# 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.

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).

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.

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.

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…”

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…

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

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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:

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!

# How to Learn Math #12days Post 1

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As part of my LTL leadership project, I decided to focus on Growth Mindset in the Mathematics Classroom.  I went through Jo Boaler’s How to Learn Math course a couple of summers ago and it really impacted my personal view of the classroom.  It, along with Max Ray’s Ignite talk 2>4 and the More Good Questions (Marian Small) book, encouraged me to seek out more opportunities to offer open-questions to my students and to become a better listener in my classroom.

Two things happened in one particular class this week…  After a discussion on various ways to multiply 18 and 5, students were asked to multiply 12 and 15 without using the traditional algorithm.  I turned around and 16 students had their hands raised wanted to volunteer their approach.  Most were not the 5 or so who typically want to answer.

What made me most proud.  A couple knew their approach was wrong, but they wanted to know why it didn’t work.  And they were comfortable putting themselves out there to allow their classmates to comment.  That’s a win – coming from a group who plainly stated they did not like to volunteer an answer or be called on randomly because they were afraid of being wrong.

The second thing that made me smile was after a discussion of the Berkeley study about failing math students and the study groups – how talking with peers about their math work actually improved their learning.  Students were first asked to write why they thought student learning improved when you talked about your work to others.  First of all, a minute, not even two minutes was enough time for them to write their responses.  What?  They were engaged and sharing their thinking / opinions / reasoning as to why they thought this happened.  When I asked if anyone would care to share what they wrote…again, numerous hands went up.  Four students in particular who had NEVER volunteered all semester. (HaPpY dAnCe!)

This was my reminder to keep on keeping on.  I sometimes lose sight of why I chose to go a different path than just a few years ago.  And I am grateful for the nudges from my students that re-engergize me to keep moving forward.

For the past month, I have been using my popsicle sticks again to call on students – this eliminates the same 5 or 6 always dominating the class discussions.  I encourage them to wait, no hands up – during certain activities as not to distract their classmates’ thinking.

Per Jennifer McDaniel from Clay County – who led an ACT Bootcamp a few weeks ago, we get new groups/seats every Monday.  I pull out the popsicle sticks and draw for the groups.  This one thing, I believe, has had a huge impact on the climate in all of my classes.  Majority of the students actually seem more settled with new groups each week.

I have so much flowing through my head and need to blog more.  Not to share but to sort out my thoughts and help give me direction as this semester ends and I begin to prepare for next semester.

All in all.  Its been a good year.  I love my students.

#12days Post 1

# Students Making Sense of Quadratics

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I realize some folks will bash me for sharing this from an Algebra 2 class, but based on benchmarks, most of my students have major gaps in quadratics.

I began with reviewing multiplying 2 binomials on our whiteboards.  I shared the box/area model and several smiles celebrated because they “saw it” and were doing it correctly!

Last week, I pulled out a box of Algebra Tiles.  We literally explored building squares.  I wish I had taken pictures because some of their squares were like a grandmother’s beautiful quilt blocks.  I began tying it back to our box/area models -I’d rather think of it as leading (not forcing) their thinking – but they were quickly picking up the patterns.

We then began looking at the algebraic equivalents, again, with a sketch along side allowing them to “see” the process.

Our next step was to find the missing value without tiles/picture models…and then I asked them to review their multuplying with 5 expressions alongside.

“What? You think they’re the same thing?!?” I asked,  “Prove it to me. Well, by-golly-jee. You are on to something!”

The following day in class, I made a HUGE ordeal of different ways to write zero.

I explained our next few minutes were a process. But we talked about it, step by step, completing the square, adding ‘that zero’ in our expression, the separating the trinomial and 2 constants.  Rewriting our trinomial as a binomial squared.

Ok. Why in the world would anyone want to do this?  I told them we were finding hidden information.

As they arrived at this form (x+4)^2 – 9, I paused, reminding them to think back on our function transformations before Christmas break.  How would this function y= (x+4)^2 – 9 move on our graph from this one y=x^2?  Quiet. “Move left 4 and down 9!” Someone exclaimed.  Really? Are you sure? We graphed the two and yes, it did just that.  So what does this tell me about my parabola?  They didn’t say vertex. Or minimum.  They said it shows us how the graph was transformed.

I will take that.

I then asked them to move left 4 and down 9 from the origin.  What have you found? The lowest point.  The vertex. The minimum. All their responses, not my statements.

We set our expression equal to zero and solved the equation, using our inverse operations.  They made the connections with the x-value of the vertex being the “center line” of the parabola.  They realized the +- 5 were steps in either direction from the center line.

I most appreciated the questions they asked on #3, 4 and 7.  Several chose #7 thinking it was shorter, thus less work. Snafoo. No middle term. What happens?

I suggested they look at it from a transformations point of view.  Someone shared-It doesn’t slide left or right, only down.  Another student said-well, that’s the easiest equation to solve! (Yep.)

Why did #4 bother some? The middle term had an odd coefficient.  But once they shared their thinking, ok. Got that one too!

#3 was what we math folks recognize as perfect square trinomial.  But for the students, it was an a-ha.  Again, using the transformations context, we moved right 5, but not up or down.

L: But I thought all quadratics intersected x-axis twice?  I asked – did this one? No.

What about y=x^2 + 3?  It moves up 3. Ok. How many times did it intersect the x-axis? It doesn’t.   Hmmm.

A student who is rarely engaged then asked, if you can make a parabola that doesn’t intersect the x-axis, can you find one that doesn’t intersect either axis?  Me: Can we? What would it look like? S: Noooo. As its going up, increasing, it would be increasing outward, too!  More discussion, between them. Me not included. I was smiling.

And their questions were what drove our lesson today.   And I was so excited, telling them their questions make me think! And when they’re asking questions, their brains are processing the information – making it their own.

It was a good day.

# Student Reflection on HW

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When I get back from a conference, I have the best intentions of sharing, but its nearly 3 weeks later and I am just starting to get caught up…only to realize there are less than 3 weeks of instructional time before Christmas break.

Starting to stress in my Geometry blocks classes…similarity (although I tied in some with our congruence unit and they used dilations in our transformations unit…) right triangles and circles…then a super dooper quick approach to modeling via 3-d problems.  Anyone have an amazing project that ties circles and right triangles together?  Anyway, a bit off topic, because the stress causes me not to focus.

I attended a session led by @ottensam sharing different approaches to ensure we are integrating the SMPs in our instruction.  He was very engaging and shared some simple, research-based strategies.

A great idea he shared was to change up the way we approach homework.  One simple suggestion was to ask students to eflect on the problems…which were most alike? Most different?  Why? Which one did you think was easiest? Most difficult, why?  I had students to do a quick write using this idea this past week.  Once they were finished, they had to meet with someone they did not sit next to and share their responses.  Finally, I called on students, asking them to share -not what they had written- but something they had heard.

I am always amazed at student responses when I use startegies similar to this and could kick myself for not being more intentional, more often.  Several shared exact similar/different pairings but for totally different reasons.  I love it, being able to see and hear their ideas and thinking.

# Happy Birthday #Made4Math !!! Formative Assessmemt Reminder Cards

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First, just let me say a big THANK YOU to @druinok for beginning #made4math and to all of the generous folks who have openingly shared their classroom ideas, lessons, tips over the past year.  I was overwhelmed with how quickly it took off!  Still, today, I am amazed at the generosity of this community.  I have learned so much and my classroom was definitelh impacted by your awesome ideas!

My share for today was initially a result of a convo with @rachelrosales and @druinok, brainstorming ways to organize reminders for the numerous formative assessment techniques…something simple, at your finger tips.

I loved @druinok’s post today and her Student Engagement Flipchart.  Very.Nice.  It will definitely be on my to-do list for a future project.  However, I am choosing to share a similar idea, just a bit different format.  I cut down index cards to fit sports card pages… pack of 10 for \$1.  I am able to display up to 90 of these reminders ranging from formative assessment techniques to various strategies for student engagement, reflection, etc.

Front side of card has title, with some information…

Back side of cards has description, suggestions, reminders…

I have placed the pages in a small 3 ring binder which can easily hold more pages.  Currently, I am trying to include summaries/reminders of techniques I have used or see being easily modified for math class.

Looking forward to learning and sharing more FA techniques with my amazing PLN!!!

Pam Wilson, NBCT
5 Practices for Orchestrating Productive Mathematical Discussions, Smith & Stein
Teach Like  a Pirate, Dave Burgess
From Ashes to Honor, Loree Lough

# Rigor, Relevance & GPA

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A couple of weeks ago, our school participated in Instructional Rounds. Administration from our Central Office along with colleagues from nearby districts walked through a sample of our classrooms. Reading their comments / findings was interesting. 100% of teachers were doing something – working hard – teaching. 90% had agendas or I can… statements posted in their rooms. Students were doing what they were supposed to be doing – does this mean they were engaged or being compliant? 75% of questions being asked were in the bottom 2 levels of Blooms. That means 25% of what was observed was “rigorous”, I guess???

The following day we had PD – Rigor, Relevance & Congruent Tasks. We were asked to define Rigor…well, most of us felt Rigor was different for different students / tasks – but all agreed – it was just beyond the reach – some struggle may have to take place – not knowing something but making connections to what we can already do.

Using Etymology…where do these words come from?

Rigor “to be stiff” – does that mean, we may need to loosen up before we can actually accomplish a task? learn something?  It may not be accomplished on our first try???

Relevance – congruity, appropriateness, agreement, pertinent to the matter at hand – okay I get that – are the tasks I’m choosing meaningful and do they lead to the standard I have chosen?

The presenter mentioned a professor from graduate school – I shuttered – it was the same professor in my very first education class. I even shuttered aloud – uhhhhhhh and the presenter laughed.  Not good memories – I struggled in that course. It was not easy. But as I sat throughout the morning – I had vivid memories of lectures, activities and discussions. I remembered specific things from an Intro to Ed, Monday night-class 20+ years ago. Ask me about other Ed courses – I cannot say much. My last course before my semester of student teaching – we discussed a lot – student lead learning, research. But beyond those 2 courses in undergraduate, very little is recalled. But in both of the courses, my professors challenged me to think – they challenged my views / previous knowledge and experiences. I had a voice, they listened, they responded, usually with questions.  I may not have understood it then.  I certainly didn’t appreciate it then, but I now realize, I learned a lot under both professors.

In the past 3 years, I am realizing I am a guide. I am learning to listen to my students’ conversations. I am learning to ask questions beyond factual, yes/no levels. I can at least ask them what made them respond that way OR if they agree/disagree with a classmate.

Our discussions on rigor/relevance have really gotten me to thinking…wondering how I can improve, make learning worthwhile for my students.

A tweet from @AmberDCaldwell earlier this evening really resonated with me and my struggle to convince students,

Again, a post from Emergent Math last month The Struggle for Productive Struggle – take time to read/listen to the NPR link he provides.

As I look through tasks from MARS, PARCC, Balanced Assessment, Illustrative Math – its obvious a classroom of 2 examples, practice these, check, quiz and move on to the next concept is not a prescription for success. I have wonderful students – but I don’t want them to do/think because I said so.

I want them to be able to think on their own, to feel challenged – yet without feeling a need to give up. I want them to feel comfortable asking questions, sharing their thoughts / ideas – even acknowledging their mistakes.
I want them to be able to listen to others’ ideas – and decide for themselves if they agree or not.
I want them to notice cool patterns in math – that its not just a bunch of worksheets and unrelated problems in a textbook.
I want them to recall things we discussed prior to this lesson because they developed an understanding deeper than just the surface.
I want them to be able to make those connections on their own (me only as a guide) and move forward.
I want them to be able to learn/be productive without me telling them when, what and how to do so.
I want them to value success because they’ve worked hard to earn it.

I want them to value learning – and realize “they are not defined by their GPA.”