Counting Dots

A teacher asked me about the Counting Dots activity I did in her teacher workshop I facilitated a couple weeks ago. I did the Counting Dots activity as a follow-up to Max Ray's Ignite talk: Why 2 > 4. I believe we teachers need to experience how valuable it is to listen to each other share strategies. If we're going to do it in our classrooms with students and value student thinking by listening to them, then we need to practice ourselves. You know? Build that muscle memory.

I was inspired by Dan Meyer's 2014 NCTM talk titled Video Games & Making Math More Like Things Students Like. You can find his specific reference to Counting Dots at the 30-minute mark. This link includes Dan's NCTM session and references to Ruth Parker's work, who according to Dan popularized Counting Dots.

I also made a video of the exact slides I used with teachers, a few extensions to counting dots, and a behind-the-scenes for anyone interested in making their own.

Questions? Let me know.

Dots,
1200

Featured comments:
Graham Fletcher shares an insightful article on Subitizing.

Dan Kearney shares more goodness from Steve Wyborney.

How Do You Like Your Bacon (Math Modeling)?

During the past few weeks I've had the pleasure to work with and learn from teachers in various places in the country, facilitating district/school workshop trainings as they prepare for their school year. Part of our time together was working on problem-solving tasks and breaking down Mathematical Practice 4: Model with Mathematics. At some point, either before lunch or in the afternoon, I tossed up this Estimation 180 challenge and asked:

How long to cook the bacon, starting with a cold skillet?

I love this estimation challenge because it showcases many parts of the modeling process, especially the two following parts:

• Identifying variables
• Formulating a model

Here's why. Teachers instantly start asking questions like:

• How do you like your bacon?
• Crispy, charcoal, or like beef jerky?
• What type of bacon is it?
• Turkey bacon or real bacon?
• Is it thick cut or the other stuff?
• Is the bacon room temperature, cold, or frozen?
• Is it cooked on a gas or electric stove?
• How hot is the flame?
• What is the percent decrease in size of one strip of bacon?

Teachers are identifying variables and asking for information that matters to them in order to formulate a model. I love it. I have also done this Estimation 180 challenge with students before and they have asked many of these same questions too. I love it.

I had a great conversation with Joe Schwartz and others at TMC15 about state tests lacking what the modeling process demands: asking questions. Why do the SBAC and PARCC tests not have students simply ask questions about scenarios? If we're asking students to identify variables and ask/search for information necessary to formulate a model and solve a problem, why don't tests place more of a focus on this? What if we presented students with scenarios a la the Math Forum and simply have students first submit mathematical questions that could be solved. What if we then followed it up with giving students a list of three to four questions they could solve and they pick one?

Another great conversation I had with Nathan Kraft and others at TMC15 was the idea that direct instruction can have a negative connotation in the MTBoS. A similar notion is that the instructional strategy "I do, we do, you do." also has a negative connotation. With problem-solving and mathematical modeling, direct instruction is not the focus. The focus is conceptual understanding. From my experience, I've learned that timing and placement of direct instruction is what matters. I've been catching up on reading NCTM's Principles to Actions and I highly recommend it to anyone; teachers, coaches, parents, administrators, students, and more. It's about 100 pages. Get on it! I think it paints a pretty clear picture why, how, and when conceptual understanding should take place in relationship to procedural fluency.

Principles to Actions really does a great job driving the point home that procedural fluency is important. However, procedural fluency won't stick nor have significant meaning if the students lack the conceptual understanding first. When I'm done with Principles to Actions and have had a chance to let it simmer in my brain, I plan to blog more about it. I also need to explore the Principles to Actions Professional Learning Toolkit.

Last, and certainly not least is literacy. I'm glad that one teacher at a recent workshop voiced her concern about teaching literacy in math and that the use of multimedia in a 3-Act task or an Estimation 180 challenge really doesn't strengthen literacy. I agree.

Trust me, I'm all about building literacy. However, the more I teach and work with teachers, the more I believe in the importance of making the conceptual understanding accessible first as a means to transitioning to procedural fluency and strengthening literacy by scaffolding. If I don't make the conceptual understanding accessible to my students, than I'm not scaffolding both the mathematical procedural fluency and literacy.

That said, I tried to imagine what Day 185's bacon estimation challenge might look like. I still love the visual and simple question and would still start with the current setup as the introduction to the task. Once students and teachers voice their questions, Act 2 information might be presented in text. Here's what I came up with (I know it could be better):

I have 20 minutes to prepare and eat breakfast before leaving for work. I need to cook 12 pieces of bacon for my family and the skillet only holds 6 pieces at a time. We like our bacon crispy, but not like charcoal. The gas stove will be at a medium to high heat. The first batch of bacon starts to sizzle one and a half minutes after I put the skillet on the lit stove. Five and a half minutes after the bacon starts to sizzle, it is about 65% cooked. Will I have enough time to cook all 12 pieces of bacon?

I'm not sure this blog post brings much closure. However, it has brought a greater focus for me as I prepare for the school year. I am more focused on

• students asking questions
• students building conceptual understanding first
• teachers making conceptual understanding more accessible (as much as possible) 
• teachers scaffolding their classroom activities and direct instruction to strengthen procedural fluency by building upon conceptual understanding

Does this sound reasonable?

How do you like your bacon?

Let me know. I'm on my way to finishing Principles to Actions.

Bacon,

219

Until Further Notice...

This is me until further notice...

CA Teachers Summit Ed Talk

I had the honor and privilege to give an Ed Talk at the California Teachers Summit held on Friday, July 31, 2015. Across the state of California, teachers attended college campuses for breakout sessions, Ed Talks, and keynote speakers. I gave an Ed Talk at the Pasadena Convention Center titled, Estimation: A Gateway To Better Number Sense. Check it out.

I gave an Ed Talk at the California Teachers Summit on July 31, 2015.

I also had the privilege of meeting the other two inspiring Ed Talk presenters, Nicol R. Howard and Thema Bryant-Davis. The highlight of the day for me (and my son) was that I got to meet one of the Keynote speakers, astronaut Leland Melvin!

Nicol, Leland, and me

Ed Talk,
1146

Turning 3 Challenges Into 3 Charges

My next three weeks are slammed with opportunities, via conferences and teacher trainings, to work with fellow teachers, learn from them, and share resources and thoughts about what I'm most passionate about in math education. The great thing about this two-way learning environment is that it will make me even stronger and better equipped for the upcoming school year as I support my fellows in their classroom.

If we're at the same conference or teacher training, you will be hearing me drive home the importance of the following three challenges we face as teachers of students and students of teachers:

• Problem solving
• Student thinking
• Number sense

• Why are these three challenges so important?
• Why am I so passionate about lesson design and tech tools that meaningfully support these three challenges?
• What resources are available to teachers to support students?
• How do we implement said resources to support problem solving, student thinking, and number sense?

At the end of our time together during a workshop or conference, I hope you become better equipped with strategies, tools, resources, ideas, and inspiration to make them your charges for the upcoming school year.

Thanks in advance for letting me be part of your valuable PD time and for allowing me to absorb your insight and knowledge at the same time.  

Charge,

855

Open House: Week 1, Day 2

Last week was the first week of a four-week summer academy. I teach a 110-minute class, four days a week, and it's designed as an enrichment course: no homework, no tests, no grades. The class is titled Get Ready for Algebra and here's the working schedule for the four weeks.

It's a great opportunity for me to practice some of my skills; skills I once possessed and skills I learned this past year, but didn't practice enough. Initially I thought I'd be keeping some of my skills sharp, but I soon found out how rusty I am at various moves, ranging from facilitation, to instructional, to questioning. How do I know, I recorded three of the four days last week.

Before I break down an 8-minute video clip from the second day, I want to be clear:

1) It's Open House everyday in my classroom. Anyone is welcome at anytime. I observed so many teachers this year and am extremely grateful for the opportunity. I learned a lot by being an observer. Being an observer allows me to reflect on my own practices, not necessarily the person being observed.

2) Recording my Open House classes allows me to identify certain skills and improve them.
These are NOT self-fulfilling, self-righteous, self-absorbed, self-promoting attempts at my own teaching. If I am to encourage other teachers at video-recording their teaching in the name of professional development, I need to do it too.

3) What professions record themselves (or other professionals) as a means to strengthen their craft? I recently tweeted some professions that benefit from recording their craft, ranging from athletes to musicians to dancers to doctors to comedians to teachers. You can only get better, right?

What professions record themselves (or other professionals) as a means to strengthen their craft? #thinkingoutloud

— Andrew Stadel (@mr_stadel) July 1, 2015

4) I need to be vulnerable, humble, and open to improve. I intend to keep my video clips to segments shorter than 10 minutes. I picked a specific part of the class I wanted to focus on improving. Even if no one else sees your video, I hope you'll prop up a camera in your room and record 10-15 minutes of your class. Play it back and reflect. Go in the next day and make those changes (improvements) you want to make.

5) One more time: These are NOT self-fulfilling, self-righteous, self-absorbed, self-promoting attempts at my teaching. Nor do I pretend to be perfect at this stuff. I just know I can be better and video recording helps immensely. If you have the stomach to continue, you'll see my notes as I reflect on my facilitation of this Estimation 180 challenge: my wife's height.

    

1:45
I ask the class to share a "too low" for Mrs. Stadel's height.
I try to get the class' attention about writing down 4'10" and how it's confusing... because I know students typically write feet and inches using decimal notation (4.10).

Here's what I wish I did (or could have done):
If I had done a better job of checking student work while they're working, I would have known how the student wrote 4 feet 10 inches on her paper.
I wish I asked the student how to write 4 feet 10 inches on the whiteboard.
I could have had the student write it up on the board while all the students were working.
I should have asked her why 4'10" was a reasonable "too low".
Having her share her reasoning would strengthen her voice and mine at the same time.

2:10
I received 7'10" from another student for the "too low" and asked "Is that taller than me?"
Like the previous student, I should have asked her to share her reasoning.

2:20
"Did anyone put 6'4" as their 'too high'?"
I need to capitalize on why 6'4" would be an extremely reasonable "too high".
Five students raised their hand on camera and I remember about two others students (off camera) raised their hand as well.
I should have followed up with at least one of these seven students as to why 6'4" makes sense here.
*I usually do this... argh.

2:35
"Don't share with me, share with your neighbor. Share with your group."
Why did I not say, "I would also like you to explain your reasoning behind your answer."
This would have also bought me an extra minute to check in with at least one more student. When checking in with students, I typically have them rehearse on me so I can encourage them to share whole group in a few minutes.

3:15
I began to take four estimates.
"I'm totally forgetting something."
I forgot to introduce the idea of students filling out their number lines.

4:00
I like the idea of the class filling out the same number line together on the first estimation challenge.

4:20
I explain that it's a common student misconception to put their estimate right smack in the middle of the number line, even if it's wrong.
Instead of being abstract in my explanation, I could have been more concrete by showing them an example. Let's say a student thinks Mrs. Stadel is exactly 5 feet tall (5'0"), they would still put there answer right smack in the middle. I should have put 5'0" in the middle.
Ask students: What's wrong with this?
Ask students: How can we use this number line more accurately?

I was truly lucky to get a difference of three feet between 4'10" and 7'10".
I didn't explain my thinking to students: I knew it was three feet, but didn't explain that I knew that half of 3 feet is one and a half. I should have modeled my thinking for students.

4:55
"For time reasons, I'm going to help you out"... look at me, the clock watcher. Am I really helping students out? I feel more like I'm helping me out by bailing them out. For about a minute, it's me doing the math, not the students.

How silly of me, I could have simply marked 5'10" and 6'10" on my number line breaking the distance between 4'10" and 7'10" into thirds. And again, explain my thinking.

5:10
I'll start taking your guesses.
I like when I ask students to stop me when I get to their guess on the number line.
Should I take the average? or should I have students argue it out?
I think the latter.
Here's why: that initial placeholder on the number line impacts EVERY estimate afterwards. Make it count.

6:10
I just got done taking four estimates and didn't ask for one single reason behind them...
I say, "A lot of good guesses here." because I already know the answer. I can see that the four estimates I took are in the ballpark. However, I never called on one of those four students to convince me.

6:15
I tried to dig myself out of this hole by saying, "I'm going to keep you guys in suspense."
I wouldn't say this is the best move. However, it was better than completely ignoring (or forgetting) the reasoning behind their estimates.
I went with two reasons. Half the number of guesses I received just a few minutes prior. BUMMER.

7:45
Once I revealed the answer, you can hear the "ohhs." because they were actually pretty close.
I say, "If you are within one or two inches, you should be really proud of yourself."

A student says, "Dude! I was off one inch!"
Same student, "I'm so happy!"

Maybe I missed it (or forgot by now), but a couple kids had a huge sigh of relief and were even excited about this. Seriously, that's awesome. If all we did was worry about our "answer" being exact every time, then we've lost focus on the process of estimating. Doesn't it feel absolutely awesome to have an estimate within a couple inches?

For my second day, I've seen how rusty I am and what needs improving. Moving forward:
Goal #1:
I want to cheat when placing my "too low" and "too high" on my number line from now on. Once I place my "too low", I will choose a "too high" that allows me to easily partition my number line into reasonable spacings.

Goal #2:
While students are filling out their handouts, look for 8 different students and prepare them that I'd like them to share with the class (and include reasoning):
Too low: 2 students
Too high: 2 students
Actual estimate: 4 students

Goal #3:
Ask the class who agrees or disagrees with the first estimate given.
For example, "Who disagrees with the estimate of 5'9" and would like to share why?"
Continue this with the second and third estimates given.

If you made it this far, feel free to offer any suggestions to help me get better. Thanks in advance.

Open house,
816

Barbie Zip Line (2015) Part 1

Last summer I tried Barbie Zip Line and reported the experience here. I also supported a handful of Math 8 teachers interested in Barbie Zip Line during the school when they explored the Pythagorean Theorem. I have to admit, with every experience, it always felt like it could be different, possibly better.  This year, I went a different route and Part 1 just documents what I've done so far. Part 2 will be the conclusion.

First, I avoided the Pythagorean Theorem (for now). On Monday, my students already knew we would be starting Barbie Zip Line on Thursday. That was about as much information as I revealed. Everything else was structured to elicit as much student insight, information, and ideas as possible.

I started by projecting this slide:

Students discussed in their groups and a few shared whole group. I jotted down a few quick notes:

I love this informal language. Would this be an opportunity to work in slope? Maybe. I wouldn't force it as I'm confident we'll have plenty of other opportunities.

Me: Has anyone her gone zip lining before. 

A few hands go up.

Me: How would you describe it to someone in the class who has never been?

Katherine: Awesome!

Me: How would you describe what zip lining is to someone unfamiliar to it?

Katherine: You wear this harness. You ride down a line...

Mateo: You have two cables attached to you in case one of them breaks, there's a backup. Someone pushes you at the beginning and you ride along a cable...

Me: Great. Thanks. Would it help if we saw pictures or video of someone zip lining to give everyone a better perspective?

Everyone: YES!

Me: Here's what Google Images has for "zip line pictures". 

Me: A good business model will provide their customers with a safe and thrilling experience. Therefore, I'd like you all to fill out this Google Form with the following prompts and questions:

• Briefly describe the characteristics of a DEATH zip line.
• Briefly describe the characteristics of a BORING zip line.
• Briefly describe the characteristics of a JUST RIGHT zip line. 
• What information would be useful to know when building a zip line?
• If we had a small scale zip line in class, what data can we collect from the small scale? 

I'm fascinated by the results. I learned I need to truly value, trust, and use my students' intuition way more often and when launching a lesson/activity. Check out their results here. These results will help guide Desmos Part 2. However, first we need to do Desmos Part 1.

The actual zip line quad.

Desmos Part 1
Students go to this Desmos graph and quickly create three zip lines.

Once they are done, they head over to this Padlet page and post their Desmos graph for their classmates (and me) to see.

Created with Padlet

Desmos Part 2
*I will post what students do in Barbie Zip Line (2015) Part 2.
Before going outside, students begin doing a small scale version of the zip line inside the classroom. Here are the materials:

• 3 paper clips
• 2 measuring tapes
• 1 string (100 inches)
• iPad (for Desmos part 2)
• iPad or phone timer

Record their data inside of this pre-made Desmos template.

*If you go the route of the Pythagorean Theorem, adjust your table accordingly.

Here's a handout for each student. After collecting their data, students will be expected to draw a pretty descriptive scale picture of their zip line on this handout. They'll also need to predict how long it will take their doll to complete her zip line ride.

As you can see from the handout and expectations, I'm placing a big emphasis on the following:

• Scale
• Proportional reasoning
• Rate of change (or slope)
• Rate

No mention of Pythagorean Theorem. Find out if it stays that way in my Barbie Zip Line (2015) Part 2 post, next week.

Zip 1,
1011


P.S. Most importantly, my son was really excited to visit my class today and partake in the Barbie Zip Line adventure. I was really excited too. DUH!

Tacos For (almost) Everyone

Do you remember when I blogged about the Ultimate Task for Vertical Planning: Stacking Cups? If not, feel free to check it out at your convenience. I've got another task for you that can be used at multiple grade levels: Dan Meyer's Taco Cart.

When asking:

Who will reach the taco cart first?

there are so many mathematical opportunities awaiting us. Here are a few:

Math 6 (maybe Math 7)
Pass out this handout during Act 2 and tell students you will only give them one dimension. Choose wisely.

Read more about this great technique on Fawn's blog post about Mr. Meyer's Taco Cart.
It simply is brilliant. Students are measuring the dimensions (distances) on the paper and using proportional reasoning to figure out the real life distances. I recommend students use centimeters when measuring the dimensions of the triangle on the handout. I really enjoy this technique.

Math 8
If you're a math teacher and you see the picture Dan provided for Act 2, your intuition will most likely steer you in the direction of the Pythagorean Theorem. Go for it!

Geometry (HS)
Let's say you have already used Taco Cart during the year to apply the Pythagorean Theorem or Distance Formula (Desmos). How about we extend the mathematics and look for more right triangle relationships in Taco Cart. I noticed that the hypotenuse is about twice the length of the shorter leg. Let me connect that to the context of the story: Ben's distance is about twice the distance Dan travels in sand. That's right, Dan gave us a 30-60-90 right triangle. Pro skills there, Dan.
*I'm not saying the 30-60-90 relationship is the most intuitive, but we'd be helping students make connections with previous learning. 

Algebra and Beyond
As you move into the sequels provided on the website, there's a lot of higher level math. Depending on the question, the problem-solving is fun. I worked with a high school group of math teachers who found it extremely challenging to solve the question:

What path to the taco cart would take the least amount of time?

Overall, this is such a fun and meaningful task. Dan has given us a treat! Today, my students did such a great job arguing, sharing theories, identifying variables, and using their intuition even before I unveiled any information from Act 2. It was awesome! I'm avoiding the use of the Pythagorean Theorem this round. I went Fawn-style by giving students only one dimension on their Act 2 handout. So good!

Tacos por favor,
942

How Much Is Your Name Worth?

Starting next week, I'll be back in the classroom with my own roster of students. I'm super pumped and plan to be really active on this blog...  I plan to do a mixture of blogging about ideas before I use them with students and after I use them.

I need to quickly learn the names of my students on Day 1, especially since I'll only be with them for only 20 days. I'll probably do the Name Tent activity and Class Height activities found here. However, I want to establish some mathematical tones as well. For example, most tasks/activities will require students to:

• make guesses (too low, too high, just right)
• submit data
• collect data
• sort data
• use the data
• measure
• problem-solve
• make predictions
• use technology

How much is your name worth?
If each letter of the alphabet was worth its place in the alphabet, how much is your name worth?
For example:
A-N-D-R-E-W would be 1 + 14 + 4 + 18 + 5 + 23 = 65

Figure out how many points your name is and submit it here:

What name will have the lowest points?
What name will have the highest points?
What will be the class average?

If this is golf, the lowest wins.
If this is basketball, the highest wins.
If I want the class average, what would that be?

• Students will submit their values using Google Forms.
• We learn how to sort the data in Google Sheets.
• We can answer our questions.
• We can use the data to predict the value of the next person that walks into our class, or the principal, or a parent, a stranger, etc.

This should not necessarily last that long, but there will be parts of the process that will be important to being more successful and efficient during our time together.

Name value,
543

Classroom Visit in New Hampshire

Yesterday, I had the great fortune to Skype with a second grade class and their teachers in New Hampshire.

IT WAS AWESOME!

There were about 6 students who came up to the webcam to:

• share their name
• share their favorite Estimation 180 challenges
• ask me a question about Estimation 180

Here are some of their favorite Estimation 180 challenges:

• Large Chocolate bar
• Surface Area of my daughter's hand ("cuz she's so cute.")
• Coins
• Marshmallows
• Lego Transporter set
• My height

Some of the questions they asked:

• How do you say your last name?
• How do I think of the estimation challenges?
• Who takes the pictures of me?
• Will you do more Lego estimation challenges?
• How many estimation challenges are there on the site? (I was asked this twice.)
• Will I continue to make more estimation challenges?

After finishing Q&A with my six new friends, Ms. Spear asked if anyone else wanted to share something. One girl spoke up and thanked me for

"...helping my brain to think more and not give up."

This warmed my math heart and made my day. I think that's a direct reflection of Ms. Spear and her colleagues who are creating a classroom of curiosity, perseverance, and risk-taking. They're raising the bar high for all of us, so anyone who gets Ms. Spear's students in the future, please continue to carry the torch and never let that flame become extinguished.

Something else warmed my heart. Ms. Spear shared that the class used estimation challenges, mohawks, and the strength of a small school community to raise money for Levi and his fight with cancer. Woah! Cool!

I thanked them for being such a polite, mature, and respectful group. As you can only imagine second graders staying seated for longer than 18 seconds is a small miracle. They were a classy group that has inspired me.

Thank you Ms. Spear and your students for allowing me to briefly visit your classroom.

You're the inspiration!

Thanks,

930

Fastest Sticky Sticker

It's rare that I post about something I haven't tried in the classroom. Here's an idea that came to me today, inspired by:

• My File Cabinet task
• Anyone who has asked, "How long did it take to put all those stickies on?" 
• Al's Card Tossing activity
• Jon's Trashketball Spiralled Lesson

I haven't done this activity, yet. If you try it out, please report back or offer suggestions. Thanks!

Competition:

Who is the fastest Sticky Sticker?

Translated: Who is the quickest at covering a 2-dimensional shape with sticky notes?

Materials:

• Whiteboards
• Stickies
• Blue painter’s tape
• Scissors
• Timers

Break students into groups.

Have them time each other sticking 10(?) stickies somewhere (whiteboard, desk, etc.).

• Determine who is the fastest Sticky Sticker of the group.
• Use their cell phone stopwatches as timers
• Use some type of table to predict how long it will take each person to stick different amounts of stickies and write an equation.

• Have each student determine their rate.

Reveal the playing fields

• First, without the dimensions, of course.
• Muster up some trash-talking
• I bet you I could beat anyone in here with one-hand behind my back.
• I might even give you a head start.
• I could beat you blind-folded.
• Have them write down guesses as to how many stickies will cover each shape.
• Have students guess the dimensions.
• Have measuring tapes out for students to measure their shape.

Have all groups figure out how many stickies are necessary for each shape. All dimensions given in inches.

• Square (24x24)
• Rectangle (21x27)
• Triangle (27x24)
• Parallelogram (24x18)
• Trapezoid (b1= 27, b2=21, h=18)
• Circle (d=18)

*The following is where I start thinking out loud and not entirely sure what makes sense since I haven't tested this out with students. Feel free to try it out and please report back.

Have each group randomly pick a shape.

• I'm going to predict that some students or groups will complain/gripe about receiving anything other than the square or rectangle. That's where the scissors come in.
• Give each group the amount of stickies they calculated for their shape
• Include a couple(?) extra stickies for a mistake?
• Give scissors to every group, but the square and rectangle groups.
• Groups who don’t get the square or rectangle must cut their stickies to fit inside
• The Circle group(s) should maybe get a little bit of a cushion (modification).
• The square and rectangle groups need to be challenged while they wait.
• They can help other groups prepare or figure out a reasonable head start.
• Should certain shapes get a head start?
• Should the head start be:
• time?
• stickies?
• Can we modify any of our equations from above?

Ready, Set, GO!

I’d love to see each student participate in the competition. At first, it might appear as though each group picks the fastest Sticky Sticker, but I’d love to make this competition a relay race.

• Have each group divide their total number of stickies by the amount of group members
• Each group member should stick about the same number of stickies.
• Groups determine the order (strategy)
• Could we graph what that might look like?

*At this point, go back to the blog posts by Al and Jon for more tips.

Determine how the head starts will be determined.

Blow the whistle and get kids sticking those stickies.

Congratulate the winners. Take selfies. Play your national anthem...

Round 2

Who can take the sticky notes off the fastest?

Useful Math:

• Area of various 2-D shapes
• Ratio of stickies stuck to time (or time to stickies)
• Rate
• Unit rate
• Writing an equation to model the rate
• Using the rate to predict how long it will take
• Possibly graphing the data (or “constant of proportionality)
• Translate (graphically) the equation above to account for the head start
• Piecewise functions for different members of the group.
• Decompose square units in a defined area

Let me know if you're going to try this one out. I will probably test it out in a few weeks during my summer course and report back here.

Sticky sticker,

615

Should We Use the Term "pace" More?

I'm preparing to be one of the presenters at a 3-day NCTM Deep Dive Institute in July. Hope you can make it. Fawn will be presenting too.

I've come across some great tasks from NCTM, thanks to Peg Cagle (that's C-A-G-L-E everyone) that I've adapted to have a much lower entry point for teachers and students. However, I'm also looking to mix in some favorite Estimation 180 challenges and 3 Acts like Fast Clapper from Nathan.

Fast Clapper Act 1 from Nathan Kraft on Vimeo.

I really dig this task. At first glance though, it looks pretty straightforward. Act 2 could look as simple as showing students a screenshot like this:

Then tell students to use this ratio to predict how many claps this dude will complete in a minute. We could call it a day, but what fun would that be?

Here's what I think should precede any screenshot from the Act 1 video. Have students get out their cell phones and partner up. Record their partner clapping for various increments that are less than 30 seconds. For example: 0, 5, 6, 10, 12, 15, 20, etc. Keep track of it in a table...
Then see who is the fastest clapper in the class and if they can break the record. Talk about what might prevent the students or dude in the video from breaking the record.

Here's an additional place I'd like to take Act 2. Talk about the term "pace". I really like this question I'll be using from now on with students and teachers:

How often should we check to see if he is on pace to break the record?

I think this question opens up the mathematics, especially for a table of equivalent ratios and double number lines. Forget equations (proportions) here. Furthermore, it reminds me of the pace timers that you sometimes see on television during the Olympics.

Talking about the word pace, is this the same thing as rate...?
I'd really love to hear from you about the term "pace".

• How often do you use the word pace in math class?
• What context do you use the word pace?
• Are terms like rate or slope synonyms to pace?
• Tell me everything you know about pace or how you use pace in your class?

Seriously, I want to know. Teach me!

Pace yourself,

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Questioning Strategies from the #MTBoS

This past week, a fellow asked me about questioning strategies.

Okay #mtbos, unleash your favorite book/blog/video/source on effective questioning strategies. GO!

— Andrew Stadel (@mr_stadel) May 22, 2015

I'm not sure where your brain goes when someone asks about questioning, but there's at least a few places my brain goes:

1. What questioning strategies can we use to unstick students when they're stuck?
2. What questioning strategies can we use to explore student (mis)understanding?
3. What questions can we ask when designing lessons/activities/tasks?
4. What questions can we ask when launching lessons/activities/tasks? 
5. What questions can we ask...

Thanks to Twitter and the amazing #MTBoS, here's the list you all gave me. Hope this is helpful. If I missed any resources, please add them in the comments.

Questioning,
327

P.S. Here's the list.

Ketchup (Guess vs. Estimate)

I had breakfast at a restaurant this weekend and noticed the ketchup bottle on the table. You know, the bottles that are red plastic? That are supposed to appear full? I always get a kick out of these bottles. Here's why..

I immediately wonder how much ketchup is in the bottle?

Take a second to think how almost any answer is pure GUESS.

WHAT information would you want to know here to make an estimate and not a guess?
HOW would you go about getting the information to make an estimate and not a guess?

The second I do this...

I know WAY more information. It's no longer a guess.

Think of other senses that could be used to make a better estimate. 
*One scenario would be something along the lines of me watching the customer(s) before me to see how they held the bottle. How did they shake the bottle? How many people at the table used ketchup and how much?
But that's just plain weird...

Sure, pick up the bottle. Formulate an answer and be ready to back it up with a reason. Don't skip this reasoning part. The bill depends on it!

Let's now move to the answer. Let's say we have more information now.
How would you describe your answer?
How might someone else describe their answer?

Could we say any of the following?

• It's half full.
• It's about 3 squirts.
• It's two-thirds empty.
• It's about a pound.
• I could eat 5 french fries with that.
• It's about 8 ounces.
• [insert other]

Here's where specificity matters. How should we agree to quantify the amount of ketchup in the bottle? Should we agree at all?

This ketchup bottle context is one of the simplest contexts I've come across in awhile. Here's why:

• The question is straightforward.
• You demand more information to do anything better than a pure guess.
• With one small piece of information, your guess should now be an estimate!

In case you're wondering about the time of day, I don't think it really matters here. The bottle was about a quarter full and this was at breakfast time. It's not like someone went around the night before and filled every ketchup bottle. Which begs the question: 

Is it more efficient for an employee to go around lifting all of the ketchup bottles to determine if it needs refilling or should they just wait until a customer says, "The ketchup bottle is empty, can we get a new one?"

Why haven't you seen more Estimation 180 challenges that deal with weight, density, etc? They're tricky to capture. I wish I could fix that, but I digress. I'll put that onus on you. 

My charge to you is:

No matter what grade level you teach, bring in an item like the ketchup bottle. Ask a simple question where the answer is pure guess and students demand more information to make an estimate. Literally, keep track of all the questions/demands students formulate. Report back.

Classroom (or lesson design) application:

• Design lessons with less. (notice "less" is in "lessons")
• Ask straightforward questions that demand more information.
• Use information to move away from guesses and into estimates.
• Is it more effecient to go around asking our students what they're stuck on and re-filling them with information or should we wait until they realize their stuck and we help them get unstuck?

Lots for me to think about. Feel free to chime in with some advice. Thanks.

Ketchup,

1248

P.S. This reminds me of one of my favorite jokes:

A momma tomato and baby tomato are walking down the street. The baby tomato falls behind because it's going slower. The momma tomato turns around and stomps on the baby tomato, yelling "Catch-up!"

The Ultimate Task for Vertical Planning: Stacking Cups

This past week, I submitted a speaker proposal for NCTM 2016 in San Fransisco. The proposal is for a Grade 6-8 Burst (30 minutes) with the exact same title as this blog post: The Ultimate Task for Vertical Planning: Stacking Cups. I figure if I don't get accepted, at least I can share my thoughts here and you all can help spread the word about my idea if you think it has potential. If it does get accepted, I look forward to giving an update a year from now at NCTM. Here's my session description:

Who says you can't use the same task each year? Come see why Stacking Cups might be the single best secondary math task to get teachers at your school, district, or state to see the importance and necessity of vertical planning. Use tasks that utilize connections from the previous year and extend the mathematics each year. Work smarter, not harder. 

Let's first back up a bit. I attended Alex Overwijk's session at NCTM Boston a few weeks back. I had already read his awesome blog post "Open Strategy Cup Stacking" and knew there are multiple teaching moments with Stacking Cups. I remember teaching Math 8 a few years ago and getting a lot of use out of Stacking Cups as you can see a couple times here and here. I was preparing for a training with math teachers from grades 6-12 and THAT's when it hit me: I could have a room full of math teachers from grades six through twelve and they all could:

• be working on this task
• see the different skills and tools necessary for solving
• know the expectation of each grade level

I've heard comments from teachers numerous times like, 

"Well, if they do File Cabinet in 6th grade, I can't do it in 7th grade with my students."

"If they've done Stacking Cups in Math 8, then I can't do it in Algebra."

"If the 5th grade teachers use Estimation 180 with students, then I can't." 

YES! YOU CAN! It's called vertical planning.

YES, YOU CAN! Instead, let's ask different questions like, "How can we use the same task to extend the mathematics each year?" and  "How can we make connections to prior learning from the previous grade level?"

Let's work smarter, not harder.

I will spend the rest of this blog post highlighting each grade level and suggested uses for Stacking Cups. It won't be complete or the final version as this is through the lens of one person. I'm confident, with your help and critique, we can make it even better.

Math 6
Question: How many cups do we need to stack (alternating) to reach someone's height?
We talk about rate. We organize our information on a number line, in a table, using a tape diagram, etc. We explore the rates using various models.

Math 7
Question 1: How many cups do we need to stack (alternating) to reach someone's height?
We continue the conversation started in Math 6 revolving around rates, using constant of proportionality. All of this can be represented in a table, as an equation, and in a coordinate plane.

Question 2: How many cups do we need to stack (consecutively) to reach someone's height?
We now shift our thinking a bit where there is still a constant increase with each cup, but there is an initial amount (the cup handle). Students explore how to write an equation to represent this situation and solve it.

Question 3: What would be possible dimensions of a box that would contain the cups to stack to someone's height? Which dimensions would be the most cost effective?
Imagine students understanding surface area and volume and how they're related to each other, especially if we model with mathematics, by identifying variables such as:

• cardboard cost
• delivery truck capacity 
• store storage sizes
• consumer trends with buying cups
• more

Math 8
Question 1: How many cups do we need to stack (consecutively) to reach someone's height?
Similar to question 2 in Math 7. However, we extend the mathematical understanding as we explore constant rate of change (slope), input and output, linear, and how our situation can be represented in the form y = mx + b.

Question 2: When will two stacks of different sized cups be equal in height and have the same number of cups in each stack?
We introduce students to linear systems using this task. Students can organize the information about each cup in a table. We can extend prior knowledge to represent the situation using graphs, equations, and functions.

*By the end of Math 8, it might be helpful to mention (at least informally) to students the significance of discrete functions.

Algebra
We tighten up the math (both questions) previously learned in Math 8. How can we extend the mathematics. Add more challenging situations like the stacks start on different objects like desks, boxes, etc.
Question 3: How many cups would we need to stack in a triangular formation to someone's height?
This questions really extends the mathematics for students, but we can still use the tools they've learned from previous grades. Maybe students start by organizing the data in a table. Maybe they graph the data and notice it isn't linear. Maybe we can use desmos with sliders or a line of regression to explore quadratics.

Beyond Algebra and Geometry:
I'll admit this is where I'm a little rusty and would need you high school pros to jump in and contribute. I think with the triangle stacking, it can be taken from quadratic to a divergent series. I've also seen high school teachers come up with the following representations:

Al Overwijk also stacked cups in a triangular pyramid which is awesome.

Let's keep this vertical planning going. If you would like a couple charges, here you go:

Go to your site and/or district and push for Stacking Cups to be a signature task at all sites and secondary grade levels. Help support your colleagues with vertical planning. Report back.

Look for other tasks out there like Robert Kaplinsky's Hot Dogs or Dan Meyer's Penny Circle or Mathalicious' Wheel of Fortune or Graham Fletcher's Water Boy that can be used with vertical planning. Report back.

Vertical,
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