Category Archives: SMPs

Identifying Linear Functions

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Linear Functions Organizer this does not include arithmetic sequences, which was earlier in the year, but I can refer back to our work with them to activate prior knowledge for this unit.  The next unit will be linear regression which will include correlation, describing scatterplots, finding regression equation with technology, using the equation to predict and finally introduction to residuals.

Students started with a pre-quiz similar to the one below.

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Identify Linear Functions This is a booklet with a Frayer Model for our notes, a variety of math relations to identify as linear or not and a 2-minute reflection grid on the back.  Prior to beginning our notes, I gave them 1 minute to jot down anything they thought they knew about linear functions.  Then we pair-shared before sharing with the entire class.  Then we took our notes. (as a follow up the next day, I gave them 2 minutes to jot down all they could remember about linear functions as a small retrieval practice).

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Our next task was created by cutting apart these relations and posting them around the room with a chart that asked if they agreed or disagreed with the example being a linear function.  Students received stickers to place on the chart as they visited each station.

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I was fairly accurate in which ones I thought we’d have to use for discussion, but a couple really surprised me.  These are the 4 we discussed following the carousel activity.

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I. y = 2x was the one I was not expecting.  When I asked if someone would share their thinking, one student said they thought x was an exponent.  Another shared they did see “the b” for y-intercept.  We looked at a table of values and graph to agree, and show the y-intercept was at the origin and indeed y = 2x was linear.

The other I failed to snap a picture of was graph K, a vertical line.  Yes, it’s linear, but not a function…two students got that one correct in this particular class.

Using the 2-minute reflection grid as our exit slip to see students thinking about the lesson, I was excited about some of their “I still have a question about…”

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On the reflection grid, if they have no questions, nothing is confusing, I ask them to give me a caution…something to be careful or / watch for.  Several of these questions encompass multiple students.  Some of them I only needed to clarify what was said.  Its pretty clear I was not communicating very well on a few of the.  I hear my “expert blind spot” showing up…”Of course squared is not linear, we learned it was quadratic in our functions unit!”  But so many students on the pre-quiz used vertical line test as their reasoning for linear…we had some side conversations about this misconception…that it shows functions, but does not prove if its linear.

Some of the questions, I allowed other students explain their reasoning to help clarify their understanding.

I know I shouldn’t have favorites, but in this list…

Why can’t you multiply the numbers by each other?  We tried it.  Add 2 numbers that will make 18.  Create table of values, find rate of change, graph it.  Yep, that’s linear!  Multiply 2 numbers that will result in 18.  We created a table of values of their answers, found the rate of change and graphed them.  No, that’s not linear!

If an exponent is less than 1, can it be linear?  We will try it tomorrow as our bell ringer.  But I look forward to exploring their questions more!

I told them how excited I was about their questions and posted them on our “THINKING is not driven by answers, but by QUESTIONS” board.  One student had the biggest smile and as she said, Look!  I’m so proud, my question is on the board!  Something so simple, yet, my hopes are that it will encourage her to ask more questions.

One student asked me, but isn’t it disrespectful to ask questions and interrupt the lesson?  Nooooooo.  I love when you ask purposeful, curious questions you wonder about!  Finally, a break-through to get them to start asking and wondering more…

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!