# 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…

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For years, I have tried to share a related article as appropriate with my classes.  Often times it was a news article related to a data collection lab.  However, I feel more impact for reading in math class is from informational reading with graphs/data related pieces.

One day each week, I plan to use a “Laker Literacy” article (named penned by data team in school wide iniative last spring) or  Stat Rat (Graph or Data related piece).

I asked students to number the paragraphs 1-6.  After time to read, they were instructed NOT to answer the questions on the back, but rather as they read each question, make a note of which paragraph from the article could be used in helping them respond to that question.

We will be using the article and questions next class.

# #readthree Summer Challenge #MTBoS30 Post 22

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Last summer I wrote a post after a tweet from @burgess_shelley (she’s married to that Pirate guy)…but I enjoy reading her journey and experiences as an educational learner and leader!

Time to read the blogosphere is sometimes put on the back burner during the school year.  I hate this because interacting with #MTBoS challenges me to be a better teacher.  But it happens.

So today, I offer this challenge (again) to myself, but feel free to join in.  Read posts from 3 people you follow on twitter.  If you cannot do it daily, set a goal for yourself. I am challenging myself to 4 days a week and one day to reflect and share my take-a-ways by compiling my to-do list for next year.

My intentions for next year are to incorporate more hands-on labs/data collection, focus on vocabulary and literacy strategies to empower student reasoning (writing and summarizing, @druinok!), providing more purposeful interactions/discussions (Strength in Numbers, @tchmathculture!) and planning more engaging tasks/activities.  Hopefully these goals will guide my focus in this challenge.

I am looking forward to reading from some new bloggers as well as catching up with those tried & true!

# 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.

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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…

# We Must Model What We Wish to See in Our Students

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Learning is not attained by chance. It must be sought for with ardor and attended to with diligence. – Abigail Adams

For years I mumbled about writing and focus on literacy in my class, afterall, I was a math teacher.  If I wanted to teach reading, writing, etc., I would have been an English teacher, right?

No longer, I finally get it. I realize the importance of developing strategies for vocabulary, using multiple writing opportunities that allow students to summarize their understanding, share questions.  At risk students, especially, need someone to model these learning strategies, which will give them more tools for success.

I was an avid reader growing up.  Nonfiction/biographies is where you would find my nose in elementary school.  When did I lose this passion for reading?  Through high school, even college, I was a ‘model student’ doing what I was assigned, going through the motions, but I was not passionate about learning and reading.

I give credit to @joyinlearning for reigniting this love of learning just a few years ago.  In the past 3 years, I rarely put down a book, that I don’t have another ready to pick up.  I also credit twitter book chats for keeping me on track/accountable for professional reading…seeking opportunities to really apply what I am reading to my classroom.

A statement from a colleague made me pause one day this spring, ‘I wish more would read and participate’ concerning our Library Media Science elective.  My thought, “But are we encouraging it?  If we want our students to read, are we reading ourselves?  Are we sharing what we learn from our reading? Even if its just a great fictional book to read!?! If we want our students to be lifelong learners, are we models of lifelong learners ourselves?”

My students were always teasing me when I would share an idea or new strategy ‘I had read about,’ but I believe it was evident to them, my desire was to move forward, I was searching, reading to provide new opportunities for them to learn.

A few days before our end of school, I read this tweet and loved it…

What a great way to share what we are reading!  I could totally get started with adding my reading list to my email signature!

Beginning with open house next school year, there will be a poster outside my door sharing what I am currently reading, both professional and for-fun.  I plan to include a variety of texts-books, articles, blogs.  I am hoping other colleagues will join in this effort to create an environment that encourages life-long learning with a focus on reading!

A couple of posts I have read this summer…

Developing Active Readers – Rebecca Alber

# Formative Assessment Lessons

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Its been 3 weeks since I’ve blogged.  Not because I didn’t want to – but life has just been head over heels busy.  The week following my last entry – I presented at KCTM – Literacy in Math Class.  I’ll blog about it soon.

In Kentucky, I was introduced to Formative Assessment Lessons about a year and a half ago.  I remember the first one I tried was not so successful.  But the more I learned, the more I realized, there was some good things embedded within these lessons.   At our last KLN meeting, we were asked to discuss our experiences with the FALs.  I hadn’t realized I had actually used as many as I have until we started running through the list.

My students can find some level of success as well as being challenged on the other end.  I observe student success with these lessons.  They are formatted in such a way, I am able to listen to student discussions, considering their ideas and able to pose questions that will foster more discussions.

Part of my session on literacy was to give students opportunities to talk, share and ask questions about their thinking.  Within the FALs, students are given either a problem solving task OR a conceptual development task.

In all lessons I’ve used, students respond to a given task as a pre-assessment, after completing the lesson, class discussions, they are given the opportunity to revisit the same or a similar task.

In the problem solving tasks, students are put into groups homogenously (based on similar approaches to solving a problem or even similar misconceptions/mistakes – not necessarily ability).  This allows students moving in the right direction to continue; while my time can be targeted to smaller groups of students, using questioning to guide their thinking, discussions.  Each group is given sample responses, and asked to think about the student’s reasoning – why they approached the problem as they did.  This gives the group an overview to see multiple ways to consider and opportuinties to critique the reasoning of others.

In the concept development tasks, students are usually given a task/questions and card sorts/matching activities.  Instructions will almost always require students to verbalize their reasoning, then their partners must either explain the reasoning in their own words OR why they disagree with their partner.  I feel verbalizing their thinking is a key component of literacy – helping them work through their own understanding but also listening to ideas of others, in a small group setting.  Many lessons offer extension suggestions if needed.

To complete the lesson, there is often a plenary discussion to wrap up, solidify the concepts.  Its very important to really listen to students – in some lessons, you are encouraged to scribe student comments/ideas with their names for ownership in the discussion.  White boards are a common component – seeking student responses – sharing different responses – asking questions – if others agree, disagree or have something to add to someone’s comments.

I am sharing about FALs because today, I left my geometry classes feeling good – that students were given an opportunity to think, discuss, share and learn – clear up some misconceptions.  I am looking forward to our whole-class discussion on Monday and the follow-up assessment!  Though there are still some mistakes – I think the sharing out will add/deepen to students’ understanding.

Representing and Combining Transformations was the lesson students worked on today.  I paired students based on similar responses on their pre-assessment.  I really enjoyed “sitting back” and listening to their discussions.  The particular task, they were given 6 different graphs with an L-shape and 8 different transformation cards.  They were asked to connect 2 shape graphs with a card describing the relationship between the two.

I’ll be honest in questioning the need for the transparency graphs – but after observing students, these were a key learning tool for most of them.  When they asked for help, I encouraged them to use their graphs to “see” what happens, then use what they noticed and apply it to their shapes.  I also found allowing students to place a push-pin at the center of a rotation was very beneficial to their understanding.  To observe how using different centers of rotation will affect the movement of the shape.

Recently, a colleague decided to try a FAL – Forming Quadratics with an Algebra II class.  In our last PLC, my colleague shared pros/cons observed during the lesson and that all but only a couple of students had improved / were very successful on the post-assessment.

FALs are idealy used about 2/3 the way through a corresponding unit of study.  This allows the teacher to view misconceptions and clear those up before finishing the unit.  Most lessons consist of a 10-15 minute pre-assessment, 1 hour for lesson/discussion (this can vary depending on students), 10-15 minute follow-up assessment.

Each lesson is aligned to 8 Mathematical Practices and outlines which CCS is addressed.

There is some prep-work involved, so don’t print a FAL and expect to use it immediately.  I use card-stock for the card sorts (each type of card gets its own color) – if you laminate them, maybe they will last longer.  Also, when it calls for a poster of student work, I don’t want them to glue pieces on a poster – then I’ll have to make an entirely new set next time.  I want to reuse them.

Today, I had students add a post-it note with their initials and I snapped a pic of their cards.  They can create an answer key on paper as well.

I would love to hear about others’ experiences with FALs – ways they’re using them in their classrooms!