Curve Sketching with the 1st and 2nd Derivative

I am a teacher by day, and an adjunct by night.  One evening, while looking out into the sea of college freshmen, during one of my "lectures" I noticed that their faces all had that glazed over look, they were not as engaged and all around looking miserable.  The college students humor me by laughing at my horrible jokes, and will even ask questions when they need, but I did not feel the room at all that night.  Later that week I was on Twitter scrolling through, and found an interesting Tweet by Robert Kaplinsky, "Who ever felt that lecture was the pinnacle of education?! If anything, we should be giving them our best instruction to prepare them."  Although lecturing is sometimes necessary to convey content, it should not be the sole way students of any age are taught.

Who ever felt that lecture was the pinnacle of education?! If anything, we should be giving them our best instruction to prepare them.

— Robert Kaplinsky (@robertkaplinsky) October 31, 2017

Rubric: Curve Sketching Rubric
Project Example: First & Second Derivative Test Project

With that in mind, I also remembered that last year while teaching the same topic on curve sketching.  I was so excited by my examples and how to do the problems, that after I was done with the class I quickly assigned the odd problems from the text book.  Not thinking that each problem took about 15 minutes to complete, and giving 15 problems at 15 minutes a piece meant almost 4 hours worth of homework.  I did a complete disservice to those students with the assignment, and quickly had to apologize.  After reflecting on the previous year, and thinking about how I wanted this current semester to go, I figured out a new way of teaching the problem set and I feel I came up with a more interesting and collaborative way to practice the skill of curve sketching using derivatives.

Group 4's Project and Discussion were both great! 
Read below for more on the interactions of the group and the language used. 

Students were randomly assigned into 6 groups of 4.  I created a Google Drive Folder for each group and then made some adjustments to the problems from the text and my unit review.  Each group was assigned two functions to use the first and second derivative test, creating the curve sketching information that they needed.  Groups were to find the relative maxima, relative minima, any inflection points, where there was concavity up or down, and where the function was decreasing and increasing.   The groups added comments and edits, gave each other feedback and created a great collaborative final answer for their functions.  They also were to sketch the curve with Desmos, and find all of the points that they solved for mathematically on the picture instead of all by hand.  I had each group member turn in their perspective rough draft to have students show more than just participation in the online document, but the collaboration piece was key.  A lot of the discussions were really powerful with well thought out questions, leading to deeper understanding of the tables and sign charts groups created.    They learned from each other how to successfully apply the derivative tests to the function and what pieces would ultimately help them sketch the graph.  the groups also made their sign charts each in different modes through technology or taking a picture of their hand drawn chart.

Student Sign Charts were interesting and creatively made. 

Another Group's Sign Chart

After everyone had turned in their Google Doc, I also wanted to get a feel for the students' outlook from completing their group project. I created a Google Form and asked several questions about how they felt the project went. A majority of the feedback that came back said the group project was enjoyable, and it helped that students with understanding the process. I also saw a lot of good computational thinking in the form of their summative assessment following our fourth unit, and I think that I will use this project in years to come. My new goal is to create one collaborative or group project for each unit of my course to engage my college learners in a more enriching fashion. Next year I plan to facilitate another learning experience that will enhance my college course, and make the course even more engaging.

One Hour of Code Initiative

After finishing two graduate courses on utilizing technology in the classroom, I was tasked with one for a collaborative research project.   The project consisted of classmates and I researching a technology trend in education and we chose coding in the classroom.  I was very impressed with the Code.org website and all of the resources that are available for free on the site, including their One Hour of Code Initiative.  I had students watch the five minute video from Code.org to get them excited to investigate coding.  Students then created a free Code.org account and started the process of completing the One Hour of Code initiative from December 4th through December 8th this year.

Following the One Hour of Code program, students were then given a set of Scratch coding activities that progressively increased in difficulty.  Students were tasked with creating an easy set of code that drew four different polygons.  Then they were given a second task to code a random polygon generator using some more complex coding pieces, which also had students create a block (their own code piece) to name the polygon that was generated. For the final project students were tasked with creating coding blocks to draw a Sierpinski Triangle.  (My code is attached below.)  I used these projects in my Saturday STEM Mathematics program last spring, and found that they would be great for school this year as well. 

The whole program took students about 5 days and an average of 5 hours to complete all of the coding projects.  Students were given a great opportunity to investigate coding.  To celebrate and reward their accomplishments with more than just a grade, I created a certificate of completion.  Students who successfully completed all of the projects and showed that they ran were awarded the certificate that they could then use in any resume.  My district is trying to increase the amount of experiences students are given for job and college readiness for PA State Act 339.  So the experiences in coding and computer science will give them more confidence in working on computers, and some confidence with job readiness skills.

Once students finalized the projects, they began to investigate the math and history of fractal geometry, a mathematical definition of the patterns seen in nature.  Computer science has used fractals to create more and more realistic computer generated imagery in video games and movies.  Students got to investigate logarithms to define the fractal dimension and took in-depth looks at how the math has changed how we look at the similarities in nature.  For the last four years, I have been implementing and developing this two week unit before Christmas break. Leading up to students creating a holiday Sierpinski Tree.  I hope to one day inspire my students to investigate fractals even more, because the math behind it is very interesting.

Conditional Statement Story - Post Observation Reflection

Ever read the story, "If You Give a Mouse A Cookie" by Laura Numeroff?  I had not until I married an elementary school teacher, and read the book she had in her classroom.  Several years ago after reading the book I had an idea that was going to allow me to use the story for my students as well.  What could a Geometry Teacher do with a children's book you may ask?  Well it's simple, the entire book is written as conditional statements, if p then q.  In my second unit in the school year,  Reasoning and Proofs, we delve into the mechanics of writing conditional statements, their converses, inverses, even contrapositives, and using the symbols to write these statements.  The statements are used throughout my course to write and prove postulates, properties, theorems and corollaries.  Students immerse themselves in the logic of the statements and decide whether or not there is a truth value to a given statement.

6th Period Starting the Project

When I originally designed my project I had students write their own book and draw illustrations in it.  But now as I am becoming more tech savvy and more digitally in-tuned with my students needs. I figured why not adapt the use of computers to make a PowerPoint of the book?  The end goal is to make Google Slides the tool we use. But until students can log-in to a true Google Account and collaborate with students on the use of Google Apps, I am stuck with what little true collaboration my students and I have now.   Students are going to have to write their own Conditional Statement Story Project, much like "If You Give a Mouse a Cookie."  The requirements are that the story has 10 conditional statements, 4 original statements, 2 converses, 2 inverses and 2 contrapositives that flow together much like the story I read to them Tuesday.   Students will then work in pairs to write their story and illustrate it with pictures from the web.  I have a Google Slide set up to show them the proper uses of a Google Image Search to utilize photos for their presentation that have permission to reuse and distribute the image.  I added this component of Digital Citizenship to promote the Creative Commons that many other students already know about.  

Since students could not access Google Slides from the network computers we were utilizing,  I then had to change gears in my planning and have all students create their stories using Microsoft PowerPoint.  They then uploaded their typed rough draft and final PowerPoint presentation to my Dropbox account.  My Wiki page linked here allowed them access to the links I provided for my Dropbox, where I could then add the presentations to my Google Drive and convert them to Google Slides to easily share online.  One key characteristic of being an educator is that I am very flexible and while I would like to continue my journey in learning and utilizing Google Apps, I know that there are tools out there already that help me with technically updating my craft.  

7th Period Starting the Project

Wednesday I was observed by my administrator while students were working on this project and so I am using this post as my reflection piece.  (Lesson Plan Here)  Some thoughts I had while thinking back on my activity are; I would like to change my requirements for the rough draft for next year by making students label their statements, and also have them put the labels in the notes section of their PowerPoint or Google Slide.  Although most students completed a nice cover page some needed to be asked to do so, so next year I am going to add a cover page section into the rubric.  I feel students worked really hard on their stories, had good questions about the project, and I chose some of the best ones to share below.


 Please read and enjoy the great things these students accomplished while writing, creating and learning online.

First Marking Period - Data & Committing to Life Long Learning

Recently the Keystone scores for Pennsylvania were released, and our Algebra 1 scores were not where we would have envisioned them.  Our English Language Arts scores saw growth, and a 9% gain in overall Proficiency, along with the Biology exam that is now given in PA.  Math took a 2% hit, although the numbers of students in the below basic category dropped and increased their scores, we need to re-examine the way we teach mathematics.  We believe student scores will increase if the math department increases the writing proficiency in Algebra.

In my Geometry course I am committed to help increase the perserverance in writing mathematically and technically.  To foster those skills I am utilizing more task problem sets to practice skills in Geometry.  Recently students investigated the use of areas in carpeting and tiling a first floor of  a house.  I used Carnegie Learning's Geometry course when I student taught and have found continued uses for the problem sets that are available in their old books.  I bought a student edition off of Amazon for $10 and have been using specific problem sets for my course to enhance the classroom rigor and relevance.

Carpeting & Tiling a Floor- Using Task Problems to investigate
area in a Geometry Classroom while increasing
writing in mathematics. 

I also am expanding my classroom set-up, with a new use for the white board in the back of my classroom.  After participating in a  Twitter conversation with some other professionals across the United States about mathematics and education, one topic really stood out to me.  What do you do with good student questions?  One educator said he had a Pressing Questions board, and I thought to myself, well I have a blank White Board in the back of my room, let's utilize it and discuss more questions with students. One question led us to a whole class discussion and deeper thinking while looking at a drawing on GeoGebra to illustrate what happens when polygons increase their number of sides.  My visual learners were fully engaged in the discussion and said they would like to use GeoGebra themselves.

New Pressing Questions White Board - Use questions students come up
with that cannot be addressed right away, or that are good enough
to keep coming back to. 

Students were also introduced to the use of Geogebra to start manipulating and working with digital drawings of mathematical concepts.  I enjoy teaching my students how to work with protractors and compasses, but this tool is much more efficient and easily understood, especially for my visual learners.  Throughout the school year I plan to utilize more Geogebra in the classroom to model mathematics in the real world, and maybe even in uploading photos for my students to modify and work with.   However, I'm finding that using our district network is making it difficult to continue with some of my ideas and the work I'm doing in my graduate courses.

Collage of Student Projects
Triangle Angle Sum, Interior Angles of Polygons
& Angle and Segment Bisectors

The only catch with all of the work in my courses I would like to continue with, is the fact that I cannot modify anything with my students and have them complete work in their own Google accounts, where everything can be saved to a cloud.  The school district I work for restricts the useage of student log-ins for any type of e-mail or account log-in.  Once we become a Google School I will be proud to share with them in all of the uses of Google tools in the classroom!  I even went so far as to become a Google Certified Educator, Level 1, and hope to continue this school year to gain Level 2 and even Trainer status to compliment my future master's in Information Technology Specialist.  I have a lot on my plate and continue to keep trying new things as an educator because I know my students deserve the best.  I want to inspire them in a way they have not been before, to look into technology and mathematics as a career and life-long learning process.

Topology Assignment for my Grad Class,
the computer lab system I work with, when
it works.