# 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:

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:

The first sample was Leon:

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:

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!

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!

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!

# Complicated to Doable

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This was the Visual Pattern for the bell ringer and some of their thoughts…

So we ended up with some form of these 3 equations in both classes today…

I got tired of writing step # and total blocks so I asked if we could shorten it a bit…so we defined n and x as those quantities!

And majority of students raised their hand saying those equations looked hard or confusing…

Until…we picked a step number.  Woe.  That’s doable.  We substituted 4 as our step # and all they saw was addition, subtraction, multiplication, parenthesis…  we just went from this complicated thing to something I even do without a calculator!

And the arithmetic showed each equation actually resulted in the same values.

# Barfing Monsters Day 2 & Day 3 #MTBoSBlaugust Post 18

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Here’s the version of their documents/ideas we used in class this past week Day 2& Day3.

After discussing a few of our friends issues from Day 1 – Students were asked to work alone on Day 2 – where Blurpo was burping up graphs.  After a while – I asked students to turn and discuss their responses with a neighbor – discussing any differences they may have had.  We then had a whole class debriefing utilizing desmos.  It was a great way to introduce them to desmos.

The sliders really helped when students commented the parabolas – one was wider than the other and others argued they were the same graph, only transitioned down 2 – which made it appear wider at a certain point because the original was “inside” the translated one.

I also had some pipe cleaners to demonstrate the width actually held the same.

What I like most about this day was the 3rd graph, they had to provide the burped up version – and the last graph, where they were given the burped version and had to describe the “eaten” graph.

I used this to share how their brains were processing the patterns to provide structure to apply the pattern to a new questions.  And how being given a “backwards” problem required their brain to think in the other direction as well.  When many of them continued the original pattern and was wrong, they realized their mistake and was able to correct it.  Our brains just grew!  Twice!  We talked about how it was not a misconception (they didn’t understand it) but a mistake (not paying attention) that they were able to correct on their own- not needing me to tell them “how” to do it.

Day 3 is a perfect intro to visual patterns.  Students were given the choice to work on their own, in pairs or a small group.  Linking blocks were available for those who wanted to “build” Spikey’s patterns.  I enjoyed observing their different approaches to building/drawing the patterns.  It was fun listening to their discussions of how to continue the patterns or figuring out how to find the number of blocks required to build the nth pattern without actually building/drawing it.  Again, the power of SMP at work.

These are some of the strategies I saw/heard along with some of the equations a few developed.

It was a great way to allow them to see how others were viewing the patterns.  And when asked which method was better? Silence.  Finally, a student says, well I liked mine best until I saw ___’s and it makes more sense to me now.  But we all agreed there were multiple approaches and the one we should choose is the one that our brain sees.

We only used different equations to show they would result in the same number of blocks needed for a given step.  I didn’t do a very good job of connecting their equations to the methods used to build the patterns – something I definitely want to improve in the future.

Again – I want to shout out to @cheesemonkeysf and @samjshah (was @mathdiva77 in on this as well?) – thanks guys!