# 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…

# Better Questions Week 3 #MTBoS

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I’ve pondered this challenge for a couple of days.  So many options!  But a tweet from @mathymeg07 led me to a post from @MrAKHaines blog Math Pun Pending.

The post was celebrating a variety of strategies his students had use to answer the question:

He wrote:  When I wrote the question, I had anticipated that students would use a couple of different strategies. What I didn’t know was that my 25 students would use a combined seven correct solution strategies to solve this problem.

Two parts to my post:  1. How can I make this an open question and 2. How can I use student samples to develop a better lesson in the future?

How can I make this an open question?

A.  Name a point that is NOT on this line.

B.  Name a point that this line passes through.

Thanks to @PIspeak‘s TMC14 session in Jenks, I urge students to “Support your claim with evidence/reasoning.  I want to see your thinking!”

How can I use student samples to develop a better lesson in the future for my classroom?

I appreciated the fact that he never explicitly taught “the teacher’s efficient strategy” but allowed group discussions and support to drive the lesson.  Students shared ideas.  The last paragraph  in his post says, “My students are acting like mathematicians, y’all. They’re using their toolkit of math ideas to solve problems flexibly. I couldn’t be happier.”

In the end, that’s what we all want – students thinking on their own, making sense and being confident enough to explore a problem with their own ideas.  So, how does this tie in with the Better Questions prompt?  My outline of the lesson feels a bit like those I’ve used from Formative Assessment Lessons, but I feel it lends itself to students doing the thinking, talking – I only provide the materials and support to make desired connections that will lead to the learning goal.

I’ve been following the #T3Learns chat from Wiliam’s book.  In chapter 3 of Embedding Formative Assessment, it suggests using student sample work. How might I structure a lesson, utilizing student samples of this question?  In Principles to Actions, MTP3 states Effective teaching engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

1. Begin with the same question. Allow students to write a response. (3-5 min)
2. Place students into small groups and allow them to share their approaches. (5-7)
3. Allow groups to explore student samples, making note of different ideas, what they like/ways to improve, questions they’d like to ask the student. *maybe as a gallery walk? (15-20)
4. Discuss their findings as a whole group. (10 in)
5. Connections to/between the different mathematical representations. (5-10 min)
6. Written reflection:  (3-5 min. possibly use as a start up / bell ringer to begin class with following day – providing an opportunity for retrieval of previous days information?)
• my strategy was most like:____
• the strategy I liked most was ____ because ___
• the strategy I found most difficult to understand was ___ because ___
• Which approach was most efficient?  Why?
• What do you think was the BIG IDEA your teacher intended for you to learn/understand?
7. Transfer…provide a few, different contextual problems that allow students to connect the mathematics to something tangible, maybe in a problem posing situation (should this be small group?  individual? ) (5-15, would this be better as follow-up the following day?)

Timing is often an issue for me.  I want to provide students with enough time to make sense/discuss, but not so much time it feels long and drawn out.  Are the times I have listed appropriate?

Please offer suggestions.  How have you used a similar approach successfully in your own classroom?

# #tmc14 My Favorites

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Black Lights and Highlighters

Ghosts in the Graveyard, Kim Hughey, Math Tales from the Spring

Grudge, Nathan Kraft’s I Shall Never Play a Review Game Again
Used most Fridays, either first or last 10-15 minutes of class to highlight our big ideas/skills from the week.

Plickers all you need is paper and a smart phone. New app for me, so I am still learning. What little we’ve played around, it has some great potential!

# #5things to Do with Sticky Notes #julychallenge

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2-Minute Assessment Grid ideally is for the end of a learning task, but is a great reflection tool used toward the end of an entire unit.  Each student gets 4 sticky notes to respond on for each prompt as seen in the picture.  I like it 3 or 4 days before a unit assessment.  I am able to create a chalk talk with the questions they still have-which allows students an opportunity to respond/learn from one another before I intervene.  Read post here.

12×12 Sticky Notes These were a treasure from our local Mighty Dollar store.  25 large sheets for \$1.  Yes, I bought all 10 packs!  I basically cut apart a pre-assessment and tape one question to each giant sticky then distributed them to pairs of students.  They responded to the question, then hung the sticky on the wall.  Students carouseled around…responding they agreed or disagreed with suggestions.  I believe this particular one had 9 stations and I asked that they visit at least 5 or 6 in the alloted time.  We then discussed their responses and arguments as needed. Full post here.

Post-It Note or Stop Light Quiz has been around for several years, post here.  The basic idea is for students to place their name on the back side of the quiz.  They respond on the front side, self-assess to determine their level of understanding/confidence and place it in the corresponding space.  Its a nice visual for me yo scan as they leave the room in determining what’s next the following day.  I have RYG folders for them to drop their papers into when we aren’t using stickies.  Red – needs some help, most of the time these are the students who have been absent.  Yellow – still lacks confidence, maybe a little more practice.  Green -Got it! Ready to move on.

Flip for Answers -I like having sttudents create their own problems.  When they enter class the following day, they can exchange, work each other’s problems, then check.  The sticky can serve as a cover-up for the solution.

Notice & Wonder The last suggestion came during our ppschat last winter Powerful Problem Solving by Max Ray, his post here.  If you aren’t familar with it, you need to look it up!  His Ignite talk is great too!   With student work displayed, either patterns, data collection, graphs, various models or solution approaches…give each students 2 stickies, preferably 2 colors.  One is for something they notice, the second is something they wonder while viewing other student approaches, etc.  They attach it to the samples.  Continue to visit each station, reading others notice and wonder postings.  This should be a nice springboard for class discussion.

# Gallery Walk #ppschat Challenge

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A common theme in many chapters of Powerful Problem Solving is Gallery Walks. Several techniques are offered throughout the book, but the common goal is to allow students to view their classmates’ approaches to problems.

One of my faults with online book chats is lack of follow-through. I can sometimes use an extra nudge of accountability. There are often so many great ideas and strategies in the books we are chatting that I get overwhelmed and not sure where to begin. Advice: pick 1 thing. Try it. Reflect. Revise. Try it again.

So here is my attempt at a gallery walk. I simply cut apart a pre-assessment for a Formative Assessment Lesson and each pair of students taped it to a large sticky note, discussed and responded. I was confident in many of the questions, but my goal was to identify the few some students were still struggling to understand completely, mostly questions involving transformations.

1. The large majority are fine with creating a possible equation, given the x-intercepts.

2. Initially these students tried -6, -4 and 2 as their intercepts. I asked them to graph their equation then reread the instructions. Oh. They had read write an equation, looked at the graph for possible intercepts and failed to read the y-intercept of (0, -6). One quickly stated the connection between y-intercept and factored terms and was able to adjust their response with ease. I believe it happens often to see a graph skim question and think we know what we’re supposed to do, only to realize skimming sometimes results in miseed information.

3. Within the lesson, many students quickly realized when a factor was squared it resulted in a “double root” and the graph would not actually pass through the x-axis at that point.

The 4 transformations seemed to causes the most disagreements. These were the ones we discussed folowing our gallery walk. However, it was during the gallery walk most students were able to adjust their thinking.

4.i. Listening to students as they were at the poster helped me realize there was not a solid understanding of the reflection across x-axis and maybe we needed to revisit. Possibly, they are confusing with across y-axis?

ii. A few students disagreed initially, but the convo I overheard was addressing that changing the x-intercepts was not sufficient, they looked at the graphs, then said, the functions needs to be decreasing at the begiining, that’s why you have negative coefficient.

iii. Horizontal translations always seem to trick students up. One disagreement actually stated ‘they subtracted and did not add.” Of course, we definitely followed up with this one.

iv. This pair of students argued over which one was right. The expaned version or factored form. Simple, graph the new equations and compare to see which one translates the original up 3 units.

A1 & A2 I believe they’ve got this one.

B1 & B2 some confusion here due to the extra vertical line in the graphic. This student was also interchanging graph & equation in their statement.

I thought the gallery walk was a good task to overview some common misconceotions. It was not intimidating, students were able to communicate their ideas, compare their own thinking to others. I truly tried to stand back and listen. They were on task, checking each other’s work. Each station allowed them to focus on one idea at a time. They were talking math. Most misconceptions were addressed through their discussions or written comments.

Having a moment to debrief the following day highlighted the big ideas students had addressed the previously and reinforced the corrections they had made. This was so much more valuable than me standing in front of the room telling them which mistakes to watch for. Their quick reflection writes revealed majority have a better ability to transform the functions, which was my initial goal for the gallery walk. A few still have minor misgivings that can be handled on an individual basis.