#WTPW Simplifying Radical Expressions-Rationalizing Denominators #tlapmath

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I am not sure how exciting this lesson is, but I believe the idea beats the run of the mill take notes-practice on a worksheet.  It gives students opportunities to notice patterns on their own, a chance to share and discuss those ideas as well as consider ideas from their classmates.

I appreicate Math Equals Love Walk the Plank Wednesday post and will definitely use some of her ideas with the “why” we do this.

My goal is for my students to be able to determine if expressions are equivalent, so I am beginning with a simple card matching task.  As students enter the room, they will receive a card with a radical expression either simplified or not (similar to set A).  As we begin class, they will be asked to find their match…without verbal communication…while I post attendance, etc.  They will come to me with their match and I will confirm if they are correct.  Yes, I will allow calculators.  I know, not too high level on the thinking scale.

I will have several sets of cards similar to those they matched.  Each group will then be asked to complete an open-card sort.  This simply means, I do not give them any direction on how to sort their cards.  The only stipulation is they are ready to explain why they chose to sort them as they did.  When the timer goes off, we will share sorts (both volunteers and any I find that are interesting to me).

Part C, I will have concept attainment cards placed around the room.  Each card will contain examples of radical expressions labeled simplified and expressions labeled not simplified.  Students will carousel to different cards, noticing patterns, trying to develop their own rules.  After a set time, they will do a quick pair-share to summarize their findings before we have a whole class discussion.

Hopefully their ‘rules’ will encompass all we need to know, but if not, I can always use their ideas to lead us to our goal.

We will create a set of notes for our INBs.  Part of their HW will be a LHP assignment to give examples of expressions that are simplfied and not simplified from their earlier carousel work.  Ideally, they would create their own expressions.

If students need practice with skills, an idea from a workshop several years ago…on a page of say 30 problems, I pick 5 I want them to do, then they pick another 5 or 10, whatever I/they feel is necessary.  By giving them this option, I have more success getting them complete the practice.  I would much rather have 10 complete than 30 incomplete or not even attempted.

An idea for formative assessment…return to card sort from Part B.  They should sort into groups of simplified/not, even match up equivalent expressions.  One person stays with the sorts, while others go to different groups to peer assess.

Possible written assessment questions, a) give a bank of expressions to match equivalents, noting simplified terms; b) given a simplified expression, create an unsimplified, equivalence.

This is a very generic layout, but I can use the sequence with whatever level of Algebra I am working with.

I will post again when I have sets of cards completed.

Feedback to move forward, ideas  for improvements are welcomed.

Pam Wilson, NBCT
5 Practices for Orchestrating Productive Mathematical Discussions, Smith & Stein
Teach Like  a Pirate, Dave Burgess
From Ashes to Honor, Loree Lough

Happy Birthday #Made4Math !!! Formative Assessmemt Reminder Cards

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First, just let me say a big THANK YOU to @druinok for beginning #made4math and to all of the generous folks who have openingly shared their classroom ideas, lessons, tips over the past year.  I was overwhelmed with how quickly it took off!  Still, today, I am amazed at the generosity of this community.  I have learned so much and my classroom was definitelh impacted by your awesome ideas!

My share for today was initially a result of a convo with @rachelrosales and @druinok, brainstorming ways to organize reminders for the numerous formative assessment techniques…something simple, at your finger tips.

I loved @druinok’s post today and her Student Engagement Flipchart.  Very.Nice.  It will definitely be on my to-do list for a future project.  However, I am choosing to share a similar idea, just a bit different format.  I cut down index cards to fit sports card pages… pack of 10 for \$1.  I am able to display up to 90 of these reminders ranging from formative assessment techniques to various strategies for student engagement, reflection, etc.

Front side of card has title, with some information…

Back side of cards has description, suggestions, reminders…

I have placed the pages in a small 3 ring binder which can easily hold more pages.  Currently, I am trying to include summaries/reminders of techniques I have used or see being easily modified for math class.

Looking forward to learning and sharing more FA techniques with my amazing PLN!!!

Pam Wilson, NBCT
5 Practices for Orchestrating Productive Mathematical Discussions, Smith & Stein
Teach Like  a Pirate, Dave Burgess
From Ashes to Honor, Loree Lough

All Student Response Cards #made4math Monday

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In reading Embedded Formative Assessment (Wiliam, 2011), there have been several practical techniques presented in each chapter.  While discussing chapter 4, @druinok suggested creating response cards this summer, based on the technique All Students Respond.

I had seen a set made by an elementary teacher in my leadership network.  She had several cards labeled with letters, hole-punched and attached to a 3 inch ring that could be opened and placed around the metal frame on student desks. She explained students always had access to them.

I kept thinking about how to accomplish the same idea for my classroom.  I had a package of name badge holders I had picked up at our Mighty Dollar in town, but never found a use for them.  Basically, I put this example together quickly, to have something for #made4math today. Its not innovative, but for anyone who does not have a “clicker system” or devices to use with Poll-Everywhere, etc., its an option that I believe could prove as a useful tool.

My idea is to have a single card, with all responses.  I would need to ‘train’ students how to hold their cards allowing me to see their response clearly.  Mine is double sided, this could easily be accomplished with cardstock printed, then laminated if you didnt have the badge holders.  Each student could clip one into a pocket of their INB and have them on hand when its time to use them.  Or they could be clipped either to a hanging ribbon or the side of a magnetic cabinet, even placed in a basket if you only had one classroom set.

The first side includes a favorite of mine…always, sometimes, never…color coding green, yellow, red, respectively.  The student places their hand, so only the response they choose is visible and located at the top of the card when they hold it up for me to see.  I didn’t have the color circle stickers here at home, but I believe they may help in the visual for me to see.  By keeping responses color coded, I can quickly scan the room to see where students are, then make a decision as to what type of question follows or if we should procceed with discussion of why they responded as they did…supporting their claims with mathematical evidence, of course.

Notice, the QUESTION response.  A student may have a question or require some clarification, this choice doesn’t allow them to opt out, but provides a way to say, I need some help.

On the back side, there are simply color-coded (different from other side) multiple choice responses, again to allow a quick scan before deciding how to proceed.  If multiple answers are chosen, begin by asking students to give possible reasons why a student may have chosen A or D-the other answer, if I chose A, could I figure out how someone else would have chosen D?  I also like to ask, noone chose B or C, what is a possible reason why someone would not have chosen  ___?

Like I said, I plan to use color circle stickers which allow me to see student responsesmfrom across the room.  I am debating on howmto do true/false.  Would
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Evaluating Statements About Length and Area

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This lesson can be found http://www.map.mathshell.org same as title of the post.

This is one of six cards students discussed within small groups today. A student stated, “this is going to be a thinking day,” as they began removing the clips to start reviewing their cards. Most students would quickly come up with an always, sometimes or never true. However, to create their own examples or counterexamples to either justify or refute the statements was a struggle for some of them. Several groups had similar statements for this particular card. It was when a student asked, “do they have to be triangles?” that a turning point came for some.

Within our share out as a whole group, a student shared examples of reducing area, same perimeter and less perimeter. A question they wondered…can you reduce the area but increase the perimeter?

I really enjoy days like this, students are giving me the information, I am their scribe and I am slowly learning to let them determine if they agree or disagree with each others’ claims. I’m not even sure where the key is, that way I am actively having to listen to their arguments to determine if I agree or not. (Shout out to Max @Math Forum, I am listening to my students, not listening for the answer!) I go through the cards myself prior to the day of the lesson, just like I require them to do. But I am still closed minded in my own thinking at times. Why would you limit the example above to only triangles? Because that is what shape was presented on the card. However, does it state triangles only? Nope.

A task like this may drive some teachers crazy. Once you start considering different shapes, you begin to see what works for one, may not work for another. I had students cutting scrap paper, tracing patty paper, measuring side lengths…without me telling them to do it.

The classic question, a square and circle have equal perimeters, which has the larger area? I will do my best to share more reflections as we wind up tomorrow, if we wind up tomorrow…depending on their questions, discussions, claims and supporting evidence.

Always, Sometimes, Never – #75FACTS

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I’ll be honest, I’ve only truly dug-in to reading the first 6 FACTS of Keeley & Tobey’s book over the past 2 weeks.  Through KLN – Kentucky Leadership Network, I’ve explored several others over the past year.  But I’ve gotten very drawn in to processing the descriptions, suggestions given on the first 6 (by the way, they are listed alphabetically, didn’t know that until someone pointed it out in twitter chat).

This past week, of these 6, I’ve attempted some form of Agree/Disagree (#1), Always Sometimes Never (#3) and Comments Only Marking (#6) in my classroom.  I’ll share more later on A/D and Comments.

Last year, I began experimenting with the Formative Assessment Lessons from the MARS site.  Sorting Equations and  Identities lesson asked students to sort mathematical statements into categories – always true, sometimes true, never true.  Part of the task was to justify their choices.  After using this lesson, I realized students really struggled with these statements.  In fact, they hated them – moaning/groaning each time one would pop up.  Which said to me – they were having to think.  I began embedding them in lessons/notes – class discusses/questions – especially in assessments.  By the end of the year, students were “not afraid” to face ASN questions as before.

This week, I gave geometry students 15 statements about quadrilaterals/polygons, in which they had to answer ASN.  When they arrived in class the following day, I had areas of the room designated A, S, N.

Depending on the FACT, it may help to explain to students why you are using the new strategy.  Part of this discussion was that when someone makes a statement, it may seem true, but we should check it out to determine if in face it always applies, sometimes applies or never applies (page 57).  Through the activity, students were able to share counterexamples if they disagreed with another student’s statement.  Great discussion (even a few semi-heated arguements) occured!

Mathematical Practice – #3 Construct viable arguments and critiques the reasoning of others.

Were students engaged?  Definitely – from the time they walked in, they saw the A, S, N posted and KNEW what was coming.  Most were engaged during the activity.  At least those who didn’t want to think – had to at least choose an area to move to in the discussion.  I used my “name cards” to call on students to ensure everyone needed to be ready to share their justifications.

Were you confident/excited about using the FACT? Yes.  I’ve found a new love for always, sometimes and never statements – though I remember detesting them a particular college geometry course – now I realize what a great learning tool they can be.

How did use of the FACT affect the student-to-student or student-teacher dynamic?  I tried to allow students to share their own counterexamples – but when one was stuck, I would question – referring back to properties we had investigated, drawing figures on the board, presenting a what if… if needed.

Was the information gained from the FACT useful to you?  I realized some students still confused a few of the rhombus, rectangle, square statements.  Mostly, that students often only considered the “obvious” – but this activity was great because others were able to share their “what about…” with their classmates.

Would you have gotten the same information without using the FACT?  In the past, I would have likely made the same realizations but only after giving the unit assessment.  This FACT helped clear up some misconceptions during the learning process rather than at the “end of the learning.”

What added value did the FACT bring to teaching and learning?  Students had to think about their thinking, jusitfy their reasoning, could be critiqued by classmates’ thinking – great opportunities for discussion / sharing!

Did using the FACT cause you to do something differently or think differently about teaching and learning?  During the task, I was able to use student comments as a springboard for whole class discussion, pointing out examples that made it true and examples that made it false (great piece of learning to impact understanding of counterexamples).

Would you use this FACT again? Yes.

Are there modifications you could make to this FACT to improve its usefulness?  This FACT lends itself well to written work, whole class & small group discussions.  Follow up is key – probing students and guiding them to consider other examples – if not shared by classmates first.  Even after arriving at what seems to be class consensus, ask again – challenge their thinking – don’t settle for the first correct responses – ask why – let them justify their reasoning.

Thoughts on #75facts

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As I read SimplifyRadicals #75facts post this morning, it really got me to thinking…about things I do and how I could use “Create the Problem” in my own classroom.

I’ve given students the answer before and asked them to write a scenario that could model the problem.  But reading her refelction and suggestions for modifications helped me realize a couple of ways I could improve the way I’ve done this in the past.

The FACT reminds me of ideas from More Good Questions, Marian Small & Amy Lin.  Give students the answer and they have to come up with the equation/problem.  Example, the slope is 2/3, what are 2 points that could give you this slope?

As suggested in the FACT#11 description, providing students with an open ended task takes their thinking to another level.  Student examples generate whether they know why a computation is performed rather than just knowing a procedure.  But this FACT actually asks them, not to find the computation/problem, but to give a scenario/context where this strategy could be used to solve the problem.

The key, as with many successful strategies, is sharing student ideas.  Not just allowing them to talk about their examples and how their story matches the solution, but the teacher asking the class for feedback on whether it is a match, if not, how could it be changed/made better (pg. 81)?

This reminds me of another FACT I’ve used in class before “2 stars and 1 wish.”  however, when I first saw this a couple of years ago, it was called 2 +’s and a delta…two positives and one thing I’d like to change.  Playing off of My Favorite No, I ask students “What do I know this student understands?  Give me 2 examples of what this student did well.”  By focusing on the correct parts first, especially if I’m using a student’s example (anonymously) – the student can see it wasn’t completely wrong.

Then for the delta (wish), I ask students not to point out the mistake, but to think of a question they could ask the student to help the student realize their mistake.  Sometimes, this is a tough task, depending on the mistake that was made, but by asking a question, students, again, are having to think on a different level.

In several of the Formative Assessment Lessons from the MARS site (Solving Linear Equations in Two Variables) – the lesson format actually allows students in small groups to evaluate different levels of student work.  On a slide in the projector resources for this lesson, Assessing Student Work, students are given these questions to guide their discussions:

You are the teacher and have to assess this work.

Correct the work and write comments on the accuracy and organization of each response.

•What method did the student use?
Is it clear? Is it accurate?  Is it efficient?
•What errors did the student make?
•How might the work be improved?
My thinking, use the FACT #11 – Create a Problem as an exit slip.  Divide the responses into different levels.  On overhead, share different levels, both correct/incorrect, as well as different approaches, using the above questions as a guide for class discussion.  Then present students with solution(s) and ask them to create a problem.
Thanks to Simplifying Radicals for getting my brain to churning so early this morning!

#75FACTS week 4 – #24 I used to think… but now I know…

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This week we’ve been off from school for fall break – a road trip down south to visit Winter the Dolphin in Clearwater and a few days of warm sunshine on the beach has me somewhat re-energized.  I’ll be honest, my book is still at school.  The directions for this week were to use one of the FACTS #1-10 but I haven’t been in class to do this. (Sorry)  Before leaving school last week, I chose FACT #24 I used to think…, but now I know…  as a left-hand page assignment for my Geometry students’ INB.

#24 I used to think… Now, I know…

Eight Standards for Mathematical Practices

Practice 1 Make sense of problems and persevere in solving them.  This FACT allowed students to reflect on their learning, an opportunity to share what they used to think and what they now know after working with the concept.  Students responded to this prompt after exploring in small group investigations, pair-share processing, independent practice and finally whole class discussions/questions over Triangle Congruence.  I used to think… but now I know gave students the chance to make sense of the ideas they have been working with in class.

Facts and Teaching Goals

The goal of the lesson sequence was to allow students to recognize and determine which side-angle combos were appropriate and would guarantee triangle congruencies and finally applying those ideas with informal proofs.  By allowing them to respond in writing, I was made aware of their initial misconceptions – but also able to see they had in fact realized on their own how to prove trianlges congruent with a limited amount of given information.
I learned that the AAA and HL were the two students had struggled with most but they wrote about how the activities / discussion helped them realize specifically what was needed with each combo.  Another common error they pointed out in their reflections were that order of the included sides/angles did matter with situations of AAS and ASA.

Planning to Use and Implement Facts

One reason I chose this FACT was because I am looking to implement more literacy strategies into my instruction.  This FACT provided students with the opportunity to reflect on their learning in written format – a different type of processing that just talking/telling what they’ve learned.  By the time the prompt was given, students had explored in small groups, shared verbally with a partner, practiced individually.  The writing component seemed to complete the various types of literacy strategies.  By giving students a chance to respond to this prompt, I was able to see in-depth their full understanding of the intended concepts.

Small Steps

Were your students engaged?  Yes, I was very pleased observing students as they wrote their responses.  Most students took their time to share insightful reflections.  There were a few who tried to skim by with very vague responses, I gave them written feedback and asked they resubmit their responses.  Based on their new responses, I expect those few will give their best effort first time around next time given this prompt.
Were you confident and excited about using the FACT?   I felt it was a good opporutnity to have students share their learning in writing.  I was not as excited about the FACT until after I actually started reading their responses…
How did use of the FACT affect the student-to-student or student-teacher dynamic?  Student to teacher – I felt they were honest in their responses – and most were insightful – I was encouraged to use this FACT again because it allowed me to see into their thinking.
Was the information gained from the FACT useful to youYes,  however, I don’t think I will change my approach to the lesson in the future – students were able to adjust their thinking because of the lesson format.  The FACT let me see this as a successful sequence – what a good formative assessment strategy should do!
Would you have gotten the same information without using the FACT?  I’m not sure I would have given students the opportunity to reflect had I not used the FACT.
What added value did the FACT bring to teaching and learning? Based on student responses – I believe most appreciated having the opportunity to think about their learning – it “tied up loose ends” for them in the end.
Did using the FACT cause you to do something differently or think differently about teaching and learning?   It made me realize I’ve failed to provide students with good opportunities to refelct on their learning between lessons / practice and before “official assessment” occurs.  This is something I plan to implement more for my students!  It was quick, little/no prep and offered me the chance to really see what students thought about their learning.
Would you use this FACT again?  Yes.
Are there modifications you could make to this FACT to improve its usefulness?   I believe next time I will plan more time for students to share out their responses – maybe within a pair-share then as a whole class, possibly using the ‘Around the Clock’ appointment cards idea from Global Math Department.

Using Data from FACTs

Most students realized that AAA could only guarantee similarity amond the triangles.  There were several misunderstandings about HL I was not aware of until after I read student reflections.  I will be more puposeful in defining the included parts in the various combos, for example I shared examples with students and asked how AAS and ASA are alike / different because this was one that a few still had struggle with.  During this discussion / sharing – it was obvious some a-ha! moments occurred.

#75facts Book Chat Begins Monday 9/24

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Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learnin

Page D. Keeley (Author), Cheryl Rose Tobey (Author)

They refer to the strategies in the book as FACTS – Formative Assessment Classroom Techniques thus the hashtag #75facts.

If this will be your first online book chat – its simple – read assigned material, log on at designated time and share!  I’ve heard from several of you that you’ve gotten your books in hand – so let’s get started next Monday – September 24.  Meet up on Twitter at 8:30 cst and use the hashtag #75facts in your posts.

I know this will be a great opportunity to share and learn from others!  Several of the FACTS may be strategies you currently use – so there will always be opportunity to share what this looks like in your classroom.  The FACTS may also trigger a new idea on how to modify and improve techniques.

There are 75 FACTS which means this chat has the potential to continue the entire school year – so, if you are new – please join in!  We want you to be a part of this!

Overview:

This book is a bit different than ones we’ve used in the past, so you are encouraged to get started and read ahead – getting ready for implementation – however, we’ll begin our chats by discussing 1 chapter each week.

Chatper 1 Introduction – defines FACTS, shares research, making a shift to a foramtive assessment centered classroom.

Chapter 2 – Integrating FACTS with Instruction and Learning

Chapter 3 – Considerations for Selecting, Implementing and Using Data from FACTS

My initial thoughts are to focus on 3 FACTS each week – you can choose 1 of those 3 to implement (or any prior FACT), reflect and share during our discussions.  We can see how this goes and always modify as we see fit.

Chapter 4 – Getting the FACTS is where the 75 FACTS are presented.  Each FACT covers 2-3 pages, so the reading is not the time factor here – implementation is where your time will be focused.  Don’t let this overwhelm you – if you don’t get one implemented, this by no means implies you should skip the chat!

Each FACT follows the layout:

• Description
• How it promotes student learning
• How it informs instruction
• Implementation Attributes
• Modifications
• Caveats
• Uses with other Disciplines
• Examples, Illustrations
• Notes/Reflections

If you have not already, please enter your name in the form so we can ensure we keep you posted!

I will get a form in place for you to share any blog posts about #75facts soon!

Online Book Chat – Math Formative Assessment

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Are you interested in an online book chat?  If you’ve never participated and wonder – how does that work?  Its simple, we’ll set specific parts/sections to read; meet up online and discuss what we’ve learned; share what we’ve implemented; reflect/collaborate!

Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning

Page D. Keeley (Author), Cheryl Rose Tobey (Author)

They refer to the strategies in the book as FACTS – Formative Assessment Classroom Teaching Strategies thus the hashtag #75facts.