Monthly Archives: November 2012

Complete 180°… well, duh.


Transformations …

My students do well with reflecting across x-axis or y-axis.  Though they struggle literally flipping a transparency or patty paper across the line y=x.  During whole class discussions today, someone stated “just read what it says in the equation.”  OK. “Y becomes the x. And read it the other direction, starting with x.  X becomes the y.”  True.  Simple.

Now, its rotation’s turn.

While using whiteboards to assess our FAL from Friday, it was obvious a handful were having trouble remembering “the rules” they had developed in their pair work.   They stated as long as they had the transparency, they could physically perform the transformation.   But wondered what they could do without those tools.  One student suggested, ‘I just rotate my paper/graph as it says.’ Another chimed in, ‘that’s what I do too.’  Well, duh. An example – the point (1, 4) rotated 90 degrees clockwise about the origin, would simply be one quarter turn to the right; resulting in (4, -1).  Rotate another 90 degrees for a 180 degree total results in (-1, -4).

So, as a class, we practice their strategy.  Genius.  Why can’t I think of the simple things?  So glad I finally figured out how to listen to my students – their way is so much better than mine.

Another strategy was to draw a line segment from the point through the origin to find the image rotated 180 degrees.  To find the 90 or 270, simply take your scrap paper and use a corner edge from the origin and draw a line perpendicular to the 180 line segment.  Students went on to suggest they used “slope” to place the image.

This is a sample of what they suggested.  Rotations around the origin.*

I like this last strategy, because it will also work when rotating around a given point.  Rotation around a given point.*  Another suggested that drawing a segment from the point through the given point of rotation…is kind of like the point of rotation is actually the midpoint of the image and pre-image.

I’m happy with the fact that they’re looking for a way that makes sense of the transformations.  They’re considering different strategies and our discussions often lead to whether it will work or not.

*created with screen recordings on Promethean ActivBoard – not sure if they will play otherwise.

Formative Assessment Lessons


Its been 3 weeks since I’ve blogged.  Not because I didn’t want to – but life has just been head over heels busy.  The week following my last entry – I presented at KCTM – Literacy in Math Class.  I’ll blog about it soon.

In Kentucky, I was introduced to Formative Assessment Lessons about a year and a half ago.  I remember the first one I tried was not so successful.  But the more I learned, the more I realized, there was some good things embedded within these lessons.   At our last KLN meeting, we were asked to discuss our experiences with the FALs.  I hadn’t realized I had actually used as many as I have until we started running through the list.

My students can find some level of success as well as being challenged on the other end.  I observe student success with these lessons.  They are formatted in such a way, I am able to listen to student discussions, considering their ideas and able to pose questions that will foster more discussions.

Part of my session on literacy was to give students opportunities to talk, share and ask questions about their thinking.  Within the FALs, students are given either a problem solving task OR a conceptual development task.

In all lessons I’ve used, students respond to a given task as a pre-assessment, after completing the lesson, class discussions, they are given the opportunity to revisit the same or a similar task.

In the problem solving tasks, students are put into groups homogenously (based on similar approaches to solving a problem or even similar misconceptions/mistakes – not necessarily ability).  This allows students moving in the right direction to continue; while my time can be targeted to smaller groups of students, using questioning to guide their thinking, discussions.  Each group is given sample responses, and asked to think about the student’s reasoning – why they approached the problem as they did.  This gives the group an overview to see multiple ways to consider and opportuinties to critique the reasoning of others.

In the concept development tasks, students are usually given a task/questions and card sorts/matching activities.  Instructions will almost always require students to verbalize their reasoning, then their partners must either explain the reasoning in their own words OR why they disagree with their partner.  I feel verbalizing their thinking is a key component of literacy – helping them work through their own understanding but also listening to ideas of others, in a small group setting.  Many lessons offer extension suggestions if needed.

To complete the lesson, there is often a plenary discussion to wrap up, solidify the concepts.  Its very important to really listen to students – in some lessons, you are encouraged to scribe student comments/ideas with their names for ownership in the discussion.  White boards are a common component – seeking student responses – sharing different responses – asking questions – if others agree, disagree or have something to add to someone’s comments.

I am sharing about FALs because today, I left my geometry classes feeling good – that students were given an opportunity to think, discuss, share and learn – clear up some misconceptions.  I am looking forward to our whole-class discussion on Monday and the follow-up assessment!  Though there are still some mistakes – I think the sharing out will add/deepen to students’ understanding.

Representing and Combining Transformations was the lesson students worked on today.  I paired students based on similar responses on their pre-assessment.  I really enjoyed “sitting back” and listening to their discussions.  The particular task, they were given 6 different graphs with an L-shape and 8 different transformation cards.  They were asked to connect 2 shape graphs with a card describing the relationship between the two.

I’ll be honest in questioning the need for the transparency graphs – but after observing students, these were a key learning tool for most of them.  When they asked for help, I encouraged them to use their graphs to “see” what happens, then use what they noticed and apply it to their shapes.  I also found allowing students to place a push-pin at the center of a rotation was very beneficial to their understanding.  To observe how using different centers of rotation will affect the movement of the shape.

Recently, a colleague decided to try a FAL – Forming Quadratics with an Algebra II class.  In our last PLC, my colleague shared pros/cons observed during the lesson and that all but only a couple of students had improved / were very successful on the post-assessment.

FALs are idealy used about 2/3 the way through a corresponding unit of study.  This allows the teacher to view misconceptions and clear those up before finishing the unit.  Most lessons consist of a 10-15 minute pre-assessment, 1 hour for lesson/discussion (this can vary depending on students), 10-15 minute follow-up assessment.

Each lesson is aligned to 8 Mathematical Practices and outlines which CCS is addressed.

There is some prep-work involved, so don’t print a FAL and expect to use it immediately.  I use card-stock for the card sorts (each type of card gets its own color) – if you laminate them, maybe they will last longer.  Also, when it calls for a poster of student work, I don’t want them to glue pieces on a poster – then I’ll have to make an entirely new set next time.  I want to reuse them.

  Today, I had students add a post-it note with their initials and I snapped a pic of their cards.  They can create an answer key on paper as well.

I would love to hear about others’ experiences with FALs – ways they’re using them in their classrooms!