Tag Archives: algebra i

Recursive Models #8minreflection

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When planning my first unit with sequences, I just assumed Recursive models would wait until Algebra 2.  Last week, my students took their first benchmark for the year and what do you know, but a question about  a recursive model.

F.IF.3  Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

I’m struggling to see the difference in these 2 standards.  As I look at examples, I feel they are much the same or at least co-exist within a problem.

As I watched students during their benchmark, I was aware of the recursive formula question.  When I looked all of my classes results, only 38% got it correct.  As I looked at their response distribution, most students picked an example that at least corresponded to the give sequence in some manner.  However, one class in particular, response distribution was 35%, 18%, 29%, 18% which says to me they are unsure of the notation.  The subscripts are throwing them off.  Something I need to help them make sense of.

I briefly introduced recursive models Friday, but we worked with them more today.  I had a student ask, so is this like a function of the term before it?  Hmmm.  Sort of.  Yes.  Across the board 3 of my classes are very strong in terminology and understanding functions.  So this was a connection.  I saw students eyes widen and they nodded at our discussion.  Alrighty then.  Let’s try another.  And there ya go, the connection was made, they were able to “see” the process within the model.

Student Reflection

I wrote 3 different models on the board.

1 minute, think to yourself:  How are they alike?  How are they different? Now, turn to a friend and share your thoughts.

1 minute, think to yourself:  How do I know which model is which? What do you see/look at to help you decide? Now turn to a friend and share your thoughts.

seq models

So many good things shared.  Its amazing how I can have 4 different ways of seeing something, but yet, each is beneficial somehow.  Some of their comments: Two of them have a1(first term), some have d, some have r.  But what I heard again and again – they all have (n-1) but the location is different.  What does that location of n-a tell us?  Once, its a factor for repeated addition, another its an exponent for repeated multiplication and the recursive, its a subscript for the term before.

So, their conclusion…the math is not hard.  Knowing what the notation means makes it difficult.

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.

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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!

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…

#myfavfriday Who Is Robert Wadlow & Super Size It!

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“My Favorite” was probably my favorite part of #TMC12, literally.  The snippets were quick ideas you could easily tweak for your own classroom.  So when @misscalcul8 suggested we continue – I was excited.  That is until I started thinking about what I would share.  How do I pick my favorite?   My favorite what?  I have a whole list of things I want to share – but today…a favorite unit I’ve used many, many times successfully with my students.

Sadly (for me), with CCSS, we have shifted the ratios/proportions completely to the middle school, so one of my favorite units Who Is Robert Wadlow? is no longer included in our Algebra I curriculum at the high school.  I would leave students with the question at the end of class the day before beginning the unit “Who is Robert Wadlow?”  Several would go home and look up – find information.  The following day, we would discuss, share his measurements (most in metric units) and as a class we would determine how to convert to standard units – so it would make the most since to our American Brains.  So my question – was he unnatural?  Or just a bigger version of us?  If you research, you’ll see how he was normal size baby when he was born.  We talk about how you go to the doctor for well-child visits and they measure you – plotting your height/weight on “that curve” and discuss why doctors do that.  How if we’re growing too fast/slow the doctors can run tests to see if something in our growth hormones need to be modified…

Anyway, to end the day we all measure our foot lengths and heights and create a scatterplot…surprised to see – its somewhat correlated (yes 9th graders are growing, so its not perfectly linear…) – then we add RW’s (ft, ht) to the plot…again, surprised to see, he fits the pattern…just a bigger version.  We calculate the height/foot length ratios for the class, then split the data out to boys and girls to see if there is indeed a common ratio…once again, surprised to see how close the ratios actually are.  We talk about people who are clumsy in while growing – what their ratios would look like – if they are too tall for their feet, etc.

I shoudl note I used this as opportunity to teach students how to enter data into lists on TI-84, L1=foot length, L2=height, L3 = (L2/L1) and how to create scatterplots on graphing calculator.

One year I even had students ask if this was related to Vertruvian Man and explore if they were similar to him.

As a final project in this unit, I would assign Super Size It as part of their unit assessment.

Y, B, H with their Super Size It projects.
Special K – scale factor of 5 …125 times more cereal!
Extra Gum – scale factor of 3 … 27 times more gum!
Chocolate Pudding – scale factor of 2 …8 times more pudding!
 
You could easily modify this activity to fit high school geometry – to determine how scale factors affect surface area / volume ratios.
 
Robert Wadlow Ratios & Proportions Unit Organizer
 
A few other files I have used within this unit –

So, for My Favorite Friday – one of my favorite units – I no longer get to use – hopefully one of you can use an idea or two and keep the spirit of Robert Wadlow & Super Size It alive!