Find Your Perfect (Absolute) Match

For the final week of the Explore MTBoS challenge, I am going to share my lesson on Absolute Value Functions written in piecewise notation.

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Learning Target for Students: I will be able to write the equation of an absolute value function using piecewise notation.

At this point, we have spent many days graphing absolute value functions from tables and directly from examining the vertex form equation. We have also discussed the characteristics of absolute value functions.

When students entered the classroom, this warm up problem was posed to them:
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I chose this problem as our “throwback” because I wanted students to recall how to find the domain and range for a relation, since this skill would be necessary for today’s lesson involving the domain intervals for the pieces of the functions. After giving students a couple minutes to think about their answer, I switched up the problem so that the third segment had an arrow at the end; this allowed us to talk about how the domain and range would change if the line continued on forever.

At this point, I gave each student his/her own graph (I quickly removed extras once I had taken attendance for each class period. Only one class had an odd number of students, so I partnered up someone who been absent for 2 days with another student.) Here are the 20 that I pre-made:


I asked students to examine their graph, and briefly examine the graphs around them. What is similar about your graphs (linear, slopes, restricted domains)? What is different (slopes, y-intercepts, equations)? Students copied their own graph onto this sheet, in the box labeled: “My Linear Function.”

Students wrote the equation for their line, and determined its domain. Then I told students that right now, they were alone. Their graph was alone. But that somewhere out there, it has a perfect match. I showed students how if they folded the graph at the x-value of the domain interval, they would be able to line it up with another classmate’s graph, to create a new function. I directed students to find their match, and sit with their new partner.

Original graphs:
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Post folding:
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With perfect match:
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Students then shared everything they knew about their original line with their partner, and transferred the new information into the box: “My Partner’s Linear Function.” At this time, I asked students to recall earlier in the year when we studied piecewise linear functions, and how they were able to write the equation of each line segment. We talked about how they had not noticed that an absolute value function could be thought of as a composition of two linear pieces, until this moment. We talked about how easy they found it to write the equation for their individual piece, and it wasn’t much different adding in their partner’s piece. Students combined all the information to create “Our Absolute Value Function” and made observations about the two different forms the equations could be written. Most students recognized the the connection between the y-intercepts from piecewise notation as being the distance from the vertex (h units away from k).

When we tried another problem, I gave out yellow post-its so that students could hide part of the function, and focus on one line at a time. Some students prefer to use colored pencils to highlight the different pieces, so I also encourage this option.

I had every intention of giving students this exit ticket at the end of the period, but time ran away from us and I didn’t want to cut our earlier conversations short.

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This is the second time I’ve attempted this activity, and I would do it again next year. I love how student pairs are determined by who is their perfect match, which gets my students to work with someone they usually don’t, and they know it’s completely random. I love how it’s close to Valentine’s Day, so we had a silly time with the idea of the Perfect Graph Match. I love how each student physically started out with only one line, and then did some paper folding to pair them with another. I love how this visual representation really shows the separation of the pieces, and that it’s a shared experience we can refer back to during the rest of the unit. And I love how it starts by activating students’ prior knowledge about lines, and how they could use that information in this new setting.

Would love to hear any feedback about this lesson, and if you try it in your room, please let me know how it goes!

A Better Question

Week 3 of the MTBoS Blogging Initiative corresponds with midterm week at my school. Reviewing for midterms is not a task that I particularly like.

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It’s tough to find the balance between refreshing minds or reteaching skills. Based on my review of our last unit test, I wasn’t convinced that my students truly understand the differences between systems of equations and systems of inequalities and their solutions. So I created this basic comparison:

systems comparison

At this point, we had never placed two similar systems graphs side by side in this manner. We reviewed writing the equations and inequalities together, and then I asked students to make a list of all the similarities and differences they noticed. Students were given 2-3 minutes to write this on their own. Here are some of their responses:

I then asked students to share aloud: “What similarities and differences do you notice?” This question was okay. The responses were okay. But something just didn’t feel right. I didn’t want to put these graphs in a specific context; I wanted students to go back to the basics and see the similarities and differences for what they are; but the discussion was flat, and I wanted more energy…

First period ended and I had some time to think about how I would fix this before I taught the lesson again. I remembered the blogging prompt to write about questioning, and specifically, asking better questions. And then I remembered what Chris Luzniak taught us in his Twitter Math Camp session: “Make the question debatable.” It was my own a-ha moment!

I did everything the same the next period, except for one key point. After giving my students individual writing time, I asked:

“What is the BIGGEST similarity you noticed?
What is the BIGGEST difference you noticed?”

That slight change in questioning is all it took to completely change the dynamic between class periods. All of the sudden, I had at least half the class waving hands in the air to share their opinions. The gist of what students were saying was the same between the different periods, but this time the students were more convincing and provided more evidence for their statements. I wondered if this reaction would continue throughout the day, and it did. All of my other classes had the same level of enthusiasm when I asked them for the biggest similarities and biggest differences.

Today’s experience reminded me that one easy way to ask a better question is to make it more debatable. Check out Chris’ Global Math Department Webinar for more strategies on how to do this!

Favorited Tweets #3

For Week 2 of the MTBoS Blogging Initiation, I’ve decided to write about my favorite tweets. Or more accurately, tweets that I have favorited and quickly forgotten.

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1) In Matt Larson‘s engaging ignite talk, he wants us to seek equilibrium by teaching How, Why and When:

2) Graham Fletcher‘s “Progression of Multiplication and the Standard Traditional Algorithm” video enthralls me. I didn’t learn to multiply this way, so it’s extremely helpful to see how the earlier connections students will now be making, are going to make it easier for them learn high school math one day.

3) I am loving all the Desmos Activity Builders everyone is sharing, and these two from Laurie B look particularly fun for our upcoming unit on exponentials:

4) I agree with Sadie! This Common Core coherence map is very helpful!

5) This activity from Dylan Kane gives students the chance to examine the properties and structure of polynomials as they determine which one doesn’t belong.

Update: Here are links to two past posts with favorited tweets:
Favorited Tweets #1
Favorited Tweets #2

Day in the Life of Ms. Kohn Take 3

For Week 1 of the Exploring MTBoS blogging initiative I decided to document one day of my life. Although I’m just posting about it right now, this day occurred last Thursday (1/14/16).

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5:11 am Alarm goes off. Hit snooze.

5:20 am More snoozing.

5:29 am Get up and get ready. Orange juice is my morning power beverage. Check email/facebook/twitter while eating my cereal. Forgot to pick out an outfit the night before so I waste a good ten minutes staring at my closet. No food in fridge for lunch, that means I’m buying today.

6:20 am Put out trash and leave for work.

6:28 am Arrive at school. Check mailbox and help a substitute teacher find her way.

6:32 am Arrive in classroom. I am amped up for today! We are doing one of my favorite lessons and following it up with a Desmos Activity Builder lesson that I can’t wait to try. I make some last minute edits to the activity, and queue up all the browser tabs I’m going to need for the day.

6:50 am Students start entering the building and my classroom. I immediately get bombarded with demands to know how many jelly beans are in the container. I refuse to tell them.
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They know the winner is going to be announced today, but not until 7:25 am I say. Last minute guesses are welcome. Students hang out in homeroom, play board games, and experiment with my Eno board which is now up and running.

7:20 am Homeroom officially begins. Take attendance. Two students absent. They’re going to be sorry they miss today’s lesson.

7:25 am The bell rings to go to first period, and without speaking, I simply go up to the white board and write down the correct number of jelly beans: 1472. Students from the other two homerooms next door come in to see the answer. There is yelling; they are excited! They still don’t know why we are playing guessing games.

7:29 am This is the 1st of 5 times I am going to do this lesson today. I teach five sections of STEM Honors Algebra 1 to 9th graders. It will get better as the day goes on, as I observe and adapt to how my early students respond to it. Today is the first day of our unit on Absolute Value Functions. The beginning of the lesson can be found here, minus the project part (they’ll get this later). After we dissect the jelly bean situation, I show them this Estimation 180 problem, and we guess again. I take predictions for the shape of the graph and this time they think they have it all figured it out. The shape will be a V, but skinnier! No, wider! No, a check mark! Because there are fewer under-guessing options! We are on to something:
sweetheart graph

At this point, they are ready to explore and play around in Desmos on their own/with a partner. This was my first attempt at duplicating someone else’s Activity Builder and using it in my room. Overall, I was pleased with how it went, but would definitely make adjustments for the future. Some students finished early. I wish there were more challenges, such as what happens when you throw in negative signs. I tried to throw this question in as the day went on, but it didn’t work because I had already made a class code. I also wish I had a question about the absolute value vertex form equation with h and k. So that students could be more specific when they described how the function transforms. Here are some of their descriptions:

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We ran out of time at the end of class to debrief (my fault), and we don’t have class again until Wednesday (1/20), so to be continued!

8:17 am Period A ends. 4 minutes passing time. Run to bathroom. Head to STEM colleague’s room for our common planning period. Our big Winter STEM Expo is tomorrow, so we are doing last minute prep work. Edit presentation rubrics and chat with 10th grade team about last minute details.

9:12 am Period C begins. Algebra 1 take 2.

10:05 am Period D begins. Algebra 1 take 3. Four students actually leap out of their seats when the number of sweethearts in revealed.

10:53 am Lunch! Rush to teacher’s cafeteria and order my turkey wrap. Ask for cookies too. Get a slight look of disapproval when I reject the offers for pickles, apples or carrots as an additional side. No, just cookies please. I already have lettuce and tomato in my wrap. Eat lunch with math colleagues. Discuss our losing the powerball last night.

11:22 am Period E begins. Algebra 1 take 4. By this time of the day, we have some cheaters. Students from earlier have given away the answer, but I weasle out some confessions and we move on with the activity.

12:21 pm Period F begins. Algebra 1 take 5.

1:08 pm Period G begins. My prep. Finally. It’s been a great day but I’m exhausted. Today’s lesson was a high energy one. I always try to show the same enthusiasm with my last class as the first, but sometimes it can be difficult. Check personal email. Finish editing STEM rubrics. Take care of emails. Chat with STEM Director.

1:55 pm School day ends. Go to advisor meeting about upcoming school-wide dance. Return to classroom. Approximately 40 students have elected to stay after school to make trifold posters and last-minute changes to their STEM projects. My coworker has been supervising all of them while I was at the meeting. Spend afternoon giving advice on projects and printing, printing, printing for them.

3:55 pm Write passes for the late bus and start kicking kids out. They are nervous but ready for tomorrow. Here’s a video released after the Expo!

4:15 pm Say good-bye to final students. Clean classroom.

4:30 pm Pack up and head out. Run errands. Sit on couch. Breathe. Check email/facebook/twitter. Make dinner. Have plenty of time to write this blog post… but don’t do it. Take the night off. Watch an episode of the Blacklist (okay, okay, three episodes).

10:00 pm Bedtime.

If you want to read about other past days, check out these posts:
Take 1 – November 15th, 2012
Take 2 – November 18th, 2013

A Question and A Choice

On Friday, I was having an “off” day. I was absent on Thursday dealing with a stressful family situation, and didn’t feel like myself upon return to school. One of my students started to explain, and then argue, about reasons why he didn’t have the work from the day before.¬†My mood was making me impatient and easily agitated, and I didn’t respond well to the student. I continued around the classroom checking the work of other students, and a conversation so short, but so powerful, occurred; and I’m still thinking about it two days later.

Student: “Ms. Kohn, how is your day going?” He knew something was wrong.

Me: “Okay.” I lied.

Student: “Have you been giving high fives today?” He knew I needed one.

Me: I paused. The student already knew the answer to this question. He knew I hadn’t been in the hallway giving high fives. It was High Five Friday. And he hadn’t gotten his high five yet. In that brief moment, I knew I had a choice to make. This student reminded of this choice in a moment that I needed it the most. I could choose to turn my day around. I could choose to shake myself out of the funk I was in. So I did.

I smiled. I raised my arm and high fived that student. Then, I high fived some of the students around him. During the next passing time, I resumed my usual post in the hallway and high fived everyone that walked by.

Those seven words, and a simple motion, completely turned my day around. Thanks, student.

Note: If you want to read more about the power of high fives, read this.

NCTM Regionals Nashville – My Favorite

After a whirlwind trip to Nashville attending the NCTM Regional Conference, I was able to check two items off my bucket list: 1) Go to Nashville 2) Present at an NCTM Conference. I had a great time presenting on strategies for teaching English Language Learners in math class, and will be posting more about that later. However, I want to first share my favorite moments and takeaways from the rest of the conference.

Favorite Airport Art: The Dancing Sound Wave
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Favorite Robert Kaplinsky Quote:
It’s actually impossible to pick just one. I’ve been following Robert’s work for years, and I was very excited to finally get a chance to see him in action. Robert’s session was “Motivating Our Students with Real-World Problem-Based Lessons” and we worked our way through the famous In-N-Out Burger problem. I did this activity with my students a few weeks ago, and they are still talking about it. It was so rewarding to see Robert lead us through the activity, and explain all aspects of the lesson. He stressed that you need to start with application (the burger), and then marry the context with the math content. My favorite quotes:

  • “Convince me that you’re right or convince me that I’m wrong.” – Math Practice 3
  • “You should be spending most of your time figuring out how to implement your lessons rather than what the lesson is going to be.” – So true. Robert stressed the need to anticipate what the students are going to do and think so that you are ready to react and respond.
  • “My goal in life is to be the least helpful teacher ever.” – This is something I know I need to work on. Students need to struggle, and I need to let them. It’s a necessary reminder to focus on the hints we can give our students (because we’ve planned for the lesson implementation) that are just enough to keep students going, but not enough to deny them of how they’ll feel after successfully solving a tough problem.

Favorite Teacher Move:
Robert demonstrated how he gives enough wait time after asking a question. He physically counts down five seconds by putting his hand in the air and then bending one finger down at a time. Wait time is so important; this move is easy to implement and it makes you accountable for all five seconds.

Favorite Meal: Brunch at The Pancake Pantry

Several people, (and Taylor Swift!), highly encouraged me to visit the Pancake Pantry. I was told I would need to wait in line, but that it would be worth it. I waited for one hour, and then treated myself to Banana Nut Muffin and Caribbean pancakes. Very worth it.

Favorite Card Sort:
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Kimberley Williams presented a session titled, “Are We There Yet? Increasing Rigor in the Math Classroom.” She discussed Webb’s Depth of Knowledge (DOK) and explained how you could increase the rigor of a task depending on the type of question you ask, using the DOK chart. We looked at several examples of how one topic could be portrayed at each level:
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But the most helpful part to me, was when each group was handed a set of cards and asked to sort them among the different levels. It can be difficult to differentiate between them, so it was helpful to discuss with my table. I could see this activity being done with staff members at school to help everyone figure out the levels.

Here is the card sort:

Favorite Non-Session Activity:
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I loved volunteering at the Math Twitter Blog-o-sphere booth and helping newbies learn about all the awesomeness the MTBoS has to offer. Check out the Exploring MTBoS website here!

Favorite Task Progression:
Brian Shay ran an excellent session titled, “How to be a Super Model-ing Teacher.” He led the crowd through this Illustrative Mathematics Task:

The session reminded me to check out several great sites for modeling tasks that I don’t often check: Illustrative Mathematics, Mathematics Vision Project, and NRICH.

Favorite HonkyTonk:
Line Dancing at the Wild Horse and practicing the Texas A&M Yell Chant with some fellow MTBoS-ers!

Favorite Estimation Activity:
How old is Athena, the goddess of wisdom? How old is Nike, the goddess of victory, perched in Athena’s right hand?


Found at the Parthenon

Favorite Session: Kate Nowak’s “Plan a Killer Lesson Today”
Kate started off the session by asking everyone to think of a topic we dread, and I immediately thought of Radicals. Simplifying them, adding them, everything. I dread it. And she said her goal was to find ways to adapt lessons we already have, so that we’re not throwing out all our “standard” lessons and just starting over. Kate’s suggestion is to invert the lesson: You do, Y’all do, We do.

Her strategies for inversion:
-Ask about a pre-requisites
-Ask the question backwards first
-Give sample items with the question
-Engage in MP8

I have seen and used some of Kate’s work before, but I’ve never really thought of the lessons as strategies that I could use in my classroom… until now. I really needed to hear her thought process and think about how this could work in my classroom. I’m going to re-write my lessons on radicals so that I start by asking the question backwards first. I am going to give students a set of radical statements that are true, and ask students to see if they can fill in some blanks to create more true statements. Stay tuned for a future blog write-up.

Favorite Grand Ole Opry at the Ryman Singer:
Again, impossible to pick just one. I was lucky enough to attend on a night when five (5!) Hall of Famers were performing, and they were honoring Jean Shepard for her 60th anniversary as a member. The show was simply magical.
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Favorite Guideline for Increasing Task Rigor:
During their session on the “Impact of Task Design on Students’ Attitudes about Mathematics”, Ziv Feldman and Jeneva Moseley recommended several guidelines for increasing task rigor:
-Ask students to provide multiple solution strategies
-Ask students to provide mathematical justifications
-Ask students to create their own examples and non-examples

Although I use these strategies often, it was how they asked students to provide another strategy that really stood out to me. See part c below:
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I love how simple the phrasing is, yet it creates the need for a second method.

Favorite Design Principle to Develop a Problem Based Learning Classroom:
Geoff Krall shared five design principles for developing a problem based learning classroom, and it was the 5th one that really got me: “Don’t go it alone.” He said partner up, select 2-3 tasks that will produce rich student thinking artifacts, implement, and debrief. You need to have an “accountability buddy.” So rarely do we actually have time to debrief a lesson on our own, let alone do it with a colleague. It was another useful reminder to focus on lesson implementation, and the value of teamwork.

Thanks to NCTM and all the presenters for a great conference that completely reinvigorated me as we head into the winter season!

Visual Patterns Project

This Algebra 1 project was inspired by Fawn’s Visual Patterns site, and her Patterns Poster lesson.

We do one of the Visual Patterns every Tuesday for our “Tough Patterns Warm Up” activity, but I thought this would be a great culmination activity to our functions unit. Students must write both explicit and recursive formulas in this unit, so the patterns project brings both together nicely. Instead of supplying students with the patterns this time, they created their own! I encouraged students to be creative and choose an image interesting to them.

I provided them with a planning guide and scoresheet:

Students were given one class period to brainstorm ideas, start sketching, and write their equations. Once approved, they had about one week to make their posters at home. Some worked on the posters in school during homeroom, or at the end of a period if they finished their work early.

When I do this project again next year, I will more strongly stress the array of the images. For example, the apples placed in the diagonal vs. the Olaf snowmen which were placed in simply a straight line.

I love this project because it gives students a chance to be creative, and sometimes silly, but also because it directly relates to the math content we are studying.

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