Barbie Bungee Implementation

I think that there are a million and one ways you can do almost anything.  Dan Meyer tweeted this out the other day…

I was quick to comment.

I think it’s important to note that I work most closely with 8th grade (on grade level) students.

After seeing the subsequent comments…I wanted to add that IMPLEMENTATION can be more than just here’s your worksheet, “get on with it.”

I like to lead up with this…

barbie bungee one

Can you guess what their answer is???

Then I show them this…

barbie bungee two

Of course this changes their answer.

Then I move onto this…

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And I have students come up to put a point…usually around a y-value of about 10 or so.

Then, I click to reveal additional points.

barbie bungee four

The point is…I make the point that collecting multiple pieces of data helps to make better predictions.  I also ask the question about what mathematical models can we create to help make predictions in math.

Their answers:

  • Graphs
  • Tables
  • Equations

Just what I want.

Then I show one of the many videos that you can find on YouTube.

The last thing I do is pass out a worksheet.



Noticing Subtleties…Reflection

So I wrote this post yesterday, and today we tried it out.

The purpose was to give students an opportunity to notice the subtleties in the language associated with three different mathematical scenarios they may need to represent.

We read each scenario out loud and then asked the students to compare and contrast each problem type with their table partner.

Some students began by discussing the similarities and differences of the contexts…noticing subtleties 1

Others began by making a list of what they noticed on the back…You can see that this student paid attention to more of the mathy parts~understanding what was meant by a one-variable versus a two variable equation.

noticing subtleties 2


And then you’ll see below where students were able to make sense of each scenario and the math required.  However, the first student used an equation in two variables for the first scenario and created a table to find the solution.

We didn’t get to the whole class conversation part of this lesson…I want to talk about each problem type and how to recognize the differences.  Notice that two of the students above wrote the equation for the two variable scenario, but the third student created a table.  I think we need to talk about why that is.  Also, I think I may want to do three more scenarios that would produce equations in standard form, to see if they would recognize the differences then.

I definitely think that this was a useful exercise and would do it again.

One little shout out…one student pulled out their phone because they wanted to check out the equations for the third scenario on their Desmos app!!!


Noticing Subtleties

This just happened…

I walked into an 8th grade classroom finishing up writing and solving systems of equations given word problems.  They began this learning last week.

I overheard the teacher and a student having a discussion about drawing pictures to represent the problem versus trying to write the equations.  This told me that the weekend was too big of a gap from the examples that they had done last Thursday (they had a sub on Friday).

Here is a problem from the set that they were working on:

motorcycles and cars

I briefly walked over to a different student and noticed that she only had one equation to represent this cars and motorcycles problem. She defined the variables correctly, but had written a single equation that mixed the information about the number of wheels and the total number of vehicles.  She hadn’t made sense of what her equation actually meant.

But, I thought that maybe it goes further than that.

I think the students need to see the difference between problems that require a one-variable equation, a two-variable equation, and a system of equations side by side. Here is what I came up with:

one var vs two var vs system

It’s nothing fancy or mind-blowing…I just want to see if this gives them a structure to look for when they are deciding how best to represent a given scenario.  I want them to see that there are two unknowns in the third case, and that they need to create an algebraic representation for Eli’s savings and for Lucas’ savings.

I’m wondering if we should ask them to notice and wonder???  Or simply ask…what is the same and what is different…

I’m also wondering if I should remove the headings from the table before we ask them to notice and wonder???

We’re going to try this tomorrow…so I’ll let you know how it goes.

Vacas y Pollos ~ Best Day Ever

Yesterday I was talking to our Algebra teacher.  She had asked the Spanish teacher in the building to translate the  problem below so that she could pose it to her students that take both Algebra and Spanish.

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Our population of students that are emerging bilingual has grown significantly over the past several years.  The grade 8 students are also working on systems of equations and I had the perfect teacher in mind to pose this problem (written in Spanish) to his students as well.

His last class has the greatest number of Spanish speaking students and I wanted to see the look on their faces when he posted this problem on the board…

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Priceless. I watched the biggest smile appear on their faces! The look of joy of what they knew. The boy that participates the least read the problem out loud so quickly and proudly…it was AWESOME! The rest of the students were in awe. They tried to figure out pieces of the problem. But, the students that usually have to spend time using Google translate to figure out what was going on, were the first ones to get to dig in this time.

One girl then said…”wow…they have to do this all the time.”


Best day ever.

Equal and Opposite

I don’t know if this is the right title for this post…it feels like it is.  I get to its point at the bottom…

I was supporting an 8th grade teacher that was implementing the Classifying Solutions to Systems of Equations formative assessment lesson from Mathematics Assessment Project.

The students had not yet worked with linear equations in any form other than slope-intercept form.  The students already understood the different types of solution a system might have.

The students started out by completing the assessment task.  I wanted to focus on this part below:

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I’ve encouraged this teacher to build a strong foundation for students being able to create and complete a table of values.  Creating a table should be a go-to strategy for students if all else fails.

I stood in the back of the room and I looked at the second equation…

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I wondered if I could make a connection for kids…if it would be too much, or would they follow…Here is the gist of what I did:

I interrupted the class to pose a question.

Me: What do we know about an equation that is solved for y?

Students: We know the y-intercept and we know the change in y over the change in x.

Me: What do you notice about this equation?

Students: It’s solved for x.

Me: What do you think that might tell you?

Students: The x-intercept???? (Imagine a questioning tone here…) Is that a thing?

Me: Yes it is a thing. What about the coefficient of the y? What do you think that might tell you?

Students: The change in x over the change in y??? (Imagine a questioning tone again)

Me: Yep. Let’s pull up Desmos and check it out.

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I remember being a math student that naturally made these sorts of connections~that if something happened mathematically one way, I could predict what would happen in an opposite direction.

I want the students to know and wonder about these things for themselves.  But, I think we need to make it obvious to them sometimes that they can do this thinking on their own.

My Favorite Formative Assessment Tasks

I’m a little late…but, here’s my week 2 “My Favorite” post for the Explore MTBoS blogging initiative.


The Charles A. Dana Center out of The University of Texas at Austin has put together a great set of tasks for eliciting student thinking.

One of my favorite tasks that I have used with 8th graders (for years) is called Mosaics.

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Because we spend time making sense of the patterns from the Visual Patterns site, this task works well as an independent assessment task.

I particularly like question one in that it asks students to represent the problem in at least three ways–they are not told how to represent the problem.  I like to see if they will use a table, graph, etc.

Also, it’s interesting to see how different students “see” the pattern growing and how they choose to show that thinking.

And my FAVORITE piece of student work for this task incorporated the independent use of “noticing…”


Here is a link to additional student work from this task.  We’ve used this task, along with the student work, as part of our back-to-school professional development on using a examining student work protocol.

Dana Center tasks are not a free resource. You can purchase a book of these tasks here, or on CD here.

Where’s the Algebra?

You won’t know this, but this is my 2nd draft of this post.  (I even changed the title.)  I went running half way through what I was trying to write and everything that I really wanted to say came to me so clearly. I should have stopped and recorded the words that were so fluidly coming to mind, but I didn’t.

It made me wonder, however, about why the words that I was struggling to string together for the original post came to me so easily while out running.  I decided that my mind was free of the clutter of trying to say the right things and in the right way–allowing me the ability get at the heart of what I wanted to communicate.

This–I decided–is the reason that balance problems are a great way for students to interact with the reasoning of equation solving.  They are free of the clutter of the procedures and notation and get at what is really going on.

Yesterday, I read Michael Pershan’s post about the use of balance problems (which I consider a sort of puzzle) as part of a series on solving equations.

He is able to so clearly articulate all of the reasons that I believe that these sorts of tasks are a valuable tool in reinforcing the thinking of solving equations.  (I would read his post on this because he is really good at thinking through these sorts of things.)

I used this “balance reasoning” that he referred to in order to make the connection between what students know how to do naturally and the skill that I wanted them to walk away from 8th grade being fluent in–solving equations.

8th graders have done a ton of work solving equations, but still, many are not fluent.  I used these puzzles during warm-ups without expecting any sort of equation writing/solving.  Over the course of time…I wanted the students to begin to ask why we were doing them when they were so easy.  I used the balance problems from EDCbut I also made great use of some of the resources found here.

One resource I used from this site is Mystery Numbers.  Students were able to use guess and check, along with “balance reasoning”, to find the correct value.


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The “mystery” of these problems can be ramped up rather easily and foster the need for a more sophisticated strategy.  I began having the students simply write the equation that could be represented by the mystery number equation.  The students had been wrapped up in the fact that these were at once so easy, but now not so much.  I used the mystery number problems written side by side with its algebraic representation in order to show the students how the “balance reasoning” related to the algebraic manipulation.

Students seemed to invest in this process because it was connected to something that they knew they could do.  I talked with them a lot about how math is all about learning more and more sophisticated ways to represent problems–even ones that they could do with an elementary level of understanding.  I believe it was the connection to a task that they found doable-that allowed them to find the algebra less intimidating.

Additionally, I used problems like this from Sarah Rubin.

I’m very interested in following the rest of the train of thought over at Problem Problems


#intenttalk Book Study Leads to Questions about Effective vs Efficient

I was inspired to coordinate the Intentional Talk book study this summer by a conversation between Tracy Zager and Dylan Kane.

The Intentional Talk book study began in June with various people taking the lead–including the authors of the book Elham Kazemi and Allison Hintz.

This week, Allison Hintz posted a question that lead to a conversation about effective vs efficient strategies.  I tried to capture my current wonderings in this 140 character tweet:

At the end of this past school year I came across this task.

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I worked with on-level 8th grade students this year, but as the school Instructional Resource Teacher, I spoke with our 7th Accelerated teachers and Algebra 1 teacher to see if they would be on board with giving this task to each of these cohorts.  All of the students had been taught systems of equations this year–but notice how unstructured and simply worded this task is.  I wondered what sort of strategies the students (at all of these various levels) would apply towards getting a solution for this task.  We used it to gauge the sort of student that would be appropriate for a 45 minute algebra class vs a 90 minute algebra class.

The range of approaches was very interesting to me. 

The following examples are from the 8th grade students that I worked with…I organized these student responses in the sort of continuum that occurred.  (Interestingly, the 7th accelerated and the Algebra 1 students didn’t have too dissimilar a range of solution strategies.)

Students that chose graphing…

FullSizeRender 2Student 1:  extended the lines but without precision or using any tools

Student 2:  asked for a ruler in order to extend the graph in a more precise fashion

Student 3:  asked for graph paper and created a graph using discrete points

Students that chose tables…

FullSizeRender 3

Student 4:  created two separate tables beginning at x = 0 and found the values that represented the intersection point

Student 5:  created a single table beginning at x = 10.  This student did not pull the correct information from graph to begin the table.

Students that wrote a one-variable equation…

FullSizeRender 4

Student 6:  wrote each linear equation and then used the substitution method to find the solution.  This student did have difficulty at the end because she struggled with what to do with (1/2)x = 8.  She eventually made sense of that part.

Student 7:  was able to use the substitution method to write the one-variable equation as well.  This student is not your typical top student and she was able to easily handle .5x=8.  I asked her if she used the calculator (which I had allowed) because she didn’t get stuck like a few other of the “smart” kids.  She said that she knew it took two groups of .5 to make 1 so she multiplied 8 by 2 and got 16.  It was amazing to me how it was so easy for her to flip to ratio reasoning when that last bit was a challenge for some of the “top” students.

This all leads me back to my original wondering…graphing, tables, equations were all effective methods (for some).  But, I am considering an efficient method to be one that utilizes grade level understandings and the goal should be to help students work through this continuum.

There has been a lot of discussion regarding the meaning of efficient.  Does this mean fast?  I think you are probably only able to use an efficient strategy if you have multiple strategies to choose from.  Otherwise, your ONE strategy is the effective/efficient strategy.

It’s important to give students these sort of unstructured tasks and let them figure out what they’d do on their own.  I would definitely use this task again and then use one of the targeted discussion strategies from Intentional Talk.

#WODB and Polygraphs: Lines in the Classroom

I’ve been using these sorts of problems in my classroom intermittently for many years.  My first exposure was at an NCTM conference in Baltimore in 2004.  This type of problem was titled “Puttering with Patterns”  similar to this one here:

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I worked as the math teacher in a k-21 special education facility at the time and immediately understood the benefit of posing problems like this to students with special needs…EVERYONE can notice something meaningful to contribute.

Here are several reasons why each one is different:

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I was reminded of this type of problem by Christopher Danielson when he released “Which one doesn’t belong?  A shapes book.”

I’ve been out of the classroom for many years, but this year have returned to lesson plan for an 8th grade classroom that is being taught by a long term sub (the teacher moved out of state in October).  I’ve been in this classroom more often than not–co-teaching with the sub.  This has been my opportunity to really try out many of the activities that I have come across on Twitter–YAY!!!

When @MaryBourassa created the new site “Which One Doesn’t Belong?”, I was excited that she brought this type of problem to the forefront AND gave it a more secondary sort of spin.

A good portion of the students in these 8th grade classes that I am working with are reluctant learners.  But, #WODB pulled them all in.

We’ve been working on linear equations, including graphing lines using slope and y-intercept.  So for the last day before spring break I decided to use this:

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as a lead in for this:

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The sequence of instruction worked perfectly for these 8th graders…because, really, Polygraph is just a giant “Which One Doesn’t Belong?”

Fo the #WODB task, my students noticed that the 2nd graph was proportional, that the 3rd graph had a negative y-intercept, and that the 4th graph had a negative slope.  We needed to have a classroom discussion to determine an attribute for the first graph that didn’t fit in with the rest.  We ended up talking about x-intercept for that one (a term that they didn’t have yet).

All of these observations became questions they asked when playing Polygraphs.

I’m looking forward to making the #WODB problems a larger part of our typical daily routine.