My 8th grade Pre-Algebra class was nearing the end of our linear graphing unit.  I had just given a quiz on graphing lines using our BFF (best formula forever…y-mx+b) and finding the equation of a graphed line. The quiz entailed receiving an equation and a blank piece of Cornell Notes graph paper and writing up a set of notes which would help your grandmother (who used to know how to graph, but that was 40 years ago…) understand how to graph.

As on all quizzes, results varied widely and I wanted a chance to meet with students who had similar misconceptions from the quiz to do some reteaching.

Enter…the GRAPHING GALA…

My vision was to have several graphing stations set up where students would work independently on concepts I wanted them to practice before the exam.  While this was happening, I would pull small groups of students and either reteach or extend their understanding of graphing linear equations.

While I had ideas of different activities for each station, I really didn’t want to invent them from scratch.  I definitely wanted a matching activity…equation cards and graph cards…but again, I was hoping not to make a zillion graphs on GeoGebra as I was sure someone had done this already and I could find it on the web.

I’m sure there are endless activities floating around on the interwebs, I stumbled on this site which has really wonderful stuff to use for independent stations.  It’s mostly Algebra 1 and geometry stuff, but there’s a bit of other content there as well.  Dr. Robinson is a professor at Tallahassee Community College in Florida.

I assigned students a partner based on data from their last quiz. I wanted to make sure that the pair could independently do all the stations (since I would be working with small groups all period), so some pairs were a high student with a low student, while others were students with strengths and misconceptions that complimented each other.  In addition to a partner, each student was assigned a color, so when I called groups, I called two pairs (who all had the same color).  I tried to make the 4-person color groups somewhat homogeneous in terms of needs so that I could focus on 1 or 2 misconceptions during our small group time.  Each group got 15 minutes with me and I successfully met with 5 groups during the 98 minutes.  Since I have just 20 students, every student got a full 15 min. with me in a 4-person group.

It looked like this

But in all seriousness, it was an amazing class period.

Students each received a cover sheet and in both periods I had an extra adult(s) who had an answer key and stickers to hand out when students completed a station (student teachers in 1 period and an IA and 8th grade tutor in another period).

Station #1, Stained Glass Window was practice graphing equations in y=mx+b and coloring in the geometric designed formed by the lines. This activity is from Dr. Robinson’s site.

Station # 2 was a worksheet I made (by hand as at that point I didn’t have the energy to create more graphs on GeoGebra) where students had to identify the equation of a graphed horizontal and vertical line or graph a horizontal or vertical line when given an equation.  When done, they got to play this silly game for 5 minutes.

Station #3 had 20 graphs and 20 linear equations which I had copied onto fluorescent paper for added excitement.  The cards were cut out and jumbled up in an envelope. This is from Dr. Robinson’s site with a few modifications.

Station #4 was the 15 minute session with me.  I called students, by their color groups, and used GeoGebra, projected onto the white board to pose questions to them which focused on misconceptions they had had on their last quiz.  We’d spent 10 minutes doing work together and the final 5 minutes of the group they sat together and did quiz corrections (which they could then do with ease).  They earned their sticker for that group once all quiz corrections were perfect.

Station #5, Hot Air Balloon, was a more conceptual look at a system of linear equations, also taken from Dr. Robinson’s site.  Students had to figure out their axes, scaling, etc.on their own.

It was a really wonderful day where the class completely ran itself and I got to meet with every single student.  If only I could differentiate like that every day!

But this weekend, with no John Stewart’s left in the Tivo cue, I watched a string of Colbert shows from last week.  What a stroke of luck.

Last Wednesday’s show featuring Ameena Matthews was really inspirational.  You can watch it here: ameena-matthews

In brief, she runs a non-profit in inner-city Chicago where she literally interrupts violence by walking in the middle of it, in the hope of saving lives.  She spoke of violence as a disease.  She, like many urban children, was taught by her family that when hit she should hit back.  She explains that this turns into an escalation of violence because when you hit back, the person you hit has a huge extended family ready to support and protect that person, as does the other original member of the conflict.  So violence spreads from 2 original people to hundreds as quickly as a disease can spread between all those who have come in contact with someone who has it.

The documentary about her work, The Interrupters, was co-written by Alex Kotlowitz, author of There are no Children Here, one of the most influential books on urban education I have ever read.

It’s showing this week in Oakland, but is moving all over the country.  You can find out more about showings here.

It’s also airing on PBS’ Frontline on Tuesday, February 14.

I plan to watch it then and purchase the film to show to our 8th graders.  After watching it, I”ll have a better idea of how to facilitate and structure the conversations among students about the issues raised. If you’d like to talk more about how to debrief the movie with students, I’d be happy to discuss it with you.

When I attended the Park City Math Institute last summer I returned home with several promises to myself.  One was that I would integrate technology into at least one lesson every day.  I find that it’s easy to come up with excuses of why it’s my hands and not theirs which should be on the technology (spotty wireless on the mobile Imac lab; spotty memory  issues on the downstairs PC lab; not enough time in the district pacing guide to train my students on how to use various programs, not enough patience for their questions which stop them in their tracks, etc, etc).  I’m sure you’ve experienced them all before.

But I’ve been trying hard, really hard, to get my students’ hands on technology with some regularity.  But, I must confess, that my 8th grade geometry students’ hands do a whole lot more ticking of the keyboard than my 8th grade pre-algebra students’ hands do.  Which is ironic, and disconcerting, since I could make a very good argument about why it should be the other way around.

SO…I made this packet to help guide my 8th grade pre-algebra students into learning how to graph linear equations on GeoGebra while also discovering important aspects of slope and y-intercept.  (Please note…this is NOT the way I introduced slope and Y-intercept.  Prior to doing anything formal, we have done several MARS and Shell Foundation tasks which involve real-world situations involving linear motion and I wrote a few of my own.  Students have matched cards of a graph to a t–table to an equation.  We’ve looked at slope and y-intercept as they apply to the amount of candy my son ate each day following Halloween.  We looked at slope and y-intercept as they applied to water dripping at a constant rate out of one vessel and into another one.  And we looked at varying slopes along a car trip which involved a lunch stop and speeding up not miss the start of a movie.)

But I wanted to use GeoGebra to introduce the more formal definitions and ultimately connect them to the problem solving we have already done.

We’ve only just begun…45 min. in the lab last week to be followed up tomorrow.  So far, so good, but from looking at their homework, (which was more of the same, but without the technology), it doesn’t appear that the learning has been made concrete yet.  I know there are tons of ready-made applets on graphing linear equations, but I wanted to take the time for the students to learn how to use GeoGebra beyond tinkering with a pre-made applet. The trickiest part for me, of spending a whole period in the lab, is that it’s impossible to have a whole class discussion.  I can have them stand with their backs to the computers when I want to talk (and this seems to be necessary for them to actually listen), but there’s only so much talk that can happen when everyone’s backs are to the computers.

So I’m hoping that after 2 periods in the lab, we’ll be able to have a whole class discussion about what they discovered with just 1 set of hands on GeoGebra.  We’ll see.

Today I returned chapter exams and students had some time to work together to talk through their errors.  There was one proof which only two students had gotten full credit on.

Prove that the sum of the medians in a triangle is greater than half the perimeter.

Several had very nice demonstrations using actual numbers, but only two were able to carry out all steps of an algebraic proof.  Yesterday, I asked one of these students to take her exam home, look over her work, and be ready to present her proof to the class today.  Why have her review her work at home? I have found that on challenging proofs, students often are able to write the proof, but explaining it to confused classmates presents a challenge as they realize that their own understanding is a bit fragile.

So, she comes to the board, talks the class through step by step (to several audible “Oh WOWs” and “That’s SO cool!”).  When she finishes the other student who got full credit asks to write-up her proof so that we can compare their strategies.  Her proof is met with the same excitement.

Ok readers…you’re thinking, I know, this happens to me too.  Why retire?  Is that really as good as it gets?

No, there’s more.

Then, a student asks if she can take out her phone to take a photograph of the two proofs. “Sure,” I respond.  Next thing I know, about 90% of the class is piled on top of one another, each trying to get the best view of the board for their photos.

Although her proof was quite elegant (ignore the black print as it’s part of the other proof):

the real photo was capturing the giddiness evoked by the photo session:

And the absolute highlight:

“Ms. Krasnow, can I open facebook and make this my profile photo and tag everyone from the class?”

Um…yeah

My trajectory went something like this:

Year 1 of teaching 8th grade algebra:

Me, “I have absolutely no idea how to solve these problems so I’ll never get my kids to understand them.  I’m way behind in the pacing guide, so I’m just going to skip them”

(Mind you, my year 1 of teaching algebra was my year 8 as a teacher, so the fact that I flat-out skipped teaching something because of my own fears and lack of content knowledge is actually quite bothersome)

Year 2: Me, “I know how to DO these problems, but I don’t really understand their ins and outs.  I’ll give the class 2 options: Work in a small group of students where I’ll show you something new and challenging, or stay in your regular groups and you can work on catching up on your missing work.

Year 3: I fess up to my colleague Marlo that I am terrified to teach mixture problems to an entire class of 8th graders, as I don’t understand them well myself and can’t figure out how to make them accessible to all students.  Marlo has recently been to a summer PD where she played around with using a see saw method.  It’s documented in this great video by the Teaching Channel.  I realized there was hope for me!  I still didn’t teach mixture problems to the whole class, but I had a lot more confidence in teaching them to small groups of students who wanted an extra challenge.

Year 4: I’m ready to help ALL students learn mixture problems.  After some initial work with kool-ade and water, all students understand the idea that you’ll need MORE of the solution which is closer in percent to the desired mixture percentage.  As a class, we have the routine of putting a large colored star next to the solution which we’ll need more of.  I find that this makes a huge difference in students not mixing up the quantities since they are inversely proportional.  We talk about two ways of modeling these types of problems:

I never taught systems of equations as a solution strategy to these problems and last year I’d say that at least 75% of my students could solve mixture problems correctly.  For a very heterogeneous group of 8th graders, 75% success is pretty good.

One of my 7th grade algebra students read about systems of equations in the textbook and came in to ask me about it.  He thought it was way easier, so I decided, weeks after the fact, to teach this solution strategy only to my 7th graders.  A few loved it, but the vast majority stuck to the proportions model even though they were quite proficient at solving systems of equations.

Is the way I model it just another trick to memorize?  Maybe.  It’s not as visual as David’s GeoGebra applet, but using colors (which my students always do), it felt like a decent model.  I’m curious to hear reactions.

I have a strong opinion on this, but am going to withhold it for the sake of an unbiased conversation.  Our main differences are on whether teaching students to solve equations vertically or horizontally will lead to deeper understanding and accuracy.  We’d like to teach them consistently across all 3 grade levels and are particularly interested to hear from teachers who teach algebra and higher level high school classes.  Is one method more powerful for students than the other?  Does having an entire math department teach one consistent method lead to more students understanding equation solving in more complex applications?

One new aspect which I plan to adopt is what a new teacher to our department taught us (I should mention that she’s new to our school, but taught 15 years in Oakland before coming to Berkeley).  She uses a vertical line through all the equal signs (see below) to help students ‘see’ that equality is always maintained.

With no further adieu, I present vertical and horizontal: (and by all means, feel free to critique both notations)

And I need to make a public thank you to Donors Choose who got Chevron to give a whole lot of money to fund California teacher’s grants. I just got a $2000 tablet PC laptop and Fluid Math thanks to them.

And finally, this is no longer very timely, but 2 posts ago there was a really great conversation about how to best help students see and correct their errors.

What I tried:

1. I made a spreadsheet of what % of students answered each answer choice for each question (it was an all multiple choice exam).
2. I analyzed the data and chose the 6 most common errors.  I used examples of students’ work and  handwrote a worksheet where there were 10 problems on these 6 topics.  I told students that 6 were wrong and 4 were right.  I broke them up randomly into pairs and they had to analyze the student work and find and correct the 6 errors.  We then discussed some of the problems whole class.
3. Finally, they had 20 minutes to look at their exams and correct any mistakes they found.  It was a test taking time (i.e. no talking, notes, etc). I DID NOT tell them which ones they had gotten wrong, I just wrote at the top of their exam how many problems they had wrong and how many errors they needed to correct to move 1 grade higher on the exam.

http://vimeo.com/31652592

Stay tuned for my response on my previous post about having students examine misconceptions and then have a second shot at their exam. You may be surprised at what I learned.

In our district we take quarterly exams which are the same at all 3 middle schools.  8th graders predominantly take Algebra 1, so there are no district assessments for 8th grade Pre-Algebra or Geometry students.  These are the two classes I teach, so for this, I’m on my own (which is fine).

My Pre-Algebra students took their 1st quarter exam on Thursday and I’d like a way for them to re-engage with their mistakes.  Of all my students about 5% scored below 50%, 45% scored between 50-70%, 45% scored between 70-90%, and 5% scored above 50%.  I am able to scan my students’ answers into the district data-base and create wonderful reports which tell me which questions were most missed, what the most common wrong answers were, etc.

NORMALLY, I have students do a self-reflection which looks something like this:

By circling the questions they got wrong, they’re able to see which topics they need more help in.  Then, sometime over the next week, I pull groups to work for 10 minutes on individual white boards.  I reteach the most frequently missed concepts. Test corrections are due within a week of when they get the exam back.

BUT…here’s where I need some fresh ideas…my Pre-Algebra class is filled with students who have ALWAYS done poorly in math.  They’re used to not doing well and crumpling up an exam the moment it’s returned is like a Pavlovian reaction.  I’m slowly breaking them of this habit, but DISENGAGEMENT from exams is a long, slow process to undo.

My question is…how best to set up an activity for students so that they  genuinely want to re-engage and learn the topics from the exam which are still challenging for them.

My idea, which needs refining (from YOU) is the following…

1) Let students know that they’ll have a change to revise their exam and raise their grade.

2) Spend a class period modeling common mistakes and having groups of students correct those mistakes.  I would choose which problems to model after looking at the data analysis of their answers provided by the district.  Students wouldn’t have their own exams, but we’d be doing actual problems from the exams, in groups, so they could talk through where each hypothetical student went wrong.

3) Students get their exams returned to them.  At the top, I write how many errors they have, but I do not mark which questions are right or wrong.  They get a specified amount of time (20 minutes or so), to go through their exam, under silent, testing conditions, to try to find their mistakes.

Please pick this idea apart…make it stronger…help me see what big ideas I’m missing.

I’d really like to TEACH these students how re-engagement with an exam is a very powerful skill which leads to deeper understanding.  And I know that teaching this will be a process.  I’d just like to have a clear idea about how to start…

Thanks.

And I could talk your ear off about why I believe that creating and teaching this class was an equity decision.  And some of you would talk my ear off about why tracking is bad, period.  And I would disagree.  We’d then have a really heated, informative debate where we’d both push our thinking deeper and become better educators because of it. Sometime soon, over a bottle of wine, the math department at my school will come to my home and we’ll have this very debate.  It needs to happen. And as I said, it will make each of us think more reflectively about what we do.

But for now, here’s what I’m excited about: teaching integer addition.  Mastering it is critical for success in Algebra 1.  All of my Algebra Scholars took Pre-Algebra last year and had extensive experience with integer operations.  Nevertheless, on an integer pre-assessment given shortly before I began my unit, only three students could successfully add one digit integers, and only 1 student could successfully add integers with larger addends.

I believe that they have been introduced to far too many models for integer addition, and never been given the time to learn any one model in depth.  For this reason, the ONLY model we use in class is a number line, but we have taken a lot of time to have rich activities and conversations about how to use it to model problems involving integers.

I do wonder, though, if using multiple models is useful.  I’m curious what others think.

Last year, I realized that when I do math, I love to use colors.  It helps to organize my thinking and highlight important discoveries I’ve made while in the middle of a problem.  So as of September, pencils are no longer required (though I make them readily available to students and allow them to borrow my supply at any time). Upon entering class, students take these which is a set to share with their neighbor.  They are required to use colors when we take notes. They are not allowed to use them for doing their regular work, until I tell them that they should color code. I am trying to preserve the ink for as long as possible.  They know that if I find a single dot of this extra-fine German ink on their skin, I will cut off that joint.  Ok, not really, but I have been very clear about the purpose of these uber-expensive markers.  Having these markers makes students (and teachers) not in my classes quite jealous.  And since I teach the lowest ‘track’ of students, I like it that way.

Phase 1: Comparing Integers. Using Wu’s definition that a larger number is always to the right of a smaller number on the number line, we placed integers on the number line and compared.  To escape bordom, each student had a plastic popcorn box filled with integer tiles from -10 to +10.  They chose 3 numbers from their box and compared them on the number line.  They then wrote 3 inequalities comparing their numbers.  This pushed them to consider 4>-2 as well as -2<4.

Each student made a key at the top of their paper, color coded, using three colors: 1 for positive values, one for negative, and one for zero.

Phase 2: Adding Integers.  Sticking with the popcorn boxes to generate random pairs, students then chose 2 numbers and added them.  Everything was color coded and they asked themselves over and over again, “Will I make it to zero?”

Phase 3: Discussing discoveries.  Students then took all their addition problems and sorted them onto a mat.  I had no intention of telling them rules, but instead wanted them to really be able to ‘see’ in their heads what happens when you add integers.  The colors helped to imprint the math in their heads.  We had a rich conversation about what the colors showed us on our mats.  After analyzing why positive +negative sometimes is positive, sometimes is negative, and sometimes is zero, I felt that well over half of my students had solidified their understanding of when you combine the absolute values (adding) and when you find the difference.

(Apologies for the head tweeking here):

Phase 4: Tweaking the model, but only slightly.  I realized that asking oneself what the sign of the answer would be was a critical juncture in what should be happening in their heads.  Most were really good about asking themselves, “Will I make it to zero?”  But I needed a hook, something silly, which tapped into their teen-ager-ish-ness.  This whole unit could be summarized by a single question, spoken in my sexyiest possible voice, “Hey Baby…What’s your Sign?”  Our final push for mastery of large numbers involved asking each other this question, and stiffling the giggles that ensued.   I took away the bulk of the scaffolding (most students had tired of 2 colors on the number line and ironically were begging to simply do their work in pencil).  But added one last element for those who needed it: colored dot stickers. A few students still needed a visual reminder to cue them on when to combine the absolute values (add) and when to take the difference. They all had green for positive values and red for negative.  100% of them could now quickly and efficiently respond to the question of their peers, “Hey Baby…What’s your Sign?” And before doing any calculations, they stuck a flourescent orange sticker down with the sign of their answer.  It took three weeks to get here, but we made it and I never gave any rules for them to memorize. Success.  The real goal, of course, is that they conceptually understand this well enough to use it throughout the year and in the years to come.  Stay tuned…Next week’s project is for them to design a series of lessons to teach to the students in the algebra support classes who are still struggling with integers. They can’t wait!

Today in geometry, I was reminded about what can happen when you simply listen.

We were beginning the chapter on congruent triangles and I wanted to hear their intuition on what it means for two polygons to be congruent.  Their warm up was the following problem:

Imagine that this hexagon is made from 12 toothpicks.

1) How could 6 more toothpicks of the same size be used to divide the hexagon into 3 congruent regions?

2) Keeping two of the toothpicks from part 1 in the same place and moving four, use six toothpicks to divide the original hexagon into 2 congruent regions.

Although using manipulatives or GeoGebra was a natural approach to solving this problem, I didn’t have toothpicks on hand, nor the mobile computer lab.  I  figured that students would sketch out different diagrams to find a solution. After a few minutes, I looked over and saw that a student had taken out 12 Sharpies of equal length and had constructed a solution to the first question.A student creating a scale model of the problem that was over 2 feet wide felt like the proper time to have the rest of the class drop everything and come huddle around this student’s desk.  We gathered up, some on stools, some on desks so that all could have a clear areal view.  There was unanimous agreement that this was a correct solution to the first problem.

“How do you know that these 3 regions are congruent?” I asked.

While answers ranged from, “It just is.” to “You can measure it and see.” the most interesting one to me was the student who claimed that the 3 regions were congruent because each was equilateral and congruent figures had to be equilateral.  About a dozen students joined in the debate, I mostly just listened, so I could learn more about their intuition. The idea that congruent figures must be equilateral was a misconception that I hadn’t come up in previous years.  I’m glad my ears were open.