Algebra 2

What a Difference 12 Kids Make

from map.mathshell.org

We’ve entered Spring Trimester and the volatile Minnesota weather is cooperating thus far.  If there’s a silver lining to last year’s Spring suckfest, the lack of warmer weather put off the end-of-the-year slide until closer to May.  I’m not sure we’ll have the same luxury this year.

I teach the same level of Algebra 2 that I did last year but my class sizes are a more manageable 22-24 rather than the monstrosity of 36 I had last year.  I know class size isn’t high on Hattie’s list of influences on student achievement, but providing formative evaluation (something VERY influential, according to Hattie) is much more doable with 20-something rather than 30+.

I’ve left the desks in pods because I’m convinced students interact and collaborate mathematically more often when they have multiple classmates within conversation distance.  I want to switch their groups periodically, if only I could get them to sit in their assigned seat!

One of my go-to resources is the Mathematics Assessment Project. Their lessons are robust, and provide good opportunities for students to have great conversations around the mathematics.  This lesson on investments is no exception.  The main activity is a card sort where students match a principal and interest rate of an investment with a formula, graph, table, and description.  But the everything from the pre-assessment to the closing slide makes students think and share.

Here are the openers of the main lesson:

from map.mathshell.org

from map.mathshell.org

from map.mathshell.org

from map.mathshell.org

My assumption, not being familiar with this group of kids, was that they’d go right for the obvious – Investment 3 is the odd one out because it has a 10% interest rate and the others have a 5% interest rate.  I underestimated them.  They came up with very creative, thoughtful reasons why each investment could be considered the odd one out.  I really like these questions because all three can be correct, and students have an opportunity to defend multiple answers.

The card sort was also spectacular.  I was able to have great conversations with each group about their thinking. (Yes, that’s much easier to do with 24 rather than 36).  What a difference 12 kids makes.  There is so much to this lesson to love.  If you have a unit on exponential functions, give it a try.  I’d love to hear how it went in your classroom.

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Brain Sweat

I’ve talked about my Algebra 2 class at length on this blog over the last 2 months, and as the trimester comes to a close, I want to celebrate the positives in this class as much as possible.  They frustrate me sometimes, but the bottom line is I’m willing to fight and fight hard to make their experience with math more positive.  Ultimately, they’ve been dealt an unfair hand:  crammed into giant classes and labeled incapable of high-level mathematics.  They are capable of more than they give, but they also deserve much more than they’ve been given.

The perpetual optimist in me wants to continue to celebrate their achievements and play the hand they’ve been dealt as best we can.  Today we took on Robert Kaplinsky’s Cheeseburger Lesson.  I’m not sure why I’m constantly drawn to this lesson, since the picture of the 100×100 makes me a little ill.  Perhaps it’s the constant student engagement I get from it, time after time.  The intriguing thought that someone actually purchased this godzilla-burger hooks students every time.

What I liked most about my class’s efforts toward this task was the multiple revisions they had before arriving at the correct answer.  I had many students assume that a 3×3 cost the same as three cheeseburgers, only to find that their burger only needed one bun.

Below is a student’s work that I really appreciated.  At the end of the activity, he said,

“Mrs. Schmidt, I’m sweating.  I thought so hard on this problem that I’m sweating.  But I believe I have the right answer.”

photo 5

If I’m being completely honest overall, this class has tested me, day in and day out.  I’ve worked very hard, but in the end, I’m not sure I taught them much of anything worthwhile.  I hope I have, but I’m not sure I did.  A class size of 36 seemed insurmountable, and perhaps in some ways, I never really overcame it.  Unfortunately, next year’s class size projections promise more of the same.  The silver lining, however, is that I get another crack at teaching this same course, and I’m 100% sure I can do it better the next time around.

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Etcetera, etc….

I love it when students figure stuff out.

I love it even more when:

A.  Students figure out things that, as a teacher, I didn’t  notice myself.

B.  Students who are labeled as “not good at figuring stuff out” figure stuff out.

Here’s what we did today in Algebra 2:

number pyramids

This is a SMILE resource from the National STEM Centre.  The problem I thought I would encounter is the word “etc.”  Kids don’t do well with “etc.” Etcetera is vague, non-committal, and easily dismissed.  To a student, etcetera usually means “I’ll ignore this and see if no one notices.”

It is helpful for me to be more specific with my expectations of students, especially when their mathematical well being is at stake.  But today, I was feeling a little vague and non-committal myself, so I handed out the sheet, explained what was going on and let them go…etc.

There are no words I love to hear more in my classroom than “Mrs. Schmidt, look what I figured out.”  And today was chock FULL of those statements.  Here are a few:

  • The triangles are always as wide as they are tall.
  • The sum of the base of triangles 3-wide is 3/4 of the top number.
  • As the triangles get larger, the percentage of the peak number gets smaller.
  • The percentage decrease is related to the size of the triangle
  • If the triangle has an odd numbered base, then the center number in the base is always related to the peak number.

There were lots more.  I was very proud of this class’s resolve in addressing the Etcetera.

 

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Class Commences – an hour I won’t soon forget

Recently, Michael Pershan unearthed a Shell Centre gem straight from the 80’s (literally).  This collection of materials is fantastic, and hopefully demonstrates to both students and teachers that engaging in rich tasks and high-level thinking is timeless.

I decided to give the function unit a shot in my Algebra 2 class today.  Some background on this group of students:  there are 38 juniors and seniors, last hour of the day, in a class geared toward lower-level students.   So far though, the only thing that’s been “lower” in this class is the number of empty desks I have.   I handed out this task, gave minimal directions and let them go for a few minutes on their own:

 

from: Shell Centre for Mathematical Education, University of Nottingham, 1985

from: Shell Centre for Mathematical Education, University of Nottingham, 1985

It was so interesting to watch the different ways each of them started.  Some began with 7, since that was the first you saw when reading the graph from left to right.  Others insisted to work from 1 to 7, identifying the corresponding people along the way.  A few worked the other way around, from the people to the graph.

I walked around to make sure each student was able to get started and that those who thought they had determined a solution also supported their claims.  Then, I wrote the numbers 1 – 7 on the dry-erase board, stepped back, and let these kids amaze me.
One student volunteered an answer, and then handed the marker off to another.  I intervened only briefly to make sure that every student had an opportunity to contribute if he or she wanted.  Once 7 names were completed, I knew a couple of them were out of place.  I sat and said nothing, and this entire class showed me what they are capable of.  Here was a class full of students labeled mathematical underachievers completely nailing SMP #3.  Their arguments were viable, their critiques constructive, their discussion productive.  It bothered a few of them that I wouldn’t let them know if/when they were correct.   But most of them are starting to understand that my main focus here is not the correct answer, but the incredibly rich and interesting process they used on their journey to finding it.  They came up with multiple ways to support their answers and noticed tiny details about the people that supported their findings.  For example, did you notice that Alice is wearing heels? According to my students, that is perhaps why she appears slightly taller than Errol.

I had a heart-to-heart with this group when we were done about how proud I was at how they conducted themselves throughout this task.  I’m really thoroughly looking forward to a fantastic trimester with this special group of kids.  Their work on this task gives both of us the confidence that they can tackle something more difficult next time, and they are capable of mastering high-level mathematics this trimester.

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Notice and Wonder with Gusto

My daughter was very content on the airplane ride from Fort Meyers to Minneapolis watching Frozen for the 102nd time.  I took this opportunity to read the Noticing and Wondering chapter of Powerful Problem Solving, the superb new publication from Max Ray and the Math Forum crew.  I took so many notes on this chapter since this is a strategy that I think every teacher can implement, no matter their apprehension about new strategies.  It is such an easy set of questions to ask:  What do you notice?  What does that make you wonder? Those two questions can open up an entire class period of rich discussion and mathematical exploration.  No one explains this classroom strategy better than Annie Fetter of the Math Forum in her Ignite Talk.  (Seriously, if you have not seen this 5 minute, dynamite, game-changing video, stop reading and go there now. ) 

Last Thursday was day 1 of our high school’s third trimester.  The first day of the slide into the end of the year. Regardless, the first day of the trimester always seems like the first day of school: the anticipation of a scenario that’s been played over and over in the minds of teachers and students becomes reality.  For me, this day meant the last hour of the day I would be met with 38 (you read that right) “lower level” Algebra 2 students.  My class is most likely the last high school math class that these juniors and seniors will take, and many of them do not like math or are convinced they are not any good at it.

This class has been in the forefront of my mind most of the year for a lot of reasons.   One of those reasons being that after Jo Boaler’s class this summer, I know that a huge barrier to raising the achievement levels of students in this class is the students’ beliefs that they are capable of doing high level mathematics.  And I also know that a key component to getting these kids to perform better is to give them feedback that allows them to believe that they are capable of it in the first place.

Because of the structure of some of our high school courses, most of these students have not had experience with higher degree graphs, equations, or functions.  They may have seen something similar in their science coursework, but quadratics have not formally been introduced.

I gave them the following graph along with the scenario and let the noticing and wondering begin:  Mrs. Bergman likes to golf and her golf shot can be modeled by the equation: y= -0.0015x(x-280).

A couple of them stuck to non-math related Noticings (the graph is in black and white), but almost all of them noted multiple key characteristics of the equation and/or the graph.  Some highlights:

  • The graph doesn’t have a title and it needs one.
  • Both heights are in yards
  • Horizontal distance goes up by 80.  Height by 5.
  • The peak is in the middle of the graph.
  • The graph is symmetrical
  • The maximum height is about 28 – 29 yards
  • The distance at the maximum height was about 120 yards
  • She hit the ball 280 yards.
  • The number in front of x is negative
  • The graph curves downward
  • It has an increase in height and then a decrease in height.
  • As the ball reaches the peak height, the rate the ball climbs slows.

The list of Wonderings was even more impressive to me. A lot of them wondered things like what kind of club she was using, if the wind was a factor, did she have a golf glove, how much power she used to hit the ball, the brand of her tees, clubs, glove, ball, etc.  Then one student laid out something so profound, it made the entire class stop and and acknowledge the excellent contribution:

“What distance would the ball have traveled if the maximum height were 20 yards rather than 28?” (audible ooo’s here)

After this student said that, the floodgates opened with great questions from others:

  • What was her average height for the shot?
  • What is the maximum height that she is capable of hitting the ball?
  • Is this a typical shot for this golfer?
  • If the maximum height was higher, like 35 yards, how far would she hit the ball?
  • What is the exact maximum height that she hit the ball and how far did she hit it when it reaches that maximum

There were still a few that couldn’t get passed what kind of glove she was wearing or tee she was using, but most of the students stepped up their Wonder Game when one single student demonstrated a rich example.

What I really love about this strategy is that it is so easy to implement into your classroom routine with the resources you already have.  For example, rather than starting with a procedure for solving quadratic equations, simply ask the students what they notice about the structure of the problem.  How is it the same or different from problems they have done recently?  Ask them to list attributes of the equation.  I have found most often, the noticing of one student triggers the noticings of others and the list becomes progressively more sophisticated.

I have heard from some teachers that they do not use try this strategy out of fear of students making a list of trivial noticings (like, the graph is black and white).  They will include those every time; expect it.  But by acknowledging those seemingly trivial items, that student, who would not have dreamt of entering the conversation before now has received validation of his or her contribution to the discussion.  And when students feel heard and their opinions valued, their contributions will start to become more profound.

I’m very proud of this class.  I’m really looking forward to the creative perspective that their noticing and wondering will bring.

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