Every Graph Tells a Story – A Lesson Interpreting Graphs

Every graph tells a story, but can you “read” that story?    Do you see the beginning, middle and end (if there is an end)?   Do you see how the story builds and changes?   These questions are at the heart of a unit I teach my 6th grade students.   (The unit is a Connected Math unit called Variables and Patterns).  They were also at the heart of a lesson I taught recently in which student were asked to interpret a graph, to read its story.

To begin, I wanted students to zero in on the key elements in a graph that reveal the story but I didn’t want to tell them too much.    I decided to try using a chalk talk, which is a technique in a book I have been reading with my PLC,  Making Thinking Visible .   In the chalk talk, students respond to a prompt on a large sheet of paper.    Students “talk” about the prompt by writing on the paper.   This allows them to respond to the prompt and also to the responses that other students have written, hence making thinking visible.     I decided that my prompt would be each of the six graphs that my students would need to interpret.

I drew each of the six  graphs and posted them around the room, each hanging above a blank piece of poster board.  As students entered, I assigned them to a specific graph station. I asked them to look at the graph and write what they noticed about the graph.    After everyone had completed the task, I had them rotate to another graph station.    This time, they could either write about the graph or respond to what someone else had written.

The six graphs were large scale versions of the graphs shown below.

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In this phase of the lesson, I was hoping that students would look for markers that tell the story of the graph:   what is the initial value, how does the graph change (e.g., is it a constant rate of change or a variable rate of change),   is the graph increasing or decreasing in value,   does the pattern in the graph repeat,   is there an “end”.   I wanted to draw out student thinking on these ideas as a launch into the lesson.   Since this was the first time that I was using a chalk talk, I didn’t really know what I would get, though.   It was a new process for my students, so they wouldn’t have the benefit of experience with the process to guide them and I wasn’t sure if I was presenting the task well since it was my first attempt with it.   As I looked at their work, I could see the evidence of our inexperience.   The responses varied widely   Some students gave responses  along the lines that I expected.   Some students made up a story for the graph.  A few students responded to other students’ thinking, but most used the thinking of other students to create their own elaborate stories.   Since I got results I hadn’t anticipated, the debrief of the task was a little different than I expected and didn’t really focus on some of the things I had hoped it would.   I didn’t try to force it though because I wanted to give students the chance to make meaning rather than me giving them meaning.  I liked the technique as a launch, I just need to pose the prompt a little better next time.

After the debrief of the chalk talk, I had students work in table groups to match the same set of six graphs to a set of seven different stories.    I gave them  graphs and stories on a set of cards, so it was essentially a card sort.   As students worked in their groups, I told them that they must work silently.   One person would match a story to a graph.   The next person could either match another story to a graph or could change one of the matches on the floor.     I did this to ensure equity of voice in the group, the silence and turn taking ensured that no one could dominate the group and that each must contribute.    I also did this to encourage students to analyze the choices that other students were making.  The card sort used the graphs and stories shown below.

After each group had “completed” the matches, I had the groups do a gallery walk.   During the gallery walk, they went around to each of the other groups and considered the choices that the other groups had made as they matched the stories to the graphs.    During the gallery walk, they could discuss what they saw with their group members but could not make changes to the work that they saw.   At the conclusion of the gallery walk, each group was given time to discuss the choices they had made with their own card sort and make any changes they wished to their group’s card sort.

Finally, we debriefed  the card sort. We returned to the large scale graphs on the walls that we had used for the chalk talk.    I read one of the stories and asked one of the groups to share which graph they believed it matched.   They had to identify the independent and dependent variables and explain how the changes in the dependent variable reflected the story.  They also had to give the graph a title.   Other groups then had the opportunity to comment or question the group’s response.

One of the things that I really like about this lesson is that students grapple with hard questions.   There is the potential for them to wrestle with misconceptions.  As they discuss the graphs and stories, their misconceptions are exposed.   For example, someone always suggests that one of the parabolas is the amount of daylight over the course of time.   Someone always then raises the question of what it means for the amount of daylight to be zero (either initially or in the middle of the graph).   At that point, there is always this lovely “oh!” moment in which students realign their thinking.

Next year, I do want to make a couple of changes to the way I did this.   First, I will re-work my directions for the chalk talk.   In addition to the graph, I will write a prompt asking them to respond to specific components of the graph (initial values,   how the dependent variable changes as the independent value increases, etc).   Second, I will change the way that I do the card sort a little bit.   I think I will give each group a set of dry erase markers and require them to add labels and a title to each of the graph card as they match it to the story.   I am curious whether seeing those things written on the graph will help them see their misconceptions before the whole class debrief.

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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.

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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.

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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.

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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.

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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.

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Lessons Learned from a Review Lesson on Proportional Relationships In Graphs

Sometimes, I learn more from my lessons than my students do.   Of course, just as is true form my students, I have to be paying attention.   This week, as we were reviewing proportional relationships, I noticed a few things that I didn’t expect.

Setting the Stage – The Prequel

First, this is a review.   My students learned about proportional relationships last year and this was intended as a refresher leading into work with linear relationships a little later in the  year.

This lesson was the second in a sequence of four on proportional relationships.   The previous day, we had worked with proportional relationships in tables.   It had gone about as I had expected.   After a few minutes and a little bit of discussion, students recalled the criteria as being a constant rate of change and containing the point (0,0) (even if it is only implied).   They had also recognized that one could get the y value by multiplying the x value by a constant.   I was pretty happy.

The Lesson

Students were going to build on their experience from the previous lesson using three different representations.   Each table group was given a set of two word problems.    For each word problem, they were to create a table and a graph.   Finally, they were to determine whether the relationship was proportional and explain why or why not.

Problem 1:  Evie built a robot that can put ping pong balls in a cup.    It took it three minutes to travel the course, put the ball in the cup and return to the start point.  Evie’s robot was going to compete in a 12 minute race.   Create a table and a graph showing the robot’s performance.   Determine if the relationship is proportional.

Problem 2: Sofia was really excited about all the things she was seeing in Italy.   She called Natalia to tell her all about the volcano she had visited.   It cost her four dollars for the first minute and one dollar for every minute after that.   Create a table and a graph showing the cost of the phone call.

I chose to give them two problems with a constant rate of change but with different y-intercepts because this seemed to be their sticking point in determining whether something was proportional.    They were successfully determining whether the rate of change was constant but were sometimes hitting a wall on the y-intercept.   Hence, I wanted to focus there.

As each group worked, they were to document their result for each problem on a poster.   The groups would then do a gallery walk to see each other’s work.   This is some of what they saw.

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Clearly, there are some errors in thought in this work.   However, the gallery walk served it’s purpose.   Students noticed the differences in the different solutions and they started talking.

“Time is the independent variable.   It should be in the left column.”

“What is it when you connect the points? or don’t connect them?”  “Continuous or discrete.”

“Should the points be connected for the ping pong balls?   You can’t have part of a ping pong ball.”   “We were weren’t counting ping pong balls, we were counting laps and you can have part of a lap.”  “Should it be time or should it be laps?”

The conversations were not about the question that I posed.   Everyone got which graphs were proportional.   They did so without relying on the table (which I had made part of the task as a scaffold in case anyone needed it and to also have them make connections across the two representations).   They were important conversation reviewing the finer points of making graphs that I hadn’t expected to be necessary.   They were though, and using student work to drive the conversation was the best thing that could have happened.

After a class debrief, students worked with table groups to complete a proportional relationship card sort.   (You can get the card sort here.)

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What I Learned

  • Most of my students easily translated the word problems into graphs.
  • Most of my students had a good grasp of proportional relationships in graphs.   I should have added more layers of difficulty in my problem set.   I focused primarily on the y-intercept as the tell-tale indicator of their mastery because I expected they would get the “constant rate of change” very readily in a graph.   Perhaps I should have added a scenario with exponential growth and perhaps I should have added a scenario in which part of the scenario is linear and part of it is not.
  • My students are rusty with regard to the finer points of making tables and graphs (where does the independent variable belong, what does it mean to be continuous or discrete, what exactly should the variables be).   This proved to be an important lesson for my students, but not for the reasons that I expected.   I’m thankful that I took the time to review a little bit rather than just rushing forward with new material, which is a somewhat natural inclination for me.   Taking a little bit of time to review important ideas was time well spent.

Coming Attractions

Lesson three in this sequence will explore proportional relationships in equations.   Lesson four will look at proportional relationships in all four representations.

Necessity Is The Mother Of Invention – #ILookLikeAnEngineer

Necessity is the mother of invention.   Unfortunately, the “necessity” can be all too easily forgotten as an essential component in education.    I teach what I teach, in part, out of necessity but it is my necessity not that of my students.   I need to teach the curriculum that I teach because it aligns with the standards set forth by the state but it is not a burning necessity for my students no matter how many times I tell them the essential questions and how they will use it in the future.   Knowing something only becomes a burning necessity in the mind of an eleven year old when they see a need to know it so they can do something they want right now.

So how do we create that need to know?   I think we give kids real problems that they really want to solve.   It’s not something that I can do every day, but I try really hard to find time and space to do it every year.   To do this, I  compact lessons and I accelerate where I can.   This year, I managed to squeeze out almost a month at the end of the year to do an engineering project with my students.

Request For Proposal

Students were presented with a Request For Proposal (RFP) from a fake toy company.    The proposal indicated that this fake toy company was seeking to expand market share to include more girls in their customer base for motorized toys.    The toy company wanted those bidding on the contract to conduct market research and build a toy to meet that need.    The toy company indicated that the toy must meet one of three different criteria:  travel 3 m in 3 s, climb 1 m at a 15 degree slope in 2 s, or climb 1 m at a 30 degree slope.

Creating a Team and Conducting Market Research

Students were assigned teams and formed mini-companies that would bid on the RFP.   They created a team name, logo, and slogan.   Then, they conducted customer surveys with both adults and children in the target age range.    They analyzed the data and determined the type of toy the customer was seeking.

Building Technical Knowledge

ChJmkKSUkAEA9Q-During the same time-frame, students built knowledge of how gear trains work.   They began by building gears on a frame and exploring relationships between the rotations of the gears and the number of teeth on the gears (gear ratios, teeth ratios).   Next, they added a motor and wheels so that they could calculate the rate on a 3 m course and measure the rim force on the wheel.   They repeated this process with gear ratios ranging from 1:3 up to 225:1.   As they did this, they were building important skill in construction as well as an understanding of the different kinds of performance they might expect from different kinds of gear ratios.    From there, they measured rim force on the tooth of a gear connected to the motor.    They did so for different sized gears and then learned how to calculate torque.    With this knowledge, they could explain why certain gear ratios would not move and why certain gear ratios would be well-suited to climbing.   At this point, they had built sufficient knowledge to answer the first stages of that burning question of how to build a toy that would meet each of the criteria.

Making a Prototype

Each team began construction of a basic prototype to meet their desired criteria.   This amounted to attaching the motor and the desired gear train along with the wheels on the frame structure.   Students then tested their motorized frame to see if it met the criteria.   Once they had a basic working prototype, they started constructing a body to give the toy the desired aesthetics.    As they constructed the body, they continued to test the toy to make sure the additional weight did not place them out of compliance with the criteria in the RFP.    They repeated tests multiple times and used median values in order to eliminate outlier trials resulting from poor testing technique.

Sealing the Deal – Writing a Written Proposal and Giving an Oral Presentation

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When the toy was completed, each team wrote a written report in response to the RFP and prepared an oral presentation.   The final stage of the project required each team to present their toy to a panel of judges representing the fake toy company.   I recruited 3 engineers and a soon-to-be lawyer to represent both the technical and business interests of the company for the panel of judges.   (I am lucky enough to have Sandia National Laboratories nearby and willing to provide this kind of support to encourage excellence in math and science.)   The judges selected a winning team based on the presentation and a demonstration of the toy.   (The winning team members each got a gift card to Cold Stone Creamery).

While this last stage is not “math”, it is very much a part of what engineers do and I wanted my students to appreciate the importance of being able to communicate effectively as an engineer.  Reading, writing, and speaking are just as much essential skills for an engineer as are math and science mastery

Why It Mattered

  • Students got to experience the engineering process, which is so much more powerful than hearing about it.
  • Girls had to learn how to make something and how to make it work.   It’s not that they are any less adept, but many of them are much less experienced.   This results in a certain amount of hesitancy, initially,   Having to make it work pushes them past this hesitancy and they discover just how good they are at it.    Giving girls this experience and confidence is important in leveling the playing field when it comes to engineering.
  • Students used the math that they have learned this year to do something real that mattered to them (finding unit rates, conducting surveys, making data representations, analyzing data to make decisions, finding medians, using equations to calculate torque, measuring radii).
  • Students had to find ways to work together – teams could not shift part way through the month long project.
  • Students who lacked confidence as speakers learned that public speaking is a learned skill and that you get better at it with practice.   (I made each team do a dry run of their presentation in front of their classmates and get feedback the day before the final presentations.  They took the feedback and were so much better the second day.)

Gallery of Toys

Presentations

Which Bicycle Shop Should I Use –A Lesson Introducing Proportional Relationships in Tables, Graphs, and Equations

“I’m thinking about purchasing a custom-made bicycle. Bicycle City charges $160 plus $80 per day that it takes to make the bicycle. Bike Town charges $120 per day to make the bicycle. For what number of days will the charge be the same at both bicycle shops?”

I introduced the lesson with this question (taken from the NY Institute for Learning). We began with our usual process of making sense of the problem (What do you notice? What do you wonder? What is the question? What do you know?). Then, students set to work on the task individually.

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Once everyone had solved the problem, I had three different students present their solutions (one who had used a table, one who had used a graph, and one who had used equations) to the class using the document camera. This gave us the chance to talk about how you find the solution using different solution paths.

 

At that point, I told students that a proportional relationship was one in which you could multiply the x value by a constant and get the y value. I asked them to figure out whether either bike shop’s pricing was proportional and how they saw that in their particular solution. This gave us the opportunity to discuss how they see the constant rate of change in each representation. It also gave students the chance to discover that a proportional relationship goes through the origin for themselves.

We wrapped up the lesson by completing a foldable summarizing the characteristics of a proportional relationship and how to see them in a table, graph, and equation. You can download the foldable by clicking on the link.

Proportional Relationships

The lesson is designed to give a whole-part-whole understanding of the concept. This is particularly important with students who are at the extremes on the conceptual/sequential dimension of cognitive style. I have several twice-exceptional students who are at one end of this spectrum or the other. I struggled with how to help them succeed for the entire first semester. I started building in the use of a whole-part-whole structure and the use of graphic organizers for note-taking more after reading Twice-Exceptional Gifted Children by Beverly Trail, EdD. It has had a pretty big impact for these students (test scores moved from the 50-60% range to the 98% range on the last test). I don’t know if that will continue to hold, but with that kind of improvement, I am going to keep incorporating this technique and find out.

The way that the foldable is designed, students can compare and contrast how one sees a proportional relationship in a graph, table, and equation by looking across the foldable.   The can also compare and contrast an example of relationship that is proportional and a relationship that is not proportional by looking down a column.   Marzano’s research shows that the use of comparing and contrasting can result in significant gains in student learning.

I also incorporated the use of color in the completion of the foldable to highlight the important points. This was not as intentional as it should have been. Early in the day, we worked through the foldable summarizing the ideas in the lesson. As the day progressed, I thought it would be a good idea to have kids highlight the key points in each representation (constant rate of change and going through the origin). By fourth period, I finally refined it so that kids used one color to highlight the rate of change and a different color to highlight that the relation goes through the origin. This seemed to really bring out the idea across representations, so I will do it this way from the start the next time I teach this lesson.