Unit Conversions Piece

Before #TMC16, I had asked for some help with lesson ideas for unit conversions.  Anna Vance (@typeamathland), replied with this introduction that she has used

I was thinking this is pretty awesome!  How can I make it better?  How can I use it to create a cohesive lesson on unit conversions.  I got another good idea from Gregory Taylor (@mathtans) that I would like to incorporate as well:

At TMC, I got to talking a little more with Anna, and we had BOTH been trying to find a way to make the conversions a little more manipulative for students.  I was still thinking numbers, but Anna thought shapes!  And the beauty of shapes is that I can choose ones with symmetry, so that each fraction could be turned either way!!!  This to me was the awesome part.

So with a lot of trial and a little error, I created some cards that can be used to intro how dimensional analysis needs to be set up to cancel one thing and leave another.  Here are the "conversion factors"

There are 6 of them, but they could all be flipped the other way, making 12 possible options for students to choose from.  I also made cards to be the start and end of the conversion.  Nothing too exciting to see here:

Then I played around to make sure I had enough of everything, but not too much.  And I think I do.  I like that sometimes there is only one solution, and sometimes there are three.  At this point my plan for this activity would be to show the start and end I would like on my document camera and have students work in pairs.  Then if they find a solution I will prompt, "Can you find another?"  Sometimes they will be able to and sometimes not.  Hopefully some students will be able to justify why they can or cannot find another solution.  

I played around a lot with it and I don't want to put all the pictures here, because I tried to find all the solutions, but here are a few:

Start with a square and end with a hexadecagon has at least two solutions, but start with a octagon and end with a rhombus only has one.

Obviously this is not an entire lesson, so I still have some more planning to do, but I like what I've got so far and I think it will give my students some good playing and thinking about math opportunities.  I am trying to collaborate with the science teacher on this standard, so I've got a lot to do before I can be all the done thinking about it.

I have some other notes on what the rest of the lesson might be like, but really this next part is for me, so skip to the comments and throw questions or concerns up there.  I'll post links to the docs at the bottom too!

Notes for Me:

• Me:  Shape manipulatives
• Science:  Number manipulatives
• Think Input/Output (where input/output have the same value/amount/quantity)
• Me:  discuss conversion factors need to have a value of 1
• What can we multiply by without changing the value of the input?
• Science:  look up conversion rates
• Me:  notes
• Science:  guided practice
• Mistakes?  Video?  Student created mistakes?
• Should we make an assignment menu?  Due for both classes?  Revision encouraged throughout?
• I want students to journal after doing the shape manipulating! Need a good prompt.
• Introduce new shape.  Create one conversion factor that will allow you to convert this shape to any other shape in your set.  How do you know this works?  Maybe it doesn't, but you're close.  How do you know it doesn't work?

Here are the documents:

Conversion Factors WORD

Conversion Factors PDF

Start End WORD

Start End PDF

Let me know what you think!

-Kathryn

Stacking Cups

I did Stacking Cups by Dan Meyer again.  I haven't done it in a few years and I think my intro made it more successful this time.  I let the students lead themselves to the problem.

I just held up a cup and asked them about it.  They talked about how it could hold water and food.  I shared that they were pretty dirty because many other students had touched them.  They were less eager for food and drinks then :)

Eventually they started talking about how you could stack cups and I asked for them to keep listing different ways to stack them until the stacked them inside of each other.  Then I set a small stack next to my water bottle and had them estimate how tall my water bottle was.  We discussed how even though the the cups stacked halfway up my water bottle, we would need more than twice as many.  I take a few estimates and then we calculate.  I set it up by asking what else we could measure and three (of four) classes decided to measure me :)  The other class chose a particular student and I went with that.  I actually think that was the most fun!

I also gave them some guiding questions based on standards we've been working on.  Here are the questions I asked them to answer in their groups:

1.  Identify the independent and dependent variables.  Select a letter to represent each.
2. How tall is one cup?  How tall are two cups?  Three?  How could you organize this information?
3. Write an equation to model the situation.  Check to make sure it matches the values that you stated in [2.]
4. What is the slope?  Interpret the meaning of the slope in the context of the situation.
5. What is the y-intercept?  Interpret the meaning of the y-intercept in the context of the situation.
6. What is your estimate for how many cups tall Mrs. Freed would be (closest without going over)?  Explain how you determined your estimate.

The next day we discussed as a class.  I compiled all of the equations groups had created and we talked about how they related to the cup.  We discussed how precision was important in order to calculate a correct estimate.  Some groups had calculated an estimate and then put their actual estimate down because I said closest without going over.

This is the class that decided to measure how many cups tall this student is.

After discussing we measured and then I had them complete an individual reflection on google classroom.  I asked them about an equation I had made up.  The questions I asked were very similar to the ones in their group.

What surprised me the most was how much more willing to do the math they were when they made up the question.  And they were so surprised when they saw that I knew they were going to come up with that :)

-Kathryn

Desmos Picture

My students in Algebra 1 are learning about functions, and we start with domain and range of relations.  Due to amazing #MTBoS resources, I had some excellent resources to integrate into a week long lesson, which started with pictionary thanks to +John Scammell (@scamdog) and his post here.

My one good thing, however, is about the performance assessment I assigned to students using +Desmos free online graphing calculator.  On our PD day at the beginning of the semester I attended a session by our curriculum director @montemunsinger about creating rubrics for performance assessments, so I used it to help set up this rubric based on my standard related to domain and range:

The assignment was to create a picture in Desmos with at least 10 relations.  In addition, students must restrict the domain for three relations and the range for three relations.  Then students complete a reflection where they explain one domain choice they made and one range choice they made.  The reflection is important to me because it is their opportunity to share what they learned, not just what they created via trial and error. Don't get me wrong, the trial and error aspect of Desmos is the only thing that makes this assignment at all possible for my students, but I want to make sure that through the trial and error process they are learning something.

So my #onegoodthing is watching my students create!  We worked on it off and on throughout almost the whole week.  Some students jumped right in and have created some awesome things, others wanted to copy a previous Desmos picture they saw, but could explain polar coordinates to me (shocker!) so I made them start over (aka not copy).  Some students needed a lot of guidance at first ("Try y=mx+b and substitute some things in for m and b until you get what you want.  Now what part of the line do you want for your picture?  How do we do that?") and then were able to take off and just ask me for help with troubleshooting when they made an error ("Why did my whole line just disappear when I did the domain?" *I check and see -7.5<=x<=-8*  "Remember to put the minimum on the left and the maximum on the right...").

Hopefully, I can get permission to post some pictures here, but let's just say I've seen Olaf, a Christmas tree w/star and presents, batman symbol, personal designs, etc.  My favorite part, however, is when the students learn about new types of relations.  "How do I make a circle?"  "How do I make an oval?"  "Can you help me make this rounded?" I don't usually get to share about circles and ellipses in Algebra 1, but we did this week!

-Kathryn

Rearranging Equations

To start solving multivariable equations for a variable, I have been using this task.  (Note:  For viewable files, you must download them in word.)

Here are the instructions:

 And here is an example of the cut-outs I give to each group of students:

The gist is that they have to decide which equations are derived from the "start" and explain what happened.

This is the students' first exposure to this in my classroom, so they must rely on their background knowledge solving one variable equations and with multivariable equations in the past.  Some students look for equations that have one solution in common with the "start" equation.  Some students using adding/subtracting/multiplying/dividing reasoning as we do in solving one-variable equations.  But this time I had a student use reasoning that was totally new to me, but also super-awesome :)

Her reasoning was based on comparing these equations to her prior knowledge of adding/subtracting from elementary school.  Consider the following set of equations "5 + 3 = 8" and "5 = 8 - 3"  In elementary school they were taught the relationship between these statements.  So my student used this reasoning to explain that "2x + 6y = 12" must bet the same as "6y = 12 - 2x"  ISN'T THAT AWESOME!

I feel that this is the impact conceptual understanding taught at all ages (in this case driven by common core) can be so beneficial to students.  Also can we just celebrate for a second that this student was 100% comfortable extending from numbers to algebra?  I think that is the epitome of deep conceptual understanding!

I'm excited to share this reasoning with my classes on Monday so that others can benefit from it.
-Kathryn

Number Line Task

So this is the style of many of the tasks that I do in my room, heavy on math, lots of opportunities for me to check in with students as a group and individually, reflection, and math, not always context.  But some people at #eduread wanted to see an example.

I chose to share my Number Line Task because I know I want to do it in the first week of school, but it needed some modifications first!  I did this task last year, but with less structure and it was chaotic, and one student in each class ended up sorting everyone.  Ugh!  No one got to learn!  So I made changes this year to hopefully help with the learning part of it.  It is in three parts.  Part 1 is an individual task (basically think time for each student), Part 2 is as a group, and Part 3 is both class and group.

Number Line Task:  Part 1

FYI:  Each group is getting four different numbers on their paper.  I have 6 tables of 4 students, so I have 6 different sets of 4 numbers.

I would expect students to take about 15 minutes on this.  I plan to have them share with a partner and write one thing their partner said on the back of the paper.  I am doing this because I really want all students to have a voice in my classroom, which means they need to get used to sharing and listening BOTH.  So why not start right away?

Number Line Task:  Part 2

Then as a group, students will complete the following reflection questions

[1.]  Who is the recorder?  Give a specific example of how he/she performed his/her role well.

[2.]  Who is the resource manager?  Give a specific example of how he/she performed his/her role well.

[3.]  Who is the navigator?  Give a specific example of how he/she performed his/her role well.

[4.]  Who was the facilitator?  Give a specific example of how he/she performed his/her role well.

[5.]  Who had the best “because”?  Share it with me!

As you can see these are devised to help students learn to "play their roles" and "say their becauses".  In between Part 1 and Part 2 we are going to learn what the roles of each group member are, so students will have it fresh in their heads...but another way of emphasizing it can't help!

Number Line Task:  Part 3

These are the directions students will get to read through in their groups.

Take an index card and write your number on it.

Soon all of the groups will work together to place all the numbers in order from least to greatest. Here are the rules:

• You may only address one person at a time, by comparing your number to his/hers
•  Say your becauses!
• The only thing you may bring with you is your index card
• When the class is ready, I will check your order, but I will only tell you “ALL CORRECT” or “NOT ALL CORRECT”
• We will keep working until we get “ALL CORRECT”
•  If you feel stuck, you may request “group time” where you will go back to your groups, ask your group members questions, use a calculator, ask a group question, and write notes on the back of your index card
•   “group time” will always be 90 seconds in duration
•  You may only ask me questions during “group time”
•  Remember, when you feel confused, make a mistake, or are thinking hard you are learning!—and that is the goal

      After the class has worked and received and "ALL CORRECT"  each student will complete the following reflection.

[1.]  What was the most challenging part of this task?

[2.]  What is one thing you learned?

[3.]  What is one question you still have?

[4.]  What is one question someone asked you?

[5.]  What number were you and what did you write on your index card? (Use the back if needed.)

All the documents I made for this task are in a folder here.  These are word docs, but it will open in google. Download to word to edit.  Also everything is 2/page and Part 2 should be printed front and back.  Enjoy!

If you have ideas to make this better please let me know!  I'm sure there is something better, but this is what I have :)

-Kathryn

Thoughts about Reflection

I've been working on a task for my students for the beginning of next year, and I'm stuck deciding between two different types of reflection questions for them to complete afterwards.  Essentially they will be answering some Talking Points-style questions in their groups.  I'm stuck between asking them:

• Which are you most confident about?  Why?
• Which are you least confident about?  Why?

OR

• Which was the easiest to make a decision about?  Why?
• Which was the most difficult to make a decision about?  Why?

While I think it is over-kill to ask all four questions, and it is unlikely that my students would recognize the minor differences between the questions if posed all four, I feel it is likely I would get different responses depending on which questions I pose.

Maybe I'm overthinking it, maybe not.  What do you think?

I think my goals from the reflection are:

• To help students see the value in the Talking Points structure
• To help students think about their learning
• To help me see what my students do and do not understand

Other reflection questions I plan to ask:

• How easy or difficult was it to follow the talking points structure?
• What is one benefit you saw to using the talking points structure?  One disadvantage?
• What is one thing you learned?  What is one question you still have?

What do you think?  Which set of questions will best help me meet the goals of the reflection when paired with my other questions?  Should I do a combination of the pairs (one from each), if so which ones should I pick?  Are my other reflection questions posed in a way that will help me meet the goals of the reflection?

-Kathryn

New Steps I'm Taking to Improve Group Work

I'm working a lot harder on making group work work in my classroom this semester.  I implemented some changes first semester, and expected it to work for itself from there...uh, duh Kathryn, you can't just put things in place and expect them to run themselves.  So I've been more thoughtful about it this semester, and trying to patiently look for small improvements.  Since I rarely blog, I will discuss all changes I have made this year.

First Semester

Desks in Groups of 4:  I have tables, so I've pushed two together to make groups of four.  I had this on and off last year, but I haven't even considered changing it once this year.

Group Supply Tubs:  I organized supplies into group tubs.  Something I sort of had last year, but that didn't work very well.  I also was intentional about not putting too much stuff in at the beginning of the year.


Roles:  I wrote about my role cards before school started, but I expected a little prep and some pretty cards to be enough for students to "play their role" every time...that is not enough.

Talking Points:  We do talking points as a warm-up every Thursday.  This helps give every student a voice.

Then I'd also throw around the phrase "do this in your groups" a lot and feel frustrated that they really weren't.  So I have tried to be more thoughtful.

Second Semester

Role Explanations in Notebook:  I made a half-sheet explanation of all the roles for students to put into their ISNs.  This way they ALWAYS had access to them and I could hold them more accountable.

Group Tasks:  I have been a lot more intentional about groups tasks.  One scored group task every unit is my challenge for myself.  This is super structured, and involves a significant amount of prep.  An important thing I learned from Elizabeth and our morning session at #TMC14 is to put in checkpoints for groups to call me over and see how they are progressing.  This helps a lot.  I even carry around a roster so that I can mark off what checkpoint each student has completed.  Then when I'm prepping for day 2 of the task, I have an idea of where everyone is and who I should maybe guide right from the beginning.

At this point I haven't done any amazingly exploratory/discovery task, but I think that's OK.  I'm not super-teacher, so small steps is what it is going to take.  I've turned some things I did last year (that I liked) into tasks that I now like better.  I'm sure they will morph next year into things I like even better...oh the joys of teaching!

Here is my sequence task:  Instructions, Cut Outs

Here is my function task:  Instructions, Cut Outs, Recording Sheet

Acknowledging What I Like:  I've been intentionally looking for things that I like during the group work time so that I can praise those things.  This helps me and my students, because I see the good things rather than the bad, and they feel encouraged.  It also leads to improvement because others see what I like from a group and they can work to do it in their group.

I actually just had a thought while I was blogging this that I'm going to try Monday.  I'm going to use my whiteboard to give these acknowledgements.  It is semi-public in that the entire class can see (but not other classes), but I don't have to spend the last couple minutes of class giving shout outs that they might forget by the next day.  AND other groups can IMMEDIATELY implement changes to get acknowledged as well.  I could even start with a few things on the board that I'm looking for and then write up the table number when I see that group doing it.

Group Reflections:  Every day we do any group work task work, I have them complete a quick reflection.  I'm sure I stole the questions from Elizabeth (bottom of this post) and just modified it a bit. It's just a quarter sheet, so I make a ton of copies and have them ready.  My hope is that by regularly trying to give a compliment to someone for playing their role they will become more familiar with the responsibilities of each role.  I might modify the reflection next task to involve something the individual did as well as something another person did.  Try to slowly increase the expectation that EVERYONE is playing their role.

Here is a link to what I've been giving (4 to a page).  And here is what I created for next time (2 to a page).

That's where I'm at with group work right now.  I'm proud of the changes I've made, but also disappointed that we're not farther along.  I have been realizing this year how long real change takes.  And it is a little disheartening because I'm starting to understand that it's going to take more than just me to make real change for our students.  We really need every teacher on board to make real change for students.

Let me know if you see something big (or small) that I'm missing that would make group work more valuable for my students.  I know that it is not ideal at this point, but I'm working on getting better.

-Kathryn

Group Work Role Cards

Happy Monday!  This might be one of my favorite #Made4Math Monday posts (of mine).  I've very excited to share it with you :)

I've been reading Strength in Numbers by Ilana Horn and it talks about complex instruction throughout the book.  I think these roles are based off of a complex instruction structure.  However, I've modified them some on my own and a lot in regards to Elizabeth's post here.  For the most part I stole Elizabeth's changes and them made a few small changes for myself.

Here are my cards:

I laminated them and put them as a set on a ring so that I can store them in the group storage totes.  That way they are accessible to students whenever they need them.

I really like that they will be able to keep instructions for their role in front of them as the work in their groups.  This will help remind them of their duties as well as encourage them to speak out to fulfill their obligations in whatever role they are in.

This is the first time I'm going to try group roles, and I'm pretty nervous!  But these pretty things make me a little less nervous and a little more excited :)  Let me know if you have any tips!

Here is a link to the file I used.  It has one role description per page, 6 times.  Then the last two pages are the role titles written larger, as you see on the "front" of the cards.  (Note:  the file will open in google drive, but you can download it to word if you would like.)

Thanks for reading and thanks for all the encouragement from those of you who saw the picture on twitter!
-Kathryn

5 Practices by Smith and Stein

After finishing Mindset and reflecting on it, I chose to read 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein.

I first wondered if actually naming the 5 Practices would give away the purpose of the book, so I was curious how I would blog about it without doing that.  But after reading it, I know there is SO much more to the book than the names of the 5 Practices...so here they are:  anticipating, monitoring, selecting, sequencing, and connecting.  The whole purpose of the 5 Practices is to best prepare for teaching mathematical tasks so that we have to make fewer decisions during the lesson where we are less able to make good decisions.

It helps me to look at the 5 Practices in terms of the big picture of lesson planning and teaching.  As most people who read math teacher blogs are familiar with The Three Acts of a Mathematical Story [Task], probably most famously, if not first, presented by Dan Meyer, I will present 5 Practices in terms of the 3 Acts...except I'm going to start before Act 1.

ACT 0:  LESSON PLANNING

Before any planning can occur, the teacher must "specify a goal that clearly identifies what students are to know and understand about mathematics as a result of their engagement in a particular lesson" (p. 13).  After having a clear goal, then the teacher must select an appropriate task.  A task that can be used to meet the lesson goal while engaging the students with the concepts and that "stimulate students to make connections [that] lead to a different set of opportunities for student thinking" (p. 15).  

This part is going to be a challenge for me, because I have trouble seeing what big ideas can be learned from tasks.  I know that there are a lot of tasks available from the #MTBoS, but selecting ones that are going to best fit my goals is a challenge.

Once a goal and a task have been established, the teacher begins anticipating all the possible solution strategies.  Consider strategies that lead to both correct and incorrect solutions.  Then create good questions to ask students who are using each strategy.  If it is a strategy that might lead to an incorrect solution, questions should be redirecting students down a correct path.  ALL questions should seek to reveal what understanding students have about their strategy.

Once this has been done, a teacher should complete a monitoring chart.  Here is an example I created to match what the book suggested:

I want to add a column for questions.  It will help me remember all those questions I came up with!  Here's a link to my version of the monitoring chart. [Note:  You will likely need to download into Microsoft Word to preserve formatting.]

After you have the monitoring chart completed you are ready to teach the lesson.  So here we go into Act 1!

ACT 1:  INTRODUCE TASK

Nothing new from 5 Practices for this act.  Essentially distribute needed materials, read through task, and ensure all students/groups have an understanding of what they are being asked to do.  Work to hook your students into the task so they can't wait to strategize a way to the solution.

ACT 2:  STUDENTS WORK, TEACHER GUIDES

After letting the students get to work, the teacher begins to monitor their progress.  This is where all that hard work of finding as many strategies, creating good questions, and having the monitoring chart ready is going to pay off.

As you move from group to group you have preplanned questions to ask students.  Likely the major change of this act as a result of the 5 Practices is that you will take notes on what strategies students/groups are using.  This will better help you prepare for Act 3 and it is also why the monitoring chart is so helpful.

In between Act 2 and 3, the teacher has to make some decisions.  First, select students to present their strategies.  This can be done effectively and efficiently because you have all the notes in front of you.  You will likely want to select at least one student to present each strategy.  Once you have selected the students, you will need to decide on an appropriate sequencing to best facilitate the discussion you wish to have in Act 3.  There isn't necessarily one sequencing that is best, but you must choose one that will work for you.

ACT 3:  DISCUSS AND CONCLUDE

It is time to begin the conclusion of the task by drawing the whole class together and start sharing strategies.  During this time the teacher has the important job of connecting student strategies to each other and to the learning goal.  Good implementation of this will change Act 3 from a show-and-tell, to actual mathematical discussion.  This will probably be the most challenging of the 5 Practices for me.  I definitely see it as what will turn tasks from "fun days" into "fun, productive learning days".  I also think that the anticipating I will have done with the strategies will help me to see the connections, which in turn will help me guide my students in making connections as well.

THE FINAL ACT

There is always the Final Act of reflecting and modifying for the future.  I think the first time I implement a task this year I will plan to do it a different day in each of my classes.  That way I have time to reflect and improve in between implementations of the task.  And since I teach Algebra four times a day, it won't hold the stress of four possible dud lessons on my first attempt.

I am so excited to try to implement tasks.  I think this book really gave me a way to make them a productive use of class time.  My goal will be to implement at least one task per unit.  And in July I plan to continue to develop my units, learning targets, and lessons...so I will begin selecting tasks :)  

During the year I plan to create a task binder where I keep a record of the tasks I use, the strategies we find, my monitoring sheets, and other notes.  This will help me to stay accountable and to have evidence of what I have done throughout the year.  

One more thought...the book gave a flow chart for The Thinking Through a Lesson Protocol.  I plan to use this as I plan each task, but I will likely modify it to make it more usable to me in my lesson planning.  Don't worry, I'll post again as I plan my first task!

-Kathryn

#MTBoS30

30/30