Square Dance

I recently debriefed with a fellow (teacher I support) about two activities focusing on Squares, Square Roots, and Irrational numbers. Let's build number sense. Here are the goods:

• Clothesline activity using these cards
• Square Dance desmos activity

She ran both activities with students, starting with the Clothesline activity. She used the cards linked above for students to first place the visual representations on the number line. It looked something like this:

Followed by:

Students then completed the first 7 screens in the Desmos Square Dance activity. Screen 6 includes a validator when done correctly, compliments of Nathan Kraft.

*Please note that part 1 of the activity uses only whole numbers as rational numbers. I highly recommend using the activity as a launching point for students to know that perfect squares include other rational numbers like fractions and decimals. 

Back to Clothesline:

This week she will use the next set of cards for irrational numbers. It might look something like this on the number line:

Followed by:

Back to Square Dance 

Students can build better conceptual understanding of irrational numbers in the desmos activity. Also look for teachable moments throughout the activity. 

*Please note screens 11 & 15 include non-repeating and non-terminating decimal notations. 

Screen 11

Screen 15

Just like Screen 6, Kraft-y validators are included on screens 12 & 16.

Two closing thoughts:

1) My fellow was so happy to use these conceptual representations with clothesline and desmos. 

2) She hasn't seen students making mistakes like she has in the past. Here's an example (crossed out) of a common mistake she has seen regularly in the past.

If you have time, head over to this post and have your students play War with the Rational-Irrational cards provided.

Dance,

945

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

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,

312

Estimation 180 has Lessons!

Head over to Estimation 180 and you'll see this lovely new option in the menu bar.

LESSONS!

That's right! 

LESSONS!

I've added a "Lessons" page with many lessons I've created, sorting them by their CCSS. I'd like to thank Dan Meyer and Robert Kaplinsky for their friendly suggestions (nudging) to tag my lessons in an attempt to make it easier for other teachers to find and use. Plus, I'm tired of my lessons collecting digital dust and hope that teachers can find and use them.

I was honored to give a workshop for teachers in my district today. The workshop became the motivating factor for making this Lessons page. Right now, most of the lessons are 3 Act lessons that can be found at Dan's 101qs.com A few other lessons are ones I've blogged about. However, I have added two test pages at Estimation 180 where the entire lesson is available for teachers to use. Right now. At Estimation 180.

Pay close attention to my File Cabinet and Stacking Cups lesson PAGES!.

These two full-on lessons are ready for you and your students. You'll see all three acts, teacher notes, student work, student handout (if you like/need), and downloadable videos. Let me know if you have any thoughts, advice, or questions.

I hope this "Lessons" page is useful and/or better than that silly unorganized spreadsheet I've got lingering. You'll notice a few links are under construction, but many links deliver the goods. Check in often for updates.

LESSONS!
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P.S. Thanks to Fawn, Nathan, Robert, and Eric for your feedback.

Woody's Raise

We decided to get Netflix recently and I was excited to see that Cheers episodes are available. I occasionally put an episode on in the background while I get work done. I came across this episode that literally snuck in some math (money, raises, time, rate) right before the end of the episode. Sam Malone, the owner of the bar in the tie (played by Ted Danson), is talking with Woody Boyd, a bartender (played by Woody Harrelson), about a raise. Roll Act 1:

After consulting with my man, Nathan Kraft, I bleeped out a part of Woody's last line. The two of us discussed the tendency a bleep can have in implying some profanity was removed. So if this lesson goes horribly wrong, blame Nathan! All those toothpicks finally caught up with him. Here's how the exchange goes between Sam and Woody:

Sam: We were talking about your 50 dollar a month raise.

Woody: Sam, it was a hundred a month.

Sam is caught for trying to pull a fast one on Woody. Woody appears to let it slide, but something occurred to Woody. He turns to Sam and the exchange continues:

Woody: I think a hundred a month is too steep. I'll settle for [BLEEP] a week. 

Sam (without blinking): You got it!

I anticipate students noticing that the amount was bleeped out and wondering what was bleeped. I anticipate students not sure if Woody said, "[BLEEP] a week" or something inaudible? I anticipate students noticing that the studio crowd laughs while wondering if Sam was just made a fool by Woody. I would love to first have a leisurely conversation with students about who they think just got the better deal in this exchange, Sam or Woody? Or was there even a better deal to be had? If you've ever watched an episode of Cheers, you know that neither character has a strong IQ. If anything, Woody is portrayed as a real naive, gullible, and takes-you-at-face-value type of character. Sam is about a handful of points above Woody. So what about Act 2 after you take some guesses from the class on who just got the better deal from this exchange?

This might be the first 3 Act lesson in which I don't have any additional information for Act 2. In all fairness, this might not fit my previous rant on measurable acts, but I think the 8 Standards for Mathematical Practice are rubbing off on me (in a good way), especially Practice 4: Model with Mathematics.

I posted the Woody's Raise lesson on 101qs.com with very little in Act 2 because I'd love to know where the teacher would take this with his/her class. This type of teacher discretion can't be packaged in an online portal or catalog of video instruction. Here's what I threw out there for Act 2 (the first edition):

At what "raise" amount per week would Woody "settle" for the:

1. Better deal
2. Equivalent deal
3. Worse deal

I have many questions when thinking about Act 2. Here's a few:
Over time, when does Sam or Woody begin to benefit or suffer from this deal, compared to the $100 raise per month?
Do all months have exactly four weeks? Does that matter or should we use 52 weeks in a year?
How would you anticipate students representing Woody's better deal versus the worse deal?
What would this look like graphically?
What would this look like organized in a table?
What equations could you anticipate students writing? If any?
How does this deal apply to Woody's hourly rate?
In what classroom could you talk about the tips Woody might make? Remember this takes place in a bar. Middle school students? High school students? College? A workshop with teachers? I think there's a lot of fun to be had with this video clip. Let's Roll Act 3 and see what Woody would "settle" for instead of the $100 a month raise:

I'm posting this lesson because I'm thinking out loud. More importantly, I'm curious what you would do in between Act 1 and Act 3 with your students. How would it be different in an elementary classroom? Middle school classroom? High school classroom? Teacher workshop? What would your Act 2 be? Where would you take this lesson with your students? I believe this is a multi-dimensional lesson that can take on some great mathematics. Bleeping out that weekly rate in Act 1 really opens up Act 2 for some rich mathematical discussions and modeling. Toss your Act 2 in the comments. Thanks!

Cheers,
1026

QOTW 2nd Semester - 2013

In January, my Quotes of the Week post highlighted student comments captured during our first semester of the 2012-2013 school year. I'm here to post a few captured from the second semester. It's hard to compare them to the first half of the year, so let's not. Instead, let's just enjoy the comments, observations, or questions that students gift us with, enriching the mathematical climate of the classroom.

I experimented with Kelly O'Shea's Mistake Game in Algebra about midyear. Emma is presenting to the class about identifying linear functions, given three points. Justin (from the audience) claims that Emma’s graph is NOT linear, but says there's a slope to her three points. Without skipping a beat, Emma unleashed this response at Justin. Way to go Emma! Talking smack with math vocabulary.

Students were exploring x- and y-intercepts using Desmos.com. After graphing a few lines and writing down the ordered pairs of each intercept, the wheels started to turn inside of Lisa's head. She saw the structure and pattern of the intercepts containing zero. The follow-up:

Me: So Lisa, why are you seeing all these zeros?

Lisa: Because we're learning about intercepts today. 

Me: What are the intercepts of the lines? 

Lisa: It's where the lines hit the x or y axis. 

Me: So what's up with all the zeros? 

Lisa: If it hits the x-axis, the y value is zero. And if it hits the y-axis, the x value is zero. 

Me: Bingo! 

Mark needed to graph a straight line as part of his task. That week, we had just spent an entire class period doing a Jigsaw activity so that the students could explore the 8 Standards for Mathematical Practice. Mark is proud to be using Mathematical Practice 5, Use Appropriate Tools Strategically.

Groups were given a warm-up one day where they had to collaborate and stack the highest and strongest tower using 100 snap cubes. I determined the winners by kicking the desks they stood on to see who survived Earthquake Stadel the best. While cleaning up, Logan is noticing that he could do a better job next time. This is exactly what we can do in the math classroom: learn from our mistakes, come up with a revised problem-solving plan, model, collaborate, and persevere.

The class was doing a cocktail of mixture problems one day. They were collaborating with their groups and one group had your typical coin question. Something along the lines of:

I have $9.75 from X amount of nickels and dimes. How many of each coin do I have?

Upon solving the question on their whiteboard, Sara quickly realizes that you can’t have 1.95 of a coin. She was the first to notice it and say something to her classmates about this contradiction. The group first tried looking for their mistake and then tried solving the question again. Good job girls!

Midyear, I showed all my classes the 1st Semester QOTW slides. For the first month following these slides, Charles was trying to turn every little thing he said into a quote just so he could be up on the board. As I've mentioned before, these can't be planned or contrived. These quotes just come out naturally in the regular happenings of classroom interactions.  Students were doing some individual work one day, and Charles actually let a good one slip out. He was still working individually on the task and didn’t want his efforts spoiled by someone blurting out the correct answer. How many times have we been there, either on the receiving end or the one ruining it for others?  

This is Jenna's response to the following video:

I asked the class for their interpretation of the video or any of the drawings? Was there a drawing they could relate with? Without blinking, Jenna says, "It's like graphing stories." I totally agree. It had been awhile since we explored graphing stories, so Jenna just confirmed what a great impact that concept had on her. Very cool! 

Arielle is a former QOTW winner. Students were exploring exponent rules by finding mistakes. Michael Pershan and I have done some extensive blogging about this. Anyway, we had reached the final day or two of learning through mistakes and Arielle raised her hand and shared this gem with everyone. It's a beautiful observation in my opinion about the importance of math mistakes, learning by making conjectures, and students coming up with the rules on their own instead of me spoon-feeding them. 

Two quotes in one class period! That's a first. We were playing Race Car Math and one of the review questions asked students to factor a polynomial. Eli, a boy a few words, caught his group-mate incorrectly factoring the polynomial. I love the “Dude!” part of his quote.

Also during Race Car Math that day, Kailey was so excited to be on the board after correctly graphing a parabola, following the flowchart we put up in class. Our class flowchart is shown.

Last, but not least, Chris closed out the year with, "I think my number lied to me." In geometry, we were reviewing volume of a sphere with questions similar to Nathan Kraft’s volume of a soccer ball task. Chris set up, and used the incorrect proportion when solving. When arriving at an unreasonable answer, he realized his numbers lied to him. Way to check for reasonableness!

It's truly been a pleasure having the QOTW section carved out on my whiteboard for student quotes. As far as I'm concerned, this will be a staple in my classroom for the remainder of my career. I look forward to next year's quotes!

QOTW,
717

Cent-ed Whiffle Balls

Want to know how to make Cent-ed Whiffle Balls? Here are the ingredients:

1. Bookmark this picture at 101qs.com
2. Do coin estimation with your students.
3. Go to the bank and withdraw a few dollars worth of pennies.
4. Get some Gorilla Glue.
5. Take whiffle balls from your son's collection (source of whiffle balls may vary). 

Show your students the picture from Step 1. Do the estimation task from Step 2. Show them the following slide! 

*If you don't know yet, we covered surface area of spheres in Geometry this week.

We just completed Nathan Kraft's Soccer Ball 3 Act lesson which was spectacular for volume of a sphere! (Nathan, post act 2 and act 3 for everyone NOW!) The Cent-ed Whiffle Ball is a simple task. You know you have a keeper when you hear the following come out of students:

"This is fun!"

"This is stressful!" 

Students first started this task by using a tape measure to find the circumference of their whiffle ball. Thankfully, I've finally won them over on using centimeters. Shooosh! Don't tell those people who like inches. Students then used the circumference to find the radius of the whiffle ball. Well done, kiddos! Next, students either used a tape measure or ruler to get the circumference or diameter of a penny, respectively. Ultimately, they wanted the radius of the penny. Then they got stuck.

"Mr. Stadel, what's the surface area formula for a sphere?"

Sweet! They want it. They need it. They crave it. I didn't write it on the board or give it to them on a handout. Here's where I wish I had an additional hour with these kids to explore this formula. Instead, I had a demonstration ready for them. I took our Nerf basketball we use for Math Basketball Review. I told students that I measured the circumference of the ball in order to construct a circle that has the same circumference. Before class, I cut out a second congruent circle and cut it into eight congruent sectors. I then played this game with students:

Me: How many of these circles will it take to cover the entire ball?

Student 1: Three

Student 2: Four 

Student 3: Three and a half

Student 4: Five

Me: Let's find out!

I pinned the sectors onto the Nerf ball with thumbtacks, covering a fourth of the ball.

Student 2: I was right! It's four!

Student 5: Cool!

BOOM! We had our formula: 4 areas of a circle with the same circumference as the sphere. Simply put: 4πr^2. Most groups immediately found the surface area of the whiffle ball and penny, dividing the two to get something like 88 pennies. One group of girls immediately came up to me and asked for their pennies. Before giving students their pennies, I drilled each group, asking them to explain their number and show their work.

Me: Now girls, if we've learned anything in here this year, we know that our answer on paper isn't always the actual answer. Have you accounted for everything? Look at this picture again (from the ingredients). Did you account for everything?

Devon: There's spaces between the pennies. 

Me: Yup. Why don't you go back and mathematically show me a different number of pennies, now accounting for those spaces.

I had this conversation with each group, or some variation of it. This is where the magic begins. Remember, students were allowed a maximum of six pennies. Here's what they came up with. I'll let the pictures do the talking:

 

 Chris asked for a compass to draw a circle having the same circumference as the sphere.

Elle found the area of a rectangle formed by six pennies. She then subtracted the area of six pennies to get the area of the space created by six pennies. 

Noelle used a parallelogram of pennies to execute the same idea as Elle.

Groups started coming back with revised numbers. They quietly told me their amount. Remember, there's a CASH PRIZE on the line! Im still not sure what that is yet. Groups came in with the following amount of pennies to cover their whiffle ball:
70 pennies
65 pennies
69 pennies
62 pennies

Good luck to them all. They are almost done gluing their pennies. Two groups are done and the other two are close. Here's a few pics!

This group used 71 pennies versus a theoretical 69.

I highly suggest you make Cent-ed Whiffle Balls in class! If not, here are the dimensions:

Whiffle Ball circumference: 28 centimeters
Penny diameter: 1.9 centimeters

Cent-ed,
1050

P.S. Help me make this task better.