Archive for the ‘Geometry’ Category

Mathematics Vision Project: Task Based Learning at its Best

July 1, 2014 10 comments

Recently our district has decided to pursue the Integrated Mathematics approach for high school math.  I am really excited about this transition.  Due to the lack of curriculum resources, we decided to use an online, free curriculum.  The Mathematics Vision Project is an integrated curriculum created by educators from Utah through a grant from the state.  I have looked at the curriculum online and love it, but it wasn’t until I went through their two day training that I realized it is truly the best math curriculum I have ever encountered.  Here is why I love their program:

1.  Task based learning focused on the learning cycle.

Task based learning is the rage right now with Common Core.  Our state, Tennessee, put teachers through intensive training on how to teach math using tasks.  The problem with this is teachers were left to create their own tasks or find random tasks.  In order for task based learning to work, the tasks must be sequenced appropriately and build on the previous learning.  MVP does this.  They have different types of tasks for different purposes and they are sequenced to build on each other.  The learning cycle involves Developing Understanding, Solidifying Understanding, and Practicing Understanding.  When you look at the tasks in a unit, the tasks are labeled as one of these.  This helps for both students and teachers to understand the purpose of the task.  Some tasks only develop the understanding.  Later, only after a teacher can guide a class discussion, are students expected to apply and practice the new learning.  This idea of different types of tasks for different stages of learning is critical.


2.  The have low threshold and high ceilings.

I was amazed with the multiple entry points for the tasks.  It felt as if any level of student could do something.  Often with tasks though, the mathematics is “dumbed down.”   This is not true for MVP.  The tasks are rich and have high ceilings.  If you have a group of student who finish early, there is always something in the task to stretch the learning.

3.  Story contexts throughout the module.

Take a look at Module 2 in Math 1.  It starts with a rich task about two children starting a pet sitting business.  The purpose of this first task is to start students down the pathway of thinking of multiple constraints on a variable (systems of equations).  Students will use this context throughout the entire module adding a little more information with each task.  Students should feel as if they are invested in a Problem Based Learning approach, broken into small, obtainable chunks.

4.  Not just what to teach, but how to teach it.

Most curriculum contain what a teacher should teach, but little about the best methods for teaching.  This is the first curriculum I have encountered that explicitly helps the teacher know how to teach the standards.   Each problem or exercise has a purpose:

  • Teach new knowledge
  • Bring misconceptions to the surface
  • Build skill of fluency
  • Engage students in Math Practices

5.  Meaning full homework and practice.

Practice is done by experts… Doctors practice medicine and Lawyers practice law.  Why would we send home practice when our students have not mastered the material?  This creates frustration and with Common Core, it leads to parents posting crazy math homework on Facebook.  MVP has amazing, thought out homework assignments.  They divide the homework into three categories:

  • Ready:  Things a student needs to review to be ready for upcoming work.
  • Set:  Things we did today in class that you need to practice to solidify understanding.
  • Go:  Things students should be “good to go on.”  This is review material.

Each assignment also has links to online videos to help review concepts students may not remember.  (I know in reality, that my students may not have done the homework, but I could use this as starters and exit tickets in my class.)

6.  Flexible Curriculum

Since the MVP curriculum is online, it can be updated at any time.  This means if something isn’t working or their are mistakes, they can easily be fixed.  This is not true of traditional text books.  The MVP team did hint that they are currently working to align the tasks and material to release them in a traditional math pathway.  This means that if your district does not do Integrated Math, you will still be able to use the MVP curriculum.

Overall, MVP offers a great curriculum and fantastic professional development.  I encourage you to attend an event and at the least, take some time to review the material.


Common Core & PARCC Sample Items

February 18, 2014 1 comment

I have been working on planning professional development for my district on Common Core State Standards and the PARCC assessment.  There is so much information on the PARCC website and it can be overwhelming for a teacher to navigate it.  My goal is to try and weed through the information and present only what is necessary and beneficial to teachers.

PARCC recently released sample questions in their intended environment.  This means the computer-based tools such as drag-and-drop, multiple select, text highlighting, and an equation builder are all active.  It is a great opportunity for teachers to see what computer skills are necessary and how students will navigate the assessment.  This sample assessment does not reflect a complete PARCC assessment.  The questions on the online assessment are all previously released sample items. The one frustration that I have is that  the questions are separated by grade bands and not grade levels.  In my experience, teachers want to focus on their grade level, although I think it is important to be aware of what comes before your course and where students are heading.  To help teachers and administrators, I have created the following documents to support teachers while they are looking at the online PARCC environment.  The documents address each questions content standard(s), grade level (course), and math practice.  Detailed scoring guides and explanations of the questions can be found on the PARCC website under the respective grade band.  Please feel free to provide feedback in the comments.

PARCC Computer Based Samples Grades 3-5

PARCC Computer Based Samples Grades 6-8

PARCC Computer Based Samples High School




Trigonometry and Clinometers

I know Clinometers are a pretty classic geometry activity to teach Trigonometry and Angles of Elevation, but I was really excited when the activity became relevant to my students.  I planned the activity and taught the necessary lessons.  We built clinometers using straws and protectors and practiced using them in the classroom.  The day finally came for us to go outside and measure items around the school that could not be measured with direct measurement.  I was in the main office that morning checking my mailbox and one of the Assistant Principals was showing off the new banners my school ordered to hang on the light posts  that line our school drive.  My high school is celebrating our 10th anniversary.  I asked the Assistant Principal who was going to hang the banners and if they had put any thought into it yet.  To my relief he said no.  I told him my class would be more than happy to help him.

I practically skipped to my classroom and couldn’t wait for Geometry to start.  I explained the new assignment to my class.  We needed to figure out how high to hang the banners and how long of a ladder we needed.  That was all the instruction they needed.  We spent the next 30 minutes outside seeing who could get the most precise measurements using our handcrafted devices.  I did not have to explain why someone would need to know how tall some random tree is or the height of our building.  They were hooked.  I convinced them we needed to let the janitor know by the end of the day so he could make sure we had a ladder at school that could work.  The students did a really good job calculating the height of the light poles, but the real discussion happened when we tried to decide the necessary height of the ladder.  The students wanted to know what would be a safe angle for the ladder to lean and the height of our janitor.  The more we discussed it, the more ideas and questions the students had, and I was ok with that.  We ended the lesson by turning in our discoveries to the janitorial staff.  I’m not sure if they used it or not (don’t tell my students), but my students felt a sense of pride in contributing to our campus.  The real reward came the next day when the students arrived at school and saw the banners hanging from the light poles. 

Categories: Algebra 2, Geometry

NCTM Review #NCTM11

April 21, 2011 2 comments

I had the pleasure of attending the NCTM conference in Indianapolis this year.  This was my first time attending any NCTM sponsored event.  Overall, I had a good time.  I wanted to dedicate this blog post to some of the highlights of the event.

  • NCTM Android App:  NCTM released an application for cell phones that had the complete schedule and allowed the user to build their own schedule.  This was amazing.  It is nice to see conference utilizing the technology that is available.  I enjoyed not having to carry the large book around.  The application also notified the user when a session was canceled.  There was a twitter feed on the app as well as a map of the conference center. 
  • Loring Coes did an excellent job presenting on how to make movies in the mathematics classroom.  He addressed the benefits of LoggerPro and Fathom as software packages that make building graphs and gathering data from videos very accessible.  I was sad to hear the Flip Camera is no longer going to be manufactured.  I am going to have to try to get my hands on some before they are no longer made.  Some ideas that I walked away with were tossing a hacky sack to create a parabolic curve and spinning a hacky sack on a string to create a sinusoidal curve.  Fathom was nice in that it allowed the user to build an equation to match the video and create sliders for the coefficients to discover the equation.  I think this is beneficial to the students over just running a regression in LoggerPro and having no idea how the software created the curve of best fit.
  • Angie Morgan and Gordon Wells from Ohio Valley University had a good session on Quantitative Reasoning or Quantitative Literacy.  For years the push has been reading and writing across the curriculum.  It is nice to see a push towards Numeracy across the curriculum.  They talked about getting other disciplines in our school to utilize their disciplines and bring mathematics into their classroom.  This can be as simple as analyzing data in science class and comparing local and regional global warming and looking for instances of ‘cherry picked’ data.  I know personally, I have been looking for opportunities to team up with our history department.  The study of government lends itself to incorporating numeracy.  They cited the Mathematical Association of America and their work on Mathematical Literacy. 
  • Sherrie Wisdom conducted a session on Applied Physics in the Mathematics Classroom.  I did walk away with a good activity for my Geometry Students that could be modified for different levels of students.  She suggested that we have students trace their shoes on graph paper and find the surface area of the bottom of the shoe.  They could do this by counting squares or grouping the drawing into Geometric shapes that they know.  Students then use the equation Pressure = Weight / Area to find the amount of pressure.  The students need to take their weight and divide it by the area of two shoes.  If you have students do this with multiple pairs of shoes, they can then determine which pair of shoes should be the most comfortable.  Hopefully, students will discover that as the shoes surface are increases (slippers) the pressure decreases (as compared to heels).
  • My final session was conducted by Leigh Nataro from Moravian Academy and addressed the use of Facebook with your students.  She created a closed group in Facebook and had her students join.  Facebook now allows you to upload documents and photos.  This took the place of her class blog.  She used it in her geometry class by posting a picture of a geometric figure and requiring each student to post a comment about the shape.  Students could not duplicate comments.  This means that if a student logged on later, they had to read all of the other posts to make sure they did not duplicate a previous comment.  She did allow students without a Facebook account to submit their comments on paper directly to her and she posted the comment for them.  She encouraged teachers to ‘like’ their students comments and posts.  She also recommend that teachers refer to posts in class discussions.  He entire session was based upon the idea that “You need to go where your students are.  You’ll get more traffic when you are in their neighborhood.” 

Overall, my first experience with NCTM was beneficial.  A few modifications that need to be made, in my opinion, involved the use of technology.  Each presenter is required to provide handouts.  At most of the sessions I attended, the speaker ran out of handouts.  The presenter posted their email address and told people to email them if they wanted a copy of the handout of presentation.  My colleague who presented was bombarded with emails the night after her talk and had to respond and upload her documents to each individual.  I think that NCTM should provide a website with a link for each speaker.  The speaker can then upload their documents directly to the site and people can go on their own to download what they want.  It still amazes me, the number of teachers and speakers who do not have websites or places to post their work.  I guess I’m spoiled being in the blog world and having access to excellent resources and the opinions of my fellow teachers.  I would love to see someone from the blog world present next year about blogging in education.  (Hint to my fellow bloggers…)

Construction Week

If your classroom is anything like mine, then the week before spring break is a nightmare.  The kids are antsy and several parents pull their students out early to get a jump-start on their vacations.  My first year teaching I gave a major test on the day before spring break:  Big Mistake.  It was awful trying to get students to make it up when they returned and of course, they didn’t remember anything. 

So, for the last few years I have taken a new approach to the week before spring break, at least in my Geometry classes.  I make it construction week.  No, not building, but compass and straightedge constructions.  I decided years ago not to teach them as I went because the students can’t remember to bring their compasses to class and I’ve never invested in a classroom set.  So I put all the basic constructions in one week. 

On Monday, I introduce the idea of constructions and let them play with the compass to get used to it.  I’m always amazed with how much practice it takes to get a smooth circle.  I let the students be creative and I teach them how to make a flower with the compass.  They love it.

On Tuesday, I introduce them to this Math Open Reference Website.  I print off copies of the worksheets that accompany the website and hand them out to each student.  The students are put in groups of 2 or 3 and given one laptop per group.  They use the website to complete each construction.  There is a java applet that shows them how to do each construction step by step.  There is also a list of the steps with pictures below the applet.  This allows students to go back and repeat the steps as often as they need to so they can master the constructions.  I assign 8 different basic constructions involving line segments and angles.  At the end of the week, I give them a quiz on constructions.

For the students that are absent during construction week, I assign them a project when they return.  They have to use the website to create a book explaining the constructions and provide step by step directions.  I also allow them to use YouTube for this assignment. 

At the end of the week with the laptops, I was surprised and a little disturbed with the number of students that had a difficult time following written, step by step directions.  I explained to them that this is not just a math skill, but a real life skill.  There have been and will be many Christmas Eves’ spent in tears and frustration trying to interpret directions on how to assemble a bike or toy for my children.  And don’t get me started on Ikea’s picture directions…

Categories: General, Geometry

We Built This City

March 14, 2011 2 comments

I know it has been a while since my last post.  I am teaching without textbooks this semester, and it is more work than I imagined.  In class, we don’t seem to have a problem.  I never assigned a lot of homework in the past and I very seldom referenced the book in class.  My real problem is the logistics of absent students and students who don’t pay attention in class and all of a sudden realize they want to master the past three weeks of Geometry at home on weekend, so of course they need resources.  I would send a book home, but our school has no extra books, thus the teaching without books this semester.  I have posted a ton of online resources on my class webpage, but I still have several students without internet access.  I’m learning being textfree is liberating for my classroom and limiting to students who need to work alone.

On another topic, my Geometry class completed their city projects.  Every semester, I let my students choose groups and design a city based upon the idea of parallel lines and transversals.  Several groups get very creative and develop 3D monstrosities.  Most groups of students stick to a drawing on poster board.  Either way, the ideas behind the cities are always fun.  I’ve had students design Candyland, Legoland, Classic Rock City, and even Afghanistan. 

This is not an orginal project and I need to give credit.  I use a rubric based upon this one I found on the internet.  The rubric and instructions do a great job of reinforcing the relationships of angles using parallel lines and transversals.  I normally give the students two class days (we are on a block schedule) to complete the project.  I am always pleased when they realize our city, Cleveland, is based upon parallel streets and they identify important buildings in town and their angle relationships.

Categories: General, Geometry

First Day Geometry

January 6, 2011 1 comment

Like most teachers, I struggle with what to do on the first day of class.  This year in my Geometry classes, I started with ‘Me by the Numbers’.  It worked well and the students seemed to like it, but even this activity only takes 15 minutes of my 90 minute class.  I decided to try something new.  I broke the  students into groups of 2 or 3.  I gave each group a net for a cube.  I masked each group to design a cube.  They had access to colored pencils, scissors, and tape.  Some cubes were basic and others were quite ornate. 

When all the groups were finished, I collected the cubes.  I then redistributed the cubes to different groups.  Each group now had a cube that they did not design.  I handed out another blank net and asked them to copy the cube that was given to them.  Some groups had a more difficult time than others depending on the complexity of the cube assigned to them.  The students had a good time and it gave me a chance to walk around the room and get to know them.  My goal with this activity was to get my students to realize that taking a 3 dimensional object and trying to reconstruct it using a 2 dimensional template is difficult.  Some students really excel at this and others really struggle.  I feel like Geometry is often the math for those students who aren’t as good at solving equations.  I wanted to take the fear of math and the unknown away as we started Geometry.

Categories: Geometry