Most Successful Review Day Yet!

Today we reviewed for a test on triangles and I changed the format just a bit.  For a test on multiple topics I find that working through stations works best, but students don't always work at the same pace and some students need some extra motivation, especially on a Friday after our first full week of school in I can't remember how long.  So, here was the set up:

3 stations involved kuta worksheets cut up into individual problems (angle sum rule, given labeled triangles find the congruence rule, given the congruence rule label the triangles).

2 stations were proofs (using proof cards and writing on the desks in dry erase markers).

1 station was a matching activity on the smart board (types of triangles slide 6).

I had students move the desks into groups of 4 and put the supplies for each station on a different set of desks.  Each station had a clear picture frame with the name of the station on the front (ex: Proof 1) and the instructions on the back (complete 2 proofs or solve 6 problems).

Students were to work through the stations at their own pace.  The first person to complete each of the proof stations and the smart board station got a special privilege: they were given a stamp and charged with the responsibility of stamping each person's paper who completed that station.  This did a few things: it gave the fastest students an extra responsibility which kept them busy, it freed me up from checking those stations since they didn't end up with anything on paper and students were excited to earn stamps!  I was worried going in that this would be chaotic, the stampers wouldn't take the responsibility seriously and that not a lot of learning would happen.  I was wrong.  The students with the stamps were helping other students earn the stamps, giving helpful hints but not the answers.  One student even brought problems from the other stations to his station so he was always available (this same kid said "just send me to the office" last class when I asked him to stop talking!).  They worked together on the problems and no one threw a fit when I suggested asking their question of other people at their table rather than asking me.

As students finished the stations they handed in their paper (complete with 3 kinds of stamps and the problems from the other stations) and were given the choice of doing make up work or playing Taboo.  This turned into another great motivator as students wanted to check their grades but I told them they needed a minimum of 5 stations completed before I would discuss missing work with them.

At the end of class I was shocked how quickly time had passed, how quiet they'd been and how I'd never felt overwhelmed by everyone asking for help.  This class has 28 students so even on days when they're focused it's busy.  A few students were absent today, but the ones who were absent were students who always work independently, so it was something about this activity that kept them so quiet and focused.  No guarantees it will work again, but I'm really happy with how today went.  I hope you can have an equally successful review day in the future!

p.s. This post is quite momentous.  It's the last day of November and I managed 23 posts this month, not the 30 I set out for, but more than twice as many as my most prolific month prior to this one.  This is also post #100!  In elementary school we used to celebrate 100 day with sets of 100 things, I remember bringing in 100 shells one year.  So here's 100 dots from this awesome factorization diagram.

Geometry Unit 1: Patterns

Over the summer I took everything I knew from teaching out of two textbooks and discussing the common core, then put it all together in the form of a new sequence for the courses I am teaching this year.  I had great aspirations this summer and not so surprisingly, life is getting in the way of them.  But I'm still really excited about what I've come up with and want to continue adjusting it, so I'm going back to the beginning of the year to talk through each unit I've taught.

We start with patterns.  This works as a great intro for a variety of reasons:

• it's an easy way to figure out where kids are and get their brains back in math mode 
• patterns are naturally low threshold and high ceiling
• my Geometry course is all about investigation, conjecture and (rather informal) proof: patterns hit all of these
• they're on the state test (which students take this year, so it does have to always be in the back of my mind)

The Massachusetts state test usually has some good open response pattern questions, so I pulled the questions I used from past tests and from a variety of other problems I've saved over the years.

Day 1 (a short class) I gave students a questionnaire, to be fair, I filled it out too.  One of the questions asks about your experience with math, this was my response:

"In my experience math is a fun challenge. I enjoy solving puzzles and math is often puzzling."

We start the next class with a visual pattern, where the prompt is "What questions could you ask about this pattern?"

After letting them stare at it for as long as it took me to take attendance, I suggested if they hadn't asked it yet, a good question would be "what comes next?" As students answered that and proclaimed they didn't have any questions, I asked for observations instead. When I asked the class to share we pretended it was Jeopardy and found a question that each answer could have come from. We noticed patterns (re: left, right and total faces, blank space) and pondered what happened when you got to row 6 or 7.

Then, I gave each student 1 of 4 patterns to work on. They were welcome to talk to their neighbor (who had a different pattern). This time, if they ran out of questions or observations I directed them to hint cards at the front of the room. I saw this model in a video of a Japanese classroom while in grad school and have always wanted to use it. I thought ahead of time what I would say to students who were stuck, then wrote those questions/phrases on index cards. Then students had ownership of when they got a hint and no one had to wait for me. Most students thought walking all the way to the front of the room was effort, so they didn't go until they had exhausted other options.


After a while of this, I directed students with the same pattern to gather at a white board (I have one or two empty boards on each wall, which is awesome). They drew their pattern, shared questions, answered ones the original asker hadn't been able to answer and discussed interpretations. Then, I brought everyone together for a discussion of hint card 4 (about multiple representations). By now most every group had a list of numbers somewhere on their board, so I asked them to think back to Algebra and other ways they knew to represent data. It took various amounts of prompting, but each class got to the point of "table, graph and equation." I sent them back to the boards to develop those three models for one of their questions.


The results were pretty cool.

  
 


In the last 10 minutes I had students present their findings and reflect on the class. Homework was to make up their own pattern and ask, then answer at least 3 questions.

This process (plus a quick read through of the syllabus) took 90 minutes. I was shocked that Fundamentals and Standard Geometry students could work on one pattern for an entire block! Especially on a Friday, the first week of school, at our second class meeting. I had prepped number patterns to do as well but they had enough to say about these patterns that I could push all groups into multiple representations and reach some good depth on just one pattern. Students came up with ideas, discussed, got stuck, made mistakes and helped each other.  Not everyone was engaged all the time (mastering group work is a long term goal that I have yet to tackle) but I did convince some students to share ideas with each other since I'd worked with each of them on different parts of the problem (next year I should do this on purpose!).

Next up we studied Pascal's triangle and looked for patterns in there.  The discussion led to an impromptu proof that odd + odd = even!

Homework was to make pascal's triangle starting with 2's on the ends rather than 1's.

Overall I'm happy with how this unit went.  It may have been worth spending an extra day on the numeric patterns I prepped that we never used, but it being geometry doing all visual patterns is okay.

How do you start your Geometry course?

A day in the life of a math consultant

This is a guest post contributed by Dan Allen.  You don't even need a blog to participate in DITLife, just a friend with one!  (I'm extra happy to have guest posts this month since I'm trying to blog every day of November and am starting to run out of ideas other than the big scary ones that will take a long time to write about.)  Without further ado, Dan's Day:


I am a mathematics consultant in Ontario, Canada. My board is home to 40 elementary schools
and 9 secondary schools. Although I have specific schools assigned to me via my portfolio, I have been working with all schools, both secondary and elementary, as the need presents itself. My background is as a secondary math teacher, having taught high school math for eleven years. I just began working as a consultant in September, 2012 and am luck to be a part of a great team of educators who really care about our students.

There are three k-12 “numeracy consultants” for my board and I am one of them. Our job
is to provide support at the system level. We’re often in schools to provide professional learning
opportunities for our teachers, via job-embedded learning opportunities, “lunch & learn” and other
initiatives. I am also working closely with our secondary schools, providing support for new teachers
and providing technology resources for our teachers. There is also an “Intermediate Numeracy
Resource Teacher, who is mainly based in schools (he has a rotating schedule between 4-5 schools at a time) who we work closely with. No two days are the same for me. One day I’ll be working with a
team of secondary teachers on what graphing software is the best to use in their classes, the next I’ll
be drawing rectangles with students in a grade three class and the day after that, I’ll be at a ministry
meeting, looking at new numeracy initiatives that will be rolled out in the near future. That all being
said, here is a day in my life:

Thursday, November 15, 2012

5:30 AM: The alarm goes off to begin my day & I awake in complete darkness. I went through a fairly
significant weight loss over the past two years and working out has become a hobby of mine. Last year when I was teaching at a secondary school, I was finished early enough in the afternoon that I could sneak to the gym for an hour before picking my kids up from daycare at 4:30 in the afternoon. My new position has inconsistent hours and I usually get home from work around 4:45. If I want to continue my gym habit, it has to be before the start of my day so I drag myself out of bed and grab my gym bag. I try to go to the gym before work 3 times per week. Some weeks I hit that goal, others not so much. I struggle to not make too much noise so my wife can get an extra half hour sleep before she gets up and gets the kids ready for school. I’m lucky that she supports me in my efforts to go to the gym & I know sometimes the solo mornings can be difficult. I go downstairs, force down a banana and a granola bar and head out the door at 5:50.

6:00 AM: The intermediate resource teacher & I got talking about fitness last week and he has agreed
to start going to the gym with me. I’m not a personal trainer by any stretch of the imagination but when I made my lifestyle change, I took the time to educate myself with regards to nutrition and exercise. I spend the next hour abusing his muscles and making him sweat until he has to leave at 7:00. His wife is a principal in an elementary school, so he has to go back home to help out with his daughters so his wife can go to work. I stay and get in another 20 minutes on the elliptical before heading to the locker room for a shower and to get dressed for the day.

7:50 AM: My day “officially” starts at 8:30 but that number changes depending on what I’m doing
on any given day. Today, I’m at one of our elementary schools for the entire day. Their start time is
8:45 so I stop into Starbucks to grab a coffee before making the half hour drive to the school. While at Starbucks, I check my e-mail and see that since I checked it the night before I have received 9 emails about meetings that are new, meetings that are now cancelled and some from teachers looking for resources. I reply to them then head out.

8:45 AM: The elementary schools in our board are grouped in triads based on geographic area and
school demographics. Today there will be nine teachers (three from each school), the three principals
and the three consultants in attendance. The scope of the meeting is engaging students by having them learn through an inquiry model. The class we will be going into is a grade 5 group who are studying data management. Specifically they are discussing the difference between continuous data and discrete
data. The morning begins with the teachers in the school library where we will participate in some teacher learning and then we will co-plan a lesson to address the curriculum expectations that are targeted today. We begin with a curriculum mapping activity for the teachers where we look through the curriculum from grades 1 through 8 to see where these concepts come from and where they’re going. Having taught secondary as well, I am able to provide insight into where these concepts show up in secondary mathematics (when are we ever going to have to use this?). I find these mapping activities useful because it’s important for teachers to know where the curriculum is coming from and where the curriculum is going to.

10:00 AM: After clearing up some confusion about what exactly is the difference between discrete and continuous data, we begin the discussion of what we’re going to do with the students. We follow a three part learning model with a minds-on starter, an activity and a consolidation piece. On these days, when we go to the classroom, two of the teachers co-teach the lesson and the rest of the people in the room (all 12 of us) act as silent observers. The observers make notes, take pictures and video, and record student thinking. Before going into the classroom, we actually do the activity with the teachers in attendance to help us anticipate any questions the students may have or any issues that may arise. We all take out shoes off and create a bar graph of our shoe sizes. We look at some technology supports that may help students understand the concept of continuous numbers (we used the x-axis in Geogebra, zooming in further and further so students can see that there are “numbers between the numbers”.

11:30 AM: After planning our lesson, we break for lunch. During the entire lunch break, teacher conversation continues. Discussions from assessment to activities to technology take place amongst the group.

12:15 PM: The student’s lunch break is over & the group of teachers heads into the classroom. We execute our lesson. I was acting as one of the observers so I focus in on a couple of groups on which I make notes and take some pictures. I am lucky enough to have an iPad so I am able to record student thinking via video. This can provide very useful when we reflect on student understanding later in the day. The math lesson has students create living bar graphs and histograms. We create bar graphs using their shoe sizes and histograms by sorting them according to height. The class then had the discussion about the different types of data and why the two graphical displays looked different.

1:15 PM: The teacher group returns to the library to discuss the student work and to plan a consolidation. The group decides that students would benefit from watching some time-lapse videos to show continuous growth. We find a YouTube video of a tomato plant growing and one of a guy growing his beard for an entire year. We also make note of some common instances of discrete data such as the number of siblings they have or the number of televisions in their homes.

2:00 PM: We all return to the classroom to finish up the consolidation of our math lesson. We show the YouTube videos and collect the data about siblings and televisions and by the end, the students have a strong understanding of the difference between continuous & discrete data.

2:30 PM: We return again to the library to provide some planning time for the teachers. This is a great opportunity for teachers from different schools to get together and plan their math lessons. It helps share best practices and promotes consistency from school to school.

3:15 PM: My day at the elementary school is done so I head back to the office. I have a parent council meeting that I’ve been asked to attend this evening. I will be presenting with one of the other numeracy consultants. The presentation is to demonstrate some of the different strategies that we’re using to teach mathematics. The meeting is at 7:00 and we’ve been told that we’re close to the top of the agenda. I’ve also received several emails through the day. Tomorrow I’m participating in an Eco Schools summit. Representative groups (one teacher and four students from grades 4-6) from 37 schools will be meeting to discuss ecology. I’m fairly fluent with technology so I’ve agreed to run an iPad workshop where we’ll have 25 iPads and students will be rotating through. They’ll be exploring some interactive ecology related apps. I’ll need to make sure everything is set for that meeting in the morning.

3:45 PM: I’m finally back at the office and take the time to confirm everything for the parent meeting this evening and the Eco Schools summit tomorrow. I reply to the various emails and browse through Twitter to see if there were any mathematics breakthroughs while I was creating shoe-size bar graphs. I leave the office by 4:30 & head home.

4:45 PM: I pick up my 7 year old daughter & 9 year old son from daycare & head home. My wife manages a couple of dental practices and works until 5:00 so I’m the main chef in our house. Because we try to eat healthy, we stay away from convenience foods. I get dinner prepared while the kids finish up any homework and any other things they need to tend to.

5:30 PM: My wife comes home and my family is able to sit down and have dinner together. We talk about the fact that my son has three hockey games and two practices this weekend (I coach that team by the way so weekends are no break!) and my daughter also has a hockey game. My kids lead very active schedules so there are a lot of times where we need to rely on the assistance of neighbors & friends to help us transport kids to arenas and so on. Similarly, we help them out when their schedules become crazy as well.

6:15 PM: After finishing dinner and helping clean up, I have to head out for the parent meeting. It starts at 7:00 but it’s a half hour drive and I’d like to get there at 6:45 so I can get set up and see if there are any surprises. I give the kids a hug because they’ll be in bed by the time I return. I head out again!

6:45 PM: I arrive at the school and have a quick meeting with the school principal & vice principal. They share stories of their week and we set up for the meeting.

7:00 PM: There are about 30 parents in attendance. We give them a presentation about our board’s numeracy goal and show them a video of students in our board learning through problem solving and open questions. After the video, we actually hand out chart paper and markers and ask the parents, “How much do you spend at Tim Horton’s in a year?” This is an open questions and every parent has an entry point. We get them to work in groups and give them time to formulate their answers. Rather than focusing on whether their answers were right or wrong, we focus on the processes they used to get them. We discuss the importance of group work and problem solving. We discuss the benefits of collaboration and effective communication. Our presentation was only supposed to be 20-30 minutes but the parent group was so engaged, they ended up going until almost 8:00. We finally finish up and excuse ourselves as the meeting goes on to some other school related business.

8:30 PM: I return home again. My wife is relaxing & watching television and the kids are asleep. I rifle through the fridge and throw together my lunch for the next day. I am taking an online course so I open up the laptop and read one of the required readings and get to posting and responding to posts on the discussion boards for the course. My wife goes to bed at 10:00 and I stay up until 11:00 working on my course.

11:00 PM: I finally head to bed, setting up my gym bag for the morning and trying to figure out what I’m going to wear. I need to do this the night before because I’m getting up early again and don’t want to wake everybody up. By 11:30, I’m watching some mindless show while trying to fall asleep. The alarm is set for 5:30 again and tomorrow’s another busy day…

Day in the Life: Recap and Moving Forward

THANK YOU for reading, writing, sharing your day or spreading the word.  Since the last update there have been 14 new submissions, which puts us over 50 total!  And it sounds like there are still more coming.  I'd love for this initiative to continue expanding, so I created a tumblr.  The latest submissions are below, but from now forward contributions will only be posted on DITLife.tumblr.com.  I'd never used tumblr before but now that I've set one up it seems most appropriate for sharing links.  You can still follow it by RSS and read the posts in google reader or similar, but it's also searchable by tags and maybe we will discover a new community of tumblrs who can join the twitterblogosphere!

Now that the "Day in the Life" week is officially over, what's next?  I've asked for ideas and come up with a few of my own.  I'd love to hear your feedback on these, other ideas and volunteers to kick these off!

• Re-blog, re-tweet, share on facebook and send this to big people/media (Justin Reich, Dan Meyer, Diane Ravitch and Arne Duncan were mentioned specifically)
• Continue getting new people to share a Day in their Life (try to reach different circles of educators)
• Personally I found this challenging to do, so repeating the experience of logging an entire day is unappealing, but posting a snippet like I did on Sunday is doable.  Lots of short clips is just as good (better?) than a full day.  There's a submit page if you'd like to contribute directly to tumblr.
• Record yourself reading part of your DITLife post, it's interesting to hear the voice behind the screen.
• Make a video of yourself telling a story, no longer than 2 minutes, of something that happened to you that shares some aspect of teaching; good, bad, whatever.
• Find a student to interview you, where the student asks questions they're curious to know about, and the teacher responds. Then the teacher posts a podcast of the interview. (This wasn't my idea, but I was talking to students about grading just the other day and it was interesting to hear their questions!)
• Find another teacher to interview you on whatever and post a podcast of the interview. 
• Give awards to contributors: most papers graded, most hours at work, most uses of technology...
• Compare our days to TV/movie teachers
• Compare to each other (what was everyone doing at 7 am, noon, 3 pm, 8 pm?)
• Running list of all the roles we play
• Instead of recording everything in one day, record one thing every day and create a report a la Nicholas Felton
• Link this initiative anytime you see anyone attacking teachers
• Map where you go in a day or week (I know I never see some teachers since I don't walk the same paths they do!)
• Ask people what prevented them from participating (is that you? please comment!)

I also got requests for future themes and gathered a few ideas for those:

1. The best lesson I taught this year.
2. What I want PD to look like.
3. If I was not a teacher I would be a ___.
4. Classroom tours (started in June, I want to see more photos!)
5. Teachers take a photograph of something meaningful that they've gotten from a student, and describe what that is and why it matters to them.

Thanks to Sam, Kate, Ashli, Julie, Greg, Kirsten, James, Jonathan, Lisa and Tom for their contribution to these lists.

Submissions over the weekend:

A Day in the Life: Berlin Edition on I Hope This Old Train Breaks Down

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My typical action-packed, no-room-to-breathe Fridays.


Week as Math Educator - Day 5 on Mathie x Pensive

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Entire Friday, in third person plural perspective, to wrap up five straight blog days. This last was a PD Day, but not as one might expect.


A Day in the Life: cheesemonkeysf edition on cheesemonkey wonders

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Serving students and serving the dog.


A day in the life... on crazedmummy

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Thursday, blow by blow. Aaagh!


A Day in the Life: 11.14.12 on Epsilon-Delta

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Just a normal day--nothing super exciting BUT nothing super horrible either. So, I'll count it as a good day.


Tuesday, November 13, 2012 on Bowman in Arabia

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Just a normal Tuesday in my life as a math teacher at a boarding school in Amman, Jordan.


This is It (Extended Version) on Those Who Teach

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All in a day's work: star-crossed lovers, a skeleton, and essays galore


A Day in the Life on Discovering Delta

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This is my not-so typical day of parent-teacher conferences. It's also my very first blog post. This initiative enticed me to take the first step into the blogging world!


A Day in the Life of a Mathematics Educator on Algebrainiac

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24 hours in the life of an 8th grade math teacher who has two jobs and two dogs and not enough minutes in the day!


A Day In My Life – I’m Sorry Mom and Dad on I Speak Math

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I was excited to blog about a day in my life. I'm a parent, a teacher, a 6th grade adviser and service leader, the math team coach, and the cheerleading coach. All three of my boys play competitive soccer. So, my trouble was actually finding the time to blog. I didn’t realize how much I do in a day until I typed this up. I am doing the best I can, but it never seems like enough.


My Typical Day (2012) on MathyMissC

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I'm a second year teacher at an urban school in AZ. I teach Algebra 1 & Algebra 2. The teacher next door is on maternity leave, so I look out for those kids too. Then add co-teaching to the mix, my days are usually eventful!


A Day in the (A.D.D.) Life on Approximately Normal (in the classroom)

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Just my perspective of the craziness of a random day in my life. No epiphanies, no revelations. My life isn't perfect, but it is what it is.


Pimiento Cheese Cookies (A Day in the Life of a Language Arts Teacher) on Willow Bird Baking

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I'm a language arts teacher who moonlights as a food blogger. My recipes always include an anecdote -- whether it's a memory from my childhood, a funny story from school, or an interesting tidbit from my daily life. Today, I decided to accompany my recipe for pimiento cheese cookies (they're good, I promise!) with "A Day in the Life of a Language Arts Teacher."


Day in the Life of a Math Teacher 2012 #DITLife on Technology Integration for Math Engagement

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Looking back at my day, I recognize the importance of tech in my job. I use it to stay organized, provide feedback to students, and collect assignments. Without it, I wouldn't be able to make my commute due to the paper weight in my car!

Graphing a Picture (with Trig)

I was inspired by Epsilon-Delta to assign my PreCalc students a project to create a picture using graphs.

I had a great time making a model (using Desmos Graphing Calculator).

I showed them my example on a Wednesday, then they had all class Friday to work on it (while I was at NCTM) and it was due the following Tuesday if they hadn't finished in class (we run an A/B day 90 minute block schedule).

To make the assignment clear, and to make it easy for me to grade, I gave them a checklist of required characteristics.

Graphing a Picture

My original intention was to pick a few examples to share, but there were just too many good ones, so you're getting a complete slide show!  (Cropped to protect the names of the innocent.)

Multiple Choice Tests

My school has decided to give quarterly common exams in every course (until this year we had common midterms and finals only).  The school has some scantron licenses, so the freshman team has been using scantron.  For a variety of reasons (scantron friendly, faster to grade if not on scantron, mimicking state test) the exams are mostly multiple choice.  I have mixed feelings about multiple choice, in my own tests I mostly use them as an easy way to provide a word bank or with the choices as "sometimes, always or never."  But, the state test is a graduation requirement and it's almost entirely multiple choice.

One of the my co-workers performed an interesting experiment (no claims of scientific method, just information to discuss).  She gave the quarter 1 exam and found that students did much worse than she expected based on their previous tests and quizzes.  So, she gave them another test that was essentially the same, except there were no choices.  They did much better!  The running hypotheses are that students were distracted by the wrong answers, or only read to the first reasonable answer, or just guessed because that was easier than actually solving.  Without the choices there, only the last one was a viable option, and the temptation was less.

We discussed all this in my department meeting and my department head had a brilliant idea: let's give them choices, but only after they've done out the work!  Students will have to work through all of the problems, then they will receive a sheet of multiple choice answers and a bubble sheet.  The hope is that this will get the students into the habit of solving the problem first, then comparing with the answers.  There are certainly times when checking the answers is more efficient than solving in another way, but if they practice this way often enough, perhaps students will decide wisely when to use the answers and when to solve the problem without looking at the answers.

We have yet to try this so I'd be interested to hear your thoughts and concerns before we do.

Also, a perk of reading all those Day in the Life posts was I found out about GradeCam from Ms. Miller.  I'd like to use it for students to get quick feedback on how many problems they solved correctly, then I would go back and award partial credit and make sure they provided justifications.  We'd still use the method above (solve first, answer choices after).  Have you used this program?  Thoughts on it?

An Unusual Day in the Life

Every day of a teacher's life is long, but one of the longest is parent conference night. I didn't want to choose an extra crazy day for my Day in the Life post, but I did want to share what it's like.

Thursday, November 15, 2012

I was in school for 8 hours and 50 minutes (fairly typical), went home for a bit, then returned for another 2 hours and 25 minutes. That means 11 hours and 15 minutes at work in one day.

I taught for 3 hours and tutored for 1.5 hours. Many people only think of those 4 and a half hours when they think of the duties of a teacher. What did I do for those other 6 hours and 45 minutes?

I: graded 13 projects and 21 tests, read and dealt with (via response or recording information) 25 emails, explored a new computer program to determine if it was worth trying, worked with my co-teacher to try out tomorrow's experiment and work out the possible pitfalls, typed instructions and formatted slides, made 210 photocopies and saved progress reports for all five classes (5 clicks to save each file!).

Oh, and I spoke to 26 parents about their children. Interactions ranged from "he loves your class" to "you need to provide step by step instructions" to back stories that explain behaviors and just make me wish I could 'make everything all better.' Sadly they're too old for it to be enough to just tell them "it'll all be okay."

Throughout the day I served a lot of roles beyond "facilitator of the learning of mathematics"

I: taught English (helped with a literary analysis, broke down vocab words and gave definitions), played guidance counselor and/or therapist, gave pep talks, relayed information during a crisis, played friend matchmaker (encouraged a group to work together), was both a calculator lender and calculator fixer, wowed students with my paper folding magic (we made hexaflexagons) and was a debate moderator.

I was also informed that during the pep rally next week I have to take on the role of dancer (my task is to learn Gangnam Style by Wednesday).

After over 11 hours at school I'm really not excited about going back at 7 am. Good thing I planned a fun activity where they get to cut out the pieces of a triangle and try to find counterexamples of the different congruence theorems (SSS, SAS and SSA are up tomorrow). Hopefully the students will find it engaging since I won't have the energy to make it seem extra exciting.

Day in the Life (Part 4)

It's Friday and we have 38 submissions to Day in the Life!  So we don't end up with one overwhelming list of all the posts I will share the contributions in small bundles for your reading pleasure. The first day we had 6 posts, then 12 more the next day, Thursday there are another dozen and today we have 6 more. Pick one, read them all at once or bookmark this page for later (they will also be on the #matheme page, but without the descriptions).  Don't forget to submit your contribution.


a day in the life on in stillness the dancing

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In planning the lesson it seemed like it would go well. But the instruction took too long; students had too little time for practice. And when we got to the TIC TAC TOE game too many students had too little proficiency to make the activity worthwhile! No time to reflect effectively on the lesson ... will have to build in more work on writing and graphing linear equations.


This Too Shall Pass on It's all math.

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Want to read about a crappy day in the life of a math teacher? Here's your chance!


Week as Math Educator - Day 4 on Mathie x Pensive

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Entire Thursday, in first person plural perspective, with random math times. Most typical day yet! Read this one if none of the others, maybe?


Wednesday, November 14, 2012 on Continuous Everywhere but Differentiable Nowhere

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My name is Sam Shah, and I am a math teacher in Brooklyn, New York. And this is a day in my life.


A Day in the Life... November 15, 2012 on Long Tails of Int_e^r est

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A private, all-boys, Catholic school in St Louis from wake up to bedtime. Block schedule teaching Trigonometry and Geometry.


#DITLife November14 on Musings

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A day in the life of a teacher-educator in England. Describes how one lesson observation can take an entire day.

Day in the Life (Part 3)

Three days down and there are 31 submissions!  So we don't end up with one overwhelming list of all the posts I will share the contributions in small bundles for your reading pleasure. The first day we had 6 posts, then 12 more the next day, today there are another dozen. Pick one, read them all at once or bookmark this page for later (they will also be on the #matheme page, but without the descriptions).  Don't forget to submit your contribution.


A Day in the Life of a Mathematics Educator on Finding Ways to Nguyen Students Over

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My day was remarkably uneventful (boring as hell), but I didn't know that until it was all over.


A Day in the Life... on thenumbertwentyone

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One of the least successful days I have had in a long time. A string of bombed lessons.


A Day in the Life on Numbers

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“Uh, I’ve been absent for nine days, did I miss anything?”


A Day in the Life on Take It To The Limit

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A pretty normal day of teaching Geometry and Algebra 3, but I went totally off the geek end after school when another teacher and I played with non-base-10 arithmetic.


Week as Math Educator - Day 3 on Mathie x Pensive

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Entire Wednesday, in third person perspective, with event checks. Calmer than Tuesday. Includes conic question and reflection on math v science. To be continued Thursday.


A Day in the Life of a Math Teacher Mama on Ms. Zimmer Teaches in Mathland

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Looking for keys that are already in my pocket, trying to get to the bathroom, and other musing of a math teacher mom.


A Day In The Life on The Space Between The Numbers

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Do private school teachers have it easier? Probably. Do we have cushy jobs? Absolutely not.


Tuesday ... or any other day for that matter. on 2 + 2 = 5

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Just a regular day of my life - work, kids, life!


A day in the life of a Math teacher / debate coach on Success

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Math Teacher, Department Chair, Debate coach who is hosting a tournament this week. I think I need 30 hours per day this week instead of the usual 24.


Diary of a math math teacher! Or, a day in the life on Curiouser and Curiouser

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I'm a continuous learner, especially while on walking the dog. Did you know that for 70 years Coke only cost only a nickel?


A day in the life... on Finding the Process

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Like the luge in the Winter Olympics, this day got started fast (and early!) and kept flashing around turns. I'm always amazed at how much teachers can pack into a day.


Day in the Life of Ms. Kohn on Growing Exponentially

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Typical school day of doing math, lesson planning, email checking, teacher chatting. Atypical evening with 16 parent/teacher conferences and going out on a Wednesday.

Day in the Life (Part 2)

Two days in and we're up to 19 submissions!  So we don't end up with one overwhelming list of all the posts I will share the contributions in small bundles for your reading pleasure. Yesterday we had 6 posts, here's 12 more. Pick one, read them all at once or bookmark this page for later (they will also be on the #matheme page, but without the descriptions).  Don't forget to submit your contribution.


Day in the Life: Dulce Edition on f(t)

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In which I let 14 year olds play with sharp objects.


A Day in the Life: Dictated (Dragon-style) on Hilbert's Hotel

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I used Dragon Dictation to record things soon after they happened. Actually a pretty "normal" day, but it is hectic and crazy when you look back and think about it.


Today’s Roles: IT Department, Programmer, Lecturer, Assessor, Tutor, Co-ordinator…. on Roots of the Equations

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I wear a lot of hats at my job. I wore 6 today, from time to time. Who knows what tomorrow will hold? I jot down what I did, usually while walking, because that when I had the time to take notes.


A Day in the Life of a Teacher on mrsakahoshi

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A typical day at a large inner city high school. A puppy is involved, along with Trigonometry, Precalculus, Integrated Math, and AP Statistics.


A Day in the Life - Math Teacher Edition on My Web 2.0 Journey

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I tried to keep track all day of what time I was doing what... And just reading over it all tonight made me tired! :)


A Typical Tuesday on square root of negative one

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My day in photos, or at least that was the goal.


A Day in the Life: cheesemonkeysf edition on cheesemonkey wonders

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Another glamour-filled day in the life of a middle school math and English teacher. Imagine the possibilities.


Week as Math Educator - Day 2 on Mathie x Pensive

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Entire Tuesday, in second person perspective, with time checks. More hectic with more error handling than Monday. Includes blood donation and the quadratic function worrying she has a complex. To be continued Wednesday.


Day in the Life on Infinite Sums

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I chronicle Tuesday the 13th, a day of postering for the most part. Nothing particularly crazy, but it was a good day given that I have two preps and coach a sport.


A day in the life (of a math teacher) on Sum Math Madness

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This wasn’t one of my better teaching days – since I didn’t do much (quiz day), but it was certainly busy enough and I think it shows rather nicely – a day in the life of a math teacher.


Day in the life of a boarding school teacher. on A Brand New Line

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Tuesday is the longest day of my week. Here, read about all 17 hours of it.


The Day in the Life of Hana High School's 0.5 teacher on Who's a Math Nerd? **Raising Hand**

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Quote of my job: "Getting paid half time but working full time."

Day in the Life (Part 1)

One day in and we already have 6 submissions!  So we don't end up with one overwhelming list of all the posts I will share the contributions in small bundles for your reading pleasure.  Pick one, read all six or bookmark this page for later (they will also be on the #matheme page, but without the descriptions).  Don't forget to submit your contribution.


Big Smiles & Beers on MT|SM

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Ohhhhh, today was annoying. One of those days where I was inches away from going off the deep end. At times I think I did.


A Day in My Life on An "Old Math Blog" Learning New Tricks

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How a busy wife and mother of two's day looks. Some days I wonder how I do it all.


Socrates, Basketball, and Citizen Kane: A Day in the Life of a Math Teacher on Prime Factors

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From planning a knock-out lesson in 13 minutes, to the first day of the basketball season and debating the artistic value of "The Notebook," it was a jam-packed day in the life of one math teacher in Michigan.


Welcome to My Life #DITLIFE on misscalcul8

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This was harder than I thought. Even though this was long, I didn't even go into detail with my teaching, conversations, and internal decision making. I kept wanting to go into detail with everything but at the same time trying to keep it short and sweet. This was actually a day with minimal interruptions compared to some.

I hope this helps get our point across of what it is like to be a professional educator.

What I do is both valuable and undervalued.

I hope you can understand.


A day in the life of a math/stats teacher. on Roughly Normal

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Stream of consciousness post about my day.


Week as Math Educator - Day 1 on Mathie x Pensive

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Entire Monday, in first person perspective, with time stamps. Reasonably typical despite this being the first day of secondary school strike action in the province of Ontario. To be continued Tuesday.

Day in the Life Submission Form

 It's here!  The week of November 12 is upon us and I can't wait to hear about your day.  If you missed the announcement, Sam and I want you to blog about what a day in your life is like.  For a model (that you most definitely do not need to follow) check out my day last week.  Once you've written up your post, please complete this form (if you don't see it below fill it out here instead).  Remember, if you don't have a blog I'd love to have you guest post.



Thanks for participating and I can't wait to read about this week from all of your perspectives!

Trying Triangles

Over the summer I reorganized my geometry curriculum. Matt put together a great sheet of triangles to try and I reformatted it slightly. It worked great, except I have toolkits with short rulers so I wish that some of the side lengths were smaller (plus more examples would fit on a page that way). No one was able to figure out the area or perimeter ones, but I think they're worth keeping as challenges and thought experiments. (Even if you can't draw it, do you think it exists?)

Trying Triangles

We used pencil and paper, Matt mentions GeoGebra but we have only used it once so far and I think physically manipulating rulers works well for convincing yourself something just won't reach.

When students had drawn as many triangles as they could and made their own conjectures, I used the hint cards I saw in a video of a Japanese classroom. Propped up against the white board were ten index cards numbered 1-5. Each number had a different question on it which prompted students to make particular types of conjectures. I made two copies of each card since some students wanted to bring the card to their seats last time I used hint cards. The idea is, students should be able to form conjectures on their own, but if they need a hint, the cards are available to look at. No one is allowed to claim they are "done" until they have checked all 5 cards. These are the questions I would ask if I was circulating the room and noticed a student was stuck, but this way they don't have to wait for me to arrive.

Hint Card 1: What angles work to make a triangle?
Hint Card 2: What side lengths work to make a triangle?
Hint Card 3:  What can you say about the lengths of the sides of a triangle if you know the measures of its angles?
Hint Card 4:  What can you say about the measures of the angles of a triangle if you know the lengths of its sides?
Hint Card 5: What do you need to know to prove two triangles are the same?

Friday we shared out conjectures in two of my classes and ended up with:

The triangle will be scalene if we're given 3 different sides or 3 different angles.
If two angles are the same, then the triangle will be isosceles.
If the sides are equal, then the angles will be equal.
The angles add up to 180 degrees.
The sides work when they are close, but not too close
          45, 3, 4 fails           3, 4, 10 fails           3, 4, 5 works
          3, 4, 6 works          3, 4, 7 flattened triangle

One class was just what they came up with on their own, the other class we had enough time to try some extra examples which supplemented their ideas. Next up, working to verify this list, which will remain a work in progress throughout our unit on triangles.

A puzzle for you

I mentioned in my Day in the Life post it typically takes me 4 minutes to drive to school. However it takes 12 minutes to get home. How could this be?

I'll answer questions and confirm or deny solutions in the comments.

Reflection on Day in the Life

It was more difficult than I anticipated to keep track of my day. I used a note taking app on my phone (simplenote) and several pieces of scrap paper (used hall passes are my favorite scrap paper lately).

I'm not sure how much detail is best- mine took a long time to write since I was trying to be accurate. Too much information makes it hard to read though.

Some days I feel like I was really busy, but wonder exactly how busy I was.  Reading back over everything I do in a day I realize I'm on the go, nonstop! I wish I could say yesterday was abnormal, but I've gotten home after 5 three nights this week. That was my "easy" day since I only taught two classes. Today I taught 3 geometry classes in a row, which means no break from 7:24 to 11:58. However I only had 4 students after school and napped when I got home. I brought home grading and prep work, but it will all wait until Sunday, Friday nights are for recovery and Saturdays are school free.  I also enjoyed looking back to see the bits of my day that weren't teaching math, there were moments of connecting with students, educating them on topics other than the current framework. Today I got back an email from the student whose college essay I read, he was thrilled that I read it and is planning to send along the final draft.  Those are the moments that make this job worth the exhaustion, both emotional and physical.

I'm looking forward to reading what everyone else has to share next week!

A Day in the Life of Me, Tina C.

The "Day in the Life of an Educator" project is next week, have you picked your day yet?  I chose Thursday, which will include evening parent conferences.  But, I wanted to do a "regular" day and do a bit of a trial run to see what I was asking everyone else to do.  It was hard to keep notes during the day since I was so busy!  I expect that next week will include a lot less detail, please don't feel pressured to be as exact as I am trying to be.

Thursday, November 8, 2012

5:45 am wake up, curious how much it snowed but decide to sleep for 15 more minutes instead
6:00 alarm goes off, look out the window to see snow! Check twitter, email and blogs before getting up. Get ready, make lunch and grab breakfast to eat in the car.
7:00 drive to school, it takes 6 min instead of the usual 4 due to slushy roads in my neighborhood, but I arrive before my official start time (contractually teachers start at 7:12, only a school...).  I stick my head in to say hi to my classroom neighbor and exclaim "snow!" We teach overlapping courses and students, yet this will be the only time I see her aside from lunch.  Next I stop in my classroom to grab a packet I put together yesterday for a student (he was concussed all of first quarter so now I'm trying to get him caught up) and turn on my computer.
A student stops by to leave her backpack in my room; its too heavy to carry around so she grabs the stuff she needs and leaves everything else in my room. We chat about snow. Satisfied that I'm ready for my class when I get back, I grab some grading just in case I have time and head to the study center- I need to arrive by 7:24 and it's across the school, upstairs.
Run into my head teacher on the way- she has the PreCalc book I ordered! We discuss what's next in our geometry curriculum as we walk. I greet students as we navigate the halls.  I'm a couple minutes late arriving to the study center, but there's another teacher there too so it's okay.  (My duty this year is to tutor students in math, and sometimes other subjects.  I prefer it to patrolling the halls, monitoring the cafeteria or covering the front desk which are duties I've had in the past.)
7:30 I help a student with her work for Algebra.
8:00 I switch over to working with the student who had a concussion all of first quarter. We talk through the types of lines, slope and introductory angle vocabulary. He works through some practice problems and I alternately help him and let him try them on his own while I chat with the other students - some complain of no homework because the quarter just started, the one who needed help with Algebra also asks me to read her essay on Malcom X (I learn/recall a few things!).
8:45 student completes practice problems and I award him ten minutes of rest, which means ten minutes to flip through the PreCalc book - I like it, of course, and am mad it didn't arrive when I ordered it in June.  No time to even look at that folder of grading though.
8:55 the bell rings and I head back to my classroom as quickly as I can, but always making sure to smile and say hi to any student I recognize (3 years of teaching sophomores means I know a lot of students now!).  Only one student beats me to class.
9:00 I have the word of the day up before the bell rings to mark the start of my first Honors PreCalculus class.  This group is chatty so it takes a bit of effort to get them settled.  They work on a neat SAT problem while I check homework.  Students share two nice solutions to the problem, then I show them a 'trick' and allude to using similar methods when we study sequences and series later this year.  I have to fight with the document projector a bit, but I get the homework projected and listen to some complaints about how very difficult trig functions are to work with.  "What language is that?" gets called out, and I laugh, but then we get to work figuring out some of the more complex problems.  I mention the Math Practice Standards and how those strategies are the ones mathematicians use.  We read an article about how to study math (they nearly all admitted to not studying for our last test).  While giving the instructions for that activity I had a bit of a coughing attack and had a brief panic of "I'm losing my voice!" (The Worst for teachers) but it passed.  We get sidetracked talking about college (studying and advocating for yourself, university vs. smaller school).  I give students a chance to retake part of their test (usually this is only allowed after school for honors, but the last test went that poorly) or work on practice problems.  There were several requests for students who wanted to see their final Quarter 1 grades, but there wasn't time for it.
10:22 journaling, I explain and students write down homework
10:26 bell rings! First Honors PreCalc class leaves, second Honors PreCalc class arrives
Two students are arguing, I notice one has a cast, apparently a tackle in football yesterday resulted in a broken arm.  The disagreement is mild though and when I interject with "just say you're sorry" the tackler pleads how very sorry he is.
10:31 bell rings! This class is nearly silent when I approach the front of the room, a stark contrast with the last class.  I hear really good conversations between pairs on the SAT problem.  We follow the same schedule as last class but this group finishes faster and I reward them with the opportunity to retake more than one section of the test.  I have time to work with students and still manage to get quite a few retakes completed.
11:53 journaling, I explain and students write down homework.  One student is reciting a quote and wants to share the essay he wrote about it.
11:58 bell rings!  Student with the quote opens his college essay on my computer and asks me to read it later.  Another student asks to see his grade (after class, what a smart time to ask), it was much improved since he'd re-done a project and the result "made his day" which, of course, made my day.
12:03 I get an entire 25 minutes with other adults! Lunch is a sandwich.  Veteran teachers reminisce about some of the crazy things that have happened in years past.  I eat with the math and business departments, we do lunch by subject which is nice since we can talk content, but some days we need to just share stories and not be teachers for 20 minutes.
12:28 I pop in to an IEP meeting (yearly meeting for students with learning disabilities to check progress and make goals).  Normally this would be a 45 minute affair, but because it fell during my lunch I get to stop by, share how the student is doing in my class and my plan for getting him caught up, then duck out quietly.
12:45 Call co-teacher to tell her I'm ready to grade exams.  Write a note to myself to work with the student we just met with during my prep period tomorrow.  Start to read the college essay.
12:50 Co-teacher arrives.  We grade first quarter exams, talking about individual students, the exam and what we need to do next as we grade.  We finish grading and make an outline of our new plan for tomorrow based on what we saw in the exams, rushing that conversation to finish just before the end of the day.
2:02 bell rings!  School day is officially over and students start arriving for help and/or retakes.  I had 10 students stay after today which isn't atypical.  A couple wanted their grades from the retake they took just a few hours before (I've had no time to file them in the correct folder, let alone grade them), and one was so anxious to know his grades that I finally told him to write his email address on one of the assessments and I would email him all of the grades when I got to them this weekend.
2:45 half the students are gone and the remaining half are working on the PreCalc homework so I grab my paper, pencil and calculator to do the homework with them (I do all of their problem sets since PreCalc takes me time to figure out, and then I can project my solutions for them to check their work against).
3:15 only one student is left, I remember that there's candy in the closet so we eat starburst and finish the homework
3:45 last student leaves, on his way out he asks if it's raining out - we realize we'd have no idea if anything crazy was happening outside since we were in a room with no windows for so long.
3:50 I make it to the bathroom for the first time all day!  I take a moment to breathe, I've only had 5 minutes alone since I got to the study center over 8 hours ago.  Then I get to work re-doing tomorrow's lesson plans to include going over problem areas of the exam.
4:15 finished prepping for tomorrow (except photocopying homework) so I start re-organizing my desk/classroom
4:25 finish cleaning up, go to shut down my computer and discover the college essay I never finished reading.  Read the rest of that and want to email the student.  Open email and decide to clear out the rest of my inbox as long as I have it open.
4:40 really go to shut down computer and find a form I haven't filled out. Not sure how to complete it, make a note to ask co-teacher tomorrow
4:45 actually shut down computer!  Pack up a pile of exams (I graded the fundamentals classes with my co-teacher, but not the regular class I teach by myself - I try not to grade at home during the week, but my lesson plan depends on these exams) and head out.  There are 3 cars left in my section of the parking lot, I wonder if they are people doing sports or other teachers here just as late as me. Stop by the grocery store (I only needed bread, but I treat myself to soup and tapioca since my throat is a bit sore)
5:15 arrive home. Exhaustion hits as soon as I walk in the door so I don't sit down until after I make dinner.
5:48 finished dinner so now it's time to grade tests... Oh no! I brought home the wrong tests. But, how did I not grade these yet??  Apparently I'm grading the tests from last week that got lost in the shuffle of the end of the quarter and apologizing tomorrow for not grading the exams.  Excited there's a new episode of Bones to watch on Hulu.
6:09 finished page 1. Time for twitter!
6:45 finished reading twitter, Facebook, email and google reader. Pick a new show on Hulu and start on pages 2 and 3. And open a cider, for my sore throat of course.
8:30 finish tests (and a couple TV shows)

8:45 start to blog about the day!

9:45 pm Wow did I have a long day and a lot to say about it.  No time for a final read through, it's past time to get ready for bed.  Then I get to do it all again tomorrow...

*Nov 9: edited formatting and fixed the tenses, I hope.

Quarter 1 Grades

Our quarter ended last week so teachers have to put grades and comments into the computer system by 3 pm tomorrow. Happily, I got mine done already. I spent a lot of time at school (until 5 pm yesterday and today) to accomplish that, but it's done!

The new MA evaluation system includes SMART goals and requires all teachers to make a student goal and a professional goal that are data driven. I was able to make my professional goal parent communication via postcards (which was one of my goals for the year anyway), but my SBG goal didn't fly since I don't have data to compare from last year. I was talking to the other teachers after school today and now it looks like our goal might be to try to get the numbers of D's and F's under a certain percentage. We have yet to determine that percentage, which leads me to wonder: what are the grade distributions in your classes?

This is my first time with honors in this school and I'm loving having class averages in the 80's!  I have 1 F and 4 D's out of 38 kids in those classes. My Geometry classes aren't as good though, the averages are 59, 71 and 76 (not including the 5 incompletes!). They have 33, 31 and 21% F's and 17, 15 and 11% D's (respectively).  Perhaps the first goal should be under 20% in both those categories?  Or, if we excluded students who have excessive absences then we could aim even higher (lower?).  Our policy is generally to move kids who get A's up a level, which is great because students are always in a class that's challenging for them. But I wonder if having 3 levels of Geometry is part of why I end up with so few students getting A's and even B's - they get bumped up out of my classes and into the honors (I have 2 fundamentals and 1 regular, the regular class is the one with the highest average).

Regardless of what my goal ends up being, I'm happy because I know that Standards Based Grading is helping my students.  In the last week, all of my class averages went up by 6-7% except the class that was already in the 80's and they still went up 2%.  This doesn't happen without effort on the part of the students: they are learning, going back and correcting their mistakes and misconceptions, and taking advantage of the opportunity to show me what they've accomplished.  Between retakes and resubmitted projects I had a lot of grading to do, but I'm happy to do it because it means my students are still engaged in learning even after I've returned an assignment- with the typical system that's just not the case.  I use comment only grading for projects and SBG for tests and quizzes.  Together those categories make 80% of their grade.  My students know that what I really care about is what they know, not how fast they know it.  Now if I could just get that last bit of buy in from the students I haven't reached yet (it would help if I could eliminate poverty, teen pregnancy, difficult family situations and many years of believing they can't do math...).

But, back to my original question: what are the grade distributions like in your classes?  What would be a reasonable goal for our lowest level geometry classes?

Practice Standards Posters

Yesterday I finished a course on the CCSS Practice Standards at the EDC.  Several teachers talked about how they had the standards posted around the room for students to refer to.  The EDC created icons for each standard.  Finally, one teacher's department created a cute sheet of the practice standards rewritten in "I can" language.  I took these ideas and put them together to make a set of posters.  I created them in keynote (which scribd doesn't like) so I have exported them as a pdf for you to print or a powerpoint for you to edit (but the fonts won't match).

I would love to hear your ideas for how to change the "I can" statements to better reflect the full intent of the standards.  I threw this together quickly this morning and am looking for feedback before I print them and share with my students.

Math Practice Posters

Memorizing Rules

I don't like the idea of memorizing rules but Glenn Stevens of the PROMYS program talks about learning things by heart. The difference being if you learn something by heart you've tried so many examples that you learned the pattern, whereas memorizing is just being able to recite something without necessarily understanding. I thought my precalculus students had learned to shift and stretch functions by heart. Many of them probably have, it's certainly a topic they study in Algebra 1 and 2 plus they had to do a big summer project for me that involved 8 parent functions. But, two things happened today that made me wonder if there's a better way than learning "outside the function is vertical, inside is horizontal, multiplying stretches, adding shifts."

The first thing was a comment from a student after I gave back a test (which, by the way, no one studied for! They admit it at least, but apparently the juniors need to read the article on how to study math too.) There was a question asking students to compare y=sec(x) and y=sec(2x)-3. One student said to me "I feel like I just have to memorize all these parts that don't make sense." At which point I thought "Um, yes, but also no" We have done a lot of examples, made connections and they had use of a graphing calculator to answer that question so it wasn't just about memorizing. But it was also concerning that this student hadn't figured the role of each piece out yet since we've been talking about them ad nauseum.

Then I went to a class this evening at the EDC and the awesome Bowen Kerins shared an insight about shifting and stretching functions. First: the whole "inside the function is backwards" doesn't sit well with him or with students (or with me) and, all of these rules are null when you get to something like a circle where there is no "inside the function." Instead, he suggests using a change of variable to get back to the parent function. The strategy of chunking is useful all over the place, and by the time students get to calculus, u substitution will seem obvious. So here's how it works:

    y=(x-5)² - 7
y+7=(x-5)²
    N=M²

Make a table for M, N (so much easier since the vertex is at 0,0).

Then, use these linear transformations to get back to x, y.

M=x-5     x=M+5     (Shift right 5)
N=y+7     y=N-7      (Shift down 7)

To find the point you plot, use a fancy tool some call a pointer finger: cover the M and N, plot the x and y. The shifts are right there in the linear transformations you did, and they always work, no inside, outside, intuitive, counterintuitive or solving to find the vertex. It makes sense! Now if only I'd known this before all this work on shifting and stretching.

And, it works for any type of function, let's try a sine wave:

     y=sin(πx+π)+4
y - 4=sin(πx+π)
     N=sin(M)

Even though the period of this function is 2 and the usual suspects for x values will give ugly y values, the usual suspects work perfectly for M and N.

M=πx+π     x=(M-π)/π     (Shift left π, shrink/change the period by dividing by π)
N=y - 4       y=N+4          (Shift up 4)

My students always struggle with picking the "right" x and y values to get the real shape of the graph, including the maximum and minimum.  With this method there's no need to figure out good x values to choose, they just need to know the parent function well enough and learn that one set of "good x values."

P.S. I left for school at 7 am, ran a club from 2-3 and was in class from 4-8, 45 minutes away. Over twelve hours out of the house and I still blogged! I didn't proofread, but I shared an idea. I will fill in the table and maybe another example tomorrow.

*Edited 11/6 to add tables and trig example

Mistake or Misconception?

I started this really interesting post about whether something students were doing on a test was a careless error or a bigger misconception.  But then I realized that I was the one who made the mistake!  I was glad to have started writing the post when I was only a third of the way through the tests so I didn't have to go back and re-mark all of them.

My question still holds though; when do you go from thinking "Why didn't they check their work? My students make so many careless mistakes." to thinking "Wow, they really didn't understand this."  If a couple students make the same mistake is that a clue that they have a misconception or were they both careless?  What if 10 students make the same mistake?  On this same test quite a few students used 2π to calculate the period of a stretched tangent function (trig refresher: 2π is the period of sine and cosine, but π is the period of tangent).  I'm still not sure if they just forgot in the moment with the pressure of taking a test, or if they don't know that tangent has a different period than sine and cosine.  When does an error earn the label misconception?

If you're curious, here's what I was thinking:


And what the students were writing:








And the more precise way to write out the solution to the problem:


I think mine was a mistake because I realized it didn't make sense when I started thinking about it.  Is that the difference between a mistake and a misconception?

Update on #matheme

At least once a week I update the #matheme page. I have no idea who uses it or how they do though. There are two parts to the page: weekly events and big themes that tend to happen within a specific timeframe.

The current weekly events are: Sunday Funday, Made for Math Monday, Global Math Department Meeting Tuesday and Favorite Friday. The Sunday Fundays have themes, so someone browsing the #matheme page might click through to a topic that interests them. For the others though, I'm not sure that compiling them on the #matheme page is helpful. Is anyone using them? Would a link to the hosting blog plus the name of the event suffice? What better way can we organize all the cool projects and submissions people make?

The big themes (classroom photos, writing in math, goal setting) are why I originally started the page and I'm happier with that organization. Most topics get a lot of updates when that's the current cool thing to write about, and then I add to the list if I happen upon a blog post that fits the category well. Do you see any glaring holes in this part of the page? Missing topics? Missing posts within a topic?

If you've used the page, could you please share how? If not, is there something that would make it more user friendly? Thanks for any and all input!

Running Retakes

Today was the last day of first quarter (crazy!) so this week has been filled with students wanting to retake tests and quizzes (along with correcting projects but that process is different).  I have a very fancy retake system, I know you're going to be wowed.

1.  I write 3 questions on an index card and give it a fabulous name like "Lines Quiz Retake 1"  I give a test and a quiz for each standard and make 2-3 versions of retakes (numbered 1, 2, 3 or maybe A because I did that one week).

2.  Students grab a quarter sheet of scrap paper and answer the questions.

3.  Students hand in both the answers and the questions, I re-file the questions (i.e. clip it to the other retake cards for the course) and put the answers in the folder that corresponds to the class they're in (making sure the student correctly labeled the answer sheet with their name and the name of the quiz/test).

4.  Repeat.  A lot.  I've run out of scrap paper on several occasions so I've taken to scrounging for one sided leftovers at the copiers (we also use quarter sheets to take quizzes).

It may not be a fancy system, but it works.  I don't have to make up problems on the spot.  I have a few sets of questions so multiple students can take the reassessment at once, but when I'm grading I just need to match the answer sheet to the correct index card to check their answers.  When a student is waiting for a quiz/test that's in use I just have them label their quarter sheet and put it next to my pile of cards.  I had grand intentions of creating a filing system, but there just aren't that many standards in a quarter, so it doesn't take me long to find the right one.  Maybe someday I'll type the questions and tape them down to index cards, but some of my students with testing anxiety feel much more relaxed with this casual system than they do sitting down with a large, typeset test.  Plus, if a student accidentally writes on the question card it takes a few seconds to re-write the questions on a new card.  (None of our printers have ink in them so printing something is actually an annoying ordeal that involves going to the library and hoping no one was copying when you printed to the copier.)

How do you deal with the rush of students who want to bring up their grade at the end of the quarter?

Finding Structure

Background: In honors pre-calculus I was assigning evens for homework; students wanted to check their answers before asking questions so I started doing the homework and posting my solutions on the document projector.  Some days I just post a numerical answer and we work through any problems they request.  Other times the homework is to prove something, so I project my worked out solutions.  This problem set fit the second category.

Before class I was solving this problem:

I got this problem from the website that goes along with the textbook (this matters because I'm sharing it, but it's also important to recognize that I didn't write the problem).  My first step, as it is with most trig proofs, was to re-write the entire problem in terms of sine and cosine.  This didn't yield any obvious simplifications, so I wondered if it was actually true.  I didn't want to wrestle with the thing if it ended up being false, so I checked a couple values off to the side.  For both π/4 and π/3 the equation was true, so I went back to proving it.  Looking at the equation more carefully I saw that I could do the following:

This is no easy problem, for a multitude of reasons, but I could have pretended it was an easy problem, skipped showing half the steps and posted a slick, neat solution when I projected my answers.  I wish I could tell you that I planned the conversation I had with my students today carefully, that leaving my scratch work had more to do with showing them my thinking than it did with my laziness, but neither of those would be true.  I sure did seize the opportunity once I saw it though.

In class no one had completely solved this question so they wanted to discuss it.  Looking at my solution they couldn't figure out how I had gone from all the sines and cosines to the first step you see above.  This is where I walked them through exactly what I explained above.  I told them how I got stuck, how I wanted to check before doing a lot of work, how I had an "aha!" moment that required going back to the beginning.  I acknowledged that I had to make the problem look more complicated before it simplified.  I pointed out the difference of squares, and reassured them I don't expect them to recognize that structure, yet.  I explained the math practices and how recognizing structure is one of them.  I talked a lot more than I usually do, but at least one of the students saw something and said "Look how excited she is about this!"  To which I responded, "yes!"  Thinking back about why this moment was so exciting has me realizing it's because this problem is 'real math' - this is what math looks like when mathematicians are working.  

I did have a student ask "when would we ever use this?" which is a huge red flag for me; it means the math seems overwhelming and kids want an out.  I explained that a) people really do use this so I want to prepare you for the possibility that you'll go into a mathematical field and b) we need a context to practice logic, reasoning and algebraic manipulation - today that context is trigonometry.  They all laughed when a student said "so you just rummaged around in all of mathematics and pulled trig out of the hat?" and the original questioner seemed satisfied.

Note: I've been feeling overwhelmed lately, as has most everyone who I've talked to, but I keep having ideas I want to blog about.  November is National Blog Post Month (there's some crazy acronym for it, that I can never decipher) where people take on the challenge to post every day for the month of November.  I have no idea if this will be achievable, but I'd like to write a short post each day about some cool moment or something on my mind.  This turned into a longer post than I expected, but I'm going to see what happens!