Category Archives: Differentiation

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.

tom hill

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.

tq3 fun

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   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.

dom ran

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.


Systems of Equations Unit (part 1)


So many thoughts this past week as we began Solving Systems in Algebra I which will likely lead to multiple posts…

Here is the Systems Organizer Student Assessment Tracker.  I’m not satisfied with it yet, I’ve adjusted an old Algebra 2 unit, but I know by next year, this will be one of our strongest, most purposeful units.

I’ve been using the Candy Store problems since Mary and Alex shared them at TMC-Jenks.  A great problem solving task with manipulatives to introduce systems of equations.  My only change is to adjust for the U.S. candies, Solving systems CB S (Thanks for sharing your file, Mary!).  I plan to bring in a candy treat to students to celebrate their journey when we end the unit.

Based on prior assessments, in class observations, I purposefully separated students on skill level for this unit.  I intended it to be for me to have time to focus on groups with weaker algebra skills, while letting the others move on at their own pace.  I pulled those few who tend to “do the work” into groups together which would allow for those who follow along in tasks or let someone else do the thinking, then they copy it down-be required to do their own thinking.

Here’s what I notice – my “algebra” kids struggle, my “struggling” kids soar – with the hands on task!  It just goes to show, students do have good, strong reasoning skills when allowed to think on their own.  Each group gets a cup of pennies and two different types of pattern blocks with a white board and marker – although I think next year, I will hold off on the white board and marker until AFTER they solve the first one – I really want them to rely on reasoning and number sense before trying to jump in and create equations…although that is the end goal.

The beginning was often guess and check, but I loved hearing their number reasoning as they progressed through the problems.  Let me say, I have about 15 students spread between the 5 class who are still mad at me because they did not like the struggle.  I just kept patting them on the back, asking questions and when they began to engage, I’d walk off and let them continue.

After most students experienced some level of success with the Candy Store problems, we reviewed/introduced linear combination (elimination) to solve systems when presented in Standard Form.  I had examples ready, but as we practiced them on white boards as whole class, students were asking the questions:

  • what if the terms match and aren’t opposite?
  • what if nothing eliminates?
  • why does multiplying the equation by a number work?

It was great when it was students asking and not me leading.

In their groups, they received 11 cards – with solutions on one side and system equations on the other.  The cards were placed so all solutions were facing up and a start card.  When everyone in the group had solved/verified, they located the solution and flipped the card to find a new system to solve.


Nothing more than a glorified worksheet – handwritten while waiting at my daughter’s piano lessons.  But the discussions they were having as they solved on the whiteboards were so valuable…immediate feedback, peer assessment.

It was a good day.  The first time since Christmas Break that I felt confident we were moving forward.  (I know… its March.)

I’ve been trying to be more purposeful in ending class and allowing time for reflection. Students were asked to copy 2 of the problems into their INBs, solve and verify – basically creating their own notes / examples to refer back to.

Each student received a sticky note and was asked to complete the sentences:

  • I used to think…
  • Now, I know…
  • Caution…watch out for…

And they placed them according to their level of confidence as they exited the room:


This was on Wednesday.  I felt that they had built confidence, addressed common errors and misconceptions and had seen how the algebra could offer an efficient model in problem solving.  Yet, I still had a few groups who were strong/quicker with number reasoning when solving them.


Card Tossing & Spiraling Curriculum #tmc14


Awesome session Mary and Alex!  Thank you. Thank you. Thank you.

The session focused on their experiences with Grade 10 Applied students ( Canada).  The entire course is activity based which allows students to not miss out on big ideas as they would in a traditional unit by unit aligned course.
Students have repeated opportunities to experience big ideas. The tasks are rich  with multiple entry points and different approaches to solving.  It’s a collaborative environment with accountable talk.  There are fewer disciplinary issues with increased engagement.

Each 6 weeks a mini – exam over entire course up to that point takes place.  Questions are in context and tied to activities they have completed.

We began with beads and pennies on our desks and this task… Cole has 2 smarties and 3 juju bed for $.18 while Noah has 4 smarties and 2 juju be for $.20.  They shared that systems are presented this way – no algebraic forms- for the first several weeks of class.  I, personally, can see how effective this strategy could be.

The next activity shared was Sum of Squares (he doesn’t refer to it as Pythagoras Theorem, yet – or did he say ever?)

Students are asked to cut all squares from side length 1 to side length 26.  Each square is labeled with side length, perimeter, area.  Then they build with them.

Basically students explore and eventually they focus on triangles formed with question, are there 3 you cannot make a triangle with?   Which combinations form different types of triangles. Begin looking at 3-4-5 triangle families, similar triangles (Kate suggested dilations here), discuss opposite side and adjacent sides, then give them a TRIG table and allow them to figure it out.

Compare side lengths with perimeter, or side length with areas.  The possibilities of math concepts are endless.
We ended the day with Card Tossing by collecting data, then using rates to make some predictions.

Video of Alex & Nathan picture below is only a screenshot.


@AlexOverwijk downed by @nathankraft 75 to 72

Each person in the room completed several trials of tossing our cards for 20 seconds.  We found our average rate of success, then determined who we thought might beat King Card Tosser.

Alex asked us to predict how long they needed to toss if he gave Nathan a 35 (?) card advantage so it would be super close and exciting.  Our prediction 38 seconds about 75 cards. Many ways of making the predictions were possible. Not to shabby, huh?

This task was fun, exciting, engaging.  Definitely on the to-do list.

This approach is definitely something I would like to consider, if administration will allow it!

#WTPW Simplifying Radical Expressions-Rationalizing Denominators #tlapmath


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

An Idea on Differentiated Learning



What do I do when I have students on all levels of learning?  How can I successfully help each one at their individual level.  Its not going to be easy for all units of study, but I wanted to share an idea I got from watching Jessica Addison from Todd Central High School in Kentucky.

Jessica Addison – Todd County Central HS, KY  10 minute video with student interviews on using learning targets.  I like the way she’s divided the group into different groups to practice on various learning targets – based on their assessments.

I followed this format during a unit on solving equations in my Algebra I.  It was a pretty simple set-up.  I gave a pre-assessment based on learning targets.  Based on students results, they were placed into 1 of 4 learning groups.  On my board, I had sets ofskills practice sheets (with solution/keys attached to the back) for each type of equation to solve.  Collecting Like Terms, Distributive Property, Variables on Both Side, Multi-Step, Special Situations (identity, no solution).  I also had sets of absolute value equations/inequalities, Simple Radical and Rational Equations.

Students in each group were allowed to work on a set.  When they completed a practice set 6-10 problems, verified their solutions, they were allowed to move to the next set / group.  I have a list of the order for students to work through sets – including text / online resources.  Students in the more advanced groups, work within the small group.  I am available to help/answer questions with students who are struggling.  I walk around observing / spot-checking their work.  It was a very successful unit for my students.  Students appreciated getting more one-on-one help or being allowed to “work ahead.”

It’s not perfect, but maybe it can help you generate some ways to differentiate in your classrooms.

Video was part of this presentation/ PowerPoint Slides from Setting Clear Learning Targets 12/8/10

#75facts Book Chat Begins Monday 9/24


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

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!


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
  • Design and administration
  • 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!

Algebra: A Reunion of Broken Parts, from Arabic al jebr


Although I know the importance of integrating all parts of literacy in math – I have just never been strong consistently implementing good strategies with vocabulary.  I despise the days of teachers “making” me copy the definitions out of the back of the book.  My, now retired, Curriculum Specialist was an amazing resource.  I wanted to share a couple of tools she introduced me to:

Frayer Model , a great tool, though, sometimes can be difficult with a few math terms.

LINCS – (outline of strategy) students sometimes get hung-up in Creating the LINCing picture…my thoughts are – why not draw an example of what it looks like?  The research using this tool is pretty strong.

Just the other day, I shared with a colleague how I need to make this a priority for professional growth.  I know improving vocabulary/literacy in my classroom will lead to greater student success.  I have Literacy Strategies for Improving Mathematics Instruction from ASCD on my reading list for the summer. (Any takers on a twitter book study?)

Tuesday morning, I ran across this tweet by @davidwees

Loved this => MT @jesslahey: Because math is hard for some of us.

I Scream! was about @jesslahey ‘s first day in Algebra I class.  Ok.  But as I read through, I was intrigued and had to go back to an earlier post  Quantifying the Unknown to get the full grasp of her experiences.  Basically @jesslahey teaches English, Latin and composition, writes for a New York Times blog and is a contributor to several newspapers and magazines.  Her talent is quickly evident – very enjoyable to read.  She first shares how she always encourages her students to  “be brave, diligent, and never back down from an intellectual challenge.”  She said it was time for her to put her money where her mouth is…and face her aversion to math!

I was short on time so only skimmed I Scream – but pulled it up later that day.  Amazing!  An English teacher facing her biggest fear…Math Class!!!  Quickly, the wheels in my brain were turning…she is writing from the viewpoint of many of my students.  Her approach was to connect what she was hearing/seeing to something she is very familiar with – etymology, the history/source/origin of a word.  She was making connections with the math vocabulary to roots of other words for which she knew the mearning.  Hmmm.

Frayer & LINCS are tools to help me to this.  But I’m thinking about really , delving into learning some of those Latin roots, myself.  By deepening my understanding of word roots and helping my students make some stronger connections – it could only be win-win! Right?

I’ve used the Frayer Model before, some of the LINCs diagrams – but by helping students understand the roots – this would also have a positive effect on their reading and possibly facing unknown vocabulary in future courses?  So, to my summer list, I plan to start with my content’s critical vocabulary and search for some quality resources to help me learn the etymology of my lists.

If you have vocabulary strategies, games or great resources – please share in the comments.

Lastly, I look forward to reading more of @jesslahey’s adventures in algebra class – simply because, she’s giving a voice to some of my students – I’ll be able to think about ways to address their anxieties, looking for ways to be a better teacher.

From an online etymology dictionary

algebra 1550s, from M.L. algebra, from Arabic al jebr “reunion of broken parts,” as in computation, used 9c. by Baghdad mathematician Abu Ja’far Muhammad ibn Musa al-Khwarizmi as the title of his famous treatise on equations (“Kitab al-Jabr w’al-Muqabala” “Rules of Reintegration and Reduction”), which also introduced Arabic numerals to the West. The accent shifted 17c. from second syllable to first. The word was used in English 15c.-16c. to mean “bone-setting,” probably from Arab medical men in Spain.

Literacy for Decision Making

More Good Questions


More Good Questions has really gotten me to start thinking about the questions I ask my students.  Its so easy to stay with the traditional skill/drill I grew up on – but I am encouraged by some things my students are coming up with when given the opportunity of an open question.

Yes, a few of them are still struggling with “where to start” but now they are at least giving me something – whereas the first couple of attempts – blank stares, blank papers, blank looks.

Today, students were given the opportunity to retake their pre-test to look for any areas they may not have mastered yet.  As a review, I placed a graph of a line on the board.  I asked, “What can you tell me about this line?  How many different ways can you write/describe this line?”

I will add my slides from their discussion tomorrow.  If you can picture a line through (0, 2) and (5, 0).

Timer set.  Go.

Here are some of their responses.

  1. Its a linear function.
  2. Its decreasing.
  3. Slope from the graph is -2/5.
  4. It will never be in the 3rd quadrant. (ok, didn’t expect that one).
  5. It has intercepts at (0, 2) & (5, 0).
  6. y=-2/5x+2
  7. 2x+5y=10
  8. y-4=-2/5(x+5)  which led to a student stating, “I wrote point-slope for too, but its not the same as ___.”  Discussion.
  9. A table of values was given with intercepts as well as (-10, 6) (-5, 4) (10, -2).
  10. Another verified the rate of change with the table of values.
  11. The inverse is y=-5/2x+5 (nice surprise)
  12. A line parallel is y = -2/5x + 4
  13. A line perpendicular is y = 5/2x+2
  14. It could model a budget of $10, Candy(x) is $2, Coffee(y) is $5…how many of each can I purchase and spend exactly $10?

Many students were able to be part of the sharing/discussion.  I felt it was a great review of topics – with one simple graph of a line as the starting point…I attempted to follow #sbarbooks suggestion to look for students with fewer things on their list and call on them first to allow for their participation.

I hope to continue to gather ideas from More Good Questions that I can easily incorporate into my classroom.  I am looking forward to a book study with my department as well and seeing where it leads our students.

I highly recommend this book by Marian Small if you do not have – its worth the purchase!


Open Questions: My First Attempt


I’ve been reading More Good Questions and am so excited about this book!  #sbarbook study Monday nights 9:30 est. on twitter  – the big ideas so far, have been defining open questions and parallel tasks and how easy it is to create them. 

@druinok that’s what I’m enjoying too – very low stress, but HUGE dividends! #sbarbook

@druinok @jmalpass totally agree!!! I think what I’ve gotten out of it the most is rich questions don’t have to be hard for the teacher to do 🙂

Such simple, quick changes – yet great opportunity for thinking at ALL levels!

Today, on a target quiz on slope/rate of change – I made my first planned attempt to use an open question. The last question (no discussion on the first 4 ?s) took us into a great in-depth discussion.  The question was this:

Slope is 4/5. 

Give me two points on the line.

A student asked…does it have a y-intercept?  My response, Does (voice inflection) it have a y-intercept?  When I asked students for responses – I called on this student because I wanted to talk more about his question.  Student stated – its not vertical, so it has to have a y-intercept – even if its (0, 0) – the y-intercept is zero.  Good point.

While students were working – I observed their various stratgies for getting their coordinates- THIS is the part I *LOVED*!!!!  There were graphs, tables of values, slope formulas, and other strange strategies I would have never been aware of – if I hadn’t given this question!  I attempted to call on students with different strategies for getting their solutions.  Even calling on a few I knew had incorrect answers to allow for discussion.  I didn’t have to correct them – other students were able to ask questions.** 

One student looked confused as she asked, “How can we have so many points, but the same slope?”  My answer, “How can we have so many points, but the same slope?”  Another replied – “the lines are different but they have the same slope – so it makes them parallel.”  A concept not included in the objective – but I think it will stick.  On the board graph – students were able to quickly identify points that were not giving the correct slope and able to explain – usually inverted coordinates.

About 1/3 of the class struggled with where to start on this question – but my feeling is as the year goes on and they are given more open questions, I’ll see a higher number successfully attempting it.  One student made a comment on her way out – how ‘seeing other ways really helped (her) to better understand slope and how it works’ –

How difficult was it to come up with the question?  Not at all.  Level of cognitive demand – much higher that #’s 1 – 4; Level of discussion – much more in-depth! 

This idea may be something most of you use in your classroom.  I consider myself a good teacher – but my thinking has shifted – when I’m looking at examples / assessments – my thought is, how can I make this an Open Question???  I will continue to share my experiences initiated through this book!

** When a student sees a mistake another student has made – I encourage them to  question by asking, “What question can you ask them about their work/answer,” rather than tell them what they did wrong.  It gives both students a chance to reflect.