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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howtomakeacoordgraphpdf

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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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Pythagorean Theorem Interactive Notebook Page

I wanted  something to use in my students’ Interactive Notebooks for the Pythagorean Theorem that would start with the conceptual view and build toward the algebraic formula.

 Pythagorean Theorem Foldable 

I wanted something that would give the whole, part, whole picture of  the Pythagorean Theorem is and how it is used.   I started with an overview of when it applies, then incorporated some pieces to build conceptual understanding, some practice with the concept algebraically  to solve for missing side lengths, and a final summary of what it is.

A MarkUp/MarkDown Foldable for My Twice Exceptional Student

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Most problems  have multiple solution paths.   Some solution paths are more efficient and some less so, but forcing a particular solution path down a student’s throat denies that student the chance to make sense of it in his or her own way.   I think that part of the power of exploring problems and different solution paths is the sense-making that is inherent therein.   I also think that some of the power lies in the chance to see how someone else thought about the problem, to have the chance to think about it another way.   In thinking about a problem more than one way and trying to make sense of these divergent paths, there is also a moment when one begins to grapple with the idea of the efficiency of one’s solution path.   At that point, after he or she has had the chance to make sense of something, the student can choose a path that is the most efficient for him or her (a supposedly efficient path is not efficient if a student can’t apply it correctly because it doesn’t make sense to him or her).

This idea of allowing students to find their own best path was one of the things in the forefront of my mind as I was putting together a foldable to summarize markups and markdowns this afternoon.    I like to use foldables as my summary for a lesson from time to time because they bring together a lot of the ideas addressed in class into a single coherent document that students can use to study.   Additionally, I have a twice-exceptional student who has difficulty with the physical act of writing.   Giving him a foldable to complete makes it possible for him to be more successful.

In planning the design of the foldable, I wanted something that would give a whole/part/whole picture of the concept.   This is really important to me because one of the students who will be using it this year is a twice-exceptional student who is pretty extreme on the conceptual end of the conceptual vs sequential continuum of thinking.   He needs to see the whole of a concept to make sense of it and gets lost in the parts if he doesn’t see the whole first  In order to give him the “whole”, I decided to start with the big idea.   Next, I decided to compare/contrast markups and markdowns at each stage in the foldable.   The use of this compare/contrast mechanism seems to really help him to make sense of things.   Marzano’s research in  Classroom Instruction That Works shows that it can have a big impact for all students (27% gain).   Finally, I included two different solution methods for a markup and a markdown.   These are the solution paths that I am anticipating students will have taken as they explored the problems.

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MarkUp and MarkDown Foldable

Interactive Notebook Setup

I heard somewhere that for most people, the person they marry was their third serious relationship.   I don’t know whether this is true, it is just another one of those strange things upon which someone decided to collect statistics.   I think the rationale is that you make your mistakes in the first two relationships and have finally figured out who you are and what you want by the third one.    I can’t say much about whether this holds true with regard to most people’s marriages, but it bears the ring of truth when I think about student notebooks in my math class.   Third time was definitely the charm.

When I first started teaching, I had students use a spiral notebook.   I envisioned students doing their classwork in the spiral notebook and then doing their homework on loose leaf paper.   I liked the fact that a spiral notebook was fairly inexpensive and small enough not to take up too much room in a backpack.   I quickly discovered that it presented a few drawbacks, though.   There was no good place for students to put handouts or returned work, so they inevitably ended up stuffed in the trash or wadded up into the black hole of their backpack never to be found again.   Furthermore, instead of using loose leaf paper, students inevitably gave in to the temptation to rip sheets out of their spirals, leaving a trail of paper shreds all over the floor.   I did not love the result so that was the end of spiral notebooks for me.

With the flaws of the spiral notebook foremost in my mind, I moved on to three ring binders.     Handouts could be hole-punched and added to the classwork.    Dividers could be added to keep everything organized.   There would be no more little paper shreds from spiral notebooks strewn on the floor.   It seemed like the perfect solution.   Alas, it was not long before dissatisfaction reared its ugly head.   The binders were big and bulky.   They took up a lot of space in a backpack and added a lot of weight that was carted around endlessly.   (For some reason that is not clear to me, most middle school students carry everything around in their backpacks all the time.   They have lockers, but they don’t seem to use them.)     Furthermore, my vision of neatly organized binders divided into sections did not come to fruition for a lot of students.   Things were put randomly into the binders in a mad rush.   When it was time to find homework to be checked, the location was a mystery.    The student would swear that he or she did it and put it into the binder but simply could not find it.    Once again, it was clear that this was not for me.

Then, I discovered interactive notebooks.     They were as close to perfect as I was going to ever get.    They are small and lightweight.   They are bound so things don’t get lost.   There is a structure that keeps things organized.    They were easy for me to check and easy for my students to use as a resource.   I had found my match.

The Notebook

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I have students use quad-rule composition books for their interactive notebooks.   They are bound so there are no loose pages (or pages accidentally torn out like they had with spiral notebooks).    Each page is graph paper, so there is no search for it when we need it.  I also find that the graph paper helps students align numbers more readily (e.g., when doing long division), is useful when they need to make models to solve a problem, and is essential when they need to make graphs.   The composition book is also fairly light and compact which is really nice since most of my students seem to carry their lives around in their backpacks.    Over the course of a year, most of my students use four quad-rule composition books.

The Rubric (Inside Front Cover)

IMG_1513 I have student place the rubric for the notebook on the inside front cover for easy reference.  I grade student notebooks periodically.    This helps to ensure that they keep them.   Eventually, students realize that the notebook is a resource but that realization does not happen at the start of middle school for most of them.   The occasional notebook check goes a long way.   I tell my students that this is basically a free 16 points if they just follow the rubric.   Those 16 points can help to cushion things a little if they make a careless mistake on a quiz or a test.

IMG_1552I give forewarning for the early notebook checks.   I tell them when the check will happen and what specifically I will be checking.   As the year progresses, I stop announcing when the notebook check will be.  I print a small version of the rubric on envelop labels (the categories and the numbers 1, 2, 3, 4).   I circle the values for each category and then place the adhesive label in the notebook wherever I checked it.   My goal here is to give feedback but to do so in a way that is reasonable for me as a teacher.   I want to be able to do the notebook check in a fairly short amount of time in class because students need to take the notebook home and I need to spend most of my time with them teaching (not checking notebooks).  You can download the Interactive Math Notebook Rubric by clicking on the text.

The Title Page (Page 1)

The first page of the notebook is the Title Page.    Students tape it into the notebook.   It’s main function is to ensure that the student’s name is in the notebook so that it can be returned to them when they inevitably leave it behind in one of their classes.

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When students tape the title page into the notebook, I also have them number the pages in the composition book.   The pages should be numbered on the top outside corners of the pages.   The page numbering is important for the table of contents and finding things within the notebook.

This is also where students can tape in the directions for using LearnZillion.  LearnZillion has video lessons that students can use.   I have links to the videos on my school website.   I provide these so that students who were absent during class have a means for accessing the concepts they missed in class.    These video lessons can also be helpful when a student is studying for a quiz or a test.

Finally, I have students write their calculator number on this page.   I have a class set of calculators that students can use in class.    They each are assigned a specific calculator (this helps me to keep everyone accountable for returning the calculators).

The Directions For the Notebook (Page 2)

I provide a basic set of directions on the notebook set-up and use so that students can refer back to them as needed.   These directions are taped on page 2.

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The Table of Contents (Pages 3-5)

The Table of Contents are the next three pages (pages 3-5).   Each day, students are to record the topic covered and page numbers used in the table of contents.   This provides an easy way for them to go back and revisit work on a specific topic.

When students set up the notebook, they number each page in the notebook on the upper outside corner of the page.

You can download the Math Interactive Notebook set up pages and table of contents by clicking on the text.

The Standards of Mathematical Practice (Pages 6-9)

The Standards of Mathematical Practice have been translated into kid-friendly language.   These standards are at the heart of all mathematics.   I want students to have them readily available as a reminder of what it is to do mathematics.  The math practice standards are taped into the next four pages of the notebook.

I don’t remember where I got these pages.   I found another set of kid-friendly math practice standards here that you can download.

The Classwork Pages

Each day, students use at least two pages of their notebook for the work done in class.   The first of the two pages is for the Math Minute, the In, and the Flashback.  The page is divided in half horizontally.   The top half has a narrow section on one side for the Math Minute.   The Math Minute is a 1-2 minute task that students do upon entry into the class.   It might be a quick skill practice.   It might be an entry card.   It might be a vocabulary foldable.     The larger section on the top half of the page is for the In.   The In is a slightly longer entry activity.   It might be a vocabulary foldable, a longer entry card, or a task to introduce the day’s lesson (the Launch in a CMP lesson).   The bottom section of the page is for the Flashback.   The Flashback is a review activity or task during the last 5-10 minutes of class.   It is often differentiated instruction.   It might include playing a game to practice a skill, completing an exit card, or some other type of review activity.   Students are supposed to set up this page for the next day as the final homework task.

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The second classwork page is for the lesson.   In a CMP lesson, this would be where students complete the lab.    On days when students take notes (which happens every once in a while), this would be where they take the notes and do the classwork following the notes.   I always have students set this page up as Cornell Notes.   They use the larger right side of the page for completing the lab or doing the classwork.   If they have any labsheets for the day, they tape the labsheet into the notebook on the classwork page(s).  (That is what the white paper in the photo is.)  They leave a column on the left side of the page to record the main ideas.   They can complete the main ideas column during class or as part of their homework.   It should just be done within 24 hours of the lesson.   Early in the year, we come up with the main ideas together.   Over time, I use a gradual release strategy in which students eventually must do the main ideas independently.    At the close of each lesson, we do a summary.   This is a chance to bring together the important ideas from the lesson and make sure that everyone walks away from the lesson with the big idea.   I try really hard to let students think for themselves and build a solid understanding but I always have a summary to bring things together at the end of the lesson.

The Homework Pages

IMG_1526On the pages immediately following the classwork, students do their daily homework problems.   If it is a review day and the homework is a handout, students tape the handout into the notebook on these pages.   If it is ACE problems, they do the work on the page itself.

The Inside Back Cover

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Students tape the PARCC reference sheet on the inside back cover of their composition book.  You can download the PARCC reference sheets for all of middle school here.

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