Thursday, March 28, 2013

Day 122: Unit 8 Assessment

I remember a day when I didn't dread giving assessments, because I had no reason to fear an altercation arising that could possible lead to a suspension. One of my classes has so many students that have completely given up (not just on geometry, but on school in general) that the thought of corralling them during a test terrifies me. 

It's constant giggling and whispering, but the administration simply says "give them a zero" for violating test procedures. Regardless of the fact that zeroes don't exist in SBG and that's a ridiculous way to treat grades, these particular students could care less about a zero. A few of these kids don't even turn the tests in! (I don't worry too much about test security because I don't think I've ever given the same test twice in the 5 years I've been teaching). 

So my only alternative is referral to the SRC (Student Responsibility Center) which involves a call home and a reinstatement meeting with me. If students are quiet and have given up, I have no recourse to ensure that I am not being measured by their scores. 

Oh well, tomorrow is spring break!

Wednesday, March 27, 2013

Day 121: Review

Today was basically a mop-up day as we prepare for the Unit 8 Assessment tomorrow and spring break on Friday. Students continued to work on the worksheet from yesterday, but I did spend some time ensuring that they at least saw proper setups for a few of the problems. I also handed out a brief summary of the unit so students could have a reference to help them as they create their note sheets. 

Tuesday, March 26, 2013

Day 120: Examples & Practice

I have adjusted my approach somewhat since September in that now I will do more examples for the class before we begin a worksheet. My rationale had always been that if I ensure that the class is properly suited, that I shouldn't need to show them how to do a problem. The constant hand-holding in high school is something that really bugs me. Additionally, I've learned through trial & error that the students will never be satisfied. Don't do any examples and they'll be upset. Do some examples, and they'll simply postpone getting upset until they encounter a problem that doesn't look like the ones you showed them how to do. Show them those and they'll just get upset on the quiz that looks different still. 

Even more frustrating through all this is how many students use this experience as fuel for their negative opinions of me (that's not a guess, I've spoken with some students and given evaluations across my classes). It's about more than what they *want* school to look like, it's sad to see that they think this is what school *should* look like. "Show me how to do X, I'll do X in front of you, tomorrow I'll do X on a quiz, next week I'll do X again on a test, and in a few months I'll do X on the final." 

But what about Y & Z?

Anyway, after another example of AoRP, this time given the apothem and solving for the side length, the students worked on U8 WS2. 

Monday, March 25, 2013

Day 119: Area of a Regular Polygon

Call me old fashioned, but I think the derivation for the area of a regular polygon formula is one of the most amazing aspects of high school level Euclidean geometry. It encompasses so much knowledge - it marks the the point when everything we've been learning finally comes together.

Which is exactly the reason why the students HATE this unit. 

Worst of all, most of the content that students need to get through this unit (as well as most of 2nd semester) came from 1st semester, and too many students have the "out of sight, out of mind" philosophy when it comes to the semester breakdown. It all gets back to the fixed vs. growth mindset ideas, but SBG throws a nice wrinkle into the mix. I finally settled on just saying that students will be assumed to be proficient on ALL standards covered in 1st semester. So if you need to know how to use Pythagorean Theorem to solve for the missing side of a kite, then it's on you. I will still give a proficient score of 2 (the passing line) to students who can demonstrate the proper setup (showing that the diagonals of a kite for a right angle) even if they can't solve it. 

In any event, I work through the discovery for the area of a regular polygon (how about AoRP from now on?) on my tablet as a derivation that the students are expected to follow. By the end, they're exasperated and are hoping that there's a shortcut that I just didn't show them at first (they know that is something I would totally do). Unfortunately, the only "shortcut" (after the central angle, right triangle trig, interior triangles and multiplication) is to catch the perimeter popping out. The students are generally upset that they'll be expected to do all that work for just ONE problem. Babies. 

One pleasant surprise was how many students still have their laminated trig tables that I gave them back in 1st semester AND they have a basic memory of how to use them. Awesome!

Friday, March 22, 2013

Day 118: Catch up

I lost 2nd hour due to testing yesterday, so 5th hour went to the computer lab for independent review/practice, and I thought I'd do something 'fun' for 6th hour to intro the next unit to them. Big mistake. 

Since Unit 8 will be about polygons in general, I figured we'd give constructions with a compass another shot. I had step-by-step flash animations from mathopenref on the screen and I switched between that and a document camera showing my own construction as I went and yet it devolved into total chaos. Students immediately become distracted and talk with their neighbors which forces them to miss a key step, so they'll interrupt and ask to go back so they can see the last step, which doesn't make any sense without the explanation. And of course if I don't kowtow to their needs, they'll simply talk and act out more causing more of the class to go off track. Fun times. 

In a perfect world, I would teach the ENTIRE class with a compass. It's such an incredible tool and it's been around for thousands of years. And I'm forced to resort to the SAFE-T compass with no points because the students can't be trusted with sharp objects. I'm also down about 10 compasses since September between students breaking them or stealing them. 

Sorry, I need to complain less here, but this lesson really got under my skin. 

Thursday, March 21, 2013

Wednesday, March 20, 2013

Day 116: Exterior Angles

I did a demo of the sum of the exterior angles in a polygon, recapped the exploration from yesterday, and set students to work on the first worksheet of Unit 8. 

If you've never seen the demo, it's both simple and powerful. Draw a convex polygon on a piece of paper. Create one exterior angle at each vertex around the perimeter (be sure to work in the same direction around the exterior). Number and cut out each angle, making sure to note where the angle was on the cutout. The pieces easily reassemble into a central circle, showing the sum total to be 360. 

This could have been done as a student activity and with more shapes to emphasize the independence of the relationship, but for time considerations I simply showed other examples in Geogebra and set the students to work. 

I should note that we also pointed out how to determine the measure of a single interior or exterior angle IF the shape is regular. For some reason, this equation gave students fits for days. 

Tuesday, March 19, 2013

Day 115: Unit 8 - Polygons

To kick off the new unit, I randomly assigned students to groups and had them draw a shape that had either 5, 6, 7, or 8 sides. Once drawn, they labeled the vertices, measured the interior angles and added them up to find the sum. 


  • Using a protractor is a still HUGE hurdle. It really shouldn't be, but results were off by over 100 degrees in some cases. 
  • Allowing any shape seems like it makes the task easier, but students had no idea how to measure a reflex angle, which most likely contributed to our error problem. 
  • In the future, I might direct students to round measurements to the nearest 5 degrees in the hopes that it will help.
The pentagon had decent results all day, but as the number of sides increased, the errors are compounded, and the results get further and further off. I used Geogebra to try and fill in the gaps and show the ideal case. Some students either noticed the pattern or reasoned through what the relationship between # of angles and total has to be which was surprising. 

In the end they all seemed to think it was easy, which is usually a sign that something went terribly wrong. 

Monday, March 18, 2013

Day 114: Unit 7 Assessment

I might have set a new record for "Students who turn in a blank test with their name on it" today. In light of that, it was actually fairly amazing to see how "well behaved" they were. I use quotes because they weren't actually well behaved, but considering that they did literally no work for 55 minutes, I'm surprised they weren't worse. 

Overall the results of the test were very poor. It was also very difficult to grade as I tried to implement a 'justification' standard into SBG. 

If a student graphs the 4 vertices of a quadrilateral and says it's a parallelogram because the opposite angles are congruent, how do your grade that? They are demonstrating some understanding of properties of parallelograms, but there is no way they can actually tell that their assertion is valid. What about students who say the sides are parallel, which can be shown on the graph, but they don't actually determine slopes of opposite sides to demonstrate the validity of their claim? 

This test did take a while to grade, but as has been the case with SBG, I'm very happy with the detailed information about what students know vs. what they don't that comes out of it. 

Friday, March 15, 2013

Day 113: Review

We spent the hour in the computer lab giving students a chance to prepare for the U7 Assessment on Monday. 

Students had the option of working with the Carnegie Learning Tutor (each student was moved to the current unit), watching/working with Khan Academy, or making a note sheet to be used on the test. 

Thursday, March 14, 2013

Day 112: Area & Perimeter

Today was partially lost to more (optional for our school) standardized testing, so the Unit 7 Assessment had to be moved to Monday. I generally hate Monday assessments, but in order to get Unit 8 fit in before spring break, sacrifices must be made. 

The only topic left in Unit 7 is area & perimeter of quadrilaterals  so I used Geogebra to show how we can find shortcuts to solve for the area of various shapes. I don't bother with formulas for perimeter - as long as students understand what perimeter is (and how to use the Pythagorean Theorem to solve for diagonal distances), I assume they'll be able to figure it out. 

In general, our theme for finding area is based around the idea of making a rectangle either from the quadrilateral directly (parallelogram), or making the rectangle around the shape and noticing that it's area is double the shape's area. 

In my head, understanding how to determine area is more useful than memorizing a bunch of area formulas, but students are unfamiliar with a general approach and generally feel more comfortable with the formulas. Except, most students cannot reliably use a formula as a result of very weak algebra skills. Or, they lack the ability to differentiate horizontal/vertical distances on a graph from diagonal lengths. Or they have no idea how to identify a base and a height.

Wednesday, March 13, 2013

Day 111: Formative Assessment

Students spent a large portion of class on the second quiz in Unit 7. This quiz was slightly longer than normal, so more time was allotted. It doesn't appear to have increased results any. 

In general, the correlation between student engagement and content is based primarily on ease of understanding the material. What I mean is that students in large part actually worked diligently on the first worksheet of the unit on parallelograms, but refused to work on the second regarding trapezoids and kites. If anything, we actually spent more time "learning" about trapezoids and kites when the computer lab time is factored in, so all I think of is that these shapes are less familiar to students, so their fixed mindset becomes an obstacle. If something seems easy (most likely because it's familiar), then students will given it a shot. If something is new and appears difficult, students take the "if I don't try, then I didn't really fail" approach. 

I'm guessing if I looked back across ALL of my standards since September, I'd find proficiency across the 'easier' ones (measuring segments & angles) and a marked lack of progress on the 'hard' ones (right triangle trig for example). 

Dealing with fixed vs. growth mindset is one of the biggest frustrations I have. There is a lot of pressure to make the content relevant and engaging, but I cannot do that if students refuse to move beyond the first (and easiest) step. To me, right triangle trig is amazing for its (apparent) simplicity and its almost endless applicability. It's just a bunch of proportions after all. But as soon as you give most students a glance at the endgame, they quickly state "oh well I'll never do that" as justification for why they don't need to learn it. 

This is a pervasive problem outside of just geometry, but it becomes incredibly difficult to teach 2nd semester geometry to students who couldn't be bothered to learn the content from 1st semester. How do you teach surface area and volume of polyhedra to students who never learned how to: name polygons, determine the area of a regular polygon, use trig to solve for missing sides of a right triangle, determine a central angle from dividing 360 into equal parts?

Tuesday, March 12, 2013

Day 110: Discussion

It's getting to the point that a class period set aside for going over a worksheet ends up just being more time spent working on the worksheet. So, instead of simple cycle of the EDGE method (Explain Demo Guide Enable) taking 3 days at most, a simple lesson takes about 5 days. 

Here's reality right now:

  • Day 1: Investigation. Some student centered activity in the vein of "organized play" in which students can discover the upcoming relationships without knowing what's coming.
  • Day 2: Formalization. Class works together to take notes and structure the discoveries just encountered. Examples given. 
  • Day 3: Practice. Students work on practice problems to become better acquainted with the material.
  • Day 4: Review / more practice.
  • Day 5: Quiz. 
Here's what I would prefer:

  • Day 1: Investigation. Some student centered activity in the vein of "organized play" in which students can discover the upcoming relationships without knowing what's coming.
  • Day 2: Formalization. Class works together to take notes and structure the discoveries just encountered. Examples given. 
  • HW assigned on Day 2, due on Day 3
  • Day 3: Quiz
The issue is that students won't do HW because they're either incapable or simply refuse to work independently. If they come across a problem with a solution that is not immediately apparent, they will do nothing until I guide them to an answer via a step by step procedure. 

If they won't work at home, I have to give them time in class to work. But there are 35 of them in class, so if I can't get to each and every one of them in class, they still won't work. Which means I need to give more time on the 2nd day to make sure everyone gets something done. This means there is often little or no time to actually go over the practice, which in turn creates a mentality of "the teacher never showed me how to do every problem and never gave me the answer to every problem, therefore I cannot be held accountable for demonstrating understanding on the quiz."

I honestly have a non-insignificant number of students who posses this mentality and will simply refuse to take quizzes in protest. 

Monday, March 11, 2013

Day 109: Practice and justification

Wow, a complete & uninterrupted week of school lays before us. That shouldn't feel weird, but the 3rd Marking Period is always like this. 

Students were given the hour to work on U7 WS2 dealing with trapezoids & kites. We did address the idea of isosceles trapezoids at the start of class, and outlined everything we know about quadrilaterals to this point. 

As part of the SBG implementation, I created a general standard for reasoning because I was having a hard time separating a demonstration of knowledge about an idea (like opposite angles congruent in parallelograms) from being able to explain WHY that is true. So at least for this unit, S7.6: I can appropriately justify my reasoning will be a part of almost every question. 

Friday, March 8, 2013

Day 108: Formalization of Conclusions

We reviewed the conclusions sought after from the computer lab investigation and developed a set of theorems that seem to hold true for trapezoids and kites. To avoid an information overload situation, we left any mention of isosceles trapezoids until next week. 

So far, we know:

  • Trapezoids in general have very little to offer us (One pair of parallel sides? Consecutive supplementary angles? That's it?)
  • Kites are neat, but incredibly confusion to describe in words (Two pairs of sides are congruent, but not that pair. No, not that pair either. Just take a guess). 

Students were then given the remainder of the class period to begin working on U7 WS2.

Thursday, March 7, 2013

Day 107: Killing time

It's ACT/MME test week, so the schedule is a mess. I used yesterday/today to give students more time to finish the trapezoid & kite investigation in Geometer's Sketchpad. 

Sketchpad is an amazing piece of software, but it's not terribly intuitive or user-friendly, especially dealing with high school students that aren't especially tech-savvy. As hard as I try to write directions that are clear and explicit, many students become frustrated and give up before reaching final conclusions. 

The focus of the activity was to help encourage the need to empirical evidence to justify conclusions. Rather than use a ruler and protractor to make measurements, we can use the computer to do the hard work for us, but we still need something to back up a statement like "the diagonals of a kite are perpendicular." 

Moving forward, I've begun to use Geogebra in place of GSP. Very similar functionality, but 1) Geogebra is free, and therefore can easily be installed on the school's computers, 2) Geogebra has a much more intuitive interface (IMHO). I've also begin thinking about interactive java applets I can make and upload to my class website for student exploration while at home. I know you can do that with GSP, but I lack the most recent version, so I was having difficulties getting it done. 


I going to start the arduous process of posting everything I've made for this curriculum online. They'll be uploaded to Google Docs, so feel free to download and do what you wish with the materials. The only things I will NOT upload are assessments, but if you're a teacher and would like to see them, send me an email (bfuller181 [at] gmail [dot] com and I'd be happy to oblige. Oh, and same goes for any requests for the original Word 2010 documents - I know that sometimes the conversion to Google Docs causes problems. 

Wednesday, March 6, 2013

Day 106: Continuing Investigations

The ACT/MME is wreaking havoc with our class schedule this week, so I planned the trapezoid & kite investigation to take two days. The fact that students don't have access to GSP at home also factored in to my decision. 

I actually was out sick today, so a sub covered my 2nd hour class. I hope to be back tomorrow and oversee the 5th and 6th hour classes as they finish up. 

Friday we'll reconvene as a class and formalize the results of the investigation, and start practicing. Which reminds me, I have a worksheet to write. 

Monday, March 4, 2013

Day 105: Investigation of Trapezoids & Kites

First stop was a quiz on last week's worksheet, and then off to the computer lab so the students could explore properties of trapezoids and kites. 

My hope was that if students now have a firm understanding of *how* we test ideas about shapes (use slope for parallel/perpendicular, distance formula for length, use midpoint for bisecting, etc), then then can use computer software to readily create shapes that would be problematic to explore with a pencil on graph paper. 

I chose to use Geometer's Sketchpad because that's what I've been using since college geometry, but after spending the last few days exploring Geogebra, I'll probably switch over to that in the future. 

I wrote up VERY explicit instructions, detailing what students should do, step by step, to achieve a proper shape. "Use the segment tool to draw a line segment, select a new point and create a parallel line by clicking here," etc. Of course, my best efforts were thwarted by students who absolutely refuse to read directions and immediately claim they did, but couldn't understand them. Students would call me over to ask why their quadrilateral didn't look right, and I could quickly see that they skipped step #4. "No I didn't!" they would emphatically deny. "Well actually yes, you never placed the two new points as instructed." I would reply. Then they'd get sheepish and laugh and admit the must have missed that part. 

That, in a nutshell, sums up most of my difficulties teaching 10th graders (the probably isn't generally as bad with the 11th and 12th graders in my physics and astronomy classes). A complete and total refusal to even bother paying attention to given instructions, no attempt and determining what went wrong on their own, and complete denial of any wrong doing when confronted. I understand this is simply a matter of maturity, but this strikes me as behavior one would encounter with 6th graders who might be 11 or 12 years old. I have some experience working with the same age level I do now in other school districts and I never saw maturity levels like I do now, which leads me to believe the issue is somehow correlated to community and socio-economic status. It provides a launching point for a very interesting discussion of fixed vs. growth mindsets, but that's a discussion for another venue. 

Friday, March 1, 2013

Day 104: Discussion

Again tried to make whiteboarding happen. I'm torn between wanting to give students a chance to confirm answers to the worksheets (I generally refuse to do that directly myself) and wanting to make the best use of class time. 

In 2nd hour, most of the class worked earnestly to get their assigned problems on whiteboards, but they chose for presentation style (over whiteboard parade style) and then couldn't keep quiet long enough to get through more than 4 problems (out of 20 total). 

I didn't give 5th hour the option, because I know they can't handle presentation style, so I had them stick with whiteboard parade. While the task eventually got accomplished, you can see the bulk of the work being done by the few (even though the worksheets are now graded). 

6th hour also got whiteboard parade, and for similar reasons.

Results will show on Monday's quiz.