Category Archives: Polynomials

Combining Polynomials

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This is like something that others have done often.  Maybe I have even used something similar at times in the past, but today a group of students had a big a-ha moment.

We have reviewed the basic polynomial operations and like robots they can do it well for the most part.

The last 10 minutes of class, I popped the graphs
f(x)=x+2 and g(x)=2x-1 on the board.
I pointed out specific points, asking for their sum and recorded on the board.

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2+-1
1+3
3+4
8+8…
This graph only shows part of our discussion.

Next, I graphed a purple function f(x)+g(x).

“Holy shoot” was exactly what a student said. Yep, they edited the version just for me.  But there were several ah’s and oh’s. 

Thanks to Desmos…they could see the math they’d been doing on paper.

We combined f(x)+g(x) on paper to get 3x+1.  I graphed it and they were amazed it hit our points.

I know this isn’t so amazing for many, but the look on my students faces today…they got it.

I am looking forward to more…

Number Talks 12 x 13

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I was cleaning up some files this evening and ran across these snapshots from a Number Talk in class a while back. 

Are all of them the most efficient?  For me, no.  However, if it made sense to that student at that particular moment, then it was most efficient for them.  I appreciate the various ways they consider “building” the product.

  I wanted to see their thinking on 2 digit multiplication to link back our unit on polynomial operations.

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Yes, they all had a calculator available, but my question was, how do you know?  As I learned from Steve Leinwand, “Convince me.”

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.

 

 

Made4Math #5 Polynomial Station Activities

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Its been one of those busy weeks, so I’ve not actually created anything “new” but decided to share something I used last spring.  The idea developed after @lmhenry9 tweeted a need for ideas to use with polynomial stations.  A month or so later – I decided to use a similar idea.

I purchased a bag of 8 wooden blocks from Hobby Lobby ~ $3.  Used my sharpie to add expressions to the blocks.  Created instruction cards for each station.  Based on a pre-assessment, I grouped kids by similar struggles – those who were a step ahead could “play” more game-like activites – while I could spend time with groups who needed some extra support.  We spent a couple of days in class rotating activities.  I think most pictures are self explanatory.

1.  Collecting Like Terms

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2.  Adding / Subtracting Polynomials* – let students know which “color” block is the first polynomial.  For a little discussion, ask if it really matters?  If so, when/why?

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3.  Multiply Monomial x Polynomial

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4.  Binomial x Binomial

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5.  Factor Match – I didn’t have orginal copies with me to scan – but will get them posted here asap.

 

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I also had a station utilzing a Tarsia-style puzzle with variety of polynomial multiplication expressions.

Tic Tac Times – Students pick 2 factos listed at bottom of the page and multiply.  Place game piece on the product.  First player to get 3 or 4 (you pick the rules) in a row, wins!  For more challenge, each player must use one of the factors just used by their opponent.

* A sidebar – while creating my blocks – my daughter asked what I was doing.  I replied – making a game for my students to play.  She asked – can I play it to?  My first instinct was to tell her No – but I bit my tongue.  And then I remembered a problem she had left on my board one day afterschool and my students had asked me what it was… (After school, she and a couple of other “teachers’ kids” hang out in my room and play school.) I realized it was very similar to how she had been adding and subtracting 3 digit numbers in class this year.  So I explained how the x^2 was like her 100’s, x was like the 10’s and the # was just one’s.  She rolled the blocks and did a few problems…I’m thinking – if a 2nd grader can do it – so can 9th graders, right?

So I went in the next day – and shared “her lesson” with the class.    I gave an example like the one above – referring back to the problem they had seen on my board.  They understood the process of decomposing the numbers to add/subtract.  I connected the example to (3x^2+4x+2)+(2x^2+3x+5) to get (5x^2+7x+7) – good to go.  Then I asked, WHAT IF we let x = 10…  you know – not one student missed these problems again…