do it anyway.”

It’s my hope that, even if overzealous or skeptical teachers channel Jeremiah on a no-meds, skipped-breakfast, mom-yelled-at-me-on-the-drive-to-school day, all of the attendees still get to see the teacherB take a risk and be vulnerable, maybe even look silly.

That’s kinda the last few chapters of Classroom Chef.

I want y’all to see me reach for something ambitious, even if I fall on my face afterward (which happened in Salinas).

Resources

If you want to take a bold risk with your staff, click here to get your own copy of my yellow cards, which I printed twice and laminated, so I have 40 cards.*

The quick lesson that we did together was Day 28 from Estimation 180.

Also, special thanks to Josie forB really going for the gusto. I’m almost sorry I sent you out of class on an “errand.”

Role I loved playing!! Thanks @MrVaudrey for the chance! Learning about creating a safe culture. #cuerockstar pic.twitter.com/5P5T3lUrf8

b Josie Chia (@ChiaClass) April 29, 2017

~Matt “Josie, can you take these Post-Its next door?” Vaudrey

*Dang, that means we had 36 people channeling Jeremiah today. Cool.

April 30, 2017
Guest Post – Taking a Cut at the Mullet Ratio

The following is a guest post written by Dianna Gallagher. Links and formatting added by me, all else is her work, including the cute pun in the title and this entire Google Drive folder.

Enjoy.

Hi! Ibve been asked by Matt, a guy I sort of met virtually last month, to write a post for his blog on my experience presenting The Mullet Ratio to my seventh graders. I learned of this lesson from a fellow #MTBoS member, Kathy, who was generous enough to share ideas with me as I planned this current academic year. I wasnbt really sure what The Mullet Ratio was, but the name was certainly enticing and upon a quick google search, I couldnbt imagine a more entertaining and hilarious way to pass the 100 minute block period with my squirrely 7th graders on a very rainy Tuesday.

Intro

The lesson started with the intro question, bhave you ever had a bad haircutb. Kids love tangents, so that was a great way to reel them in. I opened the Mullet slideshow with the first of our mullet-y friends, Bayley and Vladimir. One glance and the kids were howling. We flipped through the first few slides and giggled together, laying down the vocabulary and idea of bbusiness in front and party in the backb.

When we arrived at the John Stamos and mustachey prom date slide, the first of the slides with the displayed measurements, we recognized that the party and business were not measured in inches or even centimeters, but rather bdreamy eyesb and bmustachesb. This threw the kids. I found this a perfect chance to discuss units with them. We measured Dreamy Eyesb party and business in different units (cm, inches, thumbs, post-it notes), and explored the ratios from those measurements as well as a cartoon character I drew on the white board. Students discovered that as long as the unit is the same for the party to business ratio, the mullet ratio will be the same. This was a big and very important discovery for most of the kids.

When we arrived at the Dwayne Schintzius/Andre Agassi slide, not only was I reminded that Andre Agassibs mullet was actually a wig, we all learned that the mullet ratio of these two athletic icons was nearly identical. This was a great place to stop and let the students figure out why and how. We then went back to my cartoon figure drawn on the board and I asked how I could make another figure with the same ratio. They looked at me with a blank stare. No clue. So I drew a head and some party. We knew the first cartoonbs MR was 5. They told me to measure the new party b I got lucky with 10 inches. Ding! Ding! They got it b business needs to be 2.

So far, good progress with the units discovery and now equivalent ratios. Added to the slide deck, the grand finale was the newly trending bPresidential Reverse Mulletb b Donald Trumpbs party on top and business in the back. The hysteria was as alive with this slide as it was with the first 10.

Gallery Walk

I redesigned the gallery walk handout for the kids to use as they worked in partners (with calculators). I passed the handout out at this point, when I should have passed it out at the start of the opening presentation because instead of them working on white boards to find the MR, they could have been recording on the spots that I had on the handout. They breezed through this part of the activity. I noticed a couple of pairs were not recording business and party, but just the ratio.

Homework

My 100 minutes was winding down, so I gave the kids a couple of homework tasks due after a long weekend. One was to measure the MR on my reformatted famous mullets handout. This task, unlike the Gallery Walk, required the students to take the measurements of the party and business. Another was the Andy Warhol self portrait with a double MR, B= MR and the ideal MR. The students took the most time with the Andy Warhol self-portrait, but unfortunately many didnbt actually figure out the mullet ratio. They only identified the party and business. The final task, done with a sub the following week, was a Mullet Extension activity, which was a nice way to take their thinking one step further and btie upb Mullet Madness.

Thoughts

To say the Mullet Ratio lesson was engaging is an understatement. The kids had a blast while learning about equivalent ratios, graphing ratios, as well as units of measure and how they relate to ratios. I would have preferred to use another 50 minute period after the initial 100-minute block period for the other activities that I had assigned as homework. Looking forward to showing off my Mullet Madness bulletin board to the parents at Open House next week, and even more excited to revisit this lesson next year!

~Dianna

February 10, 2017
Tiny Polka Dots

My daughter, Pickle,* and I got a package in the mail today.

Developed by MathForLove, it was a kickstarter… like… nine months ago, and it arrived today.

Immediately, (after we clocked the little brother with the box, denied it, got sent to time-out, confessed, and pledged to be truthful hereafter) we sat down to play the game.

The box, as you can see, is a cute size, and the rules for the various games are also printed on cards within the box. Which is good, because I wasB hella stoked to play, but unsure where to start.

First up, “Hungry Numbers” for ages 3 and up. The purple numbers will only eat cards with the same number of dots as their value.

Pickle: Hungry Numbers? That’s a silly name!

Next, we matched cards with the same values to their buddies with the same number of dots. It was a good chance to get PickleB acquainted with ten-frames, which she hadn’t seen before.

This was a good time to teach my wife why aB ten-frame (the blue cards) are important for counting and cardinality development.

I should note, my wife had surgery last week, is hopped up on Oxy, and didn’t really care about cardinality development. She went to take a nap shortly after we finished playing.

Both those games were for ages 3 and up, so I figured we could take it up a notch with something more her level (Pickle is 4).

We played PowerDot (which is essentially War, but for with a different name for children/people in nations besides the U.S.). For this game, we added in the Orange cards (circular numbers).

This led us into some great chats about what makes a number “greater,” a term she hadn’t heard yet.

Then we were interruptedB because her brother had gotten stuck by the ottoman.

Next, good-ol-fashioned Matching.

Then, we took it up a notch.

Vaudrey: Pickle, the next game is forB big kids. Ages five and up. Do you think you can handle it?
Pickle: (blows out tongue) I can handle it! I can do a hard game!

In the Dot Fives game, we matched ten-frames with other dots to make five. “This one has three, how many more to make five?” She, of course, crushed it. My wife and I were both impressed.

Then, in a moment out of a sitcom, she cheered and cried, “Yay! I can do tough things!” Then gave me a hug.

This… doesn’t happen often in my math classroom.

Letting students struggle and asking probing questions has been the focus of my career for the last decade, so I felt well-prepared to lead Pickle in this game. Even for parents who areB not math-education geeks bent on patient problem-solving, this game is a great place to start for Pre-K kiddos.

Thankfully, the designers also includedB a Guide for Grown-Ups (posted here, about 75% of the way down). This guide should be required reading for teacher candidates in college. It should be laminated and given, poster-style, to every new hire in a school district. It should be tattooed on the forehead…

Okay, you get the point.

I no longer live in the Pacific Northwest. The next time I’m there, however, I’m scheduling a visit to the MathForLove HQ. I really wanna meet these three people, who have advanced degrees in things I can barely pronounce. Here’s founder Dan Finkel on a TED Talk worth watching:

“Thinking happens when we have time to struggle.” Oh, yeah. Finkel gets me.

Since the last time I visited their website, they’ve added a free lesson plans page. Using your resources to empower math teachers for free? That — by itself — is enough to get me on your team.

Also, the team at MathForLove also designed Prime Climb. Initially, it sounds like they were trying to cram math into a game where it doesn’t belong, but after reading the rulesB to Prime Climb, I really want it.

In short, I recommend getting your own set of Tiny Polka Dot here as a PDF or clickB here to order sturdy cards early next year.

~Matt “I can do tough things” Vaudrey

*Not her real name, but what I call her on the regular.

December 30, 2016
Visual Patterns – Week 2
Dear Claire,

I stumbled (re-stumbled?) upon Fawn’s post about the first two days. Sprinkled with her usual wit and orneryB charm, the visual pattern process struck me, especially since we’re hitting that hard next week as we “create equations and inequalitiesB and apply them to solve problems.”

(You might recognize that language from the pacing guide you wrote.)

Anyway, we started with Visual Pattern #2:

The projector was on the fritz (#RealTeacherProbs), so I drew the four steps on the board.

“Look over here. Pencils down, fold your hands. This is step one. [pause] This is step two. [pause] This is step three, [pause] and this is step four. [pause] On your yellow paper, please draw me step five. Go.”

After one round of the “talk to your neighbor song,” I drew playing cards (read: popsicle sticks) and asked students to describe their drawing to me as I drew it.

“Okay, draw five squares and then four squares.”

“No, like… the four squares are connected.”

“No! Just… look at step four and draw that first.”
I smirked, “I can’t see step four. Describe it to me.”
“Ugh! The four squares are going vertically.”

“Okay, now connect them to the bottom row.”

“No! Mr. Vaudrey! Connect them to the last one!”

“Dude, Mr. Vaudrey is trollingB hard right now.”
Jayla couldn’t take it any more. “Okay! Listen and doB exactly as I say!” She stands up.

I put on my best hurt puppy face. “But… IB have been doing exactly as you say.”

Jayla holds up a hand. “Shh! Draw the line of five squares horizontally, touching each other. Then, from the last square on the right, draw four squares vertically, all connected.”

“There! Was that so hard?” Jayla drops back into her seat.

Although the class was loud this whole time, I submit that every student was … maybe notB engaged, butB invested in the problem. The discussion of which squares go where also helped the rest of the class access the problem. We spent maybe seven minutes describing in great detail how the squares were arranged.

Visual Patterns are an example ofB doing fewer problems, but making them count.

The whole class understands the structure, so when I ask them to fill in the table, and describe how they found steps 10 and 27, they can describe their reasoning.
- Step 10 has 18 squares because step five has 9 squares and I timesed it by two.
- Step 10 has 19 squares because it’s increasing by two and I just counted.
- Step 10 has 19 squares because it has a row of 10 squares on the bottom, then nine squares going up vertically from the last one.
- Step 10 has 21 squares because I increased by two each ti–wait.
- Step 10 has 19 squares because it’s like two rows of 10, but minus one.
For each of these answersbright or wrongbI erased what I had written and re-wrote what the student said. Understandably, students who were confident in their answer were upset whenB the teacher wrote a different answer on the board.

I share Fawn’s love of student struggle. If I were featured in the next Avengers film, I’d be a super-villain who gains power off of the furrowed brows of teenagers.

~Matt “For anyone who doubts my excitement at returning to the classroom, this is the fourth blog post in a day and a half” Vaudrey
September 4, 2016
Double Clothesline – Equations
Dear Claire,

In a post last week, I described wondering if I was working hard at the wrong thing. Several students were comfortable solving one-step equations like this.

“Subtract one on both sides, then divide by 3.”

They were less confident, but able to solve, problems like these.

“Add five to both sides and divide by…. minus two.”

“I minused three on both sides, then I used a calculator.”

But their methods would prove too weak by the time we got here:

$"I got stuck. I don't get fractions."$
“I got stuck. I don’t get fractions.”

Claire, it would have beenB easy to praise their standard-algorithm style on Monday and Tuesday, then give them another standard algorithm on Friday.

Soon, math class is a tool box with a bunch of tools, but students are unable to match the tool to theB function.

For most students, it’s just a big box of metal.

image: teresaphillips1965

Instead, I wanted them to seeB why we solve equations the way we do, and Double Clothesline seemed to provide method to the madness.

Naturally, students who had been praised for their use of the standard algorithm were hesitant.
- I don’t get this way.
- Do I have to draw the number lines?
- I like the other way.
The Desmos Activity on Thursday seemed to make some more connections. Some students blew through it quickly, but the questions they asked betrayedB the appeal of aB formula or rule in math class.

Then on Friday (with no devices, #RealTeacherProbs), I used Desmos Activity Builder to structure the lesson (pulling down those equations as PNG from Google Draw).

“Look, class. Two thirds of x is six. There are twoB chunks between two-thirds x and zero. How wide is each chunk? Yes, so one-third of x is three, where does x go?”

“If one-fourth of x is 12, where is x?”

“Give me a number, just call it out. [Listens for a number divisible by 3.] Okay, I heard nine, so three-sevenths of x is 9. Talk to your neighbor, where does x go?”

Claire, hopefully you see what I’m trying to do here. Offering aB visual cue for equations with fractions. The language that students used to describe their work went like this:

I can see that each of the three chunks takes up three on the number line, so one seventh is three. Then I multiplied that by seven to get 21.

Fabulous. Now, when I show them this:

It makes sense.

I consider it a good day when the standard algorithm makes students say out loud, “Oh! That’s way less work.”

~Matt “Yes, it is. And now you know why.” Vaudrey
September 4, 2016
Math Makeover: Spencer’s Soccer Ball

Bonita High School’s Teacher of the Year for 2015, Kari Redman, asked me into her class during prep period.

So, once I texted Kaplinksy to say, “Your training went so well, you’re now a verb,” I scooted over to room 202 and she handed me this:

“A bunch of my students don’t get it. Every year. I wanna make it more… graspable,” Kari said with a grin.

We chatted for a bit about how similar it is to the Performance Task on the CAASPP test, and how one of the goals is student perseverance through the steps. Kari wants her students to have big Performance Task muscles.

I am a flawed individual. It’s a challenge for me to keep from curling up my nose at these types of problems. A rag-tag band of math teachers are creating great math tasks that are accessible online for free, yet this pseudocontext is prominent throughout math education.

That mean voice in my head now scoffs when I see these problems, since I so enjoy prepping, teaching, and debriefing deep, rich tasks.

There’s a condescending math-task hipster in my head, and it’s a bummer.

$image by zeradodich on Pixabay$
image by zeradodich on Pixabay

Kari is an awesome teacher with a safe class culture for experimentation and learning, so of course I wanted our first task together to go well so she’ll let me mooch her class sometime next year.

We discussed the difference between an open-ended task and one with a clear expected flow. I debated in my head turning this into a tailless problem, but instead, we went to YouTube and found this.

Now we’re talking. While Kari thought about how that video changes the flow of the task, I built this and walked her through it.

“Let’s start with giving the students very little; let’s show the video and ask, ‘What questions do you have?’ or ‘What do you notice? What do you wonder?’ They will get a lot of the framework down for us. Instead of us directing them, they’ll lead and we can steer.”

“Yeah…” Kari says, staring into space, her wheels turning. “My students who don’t catch on to the problem quickly will be on board if we start with something like that.”

“Exactly,” I agreed. “It’s often the students who don’t know how to start that get stuck, and something like that will offer a low door for entry. Here’s a possible flow for the lesson.”

1. Show them the video and this graph.

Ask them, “What do we know so far?”

They’ll likely mention the soccer ball and the hoop. They might mention that the hoop is lower.

2.) Ask them to draw what we have so far.

They’ll likely draw a parabola-shaped path of the ball, draw the ball, draw the hoop, draw the grass. If this is in pencil, they can make revisions later. Or, even better, give a fresh graph at the end and let them make a “pretty version” once they know everything.

3.) Ask them, “What do we still need?” and let them ask you for information.

They’ll probably ask for how steep the hill is; what a great time to remind them of slope.
They may ask how tall the arc of the ball is; we don’t know, but what a great time to remind them of vertex form of a quadratic.
They might ask how tall the hoop is; what a great time to say, “I don’t know. Google it.”

4.) Once they have the stuff they need, assist them in plugging it in.

All of those steps are helpful inB understanding and defining the problem, and this part of the process is often ignored or under-represented in math class.

Here’s where the math hipster has to bite his tongue; the equation still appears out of thin air.

Maybe next year, this is where a teacher could detour intoB Will It Go In The Hoop?B or something similar. In this case, that’s not the point of Kari’s lesson, so the math task hipster can keep quiet.

$image by zeradodich on Pixabay$
image by zeradodich on Pixabay

The equation and Desmos offer a visual to understand the problem, but the Performance Task is still focused on theB algebraic solution to the system of equations.

I’m stoked for the moment when they realize that the equation for the grass lineB doesn’t include the basketball hoop, and they have to add something on the end of the linear function.

What do you think? What’s missing?

~Matt Vaudrey

P.S. Make your own math task hipster meme here, then tweet me about it.

UPDATE 19 April 2016: She’s also the one who has her Algebra II Students build these after testing:

Adjustable Unit Circle in 2015 Bonita High Teacher-of-the-year Kari Redman’s class.#bonitians pic.twitter.com/ot8UWvjhE4

b Matt Vaudrey (@MrVaudrey) April 20, 2016

UPDATE 29 April 2016: Kari taught sixth period, which went much better than my lesson, first period. We realized (during a debrief third period) that I was attempting to warp math around a context that didn’t demand it, something my inner math hipster accuses textbooks of doing. Instead, a more guided lesson went pretty well when Kari slid into what was a more comfortable class flow.

April 21, 2016