Category Archives: #75FACTS

Generalizing Patterns: Tiling Tables

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Last fall after instructional rounds, one of the observers asked me if I would mind having some folks visit my classroom.  Sure.  They were most interested in questioning, interactions with students and use of Formative Assessment Lessons (FAL).

When they emailed to set up a date, we agreed on January 10.  Oh, wait.  This will be the beginning of a new semester with new students.  I won’t really know them.  They won’t really know me.  Great.  Now, I’m scared.  Oh well, let’s look at the positive – this will give me a chance to try out a new lesson.

I printed off 3 lessons to look at the evening before students returned to school.  I liked all three.  Building complex equations seemed perfect, so I began to prepare for it.  We were out for weather our second day back.  As I began looking over my lesson plans, it seemed the Tiling Tables was a better fit for the upcoming unit, so I switched gears.

I had done this lesson a couple of years ago, but never taught it in class.  As I began to revisit the task, I knew I liked it.  I knew it would offer some good discussion on ways to extend the patterns.  But wait.  These students barely know what a parabola is.  Would they have a clue as to how we would write an expression for a quadratic relationship?  Would I have a clue as to how to introduce it, this early in the semester?  No.

So I pondered for a while.  I would simply use the task as a way to say, we have the knowledge and tools to do parts A and B, but part C, well – that’s what we will be learning later in the semester.  It would give us a reason to learn it later, right?  Goodness.  What a canned comment.  By now, we had another snow day, so our visitors would be in our classroom on the 4th day of instruction.  I was stressing just a bit.  What was I thinking?  Starting off a new class with a FAL I had never used before?  We needed time to build some rapport.  Too late.  Let’s go with it.

I gave students the pre-assessment:

table tiles 1table tiles 2

The class was divided pretty much 3 ways – Those who doubled the number of tiles, after all – a side length of 20 is doubled to get 40, right?  The second group had sketched the designed on the the grid paper which had been provided, however, they wrote answers for the 30 cm table instead of the 40 cm.  And finally, several had the correct number of tiles by extending the pattern on the grid paper.  But I ask how efficient this strategy would be for, say 300 cm table?  Hmmm.

We began the lesson the following day by giving 3 samples of work.  Last school year, I figured out, I could save paper by having them use the shop ticket holder sleeves to hold the sample work – allowing them to draw, sketch, etc with dry erase.

These instructions would help their discussions:

samples discuss

The first sample was Leon:

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After some small group time, we shared our thinking with the whole class.  There was one student in particular who had confusion all over their face.  I encouraged them to ask the person sharing for clarification (using our starter stems).  I believe this is important to model and have them do early in the semester, so they become more comfortable with it.  Even with more explanation, they were still not seeing the pattern.  So another student shared.  Still no help.  Finally, a third students explained how they saw the pattern.  The confused student nodded and said, “Okay, I got it.”

Now, years ago, I would have said – great and moved on.  But I’ve learned…ask them to explain it to you.  They may say they’ve got it – just so you will move on, but how do you know they understand?   This student, however, could explain their thinking and were correct – they could even extend it to the next table size.

The next student sample was Gianna:

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So many more of the students picked right up on Gianna’s approach.  The confused student – smiled stating they liked / could see this one better.  For me, it was listening and watching the students discussing – that brought me an a-ha!  This is the example we will use to generate the quadratic expression I was worried about!  The total whole tiles would equal (step x step) + (step – 1)x(step-1)  Yay!

Finally, we had Ava’s sample:

ava

Many of the students could not make the connection with the side lengths on Ava’s sketches in the beginning.  Then they began going back and looking at their own sketches to verify the numbers Ava recorded in the table.  They noticed the same patterns and agreed with them.

After this final discussion – we went back to see if each student had answered the task fully.  We quickly realized though there was some good, correct thinking going on in their work – they had not addressed the questions completely.  The class agreed that Ava’s was the most complete with her table.  And it was interesting to hear their discussions of how they would explain to the other students how they could expand their responses to be better and more thorough.   One student even brought up it was challenging trying to figure out their thinking since there was no written explanations of what they were doing.  (I thought – yes, this is what I feel like sometimes too.)

As we continued discussing having thorough answers – I shared Ava’s data in a graph…  they were quick to see the quarter tiles always remained four and the half tiles being linear, a focus from 8th grade.  But what about the total tiles.  How can we write an expression to model that data?  And I took them back to the slide with Gianna’s work to look for patterns between the table size/step number and the total whole tiles.  We test our thinking with different sizes and it worked.  We tested our expression in Desmos…and what?  It hit all of the data points!

desmos-graph tiling tables

They had some experiences with the visual patterns – and good feedback to me about liking them, but still having to think.  This task reinforced some of those ideas.  IN their reflections – though many may have preferred someone else’s sample work – they “saw” how Gianna’s work led us to a more efficient expression or even Ava’s approach to orgaznizing the data in a table was pretty helpful to see the patterns so we could find describe the expressions.

Total Tiles = 4 quarter tiles+ 4(n-1) half tiles + n^2 + (n-1)^2  whole tiles.

I will definitely be using this lesson in my future.  It brought just enough confusion, but great opportunity for sharing and discussion.  And the observations were great.  Students were not shy.  At the end of the day – I was amazed we had only been together for 3 or 4 days… wow, this is going to be an outstanding semester!

Look Back… formative assessment strategy

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Not sure that this counts as a #MTBoS12days post since I actually returned to school today.  I appreciate having a day back before our students return.  It requires me to be in the building and I get a moment to refocus.

As I drove home this evening, what a beautiful moon!  So glad it was full on a day that we didn’t have students. ha.

Anyway, I pulled my Keeley & Tobey blue book over my shelf and began digging to see what strategies had I never tried – with the idea, I would pick a couple to try over the next couple of weeks.  Well, as I went down the table of contents, I realized I utilize many more of the strategies than I initially thought.

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When I came across the “Look Back” approach – basically students account for what they learned over a given period of time.  They think of specific examples of things they know now and describe how they learned them.

I have used “brain dumps” before to list as many topics / ideas as they could about a given unit in a set amount of time – then they get up and meet with a partner for give-one, get one – but this basically just creates a review list of topics / skills.

However, what makes the look back strategy intriguing to me – having students to tell how they learned the concept.  This idea helps students think about their own learning.  It allows teachers to look at the sequence of their instructional plans and determine why students got the most out of certain aspects / activities.  Interesting.  Look back can also provide the teacher with ideas on how to plan differentiated learning in the future for specific students.

We must remember that this is allowing students to share what stood out to them in the learning cycle – but not necessarily how much they learned.

I look forward to using “Look Back” toward the end of a unit – maybe even use this feedback, compared with the unit assessment to see if students in fact showed proficiency on the concepts they listed.

Revisit of Two Books #MTBoS12Days Post 5

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Since much of my goal this spring is on Formative Assessment, I felt a need to review some of my past readings…

Looking forward to a revisit with these books over the next few weeks.

Looking through my notes/markings in the #75FACTS, I have used about 30 of the strategies outlined in the book.  It has been about five years since I really dug in to this book and the conversation about a volume 2 released earlier in the year, makes me want to revisit this one.  I plan to skim those not marked to see what ways I might be able to implement in my planning this spring.

The #EFA book – seems I made it to Chapter 5 and never quite finished.  I believe this was the year were we on 7 period day and I had 5 sections of Algebra I (3 levels) and AP Stats, the semester got a hold of me and wouldn’t let me go. ha.  After skimming the TOC and my notes, I feel this is a good book for me to be accountable to better quality FA this spring.

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While looking for the Wiliam/Leahy book, I paused to look through the stacks I have at my house…  so much good reading in recent years thanks to the encouragement of my #MTBoS friends!

 

 

#MTBoS My Favorite: Open Questions & Level-Up Quiz

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myfav

Goodness.  I think this is where I fall apart.  I have so many favorite things I’ve used in my classroom, at times I cannot focus and choose one.  I become distracted, thinking I have to use EVERYTHING.  I have to pause, think about the learners in the classroom and what will be best, most effective for them.

Our second week back after Christmas break was very productive.  I chose to combine 2 ideas and focused my energy with them.  One goal I had set was to use open questions.  (Older posts – first attempt, more good questions – about strategy from Small / Lin).  Rather than giving students more inequalities and asking them to graph.  I gave them a point and asked them to create an inequality whose graph would “capture” the point.  Students had to think differently in order to create their response rather than following a procedural step by step or copying a classmate’s work.

The other was an idea someone had tweeted that caught my attention and I wanted to see how it would work in my classroom…level-up quizzes.  Since the target involved graphing inequalities, I gave each student a paper with 4 empty graphs and space in margins to write inequalities and verify.  Here is a sample of the criteria I gave them:

level up quiz

I told students I wanted everyone to be at level 3 by the end of the week – Level 4 was using multiple measures to verify their responses.  If students were at 3 or 4 early in the week, I posed a challenge to them to create two inequalities that would both capture the point.

This task accomplished several things for me.  It was obvious where students got stuck, it allowed me to give feedback or have a conversation about the symbols, which direction to shade, helped point out when/why to use the = if the point was on the boundary line or not, could quickly address issues with graphing key points of the line.  It allowed students to move on without waiting on their peers.

There were a couple of students in each class who continued to struggle-mostly students who had chosen NOT to put any time/effort into practice the prior week or who had been absent, but the rest of students made gains and improvements with this skill.  By the end of the week, majority of students were at or above the level 3.

The big thing with verifying I saw was students using (0,0) to test in their inequality algebraically as opposed to the actual point we picked.  I feel this was due to us graphing inequalities the prior week.  This year, I opted to encourage evidence of their claim by having them test a point to determine direction of shading as opposed to just saying above/below.

With only 1 response for every student each day, I was not overwhelmed, but able to give feedback.  I made notes of most common errors and addressed them as a whole class prior to passing the quiz back.  For many, I simply wrote a number corresponding to the Level-Up criteria.  Students knew the first couple of tries “didn’t count” but were opportunities to learn and level up by the end of the week.

My concerns after reading about Rubrics in Embedding Formative Assessment –  have I made it more of a skill-ckeck list?  By presenting it as an open question, is that enough to allow for student thinking?  Thoughts on how to improve are welcome!

Representing Polynomials FAL & Open Card Sorts

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After an assessment last week, it seemed to me what I was doing wasn’t sticking for my students with polynomials.  So let’s just scrap plan A.  Plan B – I pulled out my Discovering Algebra book, came up with a box-building data collection that lead into the FAL I have linked  below.

Formative Assessment Lesson – Representing Polynomials

Thursday, students were given a 16 x 20 piece of grid paper and asked to cut out square corners and create a box with the largest volume possible.  We combined our data as a class.  Recording the corner size removed, length, width and height.  Students were asked to observe the data and respond I notice…  & I wonder… and that’s where our class began on Tuesday.

We shared out our responses, some adding ideas as we continued the discussion.  Work with our data on TI84s – we saw a connection between our constraints 0, 8, 10 and the graph of the regression equation.  This was not new, during the discussion, a question was brought up about what values would result in a volume of zero.  Students were able answer that with confidence and a reasonable explanation.

The FAL pre-assessment confirmed my students weren’t quite ready for the full blown lesson.  With discussion of rigor and relevance the past few days, I wanted to offer students something engaging but not so over their head, it was a flop.

I backed up and did a bit of prep work yesterday – with the following discussions in class:

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Following with whiteboards / sharing for this slide from FAL:  FALreppoly3

and a simple practice set to ensure they were on track.   FALreppoly4

 

We began class today with a quick check of the 6 practice – with a focus on similarities / differences.  Noting the double root of #5.

Prior to the actual FAL, I decided to use the same equations and graphs they were to match during the FAL, except I would have them do a card sort.  Originally, I had planned to ask them to sort cards into 2 groups.  While pondering how I could make it better, I recalled a colleague sharing ideas about open card sorts from a John Antonetti training she had attended.  So, this is what I did.

I told students I wanted them to sort the 11 equations – any way they wanted – they just needed to be able to share out their reasoning behind their choices.  After a few moments, I called on different groups and we looked at their sorts.  I should have snapped pics / documented their responses.  I was amazed – not that they did it – but how well they did it.   The things they were looking at – were much better than my original idea to sort in to 2 groups.  Students were asking students – why they put one in one group instead of another. Pausing after we had the cards sorted on the board – giving other opportunity to look others’ groups…some were obvious, others were not.   I even had groups who had the exact same sorts, but with completely different reasoning.  Wow.

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At some point we began talking about “What does that tell us about the graph?”  Almost everyone was engaged and comments added to the discussion.  Next we went on to the graphs to sort.  Again, any way they wanted…just be ready to share reasons.

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Most of the sorts were better than ANYTHING I would have suggested.  My eyes were opened – I could see their thinking.  And others did as well – it was obvious in the eye brows raised and head nods.  In both classes, there was one equation that never seemed to “fit in” the other sorts – but students were confident suggesting it belonged to a particular graph (& they were correct).

When I realized the sharing took more time than I had planned – I ran copies of the equations and graphs to send home with students and asked them to match on their own.  My plan is to put them back in their pairs for the actual pairing of the FAL.  They also had blank graphs for any without a match.

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I learned so much listening to my students today…  I am looking forward to the assessment of this standard.

I didn’t feel like I taught anything today…

…but I did feel like my students left with a better understanding…because I chose to step aside and give them the opportunity to share their thinking…

It was a great day.