My plan for next year….

Sorry for the reblogged posts of late.  I am researching what kind of practices I want to incorporate into my classroom for next year and am “compiling” blogs about the pieces I need. 

As I work more and more with my 8th graders, I am of the same mind as Jim Scammell.  The students who are taking homework home and bringing it completed the next day are the ones who are good students and driven by grades.  Those students who truly need the practice leave their materials at school because they either go out and hang with their friends all night, or go home and take care of their siblings because their parents are going out and hanging with their friends all night.  Our district uses a Developmental Design behavioral strategy, and with that is a lesson plan layout of Spark/Lesson/Reflect- which is similar to Jim’s model of class.  I have 60 minute classes so my breakdown is normally 15/30/15 (For 3Acts, my time is typically 15 Act1, 30 Act2, 15 Act3).  During this time, I try to allow for as much in-class work as possible.

With that, formal Direct Instruction has gone out the door for me.  Typically I present student with some type of problem, or more recently a picture or video (in a 3Acts format).  I allow students to work on the problem, walk around and check in on students, and offer quick help.  One thing that my students are not used to when they first enter my class is the type of help I provide them.  I ask questions (normally those I listed in my previous blog), I never give answers and that throws students for a loop.  After a few minutes of good student struggle, I discuss things about the problem as a class.  We list strategies to try, and work through them.  One thing I don’t do during this is erase any strategy, just because it didn’t work the way the class is progressing through this problem doesn’t mean it won’t be helpful in future problems.  We get a problem done, and they are given another that becomes student led.

dueling+discounts

When that problem is finished, we hit the 30 minutes of practice and this is where I would look at splitting students into either similar solver or mixed skill level groups- depending on what activity I had planned for the day.  This is my time to give differentiated instruction to my similar solver students, and to support student discourse in the mixed level groups, similar to what was posted on the Life of Mrs. Rilley.  Which group students would be placed in would depend on whether it was a discovery day or practice/application day.  I really want to be careful of creating a “tracked” theme with myself or the students as evidenced by Fawn Nguyen.  There will be days where I will utilize both groupings.  When students are discovering or applying mathematical skills I want a mixed group so students approach the problem from all angles.  This also provides the opportunity for a lot of mathematical conversations about what method students should implement and whether their answer is correct.  During practice time I want to correct student mistakes or misconceptions and would provide work appropriate to their challenges with the mathematical concept (group students with a similar misconception, group by a missing skill, group students who have a great handle of the topic and provide them with enrichment activities).  With Ashli’s approach to grading, students would not see this as any type of leveling of “smartness”- and would realize the grouping as a result of need.

After we have the problem worked out, students would get either the big reveal or an exit pass.  When students receive the big reveal they would be expected to discuss the similarities or differences in their work and answer.  Students would show their work (via a doc-cam) and work through how they solved it and what troubles they encountered (if any).  Since I would have them in groups, there would be 4-5 presentations and students would share out reporting duties.  Before students left class in this scenario, they would be expected to complete an exit reflection form- giving me information on how they perceived the activity, what things were good/bad about it, and what they now know and still need help with.  This will help me “tweak” the activity to my student’s needs.

Reflection form

If my class is comprised of practice/application then all I really want to do is give them an Exit Pass that has problems from the practice on it, as well as 3 Reflection questions so that I can better address these problems for the next class period.  The problems on the Exit Pass/Reflection form would be graded in my gradebook, but not for students.  Instead, questions about their work will greet students to create conversations about what they need to correct.

Exit Pass

Using these forms will allow me to differentiate instruction for the next day, which will improve student performance in class.  I am still thinking out formal end-of-unit assessment, but this plan is really taking shape in my mind and I am excited to implement it for the upcoming year.

 

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Binomial Powers

Act 1:

What questions do you have when you watch this video?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

Any of these type of questions will lead students down the inquiry I hope to explore with them.

  • Is there a pattern?
  • What is changing from one frame to the next?
  • Is the pattern constant?
  • What are the different different pieces called?

How do you square binomials?

ACT 2:

Once again, this video can create a few different paths of exploration.  We can explore:

  • Patterns
  • Squaring Binomials

These are both excellent topics and students generate a lot of classroom discourse discussing each one.

  1. Patterns:  Students will notice that I first make a square, each side the same length as the initial picture.  The thing that switches up is the replacement of a “flats” for  “longs”, “longs” for “cubes” and then fill the square with “cubes.”  Once students get the idea of the first couple of squares, then I will introduce math vocabulary for these: “flats” = x^2, “longs” = x, and “cubes” = constant.
  2. Squaring Binomials:  Students will make the same observations as the pattern students.  Using the math vocabulary, they should create expressions of what they are shown, and what it becomes.  Students may make notes of individual changes; x^2 replaces the x, etc.  This should lead them into the idea of squaring, or raising the binomial to the power of 2.  Discussion should happen about how the constant value is added at the end, wasn’t the constant replaced by x?  Some students will even try to physically measure the cubes versus the long to create a ratio.

ACT 3:

  • 25 cubes (25)
  • 4 longs (4x)
  • flat, 8 longs, 16 cubes (x^2 + 8x +16)
  • 4 flats, 8 longs, 4 cubes (4x^2 + 8x + 4)

Extensions:

  • Patterns: I ask students to create their own binomials, their squares and sketches of diagrams of both.  I will then ask what happens if there is a negative on either the variable or constant and how that effects our pattern.
  • Squaring Binomials: I ask students to try a quadratic and see if they can square it.

Reflection:

  1. Ask students to summarize what the activity was about.
  2. Ask students what mathematics they explored.
  3. Ask students what they understood really well.
  4. Ask students what they felt they needed more instruction on.

Stacking Styrofoam Cups

Today I wanted to revisit Algebraic equations with my students.  I decided to use Andrew Stadel’s Styrofoam Cups 3 Act lesson.  The thing I like most about this lesson, is that it has a lot of “hidden math” conversations available- more on that later.

I showed my students the video, and asked them to write down 3 questions they had about it.  I really stress doing this with 3 Act lessons, student always hit upon what you want them to learn and they have the credit for it!  I also like it when I throw up questions and students start to evaluate them and can tell me which ones are similar or even which questions actually answer other questions submitted.

One of the hidden math conversations that I use during this time is questioning: what makes a good question?  Since my students have not seen Mr. Stadel before, I can’t even begin to tell you how many questions I got on his appearance; where did he buy that shirt, why is he wearing khakis, who is he, where is he at, do you know him?  I am a believer that all questions are good as long as they are asked with a desire to really obtain knowledge.  Many of my students were asking these questions because they wanted to gain attention from the class, and we talked about if these were good math questions.  Students defined good math questions as ones that focus on what we are trying to learn and how we can solve problems.  I was OK with that- I may even make that a mini-lesson next year and create yearly posters on questioning.  After we got through the silliness, we got down to business.

Stacking Cups

This is what the board looked like today.  Even though there is teacher writing, this was all student driven ideas.  I have found that I like students showing WORK on my whiteboard, but for idea expression I need to regulate it to be efficient and on task.  The ?’s were ones they finally decided on to be good math questions for the video.  They then wanted diagrams showing the measurements of the cups and door.

The first great conversation that came from this video what what measurements we needed to solve this.  Students needed to know the height of the door and height of the cup, nothing more.  When I asked them to take 2 minutes to come up with ideas on how to solve this, a great discussion ensued.  I asked my students to relate what their concerns were, and they were confused about the stacking process and how to properly address it.  One student showed me this model to represent the difference in stacking cups and the data they were using.

IMAG0482

This represented the height of 2 stacks of 5 cups, and students decided they needed the measure of the lip of the cup to continue with the problem.

After this discussion another great discussion came about when we looked at units.  The cups were in centimeters, the door inches.  Students knew they had to do a conversion, but many could not remember what the conversion rate was.  One student remembered that it was 2.5cm = 1in so we rolled with that.  I knew this would only serve for great discussion for Act 3.

The next task was to figure out how many cups would fit in the threshold.  As you can see from the whiteboard, there were a couple of variations that students wanted to try.  We worked on the first idea for a while, but students realized that this was a guess and check method.  They quickly decided that was too much work and I gave them another 2 minutes of partner thinking to come up with a different solution path.

Idea 2 is one that I have struggled with all year.  Students take formal algebraic equations and convey them in simple terms using basic operations.  None of my students considered writing an equation to model the situation, most just knew what steps they needed to solve the problem.  We have struggled with this concept all year, they are great with “fill in the blank” or “empty box” expressions, but those darn X’s.  Cognitively some students are not ready for that type of representation.  After they solve this problem, as a class we go back and formalize the notation- and I will also show them how their solution method connects to the properties of equality in generating a solution.

One side-note here, when we start in the solution process many students start complaining about the long decimal in the height.  This generates another good conversation about precision and what is appropriate to use.  Even though students may not believe it, they have a great feel for what precision is.

Here are some student samples of work.

IMAG0481 IMAG0480 IMAG0478

A couple of things to note, students forgot to add the last cup on the stack back into their answer.  When they did the math, they envisioned the cups at the pattern, or 1.3.  Out of all the errors to find this one proved to be the most elusive.

One thing I feel needs to be brought to 3Acts is reflection.  While we can reflect on why our answer was different, it is a totally different thing to reflect on what we did for the project, what math was involved and why we did the activity.  Right now I am reserving 15 minutes of class for good reflection time, and for this activity I put up an reflection outline for students to model and help organize their thoughts.  I hope that after a couple of Lessons with 3Act, I will be able to just hand them a blank sheet and they can write away.  Here are a few examples:

IMAG0483This really gives me an idea of what students took away from the lesson.  Some were very superficial, claiming the height of the door and cup height were key ideas.  Others connected the rim as a slope or pattern of the activity.  Other students focused on the rounding and unit conversion as important things.  Even though the main focus of this lesson was linear models, each student gave me a different perspective of what they noticed during it.  I will go back tomorrow and have a review of what we did and the mathematics involved, and as a class we will fill out a reflection form.  We will practice linear problems, and I will hit them up the next day to see what things truly stuck in their learning.

Conference Seating -3Act Math

ACT 1:

Recently I had to go to a conference and when I walked in, this is what I saw

2972-conference-picture

What questions do you have when you look at this picture?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

  • How many tables are there?
  • How many people are there?
  • How many people are in a row?
  • How many people can sit at a table?

Any of these type of questions will lead students down the inquiry I hope to explore with them.

There are 512 attendees at this conference, how many tables are needed to seat them all?

How would this change if I needed to seat 900 attendees?

 

ACT 2:

Linear Equations is the topic I wish to address with my students.  Students will need to think of counting strategies needed to determine the number of people at the conference.  There are many different combinations of ways to figure this number out, but here are the stats:

There are 8 people seated at the head table, but they needed to use 6 tables (extra space was used for the podiums). Otherwise there are 4 sections of tables, 9 rows of tables to a section, 14 people to a row, and the table size is 6′. 

These are hard to identify by the picture so the students really work on the following pieces of information: 6′ tables, 9 rows and 512 attendees.

I also ask students to create an equation for the number of tables that are needed.

 

ACT 3:

For comfort reasons, 2 people are seated per table.

You will need 7 tables per row.

4 x 9 x 7+ 6 = 258 tables.

Students will typically write this type of equation based on attendees:

512 = 2x + 8

504 = 2x

252 = x

Then I typically ask them how many tables per row, and what type of configurations makes sense for the conference (does one LONG row really work?).

 

Extensions:

After we decide the general equation for the conference, I then ask the students how many tables are needed for varying amounts of attendees: 200, 900, 1200, 2500.  What type of configurations would work for these new conventions?

 

 

PowerBlocks -3Act Math

ACT 1:

This video was inspired from comments made in Dan Meyer’s My Opening Keynote for CUE 2014.

What questions do you have when you watch this video?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

Any of these type of questions will lead students down the inquiry I hope to explore with them.

  • Is there a pattern
  • What is changing from one frame to the next
  • Is the pattern constant?
  • Is this an Arithmetic Pattern?
  • Is this a Geometric Pattern?
  • How did you make a cube?

ACT 2:

Once again, this video can create a few different paths of exploration.  We can explore:

  • Patterns
  • Rules of Exponents

These are both excellent topics and students generate a lot of classroom discourse discussing each one.

  1. Patterns:  Students will notice that I first make a square, each side the same length as the initial picture, and then a cube with each side the length of the number of cubes in the square.  The cubing part is where most classes will struggle, many will just try to create a cube out of the preceding square.
  2. Rules of Exponents: This is what I designed this video for, in an attempt to create a visual representation of (x^a)^b = x^(a*b).  Using the unifix cubes created a quick, easy way for students to quickly see and do the mathematical calculations.  The powers I started with were easily recognizable visually: (x^2)^3.  One thing my students start to see is how the base figures into all of this, we normally pause the video and use the SMARTBoard to draw lines to create the pattern of multiplying the base.  For example on slide 3: we circle the bottom left 2 blocks, one stack represents 2^2, two stack represents 2^3, four stacks represent 2^4, etc.

ACT 3:

  • Frame 9: (4^2)^3 = 4096 => 4^6
  • Frame 12: 5^2 = 25
  • Frame 13: (5^2)^3 = 15625 => 5^6

 

Extensions:

  • Patterns: I ask students to predict what a Step 4 figure would look like
  • Rules of Powers: Have the students determine a third power and sketch what their figure would look like & how many blocks it would take to create it.

 

 

40 Yard Dash -3Act Math

ACT 1:

This video was first brought to my attention from Dan Meyer’s My Opening Keynote for CUE 2014.  Turn the volume on mute when you show this to your students.  I also do not show the individual run, I start the video 15 seconds in.  I show the clips of Jacoby Ford and Terrence Cody ending at a minute in.

What questions do you have when you watch this video?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

  • How fast are they running?
  • How much of a lead does Rich get on run 2?  run 3?
  • How much does Rich lose by each time?
  • How big of a lead does Rich need to tie? to win?

Any of these type of questions will lead students down the inquiry I hope to explore with them.

 

ACT 2:

Once again, this video can create a few different paths of exploration.  We can explore:

  • The rate of the runners
  • Graphs of the runners
  • Equivalent equations

These are all excellent topics and students generate a lot of classroom discourse discussing each one.

  1. Rates:  This is one that creates a lot of arguments about precision.  Students normally start trying to time Rich by using the clock on the wall, or their wrist watch.  Some will break out the stopwatch feature.  I have other students use the watch feature on their phones.  Timing issues like accuracy starting or stopping the time, cause quit a disturbance with the students.
  2. Graphs: I love this part.  I normally show a clip twice and have students graph the race.  Independent and Dependent variables, scale factor on axis, and the solution of two lines are great topics to discuss.  Students really enjoy graphing the races and are really good at evaluating work and refining the process and answer.
  3. Equations:  This normally involves at least one of the first two processes and builds upon that.  Finding the rate of each runner (their slope) and setting their expressions equal to each other leads to when Rich is overtaken.  Having these expressions will also allow us to find the exact time Rich is passed and how much of a head start he would need to either tie or finish first.  I can’t think of a better introduction into solving systems of equations.

ACT 3:

I normally show the opening video to answer how fast Rich runs, and turn up the volume to allow students to know how much of a head start he is given in the other races.

 

Extensions:

Show them the 3 man race (1:10 into the video) and let them loose, it’s fun to watch.