# 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…

# Personal Reflection 3-2-1 #MTBoSchallenge

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Our school district will begin using a new Certified Evaluation Plan this year.  The CEP has 2 major components: Professional Practice and Student Growth.  As part of the Professional Practices, each teacher is asked to consider various pieces of evidence and complete a self reflection which eventually leads to their individual Professional Growth Plan.

I will be completing my self reflection this upcoming week, which has had me wondering this weekend, what are my goals for this school year?

3 things I want to learn, incorporate, practice:
I have read about Flipped Classrooms since before I began blogging.  Watched a couple webinars, read several blog posts, articles.  Its always been of interest, but I just didn’t have take the time.  I have recently begun my first Flipped Unit in my Algebra 2s.  It nothing major, I have linked to videos readily available on You Tube, but have quickly learned if students are accessing on their school accounts, YouTube is blocked.  So I am now looking for possible places to host my own videos (eventually, I want to use my own).

My interpretation is either introduction or skills needed for problem solving which in turn allows students time in class for real application of math.  Following each video, I include 3-5 questions of the big ideas/takeaways for student self-assessment of the video.  When I begin creating my own, I intend to keep them around the 4 minute range, continue including self-assess questions.  For student who dont have access, they can come to my classroom prior to school/class and complete, but they are not allowed to participate in the days activities until they’ve completed the video or shown understanding to me.

Lesson Study – I have read some posts, been involved in a few informal twitter chats, even discussed the process with colleagues at TMC14.  I have located some resources through our PD360 I intend to utilize, but now, I have to find a friend and convince them its worthwhile to journey with me.

Talking Points -I want to ensure that every student feels like they can share their ideas and be heard.  Talking Points is the key for me developing this culture of learning.  I look forward to learning more, sharing with my students and implementing this as a classroom norm.  Here is a place to start.  Severval MtBoS have implemented them as the school year began.  I will share my experiences soon!

2 things I want to continue improving:
Literacy in Math Class- Whether reading, interpretting/deciphering informational text, writing, reflecting on their learning, verbally communicating or strategies to help studentsconnect vocabulary to prior knowledge…communication is a key skill they can use elsewhere.  Last spring, I participated in a webinar based on the book Vocabulary Their Way.  I sincerely feel providing students with similar tools will enhance their learning across all discilpines.  I plan to use some of the structures I’ve learned from Kagan resources and develop some of my own activities for student interaction with peers.

Standards Based Grading – about 5 years ago, I became very interested in aspects of Standards Based Grading.  It just made sense.  I had read, researched, even implemented some successful approaches.  I have heard through the grapevine, theres a possile push for our district to move this direction.  Even though it has not come from an official administrator, I’ve heard teacher conversations outside of vertical meetings that sounds like it may be on it’s way.  I am uber excited.  I have been looking for some good quality resources to share, should the time arise.  @mpershan shared a link this morning for a couple of good resources.  Scroll down to Garry Chu SBG.  Although, I think the Jeff Harding’s video following it gives a fun analogy to show how ridiculous some of our grading practices are-supporting Why we should consider SBG, then Mr. Chu shares some great ideas on How to implement.  I look forward to getting to move on this journey again (finally).

1 thing that’s Imperative in My Planning…
Standards of Mathematical Practices Yes, I am very familiar with them, yet I have not been so intentional in my planning and inclusion of them.  I had a major a-ha last year that I had missed the boat when first becoming familiar with CCSS.  The SMP should have been the anchoring foundation prior to transitioning to CCSS.  As I plan this year, I will be intentional and very explicit in providing students opportunities to use them.  But also in asking students to reflect on their uses of them.  I look forward to reading NCTM’s Principles to Actions, hoping it will guide me in this goal.  Another resource I plan to revisit is Making Thinking Visible.  I read it a couple of years ago, but feel it provides quality routines to enhance student learning that support the SMP.

# Adventures in Geocaching

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This week my nephews are in town.  They love playing outdoors, have crazy fun imaginations and are loads of laughter.  My 10 year counts down the days each year until they arrive at gramma’s!  Its more exciting than end of school countdown.

On Tuesday I was brainstorming with mom and looked into geocaching.  I had never been, only heard it mentioned a few times and needed to learn how it worked.  So I went to geocaching.com registered an account, downloaded the app and yesterday, we set off on our first adventure.

Though most were the grab and go – the kids enjoyed it and were super excited when we found a cache filled with kids party favor swag.  Only problem, we had to scavenger the car to find some trade items…a creative venture in itself.

Each find has a log where you date and sign to prove you were there.  There are hints, clues on the app if you have trouble locating it.  For example, one clue was Australia.  We looked around trying to connect something with that and finally “down-under.”  Sure enough, the cache was underneath the landmark.

We found 10 on our first day and learned some great tips.  We even saw a robin’s egg that had fallen from the nest, a double rainbow as we drove from place to place, played on a playground and visited the gravesite of their great grandparents as one cache was near the cemetary.

Some very clever hiding techniques!  One cache was so tiny, we thought it was only a magnet that had lost its container somehow!  It took a while to roll the log back small enough to place in it again!

Some where along the way, someone asked if I could spell supercalifragilisticexpealidocious (sp?).  My oldest nephew said, well, it shouldn’t be that difficult if you just break it apart.  True.  My response, most words are the same way.  For example, geocache.  Geo is earth, cache, a hiding place.  Which led me to geometry…geo, earth; metry, to measure.  I went on to explain how people wanted to measure distances, areas of land, patterns they noticed in the earth, so they created geometry.

I explained how people noticed patterns with their shadows at different times of day.  I asked, when is your shadow longest, when is it shortest?  Without a hesitation, someone popped up, when the sun is on the horizon, your shadow is longest, when the sun is above you, your shadow is shortest.  I asked how do you know?  One said well, we learned that in science, but if you think about it, it just makes sense.

Again, I am convinced we force too many ideas on our students rather than just letting them think about it and develop their own sense of reasoning.

On our earlier finds, I allowed the kids to look at the GPS readings on the app.  They had to decide which direction to go -are we too close or too far north, and how far to walk.  We needed to moved west once and they could use the compass corectly, but I asked, what if we didnt have it pointing out the directions for us?  I explained how the sun sat to the west and each time after that, they immediately gauged direction based on the sun.

When I returned home after dark, I tweeted of our adventures.  In a convo,

So, now I am curious…how can/are teachers using geocaching as a context for learning?

# Thinking About Etymology, Again…

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If so many of our ‘at-risk’ students struggle with literacy, can we as teachers be smarter in how we present vocabulary, reading choices in class and providing better tools for students to develop their understanding of concepts and terms?

We are getting ready to take a look at polygons, interior/exterior angle sums…in the past, I have always listed names of polygons based on number of sides…but is that really correct? Afterall, polygon is many-angles…  Should I change and reference the list names based on number of angles?  Doesn’t that make more sense when we look at the history of the word?

Triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, 11-gon, dodecagon…the only one I see in the list that actually references sides is quadrilateral…  I remember seeing the term quadrangle in one of my daughter’s elementary assignments a few years back…that makes more sense, right?

So then that lead me to diagonal…dia, something to do with “-gonal” angles… maybe connectiong angles?  And there’s diameter…dia, something, to do with “meter” a measurement.  And diagram…dia, something to do with “gram” something written…

From etymonline.com

diagonal (adj.) 1540s (implied in diagonally), from Middle French diagonal, from Latin diagonalis, from diagonus “slanting line,” from Greek diagonios “from angle to angle,” from dia-“across” (see dia-) + gonia “angle,” related to gony “knee” (see knee (n.)). As a noun, from 1570s.

diameter (n.) late 14c., from Old French diametre, from Latin diametrus, from Greek diametros (gramme) “diagonal of a circle,” from dia- “across, through” (see dia-) + metron “a measure” (see meter (n.2)).

diagram (n.) 1610s, from French diagramme, from Latin diagramma, from Greek diagramma”geometric figure, that which is marked out by lines,” from diagraphein “mark out by lines, delineate,” from dia- “across, out” (see dia-) + graphein “write, mark, draw” (see -graphy). The verb is 1840, from the noun.

So dia-is across, -gonal is angle, segment connecting angles…

I believe I will change how I present this to my students this year, which will allow them to connect this “new knowledge” to future concepts based on the history of the roots…

Curious about other’s ideas, suggestions.  Please share.

# Unit Organizer Update & Feedback Only Grading

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Its been a good, very good, no, great start to the year.  Almost scary how smoothly it has begun.  But I will take it and be happy. Very happy.  I always have amazing kids.  They make me smile.  They make me think.  They make me love what I do.

I have read about comments only grading multiple times.  A colleague shared more research as part of an action research project last spring.  Reading and chatting Wilham’s Embedded Formative Assessment this summer convinced me I needed to give up grades on student work and offer feedback only.  So far, so good.  When I pass target quizzes back, I allow some time for students to mark on their organizers where they consider they are based on feedback I have offered.  I will definitely be sharing updates.

I shared a unit organizer here a couple of weeks ago.  It’s gone well, though I knew I wasn’t satisfied with it.  But this afternoon, Crazy Math Teacher Lady shared this

My thoughts are to modify my booklet style organizer to include this on the inside.  I appreciated the graph for each target.  This goes right along with some research from #efamath chat this summer.  It reminds me of something similar I had seen on @druinok’s block a while back.  I like how Lisa has a place for students to record multiple assessments.  This is a great layout!

I plan on keeping a vocabulary knowledge survey on the front as suggested by Math = Love.  Here is a sample of mine

And keeping the assessment grid on the back for personal reflection…

I re-intoduced an old assignment from a few years ago in my first units…students were asked to write their own unit assessment using our learning targets.  Most seemed to put good effort in to interpretting what each was asking.  Offering written feedback gave me a chance to address some of their misconceptions, mostly notational issues in diagrams they had included or clarifying some vocabulary.

My intentions were for them to go back through their INBs, notes, target quizzes.  A couple of times in class, I fielded specific questions they had.  Based on what I observed, I believe it was a useful task.

I am looking forward to my new organizers!  Thanks to Lisa, Sarah & @druinok for sharing such awesome ideas!

# INB Unit Organizer

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I wanted to create a unit organizer than encompassed several aspects but would also be narrowed to one page, fiting in to the INB.  Here’s a list of what I wanted:

vocabulary
unit overview/schedule
learning targets
record of assignments
track their own assessments/learning
place to record questions/big ideas
opportunity for end of unit reflection

Here is what I arrived at for a first attempt, copied front to back and folded in half, this is the order students will see the booklet.

The vocabulary pre-assess was a great tool.  I saw this idea over at Math = Love earlier in the summer.  It went so well. It only took students a couple of minutes to self-assess their vocabulary knowledge.  As I walked around, I was able to see several terms had 3s & 4s.  We compiled a list of our 1s & 2s words.  I explained, as they learned a word or gained better understanding, they should go back and put a +.  Before the end of class, students were asking if they could go ahead and update their chart.

If possible, maybe completemthis part a day before beginning a unit, in order to make needed adjustments based on student responses.

I included the correlated CCSS # for each target.  Eventually, these may be beneficial when looking online for a resource on a specific standard.

I am not fully satisfied with this chart yet. Assignments made for specific targets can be listed, a note if completed (stamp) and place to monitor their assessment for each.  A second line has been included in case RTI/enrichment is needed.

Finally, the back side has a place to record reflection.  Ideally, I would have them complete the reflection at least 2 days prior to unit assessment, allowing to address any needs the following day, prior to assessment.

As always, this is a work i  progress, suggestions and ideas are welcomed!
PDF:  http://db.tt/DWfJUqLL
Foundations in Geometry doc

Intro to Matrices:

# 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

# Interactive Notebooks

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INBs were introduced to me at TMC12 last summer by Megan Hayes-Golding.  It was consistently the #1 response students stated on their evaluations as something I should continue doing in my classrooms.  They are not bulky, students like the conciseness of the information we place in them, they helped students stay organized.

Using CWP (color with purpose), foldables, graphic organizers for note-taking allow students to develop skills that can carry over to future coursework.

By creating assignments that require students to ‘interact’ with information helps them develop connections and retain the knowledge.  Often times, we would complete an inquiry task in-class, then together create a summary of what we saw/learned. By using a variety reflection tools after completing a task, students are able to self-asess any questions that remain and we can address those misconceptions either individually or in-class.

The TOC is imperative.  For teacher accountability 🙂 and it allows students to quickly locate info in their INBs.  One change I plan to make this year, is the addition of tabs as suggested in this post by Mrs. Hester.  The only change I plan to make to her suggestion, is to use the unit title.

I really like the EOC Review glossary she shared in the post.  I believe using different techniques which allow students to interact with the vocabulary helps students develop deeper understanding of the words.  I appreciate the complete glossary, but do I dedicate several pages at the end of the INB for this?  What are some ways you incorporate literacy/vocabulary into your INBs?

A KAGAN structure I used often in geometry this year  was Developing Definitions.  Examples/non examples of each term were posted around the room and students would carousel to each, creating their own definitions.  There was a pair-share, then whole class follow up to discuss how they defined terms to ensure we were all on the same page.  After the first time we used this task, students requested that we do it again.  They said by having to come up with the definition on their own, they were able to have a real understanding of the terms.  For a left-hand page assignment, we would often play “Draw What I Say” – another task from KAGAN.  Students would play pictionary of sorts by using a prescribed statement incorporating specific terminology.

I wonder if by having purposeful assignments within each unit of study focusing on specific terminology, then as a review prior to the end of a unit, allow students to complete those entries in the glossary, if this would have greater impact?

Another idea that developed as the year progressed, were pockets.  We began by having one pocket at the front of the INB.  However, a student suggested to have other pockets throughout.  This coming year, my intenions are to have a pocket at the beginning of each unit.  Here is an example of a pocket.  You still have room to place information.  Possibly your essential questions for the unit, a concept map-brainstorm at the beginning, then revisit as a reflection and modify it at the end of a study?

This pocket is super easy to construct!  One of my favorite things I learned at TMC/Global Math!  Simply fold top left hand corner down on a page.

Place glue or add tape to the bottom and left edge of that page.

Flip pocket top to the right and adher to the back of pocket.

These pockets are much sturdier than I gave them credit to be.

Another idea I plan to develop before the school year begins are unit organizers to attach to the back of the pocket.  I just need to figure out how to modify this

into a folded-booklet style which also includes a place for students to record their own learning progress.

I am super excited about continuing use of INBs in my classroom and look forward to developing an even larger basket of ideas to make them even better learning tools for my students!

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

# To-Do List for Next School Year

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During the school year, I just don’t have the time I want to read the amazing ideas teachers are sharing through their blogs.  Sure, I can capture a great idea here and there, but time to really immerse myself, processing ideas and working through how I can tweak for my students, nope. A tweet from @burgess_shelley suggested to read blogs from 3 people you follow each day.  So I started at the top of my list and have begun to work through it by reading 3 a day…my hopes are once each week, allow myself some time for reflection, adding to my to-do list for next school year.

In Whiteboarding Wins and Fails @borschtwithanna, it was the last item noted in her fail list.  I love the idea of snapping pics of group whiteboards and posting on class page, assigning reflection as hw…even offering prompts at the beginning of the year:
I liked ____’s strategy because….
Which strategies are most alike/different…
How can we connect the ideas…
One way to improve our strategy…
A question I have for the _____ group is…

Please offer ideas for other relection prompts!

In the comments of the post, Max/Anna discussed allowing students to classify mistakes and offer suggestions for helpful feedback.  Again, another great idea on my to-do list for next year!

@mathequalslove posted Vocabulary Knowledge Survey, a strategy suggested in the book Styles and Strategies for Teaching High School Mathematics.  I appreciated the simplicity of the task, but when used purposefully, in my opinion, can have a great impact on student learning.  Vocabulary and literacy strategies can offer struggles for math teachers, but this is a very reasonable way to a) assess students’ prior knowledge b) opportunities to discuss connections to their prior knowledge c) allow them to self-assess, seeing where they are in their learning d) allow them to track their own progress throughout the unit of study.  Thanks for sharing this, Sarah!

I Shall Never Play a Review Game Again, @nathankraft1 offers his version of Grudge, a fun game of revenge on classmates for those who correctly answer a question.  What’s so great is how students who are out of the game can still participate by taking on a zombie role.  Sounds fun, engaging and I cannot wait to try it out in the fall!

# Developing Definitions

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I’m back!  Nearly 2 months? Yikes. Some fellow teachers on Twitter were committing to blogging once each week.  I think  that’s reasonable – besides, usually my best reflection comes during the moments I blog.  Reflection – seems to be the first thing I push aside when I just don’t have the time.  Yet, its the most valuable use of my time.

I’m sharing some successes from Kagan Geometry (one of my favorites by the way).

I was going to be out for a number of days due to being seated on the jury for a trial (give me 100+ teenagers over the courtroom anyday!).  I wanted to leave something productive.  I did short videos (<10 minutes) filling out certain pages in the INBs in addition to other activities.

The first Kagan activity was for vocabulary.  Each strip of paper included examples and counter-examples for each term.  I modified from the round-table recording it suggested.  Students were asked to pair up (a new partner for each new term) and develop their own definitions.  I loved it simply because most were terms students had previously been exposed to in middle school.

When I returned to the classroom, I ran through all I had left during my absences to address any concerns/questions.  Several students commneted how they liked (appreciated) doing the definitions this way.  Their comments ranged from – ‘You actually had to think about the terms; Talking with someone about it really helped you process what it was before writing it down;  The pictures of examples / nonexamples really helped understand the word better.’

Yesterday, we developed more definitions about angles.  When I told them what we were doing – they were excited about the activity.  Listening to the conversations – I was very happy with their discussion / questions / specifics they included in their definitions.

I remember several times in the past doing examples / non-examples, especially when using Frayer Models.  I believe taking it out of my hands/mouth and giving them the opportunity to work in pairs really enhanced their understanding of the terms.  Even when discussing HW  today – they used appropriately terminology.  Yeah!

Another Kagan activity I used as a LHP activity

from Kagan Geometry

– very similar to Everybody Is a Genius’  Blind Draw.  Students were placed into small groups and given 12 cards with written directions.  Person 1 chose a card, read the directions, gave others time to think and draw a diagram with labels.  The reader confirms/coaches/praises others’ work.  A new person chose a new card and the rounds continued until all cards had been used.  One thing I appreciated about this – another card asked students to draw a ray from E through M.  This allowed students to realize differences in very similar diagrams.

Again, when I returned to the classroom, students shared how this activity was different from anything they’d done before, saying it was both challenging but helpful in that it helped to clarify certain misconceptions they had; especially with labeling the diagrams.

I have learned the Kagan strategies help students develop and process concepts.  There are “game like” activities where students must find their match and discuss.  Visual, Auditory, Kinesthetic – something for everyone.  Its not an end all – be all resource.  But the amount of HW / practice is minimal when I’ve used these strategies correctly.  I am a firm believer that they help start a strong foundation to build upon.  Hey – if students are smiling and laughing while “doing definitions” – its gotta be good.