A 3-2-1 Formative Assessment on Ratios With a Side of Surprise

A few years ago, I started trying out some new formative assessment techniques.    It has been a journey full of surprises.   I’ve learned a lot about what my students think along the way.    One of the big surprises was during a unit introducing ratios.   I gave my students this  exit ticket.


When I made it, I was just trying out the structure and thought it was a little bit of fluff.    Then, I saw the results.   I gave the assessment about a week into a unit introducing ratio reasoning to my sixth grade class.   We had completed lessons on tape diagrams, ratio tables, and double number lines.   I expected students to just plop those three representations into the exit ticket without much thought.   Instead, I got 3:2, 3 to 2, and 3/2.

I’ve given a lot of thought to why I got these responses.   Did the students just key in on the “3” part of the question and not really read the prompt carefully?   Did I fail to draw out  the connections between tape diagrams, ratio tables, and double number lines enough?   I don’t know why I got the responses that I did, but pondering the question of why was important.   I started working harder at drawing out the connections between the different representations the next year and the next.

I gave this assessment a few days ago.   As I walked around the room, I marked each response with either a green or a pink highlighter.   Most of the responses were correct.  Those students who got a pink mark, went back through their interactive notebook to find their mistakes and correct them.

I still wonder why some of my students are making this mistake, but feel a little closer to the answer.


What Does Division Really Mean

Fraction division is a messy business.   Now that we have Common Core State Standards, students must be able to model fraction division in addition to performing the task algorithmically.   This is no simple task for many of them, because it forces them to grapple with the question of what it means to divide.   In order to do it successfully, they must really understand that division represents one of two things.   It can be dividing something into a specified group size to find the number of groups. (I have twenty four cookies and I want to make packages of 2 cookies, how many packages can I make?)  It can also be dividing something into a specified number of groups to find the group size.  (I have twenty four cookies and I want to serve 12 kids, how many cookies can they each have?)  They have to be able to read a problem and figure out which of those two types of problems it is and then form groups accordingly.     Making sense of these ideas and constructing a real understanding of fraction division is hard, even for a lot of adults.

As we were wrapping up our work with these ideas this week, I wanted to do a quick formative assessment to see where everyone was on these ideas.   I took two of the ACE  questions (this is the set of problems from which we draw homework assignments)  from the Connected Math textbook that I use.   Instead of using them as part of a homework assignment, I turned them into a Vote With Your Feet activity.

Vote With Your Feet is a Marzano high engagement strategy that incorporates movement into a lesson. In the activity, students are presented with a multiple choice item.   They move to different locations in the room based on their chosen response to the question (e.g., north wall for A, east wall for B, south wall for C, west wall for D).  The activity gives students a chance to get up and still stay focused on the task at hand.   It also is a really quick formative assessment, taking only a minute or two to see what each student thinks and where their misconceptions are (if the question is well-designed).

Here are the two questions that I posed.


I had students vote on each of the two items without commenting on their choices.    Afterwards, we debriefed both questions by discussing what problems each model might represent.   Kids talked to partners and in table groups.   The class talked about it together.    As we talked, I asked students to give me two division problems for each model.   What is the problem if you are dividing by a specific group size?   What is the problem if you are dividing by a certain number of groups.

These are two of my favorite problems for a formative assessment on fraction division.   This year, I used them as a Vote With Your Feet activity.   Next year, I might take the four options and make them Quiz/Quiz/Trade cards.   I might take them and just use the model and present it as a “here is the answer, what is the question” formative assessment.   In the meantime, I will probably go ahead and make Quiz/Quiz/Trade cards with problems like this to use as a quick review of fraction division from time to time later in the year.

Making Graphs Anecdotal Records Form

Knowing what my students know is important to me, but remembering who is struggling with what aspect of a given concept from day to day is impossible.   There are just too many students and too many variations.   I absolutely have to write down what they know if I want to act upon it the next day.    Unfortunately, there isn’t a lot of time in a given class period to take those notes.   As I result, I have started creating anecdotal record forms that are specific to a big idea that I am addressing in class.    On these anecdotal records, I have a single box for each kid.   In each box, I have specific items of which I want to ensure mastery.  I use one sheet for each period.   I put a different student’s initials in each box.  Then, I make copies of the sheets so that I have enough to use for more than one day.   Alternatively, I use a different color pencil each day.  I keep them on a clipboard for ease of use.   As I circulate around the room, I circle an item in a student’s box if it is an area that I need to address with him or her.   The next day, I simply look at the sheet from the previous day and I know exactly what I need to address with specific students.

I have just started a unit in which students will be representing and analyzing data in tables and graphs.   During this unit, I will be using the anecdotal records shown below.

I use the” Ind/Dep” category to indicate whether students are correctly selecting the correct axis for the variables (independent variable on x, dependent variable on y).   The “Intervals” category indicates whether the student is maintaining uniform intervals on the axis (a very common error in the early days of making graphs).   The “Continuous/Discrete” category indicates whether a student is correctly determining whether or not to connect the points on the graph.   The “Plot points” category indicates whether a student is correctly plotting points (x,y vs y,x).  The remaining categories are more minor errors, but errors that I want students to clean up.

You can download the form by clicking on the link below the photo.



This year, I have added a small refinement.   I use red/green marking on the student’s graph to indicate the error as well.   For example, if a student has incorrect interval spacing, I use a pink highlighter on his or her graph to show the two different sized intervals.


How far can you ride in a day? – A Lesson Introducing How to Make A Line Graph

What twelve year old wouldn’t want to spend a few days bicycling along the ocean front, spending their days amid sunshine and ocean breezes and their nights under star-filled skies?   Along the way, they get to swing through Cape May, a lovely ocean-side town filled with beautiful Victorian buildings, visit Chincoteague Island to see the annual auction of wild ponies who swim to the island from Assateague Island, and swim in the ocean.   This is the context for my students’ exploration of different ways to represent  and analyze data  (tables, graphs,  and eventually equations).   The series of (Connected Math) lessons center around a set of college students who are setting up a summer bicycle tour business to earn money for school.    In the series of lessons, they explore the question of how long each day’s ride should be, whether the planned route is feasible (they test out the route and collect data), where they should rent bicycles for the tour, finding the perfect price point to maximize their income, how long the drive back from the final destination will take at various different driving speeds, and the cost of taking the tour participants on a side outing to an amusement park.

Yesterday, I started the unit by introducing the problem  context.   I began by showing a short video clip of someone on a bicycle tour through Great Britain.    I chose to begin with a video clip in order to support my English Language Learners and students from lower socio-economic households, in order to bridge language and economic divides that might make the problem context difficult to grasp.   By seeing a bit of a bicycle tour, they would have better access to the problem context.

After students watched the video clip, I introduced the problem – five college students setting up a summer bicycle tour business.    The first question the college students were considering was how long each day’s ride should be.   I asked my students what they thought would be reasonable.   This led into a nice discussion of some of the factors that might influence the answer to that question – the relative incline (uphill, downhill, flat), the terrain (pavement vs sand or gravel), the weather (riding into a wind, no wind, wind at one’s back).   I then asked them if they thought it was reasonable to expect the riders to maintain a constant rate for the entire day.    Some students thought not, but some students thought the riders could pace themselves.    This laid the groundwork for the first part of the lesson.

In order to explore the idea of pacing, I had students conduct a jumping jack experiment.   At each table group, one member of the group would perform jumping jacks for 2 minutes.   Another member of the group would be the timekeeper, marking the time in 10 second intervals.   Another member of the group would be the counter, counting the jumping jacks.   The final member of the group would be the recorder.   (I assigned tasks to group members by their seat position within the group.   If a group had only three members, I had the jumper also do the counting.)   In order to speed the process, I provided the recorder with a pre-made table so that he or she would  spend less time copying out a table and could instead just fill in the table entries.


At the conclusion of the experiment, I had groups collect data from their recorder and complete their own data tables.  Then, I asked students to describe what happened to the rate of jumping jacks as time progressed.   There were a few groups throughout the day that maintained a fairly steady pace, but most groups experienced a steady decline as time passed.   Some groups had a jumper who stopped completely part way through the experiment and then resumed their jumping after a short break.   As the class discussed this, I asked them how they saw these changes in the tables that they had created.  This gave them the chance to  explore the idea of how a change in the jumping jacks compared to a constant change in time. (I had not yet introduced dependent/independent variables.)  At this point, my goal was to begin to tie this lesson back to work they had done with ratio tables in a previous unit.   I wanted them to see that this was not in fact a ratio table because the rate was not constant.

I told students that I wanted them to look at the trends of the jumping jack data in a graph.   In order to do that, they needed to learn how to correctly make a graph.   I began by introducing the concept of independent and dependent variables in a table and talking to them about the mathematical conventions.   Then, I introduced the process for translating data into a line graph using a Flow Map (this is a Thinking Map used for sequential processes).   I provided students with the  advance organizer to complete their notes on creating a graph.  I had a regular version of the organizer and a modified version of the organizer (that is more of cloze activity) to support students with learning disabilities.





After students completed the Flow Map, I had them use the data from the jumping jack experiment to make a graph.   As they worked, I circulated among them, taking anecdotal records on their work.   As they finished, I selected graphs to share with the class.   I was very intentional in selecting graphs with errors.   I then asked the class to examine the graph to see if there were any errors.   This forced the students to think more deeply about the work they had been doing.   When someone found an error, I gave a piece of candy to both the student who allowed us to look at his or her work and to the student who found the error.   I explained that the person who let us see the mistake did as much to help us grow mathematically as the person who found the error did.   (This is a common practice in my class).   I made sure to share multiple graphs with errors in each period to ensure that no one student felt like he or she was the only one still learning how to do this.

After analyzing several graphs, I asked the students to explain how they saw the rate of jumping jacks changing in the graph.   Here, I was laying the ground work for upcoming lessons in which they will be analyzing data in graphs.

After summarizing the lesson, I had students complete an exit card in which they had to find the error in a graph.   After some thought, some of the students were able to see that the independent and dependent variables were on the wrong axis.   For those students who were having difficulty, I told them to go back to their Flow Map and work through each step to see if they could find the error.   Eventually, everyone successfully completed the exit card.


Following the lesson, I posted an anchor chart on the wall of the classroom that corresponds to the Flow Map that they used in their notes.



Pondering the GCF & LCM

Some ideas are a little bit like raindrops that have splashed onto my puppy in a summer rainstorm.    They land lightly on her but are quickly shed as she shakes her body in a happy dance.   They touch her for only a moment and are gone.   Some ideas, though are more like a piece of sticky gum on the sidewalk on a hot summer day.    They seem to be permanently attached.    For some reason, that is how Mark’s post on exit cards has been for me these last few weeks.   I keep pondering what exactly I should want to know about my students.

This question has been dancing around in my head as I have been planning the first unit for my sixth graders.   This particular unit incorporates the ideas of the Greatest Common Factor and the Least Common Multiple (among other things).   For some reason, a lot of students seem to lump the two ideas into a single muddy mess.   I think part of the problem is the language – the words are so similar and yet so different (greatest/least, factor/multiple).    Even parsing out just the factor/multiple piece is tricky.   So many students confuse the two words.    They have trouble making them their own.  I also think part of the problem is that students have some measure of difficulty determining just what a Greatest Common Factor or a Least Common Multiple is.   I don’t mean the question of how you find them, I mean the question of exactly what are they and why should anyone care.

As we explore this larger question in the unit, we will do so in a problem context using Connected Math.

  • We’ll start with a scenario in which two siblings are going to ride two different Ferris Wheels, one larger and one smaller, at a carnival.   Given the amount of time for a single revolution for each Ferris Wheel, students will consider how long it will take for the two siblings to both be at the bottom of the revolution simultaneously and how many rotations each wheel will have made.    I usually see a fairly wide range of strategies as students tackle the question.    Some students will start making a list of the times for each sibling to return to the bottom and then find the common time in each list.   Some students will use this strategy but will organize the information in a table (which makes it so much easier to make sense of the information).   Some students will really struggle to make sense of it and we’ll need to talk about what kind of strategies they might use to help make sense of the situation.   Eventually, they try something like drawing a picture or modeling the scenario using some different colored cubes.    A few will immediately see that they need the LCM.
  • Next, we’ll move onto an exploration of the life of two different cicada species, one of which emerges every 13 years and one of which emerges every 17 years.   Students will explore the question of how often both species emerge simultaneously.    Generally, the thinking will be similar to that in the Ferris Wheel problem but it will come together a little more readily because of the discussion that ensued after the Ferris Wheel problem.
  • Our next exploration will consider a scenario in which a girl is planning snacks for an upcoming hike.    She has a specified number of apples and a specified number of bags of trail mix and needs to determine how many snack bags she can make if each snack has the same amount of food and there is no food left over.    Initially, some students want to apply that same hammer – they want to find the LCM because that worked on the last two problems so it must be the way to go on this one as well.   Then, they get an answer that doesn’t make sense – how can you have more snacks than you do apples?   I like when this happens because it makes them step back and try to actually make sense of the problem – to realize that sense-making is what math is really all about.   Some students will reach for some blocks, using one color for apples and another for trail mix.   After they play around with it for a few minutes, they usually realize that they are dividing into groups and move forward without actually modeling the whole problem.   Some students will do the same kind of modeling just sketching or tallying and reach the same end result.   A few students will just seem to see it and will find the GCF.    I always try to bring out the different strategies as we debrief the solution and ask a lot of questions about why they chose that strategy and what that strategy showed them.   We follow that up with a scenario in which the canary ate some of the trail mix and the numbers suddenly get messier.   The idea remains the same, though.
  • Finally, we consider a scenario in which someone has donated some cans of juice and some crackers to the school for snacks on a field trip.    Students need to find out how many kids can be served by the donation if each kid is going to get the same amount.   This problem seems to come together pretty quickly since it is the second GCF problem and follows a pretty solid debrief of the first problem.

Following this sequence, I expect some of the students will have made sense of the concepts of GCF and LCM, but some of them will not have.    Some of them will still struggle on the next novel scenario to figure out how to solve it.   I am wondering if my question (exit ticket) needs to be something like this.

What kind of problem requires you to find the LCM?   What kind of problem requires you to find the GCF?

I want to know if they have been able to generalize the concept.    Do they realize that LCM problems are really problems of alignment and that GCF problems are really problems of grouping (dividing)?   This is a shift for me.   In years past, my exit card would have been another problem and I would have evaluated their mastery based on their ability to solve that problem.   With this question, I am asking them to think on a different level.   I want them to consider what is the same about LCM problems, what is the same about GCF problems and how are the two different.

I’m not sure if this will be my exit card question, yet.    I have a little more time to ponder.   Am I asking the right question?



Proportional Relationship Anecdotal Records

You just can’t hurry some things.   They take time.    In middle school, grasping the big ideas of proportional relationships seems to be one of those things.    Students need to build an understanding of a constant rate of change and how it presents in tables, graphs, equations, and verbal representations.   They need to build an understanding of what it means to have no “start-up” value in each of these representations as well.   They need to explore these ideas in a little of different ways in order to make sense of it.

Because there are so many pieces to this idea, there are a lot of different places that understanding can break down.   In order to know where that breakdown is happening for each student, I like to take anecdotal records.   At the end of the day, I can go back and see who knows what and make decisions about what I need to do the next day to help each of them move forward.   The thing is, I don’t have time to write a paragraph about each kid as I walk around the class looking at their work and listening to their discussion.    I need something that just takes a second or two for each kid at any given time.    To that end, I decided to create an anecdotal record form specific to proportional relationships.

I started with what I want to know my students can do.

  • Recognize a proportional relationship in a table
  • Recognize a proportional relationship in a graph
  • Recognize a proportional relationship in an equation
  • Recognize a proportional relationship in a verbal representation (word problem)
  • Be able to connect proportional relationships represented in a table and a graph.
  • Be able to connect proportional relationships represented in a table and equation.
  • Be able to connect proportional relationships represented in a table and word problem
  • Be able to connect proportional relationships represented in a graph and equation
  • Be able to connect proportional relationships represented in a graph and a word problem

If students can’t identify a proportional relationship in a given representation, I also want to know if the break down is the rate of change or the y-intercept.

I came up with this Proportional Relationships Anecdotal Records form.    I have one box for each student (I can use more than one page for a given period).   I put the student’s initials in the box and then circle the place where a breakdown is happening.   If the student is not recognizing a proportional relationship, I can write in k if the breakdown is the rate of change or the letter b if the breakdown is at the y-intercept.


I will probably use this to drive a brief review activity the following day.    I will pair students to do a card sort or to use the cards to play rummy.   Initially, I will pair them so that a student who has the idea down is working with a student who doesn’t.  After working with the idea for several days, I may place them so that kids who are struggling with the same issue are working together.     At that point, I can work with that small group to address whatever disconnect is still in place.   You can get the card sort here.

Getting to Know Students – A Survey on Mathematics Affect

It’s  perfectly normal to encounter people who will very readily tell you that they “can’t do math”.   They say it almost as a badge of honor.   I think, though, that they are really saying something else.

My question is, what are they really saying?     I’m afraid of math (or of failure)?     Math is hard (I don’t want to work that hard)?   I had a bad experience in a math class (name your reason)?   Someone told me I’m not good at math?   Someone told me I’m stupid?

As I am getting to know my students during the first week of class, one of the things that I want to know is how they feel about math.   So I ask them.   I also ask them some other questions that might help me get to the why behind those feelings.   I use the survey shown in the photo.  It is taken in part from a survey in the NCTM Assessment book and in part from a survey that a friend gave to her students.   You can download the survey by clicking on the text below the photo.


  Math Survey