Category Archives: Formative Assessment

Identifying Linear Functions


Linear Functions Organizer this does not include arithmetic sequences, which was earlier in the year, but I can refer back to our work with them to activate prior knowledge for this unit.  The next unit will be linear regression which will include correlation, describing scatterplots, finding regression equation with technology, using the equation to predict and finally introduction to residuals.

Students started with a pre-quiz similar to the one below.


Identify Linear Functions This is a booklet with a Frayer Model for our notes, a variety of math relations to identify as linear or not and a 2-minute reflection grid on the back.  Prior to beginning our notes, I gave them 1 minute to jot down anything they thought they knew about linear functions.  Then we pair-shared before sharing with the entire class.  Then we took our notes. (as a follow up the next day, I gave them 2 minutes to jot down all they could remember about linear functions as a small retrieval practice).


Our next task was created by cutting apart these relations and posting them around the room with a chart that asked if they agreed or disagreed with the example being a linear function.  Students received stickers to place on the chart as they visited each station.


I was fairly accurate in which ones I thought we’d have to use for discussion, but a couple really surprised me.  These are the 4 we discussed following the carousel activity.


I. y = 2x was the one I was not expecting.  When I asked if someone would share their thinking, one student said they thought x was an exponent.  Another shared they did see “the b” for y-intercept.  We looked at a table of values and graph to agree, and show the y-intercept was at the origin and indeed y = 2x was linear.

The other I failed to snap a picture of was graph K, a vertical line.  Yes, it’s linear, but not a function…two students got that one correct in this particular class.

Using the 2-minute reflection grid as our exit slip to see students thinking about the lesson, I was excited about some of their “I still have a question about…”


On the reflection grid, if they have no questions, nothing is confusing, I ask them to give me a caution…something to be careful or / watch for.  Several of these questions encompass multiple students.  Some of them I only needed to clarify what was said.  Its pretty clear I was not communicating very well on a few of the.  I hear my “expert blind spot” showing up…”Of course squared is not linear, we learned it was quadratic in our functions unit!”  But so many students on the pre-quiz used vertical line test as their reasoning for linear…we had some side conversations about this misconception…that it shows functions, but does not prove if its linear.

Some of the questions, I allowed other students explain their reasoning to help clarify their understanding.

I know I shouldn’t have favorites, but in this list…

Why can’t you multiply the numbers by each other?  We tried it.  Add 2 numbers that will make 18.  Create table of values, find rate of change, graph it.  Yep, that’s linear!  Multiply 2 numbers that will result in 18.  We created a table of values of their answers, found the rate of change and graphed them.  No, that’s not linear!

If an exponent is less than 1, can it be linear?  We will try it tomorrow as our bell ringer.  But I look forward to exploring their questions more!

I told them how excited I was about their questions and posted them on our “THINKING is not driven by answers, but by QUESTIONS” board.  One student had the biggest smile and as she said, Look!  I’m so proud, my question is on the board!  Something so simple, yet, my hopes are that it will encourage her to ask more questions.

One student asked me, but isn’t it disrespectful to ask questions and interrupt the lesson?  Nooooooo.  I love when you ask purposeful, curious questions you wonder about!  Finally, a break-through to get them to start asking and wondering more…

Stacking Cups… part 2 #MtbosBlogsplosion #myfavorite


I like big cups, I cannot lie.

We stacked cups in the first few days of school…


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

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

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

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


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

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


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

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

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


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

Okay, so moving on now.

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

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

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


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

VII.  For other practice, we use the expressions:


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

VIII.  Closer choices

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

IX.   Systems

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

Make This a Quiz (g-forms)


So yesterday I had the opportunity to share some online resources with colleagues in my district.  What a great day.  I enjoyed the conversations with teachers from different schools, grade levels and content areas.  It really caused me to wonder how these tools might be utilized in their classrooms and their sharing of ideas was awesome!

I’ll be honest.  I was scared.  I’ve done several sessions at conferences or for the sake of sharing information, but never to really teach, with the purpose they would gain a skill or idea and be able to walk out with the ability to use it in their classrooms.  I was very nervous.  What if it was a flop?  What if I went too fast?  What if I assumed too much?  What if I assumed to little?  What if I failed at helping them?  I value their time and wanted it to be beneficial.

The biggest goal of the day was for them to experience google forms from a student’s point of view and then learn to create one; experience quizlet from a student’s point of view and then learn to create one; and finally experience Desmos Polygraph as a student – hoping to peak some interest in learning more about this awesome tool!  We spent the first 2 hours exploring, practicing some skills and the last hour was open for them to create a task/form/stack and/or search for items they could actually use when the school year begins.  I feel this was important.  So many times, we’re given something but never time to really practice using it.

It was an awesome day – everyone was so gracious and great to work with – asking questions, exploring.  Based on their feedback, I feel like everyone walked away with something.  (Thank Goodness!)


This chick was over the moon excited about the Quizzzes tab in g-forms.  What?  I think I’d heard some talk of it, comparing it to flubaroo.  But somehow it had not actually processed until yesterday.  Here’s a follow up video I posted for my colleagues from yesterday – to show their way around, some ideas / things to do.  Please overlook the amateur screen-cast, but you can at least get an idea.

  1.  Once you’ve created your form, go to settings and choose Make this a quiz, make choices and save.
  2. Edit a question, choose answer key to mark correct answer assign points.
  3. Choose feedback to offer feedback for both correct and incorrect answers.  Even better – the option to set up a link to another resource within the feedback.  My idea is to offer questions / suggestions for incorrect responses and a link to online resource/practice/video to help with intervention.  But what makes me even happier is to offer a link within the correct answer feedback to a resource for enrichment/extension.  ***happy dance***  Yes, I realize the question I’ve included in slides is ridiculous, unrelated to links, etc but I was only playing to see what I could do!

Please share other ideas / suggestions you have or run across.  This is so cool!  Very excited about it.

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.


Better Questions Week 3 #MTBoS



I’ve pondered this challenge for a couple of days.  So many options!  But a tweet from @mathymeg07 led me to a post from @MrAKHaines blog Math Pun Pending.

The post was celebrating a variety of strategies his students had use to answer the question:



He wrote:  When I wrote the question, I had anticipated that students would use a couple of different strategies. What I didn’t know was that my 25 students would use a combined seven correct solution strategies to solve this problem.

Two parts to my post:  1. How can I make this an open question and 2. How can I use student samples to develop a better lesson in the future?

How can I make this an open question?

A.  Name a point that is NOT on this line.

B.  Name a point that this line passes through.

Thanks to @PIspeak‘s TMC14 session in Jenks, I urge students to “Support your claim with evidence/reasoning.  I want to see your thinking!”

How can I use student samples to develop a better lesson in the future for my classroom?

I appreciated the fact that he never explicitly taught “the teacher’s efficient strategy” but allowed group discussions and support to drive the lesson.  Students shared ideas.  The last paragraph  in his post says, “My students are acting like mathematicians, y’all. They’re using their toolkit of math ideas to solve problems flexibly. I couldn’t be happier.”

In the end, that’s what we all want – students thinking on their own, making sense and being confident enough to explore a problem with their own ideas.  So, how does this tie in with the Better Questions prompt?  My outline of the lesson feels a bit like those I’ve used from Formative Assessment Lessons, but I feel it lends itself to students doing the thinking, talking – I only provide the materials and support to make desired connections that will lead to the learning goal.

I’ve been following the #T3Learns chat from Wiliam’s book.  In chapter 3 of Embedding Formative Assessment, it suggests using student sample work. How might I structure a lesson, utilizing student samples of this question?  In Principles to Actions, MTP3 states Effective teaching engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

  1. Begin with the same question. Allow students to write a response. (3-5 min)
  2. Place students into small groups and allow them to share their approaches. (5-7)
  3. Allow groups to explore student samples, making note of different ideas, what they like/ways to improve, questions they’d like to ask the student. *maybe as a gallery walk? (15-20)
  4. Discuss their findings as a whole group. (10 in)
  5. Connections to/between the different mathematical representations. (5-10 min)
  6. Written reflection:  (3-5 min. possibly use as a start up / bell ringer to begin class with following day – providing an opportunity for retrieval of previous days information?)
    • my strategy was most like:____
    • the strategy I liked most was ____ because ___
    • the strategy I found most difficult to understand was ___ because ___
    • Which approach was most efficient?  Why?
    • What do you think was the BIG IDEA your teacher intended for you to learn/understand?
  7. Transfer…provide a few, different contextual problems that allow students to connect the mathematics to something tangible, maybe in a problem posing situation (should this be small group?  individual? ) (5-15, would this be better as follow-up the following day?)

Timing is often an issue for me.  I want to provide students with enough time to make sense/discuss, but not so much time it feels long and drawn out.  Are the times I have listed appropriate?

Please offer suggestions.  How have you used a similar approach successfully in your own classroom?

#MTBoS My Favorite: Open Questions & Level-Up Quiz



Goodness.  I think this is where I fall apart.  I have so many favorite things I’ve used in my classroom, at times I cannot focus and choose one.  I become distracted, thinking I have to use EVERYTHING.  I have to pause, think about the learners in the classroom and what will be best, most effective for them.

Our second week back after Christmas break was very productive.  I chose to combine 2 ideas and focused my energy with them.  One goal I had set was to use open questions.  (Older posts – first attempt, more good questions – about strategy from Small / Lin).  Rather than giving students more inequalities and asking them to graph.  I gave them a point and asked them to create an inequality whose graph would “capture” the point.  Students had to think differently in order to create their response rather than following a procedural step by step or copying a classmate’s work.

The other was an idea someone had tweeted that caught my attention and I wanted to see how it would work in my classroom…level-up quizzes.  Since the target involved graphing inequalities, I gave each student a paper with 4 empty graphs and space in margins to write inequalities and verify.  Here is a sample of the criteria I gave them:

level up quiz

I told students I wanted everyone to be at level 3 by the end of the week – Level 4 was using multiple measures to verify their responses.  If students were at 3 or 4 early in the week, I posed a challenge to them to create two inequalities that would both capture the point.

This task accomplished several things for me.  It was obvious where students got stuck, it allowed me to give feedback or have a conversation about the symbols, which direction to shade, helped point out when/why to use the = if the point was on the boundary line or not, could quickly address issues with graphing key points of the line.  It allowed students to move on without waiting on their peers.

There were a couple of students in each class who continued to struggle-mostly students who had chosen NOT to put any time/effort into practice the prior week or who had been absent, but the rest of students made gains and improvements with this skill.  By the end of the week, majority of students were at or above the level 3.

The big thing with verifying I saw was students using (0,0) to test in their inequality algebraically as opposed to the actual point we picked.  I feel this was due to us graphing inequalities the prior week.  This year, I opted to encourage evidence of their claim by having them test a point to determine direction of shading as opposed to just saying above/below.

With only 1 response for every student each day, I was not overwhelmed, but able to give feedback.  I made notes of most common errors and addressed them as a whole class prior to passing the quiz back.  For many, I simply wrote a number corresponding to the Level-Up criteria.  Students knew the first couple of tries “didn’t count” but were opportunities to learn and level up by the end of the week.

My concerns after reading about Rubrics in Embedding Formative Assessment –  have I made it more of a skill-ckeck list?  By presenting it as an open question, is that enough to allow for student thinking?  Thoughts on how to improve are welcome!

Posting Learning Targets yay or nay


Thanks to @JustinAion,  I got thinking…

It depends… on my class and the students and the activity…to determine if I actually post it.

However, when I do, I refer to it at the beginning, throughout the task – to remind students of the end goal, and again as a wrap up – whether reflection, exit ticket of discussion/summary to end class.  And I like to refer to it the following day as we begin the next lesson, just as a quick review.

I, personally, would prefer to have an overarching Essential Question for each lesson to use rather than a specifically stated target.  However, I sometimes struggle a lot with Writing EQs, would love a colleague to collaborate on these.

Here’s a section of the unit organizers I’ve used this past year (thanks @lisabej_manitou). 


And a link to this file.
Unit Organizer
Functions Overview

I give them to students toward beginning of unit, we complete the words worth knowing for vocabulary (thanks @mathequalslove). Then read through actual targets.  When quizzes are given back or practice problems checked, students have a place to reflect/record thwir level of learning as well.  Because students have this in their INBs, I can quickly refer to them if not posted on the board on any given day.