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A favorite is when students walked into class and immediately began thinking and talking about mathematics. This year, I decided to kick-off the start of each class with a rotation touching on the mathematical practices. Tied to the current unit? Maybe, but not necessarily.

How much time? Well, it’s slowly been increasing as connections back to past or current topics become more apparent (funny how this increases the more it is practiced). Perfected? Nope, some days are magical and others not so much.

The Rotation

  • Monday – Graphing Stories
  • Tuesday – Visual Patterns
  • Wednesday – Estimation 180
  • Thursday – Math Talks
  • Friday – Reflection

The schedule at my school pretends to be a block (class length ranges from 60 to 80 minutes). So, I see students for math every other day and it takes two weeks to get through a rotation.

This post is a bit long as I attempt to lay out the flow of how these warm-ups take place. The first three days have a pretty good feel to them. Thursday Math Talks are the newest for me and need the most work. Friday Reflections seem a bit standard and good likely use an injection of inspiration. All of the days are a work in progress and have changed since the year began. With that in mind, any feedback on how to improve or suggestions to try is greatly appreciated.

Monday – Graphing Stories

A big thanks to Dan Meyer and others who put this project together. This awesome site has a collection of 15 second videos of an action. Students graph the action.

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Students come in and open their notebooks to the graphing story page. First, a quick What do you notice? What do you wonder? Then, what action do you think we will be graphing?  Time is always on the horizontal axis and we discuss what will be on the vertical axis and what units will be likely. Funny, how often this is prefaced with a reminder that most videos were created in the States. Yep, that means “feet”.

Begin watching. The videos are set up to run once (15 seconds) at normal speed and then again at half speed before providing a solution. We watch the normal and the half. Stop.

Students talk for 3-5 minutes in their table groups, come up with a “sketch” of the graph and determine areas of focus for the next viewing. They know that we will again watch the half speed section. At the end of this time, students share out. This has varied:

  • What do you notice? I’ve recently begun adding this in to the graphing stories. Students often struggle with the y-intercept. Where does the graph start? Before, this came out through groans as the solution was presented or by individual checks. Adding this step of each table sharing one “notice” brings it out from the students.
  • What do you wonder? This is the focus that students will have for their second viewing. I like shifting to this question versus “What will you look for?” as it feels as if the question opens the frame for students. Again, each table group shares out and if we get repeats, great.

Rewatch the half speed segment.

2 minutes – get a line on your graph. They are free to chat but after a quick burst, the class often gets quiet.

A few solutions – a couple of volunteers come up to the SmartBoard to put down their graphs. Three colors is the max for this. When we play the final moments of the video, the solution is drawn between the student lines.

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Solution – Students use a different color to draw in the solution to distinguish from their own.

That’s it. Unless, and this is when magic begins, the students don’t agree with the solution. We go down that lane. Talk it up. Convince us that your solution is a better description of the action.

Tuesday – Visual Patterns 

A big thanks to Fawn Nguyen and others for putting together this awesome site of patterns! There are many patterns and some great ideas to get going. For this, I like to print out a copy of the pattern for each student. They mark the sheet up!

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Students get the page into their notebook and begin working. Talk at the beginning is often minimal as students process the pattern on their own. The talk steadily builds as they make progress and start trying out ideas. I ask them to draw in the next figure as well as what could possibly be Figure 0.

One of the goals of this activity is for students to work in multiple representations to build connections between graphs, tables and equations while also reinforcing skills such as table set-up and graph design. Wait, the scale has to have an equal increment? 

The various ways that students approach patterns is fascinating. The check provided on the patterns page is Pattern #43. When two students reach a solution but have different equations it’s great hearing them talk when asked, do you have the same equation as G? They get down to it, break apart their solutions, talk about equations and then realize that yes, they have similar equations.

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For the 7th graders, the concept of variable is still developing for many and having this ongoing, yearly work with variable slowly builds familiarity. At the end of the first semester, most are now comfortable with having “n” placed into their table after the first few numbers (0, 1, 2, 3, 4, 5, n).

This warm-up is a bit more time intensive. Students work at their own pace as some can quickly see patterns and others take more time to get started and make various representations. Some need hints though a nice thing about having this on a regular cycle is that students have other patterns to refer back to. They are also increasing their ability to recognize patterns.

Wednesday – Estimation 180

Another big thanks to those putting out amazing resources for the math community. This time, it’s to Andrew Stadel for the Estimation180 site. This began as a quick warm-up activity. Students came in to see the image on the screen and quickly start thinking about their estimates. As with the other “warm-ups,” the value of estimations as a doorway to lots of math creeped in more and more now it takes a solid block. The presentation is a bit different as well:

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  1. Bring in the What do you notice? What do you wonder? framework. We start slowly. Students see the image and we talk about the things we notice. Then, what do you wonder? This is starting to edge us closer to the estimation.
  2. The next question for the students – What do you think we will be estimating today? Most of the time, this question has already come up though a curve ball was recently thrown to them as I worked in negative numbers.
  3. Now, we get into the estimation framework – What’s an estimate that you think is too high?   What’s an estimate that you think is too low?   Students are off and thinking. Our focus on the too high / too low is to begin developing a range of their estimates. “What is reasonable for you? How is your confidence of the estimation related to the range of your too high/too low?”
  4. We share with lots of questions. This is a great opportunity to review. “Explain how you got to that point…What does that mean?” Now, as a class we have a range. Sometimes it’s quite wide and others fairly tight.
  5. A few more minutes for students to get their just right estimate.
  6. We share and talk some more with students giving explanations as to how they arrived at their estimate.
  7. Show the actual amount to (often) cheers and (sometimes) groans.

Thursday – Math Talks

 

This is the “day” where I would love feedback. My goal is to get students talking. I want students to realize the difference in the ways each other think. I like that students are opened up to different ways of solving problems.

A few resources that I’m looking at:

I’m not quite sure what happens on Thursday and I’m afraid to say that we often hop into other activities without having a good number talk. Again, suggestions…

Friday – Reflection

Students need time to look back over their math and think about their progress. Reflections are used to also strengthen the connection to the mathematical practices. Questions may revolve around a problem of the week and asks students to write about a mathematical practice they used. This is also a bit in development and suggestions would be great!

 

Institute #1: MSIS

Last weekend, I joined several math teachers on a trip to Shanghai. We began a series of “institutes” that will span the next two years. The Math Specialist in International Schools (MSIS) seems to be a great opportunity to talk with other math teachers, think about my practice and find ways to improve. Presented by Erma Anderson and Steve Leinwand, the weekend focused on developing number sense and a progression through the Common Core standards.

On the first day, we spent most of the time discussing systems in the early primary years. With two girls in preK-3, my attention was grabbed as the importance of thinking more in terms of age instead of grade was repeated. I can only hope that the journeys of my daughters will be full of active exploration, manipulating models and discourse.

Rich mathematical tasks quickly became the focus of the institute. Students must engage with mathematics by grappling with problems, developing solutions, revising strategies and talking about their thinking. Have I been giving enough space for all of this to happen? My personal bank of resources has grown over the years but I soon realized how small shifts in my presentation of tasks can give massive dividends in the end.

Ease into the problem. “What do you notice?”

Build excitement. “What do you wonder?”

Turn the keys over. “What is the question?”

Invest the activated minds. “What is a value that you know is too high?”

Build skills of estimation. “What is a value that you know is too low?”

But don’t simply ask and be satisfied. Question. Have students explain their thinking. Over and over and over. This is a great way to review concepts and flush out activities. “Excuse me. You said that the object is 3D. What do you mean by 3D?”

“I’m not sure that I understand. Can you tell me more about the dimensions? The units? the…”

Finally, primed minds are released to tear the problem apart. But continue to push. Convince me. Show me. (Yep, that means providing multiple representations.) Explain.

The take home for me was to slow down and question. I need to do a better job to anticipate the reaction of students and be ready for targeted follow-up questions. How does this look? After more than a decade of teaching, I’m ready to consider creating a presentation a la .ppt. So far, I’ve done more hopping into the rich task through a video, image or description but I think that I’ve lost many opportunities in the set-up. I need to slow things down, expect communicate and question more.

The opposite bookend is equally important: presentation of the solution. My work is cut-out for me. I need to expand upon my own strategies of math talks related to solutions and student thoughts. The work of students should be more directed in the deepening of understanding and demonstration of new strategies.

Typically not one to slowly invest, my first class after the weekend focused on a rich task nestled inside a presentation. I posed probing questions to responses that I often would not at and move on. We took more time setting up the problem but the conservations were rich. When it was time to start work, the class erupted in a flurry of activity. The debrief was also richer as again I focused on asking for more explanations and stopped accepting values/comments without being convinced by the student.

A final thought: How can I shift more of my class to better scaffolded, rich tasks?

On the last day of the institute, we sat down in grade-level groups. This was the first time that I had worked with this group of teachers. In fact, introductions were the first order of business. We were then asked to put-it-into-practice: create a lesson. In a relatively short amount of time, we identified a rich task aligned with our goal and designed a scaffolded plan to work with students. What if more collaboration / professional development times were spent this way…

On perseverance…

On Tuesday I took the afternoon off to finish my Japanese Encephalitis immunizations. Yes, it took the whole afternoon  – more a lesson in patience than perseverance as I waded through the Taiwanese medical system. But, this post is not about my experience. It is about what happened in my classroom while I was gone.

I’ve been going through some Park City Mathematics Institute materials and came across a Painted Cube problem. I left it for my 6th grade students.

We’ve been looking at patterns and representations through sketches, words, tables and graphs. I set the room up so that each pair of students had the problem, a large whiteboard and lots of whiteboard marker colors. The teacher who supervised my students let me know that he was impressed at how the students worked. They talked, they drew, they problem solved and they kept at it! For an hour.

At the beginning of the year, they would have put 10 minutes max into a challenging problem and called it quits. This week, they did it for 60 minutes. I told them that they are awesome!

Caffeine charge-up

As students entered the class, they were met with this poster both on the screen and in a hand-out. I asked them to write down any thoughts.

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Thanks to Byrdseed for the link for this infographic.

Moments later they quickly came up with group names for the unfolding scenarios. Acting as race consultants, their job was to determine best caffeinated drinks that could be found in local stores. From some of my triathlon days, I remember grabbing the early morning caffeine to get me going.

Groups met to determine first steps. As they began working through the data, they realized that it was difficult to compare the caffeine amounts due to drink sizes. We’ve also been working with spreadsheets so they extracted information from the poster and inputted the data. Soon, members wanted to figure out the mg of caffeine in 1 oz.

By the end of the class, good discussions about data had taken place. Many started asking me for spreadsheet sorting techniques as they delved into their data and created graphs.

The next class, potential clients responded with a few questions for them to consider:

  1. My brother always wants to be up to date with the latest trends. He stood in line for the latest i-phone and he’ll go watch Twilight because everyone else is (he’s not even a vampire). He wants to know what drink to buy if he is to get the typical or most common amount of caffeine. Could you help him out?
  2. One of my best friends never pays attention to research, labels or, well anything! I’m trying to get him to slow down and pay attention because it would really help. At the moment he just grabs whatever is the closest drink. What amounts of caffeine could he possibly be putting into his body?
  3. Wow! What has happened to drinks these days. I asked my dear Aunt Sally to pick me up a drink before my race the other day. I told her to just get a middle-of-the-road drink. You know, nothing too strong, nothing too weak. She’s really nice and came back with a tall coffee from Starbucks. Talk about the prerace jitters! Was this really a middle-of-the-road caffeine drink? Please let me know.

Students are diving into the data. This project works on a quick turn-around as students will put together presentations about their recommendations by next class. I’ve found the poster and scenario to be good entry points for the students to discuss measures of center. Instead of a “typical” reply of, “Oh, what word is it? Mode?” students talk about the data. When they move over to spreadsheets, the formula word is the key to obtaining their desired values.

Using Google Docs has let me watch their spreadsheets develop. I can chat with them but large amounts of problem solving happen on their end. This shift to working with the data instead of crunching numbers has resulted in richer math conversations.

Estimates from the Fishbowl

Immediately using ideas from a workshop helps me carry momentum forward. As I wrote in my last post, I spent the previous weekend discussing how to make learning more accessible for English language learners. Multiple strategies were modeled and I chose to use a fishbowl activity in my math class today. My goal revolved around increasing student conversations about mathematics. I wanted them to talk about their thoughts and for other students to observe how a problem can move forward through group discussion. With absences, a small class of seven students entered the room and each learner spent a round inside and on the outside of the fishbowl.

In terms of contents, I have had my eyes on Andrew Stadel’s great new Estimation 180 blog as a means to kick off regular estimation practice. Today, Mr. Stadel’s height was the first estimation with four students sitting around one whiteboard. Peering into the bowl, I eagerly joined the remaining students to observe and record the math processing.

“Do you know how tall you are?” One student was fairly confident that she measures 140 cm. A student grabbed a couple of meter sticks and stacked them end-to-end. “200 cm. Hmmm, that seems really tall. Maybe that is the upper number.”

My Mom is really tall and I’m up to her shoulders.”

“Really, well let’s see how tall you are.” I was happy to hear this check. Students made some measurements and decided that the boy’s mother was closer to 160 cm. 

As group processing slowed, I asked each student to make his or her own estimate and provide reasoning.

The estimations ranged from 180 to 190 cm. Some reasoning included:

I think the fence must measure 100 cm. I then found out that the length from his toes to his belly is longer than from his belly to head so I doubled 100 cm and subtracted 10.

I believe that the bush (located in the photograph) does not come to his hips and that his hips (student measure height of her own) would be about 90 cm. So it is a little more than double.

After a debrief where the observers shared their thoughts and the “fish” shared what it was like to work on a problem while being watched, groups switched places for the second estimation: Ms. Stadel’s height. I again heard great conversations. The observation group picked up what they believed to be key math ideas: using ratios, number lines, making calculations and explaining why, using previous (Mr. Stadel’s height from the earlier estimation) answers, and halving fractions – halves, quarters, eighths… The actual height in this case was 165 cm and student estimations ranged from 163 to 166 cm.

To wrap up, I asked students what they learned about the process of math from this activity. Many students wrote that it was important for them to see that there are many different ways to solve problems. A few also commented on the need to respect the ideas of others and to allow people to share their ideas.

I usually bounce from group to group listening to snippets of math processing and rarely find the wonderful opportunity to observe students work through an entire idea. I enjoyed it and will look forward to more opportunities to join students on the outside of the fish bowl.

Was the rant necessary?

When was the last time a company’s idea of packaging and your needs matched up? (My whole concept of packaging has been turned on end by living in Taiwan. Wow! Individual packaging galore.) It has been a long time since I watched Father of the Bride but last year I came across this clip from the movie. George Banks, played by Steve Martin, is having a rough day and heads to the supermarket to buy hot dogs and buns. I used the clip as a jumping off point of unit rate.

This year, as I looked at the clip again. Before watching I asked students to think about the following questions:

  • What did George want to do?
  • What would the cashier done if George received the outcome he desired?

I was pretty happy that students quickly brought up unit rates after a few good laughs. I then asked if George’s actions were necessary? Some student replies:

  • He can buy three bags of hot dogs and two bags of bread and it can make 24 hot dog buns.
  • He didn’t need to take 3 packs of buns. He only needed 2 packs.
  • He could have bought 3 hot dog packages and 2 bun packages, cause 8*3=24 and 12*2=24 so they are even.

As it turns out, George took four buns out of each bag of 12. He then bought three bags. I didn’t pick up on this the first time through and wonder if the movie producers even thought about the fact that the packaging actually agreed with his purchase (contrary to his rant). A few of my students picked it up on the first “blind” watching.

Did the packaging actually agree in number with the amount George put into his cart? After working out what the cashier would have done if George did not end up in the slammer, I posed the question to my students…

Less than one, equal to one, greater than one

Courtesy of Dan Meyer’s 3-Act Math resource, students began today’s class in detective mode. A brief, 30 second video clip captured their attention as two CSI investigators pulled out a sawed-off limb. The portion of the video where the percent of the mass of the lower leg to the body mass was bleeped out. Students were up and on their way asking questions about the scenario. I love questions from 6th graders and we spent some time wondering how the leg portion could have shown up on the screen before returning to the math.

I asked for an estimate that students thought would be too high and another they thought to be too low (sometimes I inadvertently let this step slip by but it is oh so revealing). The range began at 1% to 500% and after a bit of discussion was narrowed down to a range of 10-50%.

A group’s thoughts in process… We’ve been discussing ratios and are moving to percents so I was happy that groups began looking for comparisons and writing them as ratios. Today’s surprise came as I rotated through groups. A group wound up with a ratio of \frac{42}{147} and wanted to turn in into a percent but was not sure how. (A benefit of spending large amounts of time on a single problem is that opportunities for quick mini-lessons to groups of students always pops up.)

I asked the students if they believe the fraction to be less than one, equal to one or greater than one. Jumping that hurdle seems to help students position their thoughts and gives me a good idea of what they are thinking. Two of the three members in this group believed the fraction to be greater than one. Their reasoning was that 147 is greater than 100 so the fraction must be greater than one. They were convinced. Pulling a value out of a hat, I asked if \frac{13}{14} would be a smaller or larger amount. The two students indicated that the 147 would be larger though neither had a explanation as to why. Soon, they were busy breaking equal sized rectangles into portions to represent the two fractions. The student in charge of 147 pieces quickly became frustrated at having to make so many small pieces and the group had a good discussion about part to whole relationships.

Looking back, I wonder what was the foundation of the students’ original thoughts and at the moment lean towards a developing concept of the relationships between fractions, decimals and percents. They appeared to understand that a percent greater than 100% is greater than one whole. Did they transfer the thought to fractions and think that the value must be bigger than 100%? I’m working on some follow-up activities to bring out these relationships but additional ideas would be appreciated.