Stinky Sneakers #MTBoS

Hello, stranger!

Thought I’d give this site a try. I feel like I’m finally starting to move out of the 3 year funk. And I want to be the reflective and effective teacher I was when I first started blogging.

Today was a day students were being called out to meet one on one with admin and counselors to build their schedules for next school year. So I knew they would be in and out of the classroom. I wanted something productive but also flexible for this reason.

Trying some lessons from Math Medic Algebra I, I ran across a review called Stinky Feet It reminded me of the old Ghosts the Graveyard. I basically cut up a worksheet, problems numbered, had an answer key ready to go and stickers!

Stinky Feet ideas! https://pin.it/djSUHV9

Each table got a Stinky Sneaker. We used individual whiteboards for work. I had problems cut apart on a table and they’d grab one then exchange for a new one as needed.

Every person on the team had to write down the original problem and show the correct solution in standard form. If everyone was correct, they picked a sticker to decorate their Stinky Sneaker.

I set a timer to ensure we had enough time to tally points before leaving. Each color sticker was randomly assigned points at the end. Points were -5, 0, 5, 10, 20. And the team with most points claimed victory.

Yes, I know some kids copied, but it was their teammates (not me!) pushing them to complete it so the team could get their sticker. And what do you know, but a few of them picked up some skill and understanding along the way.

Yes, some groups worked faster than others. But there was discussion and debate all around. It was a fun way to get some needed practice. They enjoyed it. Good energy during those classes.

What’s a cost efficient and creative structure you use to get needed practice or review that your students enjoy?

Reflecting on Formative Assessments

Every Story has a Graph / Target Quiz

Earlier this week, I gave a short Target Quiz – just one big idea.Students were given three scenarios and asked to create a graph to model the situation.  Out of the class, there were 4 students I felt I needed to pull over to the side for some one on one time.  I found they were often drawing the “shape” of what was happening rather than comparing the distance from home to time.

The one most missed had Tom walking up a hill, quickly across the top, then ran down the other side.  Yes, most kids draw the shape of the hill.  As opposed to the distance continuing to increase as he ran down the other side.

Whiteboarding Examples / Non-examples

The second Target Quiz was on whiteboards – students had to create an example of a graph, set of ordered pairs and a table of values with a function and not a function in each example.

I laughed as one table was begging me to give “real quiz” and take a grade because they knew that they knew!!  As I walked around the room, observing, asking questions – there were 3 students with some minor mistakes and 3 who were really struggling.  Upon questioning, they were able to identify when the example was given, but unable to create examples on their own.  With some “funneling”  – they were able to get examples of each, but I have them * to keep an eye on and requiz next week.

Deltamath Practice – immediate feedback from tech;

Teacher observation & questioning

We had a very brief introduction to writing domain and range of graphs in interval notation.  We spent some time in the computer lab today practicing this on deltamath.com.   I appreciate the immediate feedback they are able to see if they miss the question.  Also, how he has programmed the many different options for defining domain and range.

Many misconceptions were cleared as we learned whether to use the endpoints or extreme values (if they were not the same).  There was discussion about the open circles and closed circles and which inequality symbols were correct to use and when.  And yes, a few realized they were mixing up the x and y for domain or range.  I look forward to practicing this skill Monday after their experiences today.

Desmos Activity – Inequalities on a Number Line – Matching Tasks

For my other class, we will be solving and graphing inequalities next week.  So while in the lab today, we worked on Desmos – Inequalities on a Number Line and Compound Inequalities.  The first task was a good review and learning opportunity for the direction of the symbols.  I still had some students exchanging those up.  Most were correct in open versus closed circles and what that meant in symbol terms.  Though I did not make it to all of the students in the second task – I was trying to catch students on the two sorting pages of the first activity as they were going through.  For some it was as simple as a brief discussion about why one was the correct choice and comparing it to their wrong match.  There are about 4 students still having troubles on the first task.  And several have not completed the last task.

I feel like looking at their responses, I can use their examples as discussion pieces while we are looking at our notes next week.

I almost feel like there were not as many issues in the second task.  However, I still have several that have not completed them yet.  But I feel like using live examples from their work and discussing maybe two stars and a wish they would have for each student – may help them steer away from making their own mistakes.

I love the real time feedback I get as a teacher and how I am able to grab kids before they move on too far and help erase some of their thinking and replaced it with correct ideas immediately.

Someday – I’ll get to have a classroom lab… I hope.  Until then, we will keep on doing what we can.

 

Generalizing Patterns: Tiling Tables

Last fall after instructional rounds, one of the observers asked me if I would mind having some folks visit my classroom.  Sure.  They were most interested in questioning, interactions with students and use of Formative Assessment Lessons (FAL).

When they emailed to set up a date, we agreed on January 10.  Oh, wait.  This will be the beginning of a new semester with new students.  I won’t really know them.  They won’t really know me.  Great.  Now, I’m scared.  Oh well, let’s look at the positive – this will give me a chance to try out a new lesson.

I printed off 3 lessons to look at the evening before students returned to school.  I liked all three.  Building complex equations seemed perfect, so I began to prepare for it.  We were out for weather our second day back.  As I began looking over my lesson plans, it seemed the Tiling Tables was a better fit for the upcoming unit, so I switched gears.

I had done this lesson a couple of years ago, but never taught it in class.  As I began to revisit the task, I knew I liked it.  I knew it would offer some good discussion on ways to extend the patterns.  But wait.  These students barely know what a parabola is.  Would they have a clue as to how we would write an expression for a quadratic relationship?  Would I have a clue as to how to introduce it, this early in the semester?  No.

So I pondered for a while.  I would simply use the task as a way to say, we have the knowledge and tools to do parts A and B, but part C, well – that’s what we will be learning later in the semester.  It would give us a reason to learn it later, right?  Goodness.  What a canned comment.  By now, we had another snow day, so our visitors would be in our classroom on the 4th day of instruction.  I was stressing just a bit.  What was I thinking?  Starting off a new class with a FAL I had never used before?  We needed time to build some rapport.  Too late.  Let’s go with it.

I gave students the pre-assessment:

The class was divided pretty much 3 ways – Those who doubled the number of tiles, after all – a side length of 20 is doubled to get 40, right?  The second group had sketched the designed on the the grid paper which had been provided, however, they wrote answers for the 30 cm table instead of the 40 cm.  And finally, several had the correct number of tiles by extending the pattern on the grid paper.  But I ask how efficient this strategy would be for, say 300 cm table?  Hmmm.

We began the lesson the following day by giving 3 samples of work.  Last school year, I figured out, I could save paper by having them use the shop ticket holder sleeves to hold the sample work – allowing them to draw, sketch, etc with dry erase.

These instructions would help their discussions:

The first sample was Leon:

After some small group time, we shared our thinking with the whole class.  There was one student in particular who had confusion all over their face.  I encouraged them to ask the person sharing for clarification (using our starter stems).  I believe this is important to model and have them do early in the semester, so they become more comfortable with it.  Even with more explanation, they were still not seeing the pattern.  So another student shared.  Still no help.  Finally, a third students explained how they saw the pattern.  The confused student nodded and said, “Okay, I got it.”

Now, years ago, I would have said – great and moved on.  But I’ve learned…ask them to explain it to you.  They may say they’ve got it – just so you will move on, but how do you know they understand?   This student, however, could explain their thinking and were correct – they could even extend it to the next table size.

The next student sample was Gianna:

So many more of the students picked right up on Gianna’s approach.  The confused student – smiled stating they liked / could see this one better.  For me, it was listening and watching the students discussing – that brought me an a-ha!  This is the example we will use to generate the quadratic expression I was worried about!  The total whole tiles would equal (step x step) + (step – 1)x(step-1)  Yay!

Finally, we had Ava’s sample:

Many of the students could not make the connection with the side lengths on Ava’s sketches in the beginning.  Then they began going back and looking at their own sketches to verify the numbers Ava recorded in the table.  They noticed the same patterns and agreed with them.

After this final discussion – we went back to see if each student had answered the task fully.  We quickly realized though there was some good, correct thinking going on in their work – they had not addressed the questions completely.  The class agreed that Ava’s was the most complete with her table.  And it was interesting to hear their discussions of how they would explain to the other students how they could expand their responses to be better and more thorough.   One student even brought up it was challenging trying to figure out their thinking since there was no written explanations of what they were doing.  (I thought – yes, this is what I feel like sometimes too.)

As we continued discussing having thorough answers – I shared Ava’s data in a graph…  they were quick to see the quarter tiles always remained four and the half tiles being linear, a focus from 8th grade.  But what about the total tiles.  How can we write an expression to model that data?  And I took them back to the slide with Gianna’s work to look for patterns between the table size/step number and the total whole tiles.  We test our thinking with different sizes and it worked.  We tested our expression in Desmos…and what?  It hit all of the data points!

They had some experiences with the visual patterns – and good feedback to me about liking them, but still having to think.  This task reinforced some of those ideas.  IN their reflections – though many may have preferred someone else’s sample work – they “saw” how Gianna’s work led us to a more efficient expression or even Ava’s approach to orgaznizing the data in a table was pretty helpful to see the patterns so we could find describe the expressions.

Total Tiles = 4 quarter tiles+ 4(n-1) half tiles + n^2 + (n-1)^2  whole tiles.

I will definitely be using this lesson in my future.  It brought just enough confusion, but great opportunity for sharing and discussion.  And the observations were great.  Students were not shy.  At the end of the day – I was amazed we had only been together for 3 or 4 days… wow, this is going to be an outstanding semester!

Interpreting Distance Time Graphs

On the 3rd day with a new group of students, I had visitors from some other districts in our classroom.  I was nervous – I really didn’t know these students yet and they certainly didn’t know me.  I had chosen Interpreting Distance Time Graphs lesson from MARS to begin our semester.  Although this is listed under 8th grade, it leads to some great discussions and uncovering of ideas and misconceptions.   The Keeley & Tobey book also lists “Every Graph has a Story” in the Formative Assessment Strategies.  This was the ideal lesson to introduce our first unit on functions, while trying to be intentional with planning FAs.

Pre-Assessment

Telling students it is only for feedback, not for a grade seems to drive most of them to really share their thinking.  After reading their responses, I had some ideas of how I wanted to change the lesson up a bit from times past.  The first time I ever used this lesson was around 2011-2012.

Let the Lesson Begin

We began our actual lesson with only the graph in this picture.  I asked students to jot down 3 things they noticed about the graph.   Pair share.  I called on students randomly with my popsicle sticks, then allowed for a volunteers (this was something @druinok and I had read in EFA2, which allows everyone to be heard).    We then read the scenarios aloud and at the table groups, they discussed which story was model by the graph.

Next I took one of the scenarios we didn’t choose and asked them to sketch a graph on their whiteboards to model it.  We had about 5 different overall graphs – I drew on the board and let them discuss at their tables which they agreed/disagreed with.  Then we shared our thinking.  Some very good sketches and great discussions.

Open Card Sort

Many years ago, a colleague shared the idea of open sorts, something she had learned from a John Antonetti training.   I instructed students to remove only the purple graphs from their ziploc bags.  (Side note suggestion- use different colors of cardstock and this allows them to quickly grab the cards they need, ie the purple graphs, green scenarios OR blue tables.  I used to have all the same color and we wasted a lot of time sorting through which cards we needed).  In pairs, they were sort the graphs any way they wished, the only requirement, was they must be able to explain why they sorted them as they did.  Again, sharing whole class led to seeing some details we had initially noticed.  If you’ve never done an Open Sort – let go and let them show you their thinking.  You might will be amazed and wonder why you’ve never done this before.  They love to think.  We should let them.

List 3 Things

A couple of years ago, I began asking students to list 3 things they noticed or knew about their graphs – anytime we were interacting with a graph.  IF you ask them to do this enough, it eventually becomes habit.  I also like this approach because it gives them a chance to survey the information in the graph before they start worrying about / answering questions.  Today, I asked pairs to label their whiteboards A – J and I set the timer.  They had to share/discuss/jot down 3 things about each graph.  Once again, I used popsicle sticks to randomly call on a few students.

Graph & Scenario Matching

Using the “rules” listed in the lessons powerpoint, students were then given time to discuss and match graphs to the scenario.  This went so much quicker than times I’ve done this lesson before.  I believe it was because they had already interacted with the graphs twice…they were not “new” to them.  I will definitely use the Open Sort and Name 3 Things before matching tasks in the future.

I gave them a chart to record their matches.  We then shared out our matches.  Each time, I neither confirmed or disputed their matches, but rather would call on a couple of other students to agree/disagree.  After some discussions, I came back to the original student to see if they agreed / disagreed with their original match.

One of my favorite graphs is this one –

And our final sorts…  And again – Scenario 2 is always up for some debate.  It reads: Opposite Tom’s house is a hill.  Tom climbed slowly up the hill, walked across the top and then ran down the other side.

Though every student did not get every match exact, there were several a-ha’s during the lesson and questions asked.  I look forward to reading their post assessment.

I’ve used this lesson as written many times with much success.  However, just making some adjustments prior to the matching made a vast difference in the amount of time students needed to complete the task.

Let me know how this lesson has gone / goes for you if you use it.

#MTBoS12days #teach180 Post 2

I failed tp blog all semester, but I am proud that I was successful with daily #teach180 posts. I revisited A Look into Learning and with the exception pf about 3 weeks, I was able to post a couple pf sentences, screenshots, picturea from a class each day. Those weeks, I fell behind I woyld […]

#MTBoS12Days Challenge Post 1

Share something new you tried this year in your classroom. We went to a 4×4 semester block this past year. Loosing time was an issue my colleague amd I worried about with Algebra 1. There was simply no way we could get it all in. So we decided to do a ‘spiraling’ unit of sorts […]

I Still Have a Question About…

We did not get through all I intended today to allow some students who wanted to watch the inauguration that opportunity.  But we did address a couple of more questions from the 2-minute reflection students had completed.  You can look back to the previous post to see the original task.

We addressed the two blue questions in the after lunch class.  Why can’t you multiply the numbers by each other?  Well, lets see.  Again, as I did with another class, I asked them to add two numbers that would give us 18.  We graphed our responses, then graphed the equations x+y=18.  And likewise, give me two numbers that would multiply to give us 18.  We graphed our responses along with xy=18.

When we added the equation to the product set, students were caught off guard with what they saw.  WHY is there a graph in the third quadrant?  Will that red curve ever cross the y-axis?  Doesn’t it eventually get to the axis?  Again, just attempting to address their question, by looking at a couple of horizontal translations and introducing them to that boundary line called an asymptote, led to even more wonderings.  Which is what math class should be about.  As long as they were on task, I continued to go with their questions.  Only 3 students were not interested, who would likely have been off task no matter what I chose to do, so I made the decision to keep going with the majority’s curiosities.

Another student asked about our statement “x cannot be in the denominator” but yet when we find rate of change with a table of values, we compare y over x. Hmmmm.  Good question.  So I gave a table of values, asked the student to talk me through finding rate of change.  When we wrote our ratios, what values did we use?  Not the actual y and x values, but the change in y over the change in x.

The entire class really reminded me that we can say something with an intent, but what our students hear is something else…how important communication is, how important is it that we allow some time to process and clarify their misunderstandings.

Finally, we addressed the question, we’ve been told x’s exponent must be 1 in the linear function.  We’ve seen greater than 1, but if it is less than 1, can it still be linear?  Let’s see.  Go to y=, type in x and choose an exponent less than 1.  What do you see.  Share with your neighbors.  So, how would you respond to this question, students?

To me, this was one of the most productive two days I’ve had in this class.  Students were engaged because we were addressing their questions.  I’m not sure I actually answered their questions, but I provided them with some examples that allowed them to answer their own questions.

Identifying Linear Functions

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…

Stacking Cups… part 2 #MtbosBlogsplosion #myfavorite

I like big cups, I cannot lie.

We stacked cups in the first few days of school…

I’ve been stacking cups since…uh.  I think my first NCTM Navigating Through…  book was around 2002 or so.  Its been a while.  I have vivid memories of discussions in classes from room 125.  Yep.  It’s been while.  Long before there were songs about Solo cups.  My guess, a few of my sets of cups may be that old.

They’re a cheap resource.  Find a buddy or two, each buy some different sizes, split them up and you’ve got some varied sets of cups.  Hmmmm. What all can you do with cups?

I.  This past week, I began by displaying a single cup and asking students to generate as many questions as they can about said cup.  Set the timer.

II.  Turn to your groups and share your questions.  Then say whether it was mathematical in nature or not.  Each group shares out 1 question with the whole class.  Then if anyone had a question they wanted to share that had not been included.

Yes, we actually looked at the etymology of cup…wondering where the name originated.

III.  a.  I went with “Why am I stacking cups?” as my transition to the task.  You guys are engineers today.  Packaging designers, specifically.  Design a box to ship a stack of 50 cups.  They needed tools, so I gave each group 4 – 7 cups (did I mention some of these cups may actually be older than some students?), each group with a different size/brand of cup and a measuring device.  Set the timer 5-7 minutes depending on class.

III b.  As I monitor their work, I usually here a few moving in the wrong direction.  I pause the timer and their discussions…attention at the board:

I need some help.  One group has a stack of 5 cups measuring 14 cm, and their height for a stack of 50 cups would be 140 cm.  Do you agree or disagree with their response?  Turn to your group and discuss.  Set the timer.

I have some varied responses usually.  When I get to someone who disagrees, I ask how tall they think the box should be and they come to the board to explain their reasoning.

III. c. Yes, believe.  You will sometimes have a class where no one disagrees with the 140 cm response.  Have them to create a table of values to record their measures for 1 cup, 2 cups, 3 cups, etc.  Set timer.  Usually during this time you will hear the a-ha’s.  Bring the class back together to discuss / share their thinking.  Modeling how the cups would be stacked.

Okay, so moving on now.

IV.  Once we feel fairly confident in our expressions. I ask them to find the height of a stack of ____ cups for their group.

V.  Well, what if I had a box that was 80 cm tall, what is the largest amount of cups could I ship in that box?

VI.  At that point, we share our expressions we’ve created for each type of cup.  I put all cups on display and ask groups if they can match the cup with its expression for  total height (cm).

This leads to some light bulb moments for a few students.  They can now see how different parts of the expression represents different physical parts of the cup.  I always thought it would be fun to list the expressions on cards and they have to match to the cups and play the Race Game from The Price is Right.

VII.  For other practice, we use the expressions:

• simplify expression
• find the total height of 50 cups
• how many cups to make a stack of 80 cm?

VIII.  Closer choices

• What’s one take-a-way from today’s task?
• Something I learned… realized… or was reminded of…
• How are the expressions alike?  different?
• Which two expressions are most alike?  Explain.  Which two are most different? Explain.

IX.   Systems

Next, have students compare their cup stack to another groups stack of cups.  When will the two stacks be equal heights?  Just using my groups’ expressions above, they get at least 6 practice problems.  You can leave it as an open task – students can choose tables of values, creating equations to solve or even solve graphically.  The key component is to ensure they interpret their solutions (x, y) = (cups, stack height) within the context of the scenario.

A Light Bulb Moment #MTBoS30

Here’s a shameful post – one of those things I knew it happened, and wanted to believe I knew WHY it happened, but in reality…I was struggling.  Until yesterday…  in after school, tutoring a student for upcoming comprehensive final.

I know how manipulating an equation will transform the graph of the function.  I can predict it quite efficiently.  I know and my students even recognize that y=a (x-h)^2 +k will translate left / right… “opposite” of what the sign / operation is in the parentheses with the h.  But why?

So one day, as I heard myself describing the transformation to a student, I thought…that sounds so stupid.  I mean, hello.  No wonder it doesn’t stick.  It makes no sense (what I had just said).  In my mind, I heard Pam, the student, asking why do we change the sign of the h, but not the k?  Why does the h move opposite, but the k translates direction of the operation?

I started really making more sense to myself when I focused on function transformations in Algebra 2 and transformations for plane figures in Geometry the past couple of years.  But I was definitely not satisfied with what I was saying.  I believe our work with equations of circles related to slope and pythagorean theorem is what started chipping away my lack of true understanding.  Because I began to explore, ask questions.  I was curious.

When I started having students create tables of values, seeing how the values changed with each transformation helped, but not to the level I’d like.

So, here it is folks… when we’re looking at the y=a(x-h)^2 + k…the h is actually NOT the x-coordinate of the vertex.  The h is the transformation back to zero (origin).  Can we look at it that way?  Does that even make sense?  The x-value is where we moved from the origin.  The h will return us back to the origin.  I know its not where I need to be yet.  But I’m open to listening to other’s ideas here.  I’m not satisfied with “it moves just opposite of what we think.”

My next failure as a teacher saga…I don’t do a good job of helping students differentiate between linear functions and arithmetic sequences.  I’m starting to muddle an understanding.  At a moment in time, they are comfortable with each idea, but they continue to mix up when its a first term, n=1 OR an initial value n=0.  The best I can do for now, verify your equation works for the values…