# Reflecting on Formative Assessments

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### 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.

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.

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.

### 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 deltamath.com.   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.

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.

# Interpreting Distance Time Graphs

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On the 3rd day with a new group of students, I had visitors from some other districts in our classroom.  I was nervous – I really didn’t know these students yet and they certainly didn’t know me.  I had chosen Interpreting Distance Time Graphs lesson from MARS to begin our semester.  Although this is listed under 8th grade, it leads to some great discussions and uncovering of ideas and misconceptions.   The Keeley & Tobey book also lists “Every Graph has a Story” in the Formative Assessment Strategies.  This was the ideal lesson to introduce our first unit on functions, while trying to be intentional with planning FAs.

## Pre-Assessment

Telling students it is only for feedback, not for a grade seems to drive most of them to really share their thinking.  After reading their responses, I had some ideas of how I wanted to change the lesson up a bit from times past.  The first time I ever used this lesson was around 2011-2012.

## Let the Lesson Begin

We began our actual lesson with only the graph in this picture.  I asked students to jot down 3 things they noticed about the graph.   Pair share.  I called on students randomly with my popsicle sticks, then allowed for a volunteers (this was something @druinok and I had read in EFA2, which allows everyone to be heard).    We then read the scenarios aloud and at the table groups, they discussed which story was model by the graph.

Next I took one of the scenarios we didn’t choose and asked them to sketch a graph on their whiteboards to model it.  We had about 5 different overall graphs – I drew on the board and let them discuss at their tables which they agreed/disagreed with.  Then we shared our thinking.  Some very good sketches and great discussions.

## Open Card Sort

Many years ago, a colleague shared the idea of open sorts, something she had learned from a John Antonetti training.   I instructed students to remove only the purple graphs from their ziploc bags.  (Side note suggestion- use different colors of cardstock and this allows them to quickly grab the cards they need, ie the purple graphs, green scenarios OR blue tables.  I used to have all the same color and we wasted a lot of time sorting through which cards we needed).  In pairs, they were sort the graphs any way they wished, the only requirement, was they must be able to explain why they sorted them as they did.  Again, sharing whole class led to seeing some details we had initially noticed.  If you’ve never done an Open Sort – let go and let them show you their thinking.  You might will be amazed and wonder why you’ve never done this before.  They love to think.  We should let them.

## List 3 Things

A couple of years ago, I began asking students to list 3 things they noticed or knew about their graphs – anytime we were interacting with a graph.  IF you ask them to do this enough, it eventually becomes habit.  I also like this approach because it gives them a chance to survey the information in the graph before they start worrying about / answering questions.  Today, I asked pairs to label their whiteboards A – J and I set the timer.  They had to share/discuss/jot down 3 things about each graph.  Once again, I used popsicle sticks to randomly call on a few students.

## Graph & Scenario Matching

Using the “rules” listed in the lessons powerpoint, students were then given time to discuss and match graphs to the scenario.  This went so much quicker than times I’ve done this lesson before.  I believe it was because they had already interacted with the graphs twice…they were not “new” to them.  I will definitely use the Open Sort and Name 3 Things before matching tasks in the future.

I gave them a chart to record their matches.  We then shared out our matches.  Each time, I neither confirmed or disputed their matches, but rather would call on a couple of other students to agree/disagree.  After some discussions, I came back to the original student to see if they agreed / disagreed with their original match.

One of my favorite graphs is this one –

And our final sorts…  And again – Scenario 2 is always up for some debate.  It reads: Opposite Tom’s house is a hill.  Tom climbed slowly up the hill, walked across the top and then ran down the other side.

Though every student did not get every match exact, there were several a-ha’s during the lesson and questions asked.  I look forward to reading their post assessment.

I’ve used this lesson as written many times with much success.  However, just making some adjustments prior to the matching made a vast difference in the amount of time students needed to complete the task.

Let me know how this lesson has gone / goes for you if you use it.

# Function Families & Why’d It Do That?

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We began our week in Algebra I with Function Families.

This old task… here are the New link to files.

We eventually end up here as a wrap up. Students come to the board and share their sorts.

The following day we summarize their findings on a foldable…descriptions of the equations and graph shapes from their groups.  The inside of the foldable contains an example of each type of function, table of values and a graph.

I began with quadratic because I see the most mistakes here. Students will use their calculators and jot a number down without pausing to ask if it’s reasonable.  We had 10, -8 and -27 for the first table value.  Hmmm? How’d they get those?  I actually used an entire set of wrong calculations and graphed, then asked, Is that what you expected it to look like?  No. So we need to check our work and find the mistake.

We completed the first table and they were asked to write about what they noticed in the numbers. And we shared.

Next, we looked at the first differences. They wrote about their noticing again.  “Oh,” a girl says.  “That let’s me see what’s happening in the graph!”

And we finished with the second differences.

I went to the absolute value next.

One student claimed, it’s doing the same thing as the first but with different numbers. Another student disagreed because the numbers were constant and not changing like the first.  But the directions were the same.  I explained that different operations would cause the graphs to look differently and we were creating a guide to help us sort through the patterns and learn to recognize them.

In both cases, I heard students mention reflection, symmetry, matched – up referring to numbers in table, not the graph.

We continued with linear and the exponential.

I began with 4^1 on this table and asked, can I write this 4*1 and it’s still 4?  Yes.  So, 4^2 would be 1*4*4 and 4^3 1*4*4*4.

Which means 4^0 would be 1* (zero 4s)…or just 1.

We had done simple function inverses prior to fall break.  I had used the -1 exponent to represent inverse.  So our discussion went back to 4^-1.  Student ask, “well, if exponents are repeated multiplication, would an inverse exponent be dividing?”  And we continue with that discussion.

We ended the day with some reflection on our learning.  They were asked to tell which 2 functions were most alike and why.  Which 2 functions were most different and why.  Very eye opening to read some of their thoughts.

At the end of one class, a couple of students we still discussing something.  He shared, “I was wondering what I’d get if I graphed y=x^-1” and he showed me the graph.  Why does it graph that way he asked.  Why does it graph that, I asked him back.

His group mate shared, well, I graphed y=x^-2 and instead of reflecting into the 3rd quadrant, it’s like it reflected across the y-axis.  Why did it do that?  I replied, why do you think it did that?

I told them both, that was my goal…to let them start asking their own questions…and to keep pondering their graphs, we would talk more about them next week.  It was a good way to end the week.

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I received this file about 5 years ago at KCTM in Bowling Green.  Kari from WKU shared it.  I apologize I cannot remember her last name to give credit.

She actually used it to play a card game style activity.  I copied sets onto different colored cardstock and laminated, I have enough sets we usually do groups of 3 people.

I do this activity along with Go Fish for simplifying radicals.

There are four different forms of each value.  Students use calculators to match cards with same value.  We create a poster as a whole class.  Then notice and wonder.

I like how students develop their own understanding of rational exponents, negative exponents and radical forms.  It’s a great intro activity.

# Modeling Systems

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Sort of a rambling post. But trying to make some sense of my thinking…

I always appreciate posts from @emergentmath.  This particular post made me pause, I had just completed the MARS task, Boomerangs, he references.  We are in the midst of our systems unit.

I used Mary & Alex ‘ s suggestions with beginning systems without the algebra.  Conversations were great, students’ strength in reasoning was evident.

I plan to use Geoff’s suggestion for a matching/sorting activity this werk for students to see the benefits of each type of tool to solve systems.

But where I struggle is with this standard:

I am experiencing some pushback from a handful of students who are able to reason and solve a system without actually modeling it algebraically.

Their reasoning is correct.  They verify their solutions and interpret them correctly.  They can sketch a graph yet “refuse” to model as a system of equations.  I struggle because “their math” is right on.  I realize places where algebraic models can help but I honestly can’t tell them my way is better…yet the standard says…

It feels almost like I am punishing them if I make them model it algebraically.

Then I have others who are not sure where to start.  The equations model provides them a tool, yet they will not embrace it.

How do others handle this situation in your classrooms?

I use graphical, alongside a numerical table of values, with solving/verifying with the equations, letting them see their own connections eventually.

My biggest goal for systems is to provide enough modeling for students to actually “see a context” to connect/make sense of a naked system of equations.

This is where I believe skill/drill has ruined the power and beauty of math.  Finding an intersection point but what in the world does in mean?  It’s a point on a graph. Whoopee.  Why isn’t it all taught in context as a model?

# Forming Quadratics lesson from MAP

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I have planned to share this lesson for several weeks but time has gotten away.  My students were not where they needed to be with quadratics, so I pulled together some tried and true tasks-framing quadratics, Wylie Coyote, et al and a new one from Mathematics Assessment Project called Forming Quadratics. You can download lesson, domino cards and assessment in that link.

No big surprises on the pre-assessment, but I did use it to place students in pairs based on similar thinking/reponses. There are 4 equations students are asked to match to 4 graphs and explain their matches.

I like this lesson for a lot of reasons. Discussion of how different forms give us different information. Allows students to seek key features from graphs, connecting them to parts of different but equivalent forms of equations. Students work in their pair but also must visit other groups to confirm/dispute their responses. The lesson outlines its goals:

This lesson should be used after students are familiar working with different forms of quadratics. This is not an intro lesson, but one I see being successful about 2/3 way through unit or as a follow-up/review activity. They will encounter standard, factored and completed square/vertex forms.

I followed the lesson pretty true to outline, changing only minor things based on my classes. After the whole class intro, pairs worked at matching dominoe-style cards including sets of functions and graphs. I was adament about them taking turns explaining their matches. Some cards had all equation forms, some had only parts. They recorded their matches on a card for the next round.

Following the initial round, one person stayed and another person moved to a different group. In the new groups, they were asked to compare responses, then discuss any differences. This took only a few minutes. Upon returning to original partner, they now had to fill-in missing information on the equations. Again, upon completing their equations, one person stayed and the other traveled to a new partner to compare. Some a-ha’s came about during this part as they maneuvered between the different forms, such as the last term in vertex form does not necessarily correspond to the y-intercept as in standard form. So if and when would they be the same was a nice question for discussion.

As an exit slip this day, students were asked to fill-in front side of this foldable for their INBs.

The following class, I pased back their foldable and gave them a few minutes to respond to my feedback. They received smaller copies of the dominoe cards to cut apart and match inside their foldable. They were asked to write any missing equations, and Color With Purpose different parts of equations and graphs.

I used the same cards and was able to offer some feedback on simple mistakes, but in the future maybe I should have a fresh set and use it as a true formative assessment to see they are able to match new sets & write new equations.

A panel on the trifold was a place to record/review other important info concerning quadratics.

Classes were still somewhat split in the post assessment. 1/3 were right on track, 1/3 had trouble with writing the equations, 1/3 seemed almost clueless- I was like “what happened?” They couldn’t correctly identify key points from the graphs. Before passing back their work, I asked what made it difficult? Even if their matches were correct, they failed to give correct coordinates of key points. Their responses all very similar “there were no values on the graphs.” I passed them back and allowed them to talk over feedack with nearby students. After speaking with them individually, I was convinced all but a couple were now moving in the right direction.

Hmm. Maybe next go, I should scaffold the assessment. Part I, very similar to their practice, including labels on graphs. Part II, similar to current assessment with no labels on graphs. Part III writing equations for given information or identified points from graphs.

All in all, I was satisfied with the discussions students were having; How they had to explain their reasoning for matches made. I see these conversations prooving valuable as we continue in our next unit on other polynomial functions.

A post of the same lesson from Ms. Rudolph.

# Equations of Lines FAL

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So this is from a Formative Assessment Lesson from MARS site a couple of weeks ago.

As I think back, the pre-assessments were very lacking, some even left blank or only minimal scribbles.  Their post-assessments were much better.  They were more confident in manipulating equations to a similar form so they could more easily compare, picking those that were parallel and those perpendicular.

However, a handful really struggled with the given graph in the lesson.  It had 3 lines without the x- & y-axes.

Part of the task asked them to place & label the axes on the graph.  Some actually drew the graph and all lines forming the rectangle outside the given graph, then transferred their work to the graph.  Interesting.  It seemed easier for them to graph the entire thing than to simply add the missing information.  I wonder why?

Several a-ha’s were noted throughout the lesson.  Students thinking opposite slopes would be perpendicular, how to find the x-intercept, in the beginning naming equations like y+4x=3 and y= 4x+5 as parallel.  It was definitely a task where I had to bite my tongue, let them struggle a little, then ask questions without telling them how I did it.

As I look over the first sort, I recall several having trouble getting started simply because the equations were in different forms.  Once they realized putting them in similar forms would allow for easier comparisons.  I gave them the categories for the sort, but I wonder how they would have sorted them had I chosen an open sort?  One reason I chose to use the lesson’s headings was because a couple served as quick reviews of checking to see if a point was on the line and how to find the x-intercept.

Would a better assessment be to create equations (not in slope-intercept form) to fit it given categories?

# Purple Circle Card Sort

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Last spring, I placed equations of circles after distance between 2 points.  The idea came from a mini-investigation in my Discovering Geometry book (formerly Key Curriculum, now Kendall Hunt).

Earlier in the semester a new colleague shared the success her students experienced with the Formative Assessment Lesson Equations of Circles 1.  I decided to use this lesson…

In my early geometry class yesterday, we literally stared at circles.  It felt like a wasted class.  No matter what example I referred back to, or what question I asked, it just didn’t work.  Thankfully, I had planning immediately following and I was able to reflect very quickly.  For my last geometry class of the day, I adjusted my sequence of leading examples.  Reviewing our previous work from last week.

The remainder of the lesson went smoothly.  A quick white-board quiz at the beginning of class today allowed me to address some small errors.  Once again, I had them create their own notes/examples in their INBs.  Yes, a few are still lacking, but the majority are very thorough in what they are including.  Asking questions about specific what-ifs, like one student brought up none of our examples today had a center located at the origin, so I asked the class if they could remind her.  Several went on to include a similar example on their page.

The lesson continued with a collaborative pair.  They were given 12 equations to sort by center and radius.  There were 4 blank blocks in their grid that required them to create their own equations.  At the beginning, some were “cheating” so I stopped them to remind them 1 person picked a card, explained why they were placing it, the other person had to agree and understand before taking their turn.  They are getting better at disagreeing and telling why when their partner is making a mistake.

Their assignment was to create an artistic picture incorporating 5 different circles and listing their equations on the back. Short, sweet, simple.  Can’t wait to see them.

# #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

# Representing Polynomials FAL & Open Card Sorts

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After an assessment last week, it seemed to me what I was doing wasn’t sticking for my students with polynomials.  So let’s just scrap plan A.  Plan B – I pulled out my Discovering Algebra book, came up with a box-building data collection that lead into the FAL I have linked  below.

Formative Assessment Lesson – Representing Polynomials

Thursday, students were given a 16 x 20 piece of grid paper and asked to cut out square corners and create a box with the largest volume possible.  We combined our data as a class.  Recording the corner size removed, length, width and height.  Students were asked to observe the data and respond I notice…  & I wonder… and that’s where our class began on Tuesday.

We shared out our responses, some adding ideas as we continued the discussion.  Work with our data on TI84s – we saw a connection between our constraints 0, 8, 10 and the graph of the regression equation.  This was not new, during the discussion, a question was brought up about what values would result in a volume of zero.  Students were able answer that with confidence and a reasonable explanation.

The FAL pre-assessment confirmed my students weren’t quite ready for the full blown lesson.  With discussion of rigor and relevance the past few days, I wanted to offer students something engaging but not so over their head, it was a flop.

I backed up and did a bit of prep work yesterday – with the following discussions in class:

Following with whiteboards / sharing for this slide from FAL:

and a simple practice set to ensure they were on track.

We began class today with a quick check of the 6 practice – with a focus on similarities / differences.  Noting the double root of #5.

Prior to the actual FAL, I decided to use the same equations and graphs they were to match during the FAL, except I would have them do a card sort.  Originally, I had planned to ask them to sort cards into 2 groups.  While pondering how I could make it better, I recalled a colleague sharing ideas about open card sorts from a John Antonetti training she had attended.  So, this is what I did.

I told students I wanted them to sort the 11 equations – any way they wanted – they just needed to be able to share out their reasoning behind their choices.  After a few moments, I called on different groups and we looked at their sorts.  I should have snapped pics / documented their responses.  I was amazed – not that they did it – but how well they did it.   The things they were looking at – were much better than my original idea to sort in to 2 groups.  Students were asking students – why they put one in one group instead of another. Pausing after we had the cards sorted on the board – giving other opportunity to look others’ groups…some were obvious, others were not.   I even had groups who had the exact same sorts, but with completely different reasoning.  Wow.

At some point we began talking about “What does that tell us about the graph?”  Almost everyone was engaged and comments added to the discussion.  Next we went on to the graphs to sort.  Again, any way they wanted…just be ready to share reasons.

Most of the sorts were better than ANYTHING I would have suggested.  My eyes were opened – I could see their thinking.  And others did as well – it was obvious in the eye brows raised and head nods.  In both classes, there was one equation that never seemed to “fit in” the other sorts – but students were confident suggesting it belonged to a particular graph (& they were correct).

When I realized the sharing took more time than I had planned – I ran copies of the equations and graphs to send home with students and asked them to match on their own.  My plan is to put them back in their pairs for the actual pairing of the FAL.  They also had blank graphs for any without a match.

I learned so much listening to my students today…  I am looking forward to the assessment of this standard.

I didn’t feel like I taught anything today…

…but I did feel like my students left with a better understanding…because I chose to step aside and give them the opportunity to share their thinking…

It was a great day.