My Favorite

I teach AP Calculus this year. Today, while exploring the trapezoid rule and Riemann Sums, a student said, “Is this going to turn into limits?” Then I smiled, nodded, pointed at him and got really excited, to which he replied, “Oh, no I said some thing to make her do that excited thing again.”

One of my favorite things as a teacher is when students discover where I’m leading them without me needing to tell them. It was a great way to end the week.

Algebra II Highlight

It’s midterm week here at CHS, and kids are stressed.  It’s hard to avoid having some of that stress rub off on you as a teacher, but I’m using to do lists to accomplish a ton this week. This includes roughly planning AP Calc through the AP test, and preparing to teach stats for the first time in Algebra II Statistics. This is a new class at CHS, and we’re trying to reach struggling learners by having an Algebra II Statistics class that meets every day, worth 2 credits. Roughly a 3/4 of the year will be spent on Algebra II and the remaining quarter on Statistics.

We’ve completed three hefty units in Algebra II so far: Functions (including trig), Quadratics and Polynomials, and Exponents and Logs. For these kids especially, Algebra II is HARD! We use Day Meyer style concept testing, so kids have multiple opportunities to show their knowledge, and they can retest outside of class even after a concept is done being tested in class, but this cohort usually doesn’t retest.

I keep a chart on the wall in my room to show what percent of students have reached the achieving level on each concept in each class.

Percentage charts 1-12

Essentially the goal is to have 80% of students at an 80% or higher. This happened in algebra II for the first time today. You can see the yellow star, with 84% of students proficient in converting between exponential and logarithmic form of equations. Of all the concepts students could reach proficiency on, I’m surprised the first one came when working with logs. I wish I could get students to see the value in reviewing and retesting their other concepts, especially as they prepare for the midterm, but if I only get one yellow star, I’ll take what I can get.

Understanding Quadrilateral Properties

I have been working with my strategic geometry students to help them understand the properties of quadrilaterals. I want them to understand more than just the basics, but the ins and outs of when a property is sometimes true, and what makes that sometimes case happen. I have a student teacher this semester, so I guided him to a great website that allowed students to explore the properties of different quadrilaterals using GeoGebra. He then designed this worksheet to guide students through the exploration.


I liked the activity, and found it valuable for students, but students got very hung up on the sometimes situations. I think in the future, I’ll rephrase the questions to be a bit more “SAT-like”, but also get at the defining characteristics of the shapes. I will ask students to identify which of the properties must be true. This removes some of the hazy gray space about the properties of kite that are true if and only if that kite happens to also be a rhombus. Then I will try to add a second phase to the activity that targets the trickier sometimes situations. Any suggestions on how to structure that?

Transformations through GeoGebra

In order to introduce the students in my Strategic Geometry classes to the power of GeoGebra, I opted to have them explore the transformation tools to reflect, rotate and translate polygons. I began by walking them through the different tools and demonstrating how to create a polygon.

Then I asked students to create a polygon of their own and translate, reflect and rotate it to create a design in GeoGebra.




Lastly I asked students to write coordinate rules that matched their transformations.


I think it was a good introduction activity to GeoGebra, while also reinforcing the concepts students are currently working on.

And so it begins…Geogebra and Proof

Tomorrow marks the first day of the second semester, and I begin teaching a course called strategic geometry, which is a support class for students taking geometry. Geometry classes at my school are heterogeneously grouped, so the students who need a little more time and help have access to that through strategic geometry. I chose to teach the course second semester because that is when geometry gets particularly hard for students, due to the emphasis on proof.

I have been a graduate student for the past 5 years, and I have decided to target my thesis research on Geometry students and their struggles with proof. It is common knowledge that students struggle with proof. Although we introduced the concept of proof to our students last semester, though triangle proofs, we will soon ratchet it up a notch by asking students to use congruent triangles to prove many theorems and properties that hold true for quadrilaterals.

I intend to ask students to use GeoGebra, a free, web-based, dynamic geometry software, to construct and manipulate shapes to prove properties by demonstration, and convince themselves that the properties they are trying to prove must be true, before they actually try and prove them. I will document my efforts through this blog over the coming weeks, and I welcome feedback through comments. Check back soon to see the status of the project.

Getting Back to Blogging

Ok. My Bad. I haven’t posted in 10 months. It’s not that I haven’t thought about blogging, but I haven’t put the pen to paper, so to speak. So here’s a few highlights of the new school year.

I’m teaching a new course this year about applications of math in Astronomy. The class was proposed, and accepted last year, and had 31 students sign up to take it, but staffing issues made us think we wouldn’t be able to offer it until mid summer, so the kids who got thrown in then were only somewhat interested. The result is that the class only has 9 students now. This is a nice number for me to work with as I develop all the materials and test them out. The students have given me feedback about what works for them, and what doesn’t, and the course is pretty fun.

Our AT system has moved from 4 days a week to 5, and I think it’s great. I’ve gotten better at managing student needs during that time and pre-booking students so the people in my room are really the people who need my support.

We’re piloting a new grading system in Algebra II this year. I should really write a whole post about it, but here’s the gist. Students receive two grades, one academic grade, and one learning habits grade. Anything that is checked for completion but not correctness factors into the learning habits grade. This includes most homework assignments, daily test prep questions, and students self evaluation on a school wide “Learn” rubric. The academic score includes concept tests, unit tests, problem solving activities, and other homework or classwork assignments that I decided to grade. If I am going to collect and grade a HW assignment, I tell students as I pass it out.

The idea behind the pilot is that fluff is removed from grades, and the academic score shows how much the student actually understands the material. The separate learning habit score shows how hard the student works. I really like the concept. My class average is lower than I’m used to, and students are still adjusting to the concept. I’ll post more about it later. And I swear it won’t be 10 months this time.

Pros and Cons of AT

This year my school implemented a new academic support time called AT. AT stands for many things…Academic Time, Advising Time, Acceleration Time…but for many students it is a glorified study hall.  We implemented it because we felt in the past teachers had so many meetings after school that if a student missed class, or needed extra help, that it was nearly impossible to schedule a common time for students to meet with teachers.  We had a program in place called teacher advisory, which was 2 days a week and provided mentoring and a common teacher for students to get to know for 4 years. However, this time was rarely used well, and often resulted in kids chilling for 15 minutes.

Here is how AT works now:

1. We have a computer program designed for scheduling.

2. On Mondays, students go to homebase, which is sorted by grade, with academic advisers. Here students meet individually with the teacher to decide what they need to work on that week, and schedule where they intend to go on Tuesday, Wednesday and Thursday.  Teachers send an email to students with their schedule for the week. AT is 30 minutes long.

3. Tuesday, Wednesday and Thursday students go to the location for which they are scheduled.  Each teacher has approximately 15 students who are working on various things. In the math classroom, this frequently means students are retesting concepts, working on math homework, or getting caught up on material they need help with. The math AT is always busy. It is a lot to manage from a teacher’s perspective because the students are all doing different things.

4. Friday there is no AT. Classes are all 9 minutes longer.

Teachers have the opportunity to pre-book students any time before the Monday AT, so if the teacher knows a student is going to need extra support, or needs to make up a test, the teacher can prebook the students.

My Thoughts: I like the idea of AT, but I find the execution to be challenging. I think this is complicated by the fact that I don’t have my own classroom. There are two teachers trying to hold AT in one room. We’ve reduced our numbers so we are both maxed out at 12, instead of the usual 15, but it is still too tight in the space. Because my number is lower every day, that means there are 9 spaces that are unavailable for me, and students find it very hard to book me.  This means I end up doing a lot of prebooking, so some weeks I have no spaces left before the Monday booking time even begins.

I’d like to use AT for more acceleration, or to meet with groups of students, but if I book students ahead of time, it leaves even fewer spaces open for kids who need help. On the plus side, I definitely think AT has reduced the number of students who come in for help after school, and it has definitely reduced the end of the quarter retesting rush. It is helping students to retest concepts as we go, which I think is having a positive impact on learning.

As a final note, the 30 minutes that we lose from the school day that go into AT are missed in my classes. It’s only 9 minutes a day 4 days in 2 weeks. So that’s 36 minutes every two weeks, with approximately 32 school weeks, that turns into about 8.6 hours over the course of the year, which is a lot.

I’m open to thoughts and questions as my feelings are mixed…

Factoring with a leading coefficient

Before Thanksgiving I posted for the final MTBoS mission an idea that I shared with a colleague about how to factor trinomials with a leading coefficient other than one. My colleague and I put together a jigsaw where we presented four different methods  to factor trinomials when a isn’t one.

The first method was the traditional guess and think sort of method.  Many of my students struggle with this method when the numbers they are working with aren’t prime because of all the possibilities.

Picture 2

The second method I’ve showed students before. It requires them to multiply the leading coefficient and the constant and then find two numbers that multiply to that number and add to the middle. Students have had mixed levels of success with this method, but they often find the factors, but then aren’t sure what to put in the parentheses.

Picture 3

Method three is the one that was new to me… I got the idea from this blog post.  It is similar to the method 2 above, but there is a twist.

Picture 4

Students really liked this method because there was no confusion about what to put in the parentheses after they found the factors. They can get everything down on paper before having to divide, and since that is down on the paper too, it makes this a bit less confusing.

The final method we shared was the factor by grouping method. Since we haven’t taught grouping yet, students weren’t really enthused by this method, but we threw it out there. Picture 5

In the past, my Algebra II students have had a very hard time factoring quadratics when the leading coefficient is not one. After seeing all four methods, students heavily gravitated towards method 3. Now they are factoring polynomials, taking out a GCF first, and then factoring the remaining quadratic with ease.  If your students are struggling with trinomial factoring, I highly recommend method 3 above.