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.

Generalizing Patterns: Tiling Tables

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Last fall after instructional rounds, one of the observers asked me if I would mind having some folks visit my classroom.  Sure.  They were most interested in questioning, interactions with students and use of Formative Assessment Lessons (FAL).

When they emailed to set up a date, we agreed on January 10.  Oh, wait.  This will be the beginning of a new semester with new students.  I won’t really know them.  They won’t really know me.  Great.  Now, I’m scared.  Oh well, let’s look at the positive – this will give me a chance to try out a new lesson.

I printed off 3 lessons to look at the evening before students returned to school.  I liked all three.  Building complex equations seemed perfect, so I began to prepare for it.  We were out for weather our second day back.  As I began looking over my lesson plans, it seemed the Tiling Tables was a better fit for the upcoming unit, so I switched gears.

I had done this lesson a couple of years ago, but never taught it in class.  As I began to revisit the task, I knew I liked it.  I knew it would offer some good discussion on ways to extend the patterns.  But wait.  These students barely know what a parabola is.  Would they have a clue as to how we would write an expression for a quadratic relationship?  Would I have a clue as to how to introduce it, this early in the semester?  No.

So I pondered for a while.  I would simply use the task as a way to say, we have the knowledge and tools to do parts A and B, but part C, well – that’s what we will be learning later in the semester.  It would give us a reason to learn it later, right?  Goodness.  What a canned comment.  By now, we had another snow day, so our visitors would be in our classroom on the 4th day of instruction.  I was stressing just a bit.  What was I thinking?  Starting off a new class with a FAL I had never used before?  We needed time to build some rapport.  Too late.  Let’s go with it.

I gave students the pre-assessment:

The class was divided pretty much 3 ways – Those who doubled the number of tiles, after all – a side length of 20 is doubled to get 40, right?  The second group had sketched the designed on the the grid paper which had been provided, however, they wrote answers for the 30 cm table instead of the 40 cm.  And finally, several had the correct number of tiles by extending the pattern on the grid paper.  But I ask how efficient this strategy would be for, say 300 cm table?  Hmmm.

We began the lesson the following day by giving 3 samples of work.  Last school year, I figured out, I could save paper by having them use the shop ticket holder sleeves to hold the sample work – allowing them to draw, sketch, etc with dry erase.

These instructions would help their discussions:

The first sample was Leon:

After some small group time, we shared our thinking with the whole class.  There was one student in particular who had confusion all over their face.  I encouraged them to ask the person sharing for clarification (using our starter stems).  I believe this is important to model and have them do early in the semester, so they become more comfortable with it.  Even with more explanation, they were still not seeing the pattern.  So another student shared.  Still no help.  Finally, a third students explained how they saw the pattern.  The confused student nodded and said, “Okay, I got it.”

Now, years ago, I would have said – great and moved on.  But I’ve learned…ask them to explain it to you.  They may say they’ve got it – just so you will move on, but how do you know they understand?   This student, however, could explain their thinking and were correct – they could even extend it to the next table size.

The next student sample was Gianna:

So many more of the students picked right up on Gianna’s approach.  The confused student – smiled stating they liked / could see this one better.  For me, it was listening and watching the students discussing – that brought me an a-ha!  This is the example we will use to generate the quadratic expression I was worried about!  The total whole tiles would equal (step x step) + (step – 1)x(step-1)  Yay!

Many of the students could not make the connection with the side lengths on Ava’s sketches in the beginning.  Then they began going back and looking at their own sketches to verify the numbers Ava recorded in the table.  They noticed the same patterns and agreed with them.

After this final discussion – we went back to see if each student had answered the task fully.  We quickly realized though there was some good, correct thinking going on in their work – they had not addressed the questions completely.  The class agreed that Ava’s was the most complete with her table.  And it was interesting to hear their discussions of how they would explain to the other students how they could expand their responses to be better and more thorough.   One student even brought up it was challenging trying to figure out their thinking since there was no written explanations of what they were doing.  (I thought – yes, this is what I feel like sometimes too.)

As we continued discussing having thorough answers – I shared Ava’s data in a graph…  they were quick to see the quarter tiles always remained four and the half tiles being linear, a focus from 8th grade.  But what about the total tiles.  How can we write an expression to model that data?  And I took them back to the slide with Gianna’s work to look for patterns between the table size/step number and the total whole tiles.  We test our thinking with different sizes and it worked.  We tested our expression in Desmos…and what?  It hit all of the data points!

They had some experiences with the visual patterns – and good feedback to me about liking them, but still having to think.  This task reinforced some of those ideas.  IN their reflections – though many may have preferred someone else’s sample work – they “saw” how Gianna’s work led us to a more efficient expression or even Ava’s approach to orgaznizing the data in a table was pretty helpful to see the patterns so we could find describe the expressions.

Total Tiles = 4 quarter tiles+ 4(n-1) half tiles + n^2 + (n-1)^2  whole tiles.

I will definitely be using this lesson in my future.  It brought just enough confusion, but great opportunity for sharing and discussion.  And the observations were great.  Students were not shy.  At the end of the day – I was amazed we had only been together for 3 or 4 days… wow, this is going to be an outstanding semester!

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.

Braking Distance and Speed #MTBoS30 Day 2

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8 minutes…here goes…

A quick retrieval from last week’s exponential growth / decay to begin our day in Algebra I. Four functions:

• y=1200(1.13)^x
• y=1200(2)^x
• y=1200(.91)^x
• y = 1200(2)^-x
1.  60 seconds:  Tell how they’re alike / different.
2. State which ones are increasing and decreasing without “doing any math / graphs.”
3. Graph each, sketch, noting the end behavior…does it support your description?
4. Let x=3, find the function value for each.  Is it greater or less than the initial value?

Texting Olympics!

Whaaaat?

That’s right, thanks to Heather Khon, we did a little sprinting and hurdles in class today.

Sprint:  staying for help in math pick me up at 4

Hurdles:  Laker baseball has a winning season!!! woooo!  #lakerproud

Recorded our times and will use them tomorrow as we look at breaking distances more…adding on to our distances how far we’ll travel in just 1 second at various speeds, then leading up to a written PSA, print PSA, recorded 30 second PSA to support ATT’s campaign, Its worth the Wait.

Here’s where we began our new content – a look at some quadratic data.

These lessons are taken from A Visual Approach to Functions, 2002, Key Curriculum.

1. Notice / Wonder about the speeds and braking distances.
2. Create a scatterplot, then run a quad reg because we don’t see a constant rate of change for linear and no common rate of multiplication for exponential.
3. Some are amused when they see this crazy function pop up and just how well it models their data when graphed.

f(x) = .0975x^2 – 2.5x + 65.25 (its pretty close to this, but I don’t have it written down with me.)   What’s that, you want to know how far you’ll travel at 90 mph, Brooke?  Let’s see.

The lessons go on with varied graphs asking students which models given sentences.  Three tables of values are shared, asking them which is likely the data for their chosen graphs.  And finally, equations – which one matches each data set.

Again, open questions because students were not limited to a single strategy.  Some students chose to use x-values as inputs into the equations and find the one that modeled.  Another group used their data to create a scatter plot and/or regression equations, and others found the rates of change within the tables and compared to determine if linear/quadratic.

All in all, I was please to see such varied approaches – and  correct in their reasoning.  Looking forward to continuing these conversations tomorrow as an intro to quadratic functions.  Maximizing Area will follow braking distances.

Function Families & Why’d It Do That?

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We began our week in Algebra I with Function Families.

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.

Real Life Water Line

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After some good old Dan Meyer Graphing Stories last week, we began our next phase of functions by predicting what our graphs (#s coops & water height) would look like for these containers:

And then we actually scooped water to see how close we were…

Walking around, listening to conversations…
“NO. It can’t go back down, you’re still adding water, so the height of water keeps increasing.”

“It might slow down or speed up, but it won’t decrease until we empty the vase.”

“So we need to reverse the steepness…where it’s steep, we need to flatten it out and where it’s flT, we need to make the graph steeper.”

I required them to complete a group graph to predict before I gave them their scoops to start data collection.

Several had to go back and finish details like labels and scales on graphs…a good reminding activity.

One student asked – do you do activities like this often?  It makes it (math class) fun.

It was a good day.

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I realize some folks will bash me for sharing this from an Algebra 2 class, but based on benchmarks, most of my students have major gaps in quadratics.

I began with reviewing multiplying 2 binomials on our whiteboards.  I shared the box/area model and several smiles celebrated because they “saw it” and were doing it correctly!

Last week, I pulled out a box of Algebra Tiles.  We literally explored building squares.  I wish I had taken pictures because some of their squares were like a grandmother’s beautiful quilt blocks.  I began tying it back to our box/area models -I’d rather think of it as leading (not forcing) their thinking – but they were quickly picking up the patterns.

We then began looking at the algebraic equivalents, again, with a sketch along side allowing them to “see” the process.

Our next step was to find the missing value without tiles/picture models…and then I asked them to review their multuplying with 5 expressions alongside.

“What? You think they’re the same thing?!?” I asked,  “Prove it to me. Well, by-golly-jee. You are on to something!”

The following day in class, I made a HUGE ordeal of different ways to write zero.

I explained our next few minutes were a process. But we talked about it, step by step, completing the square, adding ‘that zero’ in our expression, the separating the trinomial and 2 constants.  Rewriting our trinomial as a binomial squared.

Ok. Why in the world would anyone want to do this?  I told them we were finding hidden information.

As they arrived at this form (x+4)^2 – 9, I paused, reminding them to think back on our function transformations before Christmas break.  How would this function y= (x+4)^2 – 9 move on our graph from this one y=x^2?  Quiet. “Move left 4 and down 9!” Someone exclaimed.  Really? Are you sure? We graphed the two and yes, it did just that.  So what does this tell me about my parabola?  They didn’t say vertex. Or minimum.  They said it shows us how the graph was transformed.

I will take that.

I then asked them to move left 4 and down 9 from the origin.  What have you found? The lowest point.  The vertex. The minimum. All their responses, not my statements.

We set our expression equal to zero and solved the equation, using our inverse operations.  They made the connections with the x-value of the vertex being the “center line” of the parabola.  They realized the +- 5 were steps in either direction from the center line.

I most appreciated the questions they asked on #3, 4 and 7.  Several chose #7 thinking it was shorter, thus less work. Snafoo. No middle term. What happens?

I suggested they look at it from a transformations point of view.  Someone shared-It doesn’t slide left or right, only down.  Another student said-well, that’s the easiest equation to solve! (Yep.)

Why did #4 bother some? The middle term had an odd coefficient.  But once they shared their thinking, ok. Got that one too!

#3 was what we math folks recognize as perfect square trinomial.  But for the students, it was an a-ha.  Again, using the transformations context, we moved right 5, but not up or down.

L: But I thought all quadratics intersected x-axis twice?  I asked – did this one? No.

What about y=x^2 + 3?  It moves up 3. Ok. How many times did it intersect the x-axis? It doesn’t.   Hmmm.

A student who is rarely engaged then asked, if you can make a parabola that doesn’t intersect the x-axis, can you find one that doesn’t intersect either axis?  Me: Can we? What would it look like? S: Noooo. As its going up, increasing, it would be increasing outward, too!  More discussion, between them. Me not included. I was smiling.

And their questions were what drove our lesson today.   And I was so excited, telling them their questions make me think! And when they’re asking questions, their brains are processing the information – making it their own.

It was a good day.

Inverse Functions #MTBoS30 Post 20

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This week we wrapped up our school year with 2 final teacher work days.  One day our department had some time to reflect onmour year.  One of our big issue is the immense amount Kentucky requires in Algebra 2.  Its nearly impossible to ‘cover’ it all, much less allow struggling math students time to process and develop a slight understanding.

I, personally, don’t worry with getting it all in.  I attempt to make the students’ time worthwhile and hopefully they leave with some deeper understanding about concepts rather than a list of procedures only 30% may be able to recall at any given time.

Anyway, we attempted to pick out 10 Big Rocks for our focus next year..the non-negotiables in a sense and that is where we will begin.  Sure we will have the “extras” ready should we achieve at a faster pace than expected.  But the goal is to truly develop reasoning and conceptual understanding of our Big Rocks.

While looking over the outline, there were a couple of things I remember thinking we needed to ensure we focused on during our functions unit.  1) composition of functions, not just in equation form, but to make sure we look at them numerically in a table of values and graphically as well. 2) a different approach to finding inverse functions.

It was Sam’s post that made me wonder if this way would make more sense for our students.
Looking at “x” think about and list order of operations.  Then, in reverse, list inverse operations and apply to x to get the inverse function.

Inverse operations and inverse order.
Is this a strategy others use as well?  A couple of colleagues seemed to really like this.
Wondering what situations it may not work…or at least reasons I shouldn’t use this approach.  Is there anything that follows this would cause issues for my students?

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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.