Archive for the ‘Algebra 1’ Category

Model Drawing (Bar Modeling or Singapore Math)

September 9, 2014 2 comments

Last year I was introduced to a strategy known as Singapore Math.  I have since seen it referred to as Bar Modeling or Model Drawing.  I was hesitant about any “new” strategy and very reluctant about anything that would take away from the beauty of “pure mathematics”.  It only took one short training session to convince me that this is what had been missing in my classroom.  I have tried many techniques such as tables, charts, and graphic organizers to help students with word problems with little success.  This is one strategy that spans grade levels and abilities to help all students be successful in math.  There are many great, free resources to help teachers get started.

I met with a fifth grade team today to work through a story problem that appeared very difficult at first:

Meg has 120 flowers at her flower stand.  Of all the flowers, 1/4 are red,

1/8 are pink, and 1/8 are blue.  Of the other half, 1/3 are yellow and

2/3 are purple.   How many are there of each color?

If you are like most students, a problem with this much information and this many fractions scares you.  I could see the the wheels turning in their heads as they were thinking  things like:  common denominator, multiplication of fractions, can we skip this one?

We first wrote the answer with a blank for our numbers.  I love this technique for so many reasons!

Meg has __ red, __ pink, __ blue, __ yellow, and __ purple flowers.

We then decided what we were counting or talking about:  Flowers.  We drew a unit bar after the word flowers.  (Hint:  If the problem  involves fractions, use a long unit bar.  You will probably have to divide it.)

Model Drawing One



Next, we read the problem and pause to add in the information:

“Meg has 120 flowers at her flower stand.”

Model Drawing Two


“Of all the flowers, 1/4 are red.”  This means I need to divide my drawing into fourths.

Model Drawing Three


“1/8 are pink, and 1/8 are blue.”  I need to go back and divide the other pieces are divided into eighths.

Model Drawing Four


“Of the other half, 1/3 are yellow.”  Oh no, I should not have divided the second half of the bar into eighths.  I will create another bar underneath it and divide it into thirds.

Model Drawing Five


“and 2/3 are purple.”  How many of each color?

Model Drawing Six



Looking at the first bar, there are 8 sections, so each section contains (120 divided by 8) 15 flowers.

Model Drawing Seven


The second bar represents half of the 120 flowers, there are 60 flowers to share among three groups.  This means there are 20 flower in each subgroup.

Model Drawing Eight


Now, all a student has to do is count of the number of flowers for each color group.

Meg has _30_ red, _15_ pink, _15_ blue, _20_ yellow, and _40_ purple flowers.

The great thing about the bar modeling process is it can be adapted for all levels of mathematics.  I have seen fifth grade students solve a systems of equations using the bar modeling method.  I know that all students do not need to use this method, but it is important that even the accelerated learners have a method or tool to use when the math becomes challenging.

Some great sites to visit for resources include and


Categories: Algebra 1, Common Core

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




Zombie Attack and Exponential Growth (What to do after state testing…)

May 17, 2013 5 comments

After the End of Course test I am always scrambling to find activities for my students.  I did this zombie activity in my class last year, but this year several other teachers and I teamed up and collaborated to put together this very cool activity.  (To be honest, my contribution was only the worksheet.)

On day one, we showed the movie Contagion.  We discussed the r not value and how viruses spread.


We prepared small cups of water, numbering them on the bottom.  We filled the cups half way with water and put a few lemon drops in one of the cups.  Each student received a cup.


The students were then told to share water with one other student.  This involved two students getting together and one of them emptying their content into the other cup.  We asked them to transfer the water from one cup to another three times to make sure the liquids mixed.  After this, the students equally distributed the water between the two cups.  This constituted sharing with one other person.  We repeated this two to three more times (depending on the size of the class).  After this we added a few drops from this pH test kit.


If the water turned yellow, the student was infected.  We then discussed what students shared water and who they shared with.  In both classes the students were able to discover who initially had the virus (lemon juice).  We confirmed this with the numbers on the bottom of the cups.  Doing this part of the activity really helped students understand exponential growth and how viruses spread.  We handed out the following zombie attacks packet and had the students work through the exponential growth using different scenarios.

Next year, the plan is to have the entire STEM academy join in the fun.  We would like the STEM English teachers to have their students research creative stories that include exponential growth and decay or the spread of bacteria.  The science teachers will add their expertise and critique the film from a scientific plausibility stand point.  We asked the history teachers if they would have their students research past epidemics and the effects on society.  As we collaborated as a STEM academy, the excitement grew and new ideas sprouted.  I’m excited to see the great collaboration that will happen next year after the End of Course test.  This is just evidence that learning doesn’t stop after state testing.

Categories: Algebra 1, Algebra 2

Quadratic Introduction: Baseball Math

March 30, 2013 4 comments

I recently was selected to be common core coach for TN for Algebra 2. I have tried to incorporate the philosphy into my classroom. I handed out the worksheet below and put the students in small groups. I had not spent any time in Algebra 2 on quadratics and I wanted to see what they remembered from Algebra 1. I asked them to work individually first. After approximately five minutes, I let them work together to answer the questions. I was amazed with what they could figure out without me having to teach a lesson. After 10 minutes, I asked different groups to come to the smartboard and present their findings. I had some students use their graphing calculators to find all of the answers. I know this seems bad to some teachers, but it was helpful for students who could not remember quadratics from Algebra 1. It provided them with a way to be successful. I had one group who remembered the quadratic formula and used it and found the vertex by hand.  You can get the worksheet here.

Baseball Math


I love teaching quadratics becasue of the real life applications.  I need help with finding real life like the one above.  I really want questions that can be solved using different methods.  I’m trying to prepare for the new common core tasks.  Does anyone have any good resources and wouldn’t mind sharing?  Thanks.

Constructed Response Assessment for Common Core

November 8, 2012 1 comment

I recently was invited to attend training for the new constructed response assessment for common core for 6th – 8th grade mathematics.  My system has decided that the high school teachers will help the middle school teachers grade the assessments.  We are doing this for two reasons.  The first is to gain knowledge of what is to come in 2014 for us and the second is it takes a long time to grade them.  I personally was anxious to see how my own child (a 6th grader) was going to be assessed.

I knew I would love the new common core, but I had no idea how much.  As I sat and listened to a 6th grade math teacher speak about how the common core had changed her classroom, I had to hold myself back from shouting “amen” and “preach it, sister.”  Each grade has two focus standards that all lessons revolve around.  For sixth grade, one of them is “understand ratio concepts and use ratio reasoning to solve problems.”  (How many time have you said:  “If my kids could only understand fractions…”)  This teacher stated that she no longer teaches her students ‘cross multiply and divide’.  She said that is an algorithm or trick we teach them, but it does not lead to a deep understanding of ratios and proportional reasoning.  (I was crying for joy inside!) 

Common core has identified six key principles for mathematical practices.  This is what the students are assessed on.  The content is interwoven into the these six practices. 

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

The teacher I observed had a poster on her wall for each practice.  She said she constantly refers to them when working on problems with her students.  As we began to grade the sample assessments, it was obvious that the eight mathematical practices were vital to earn a successful score on the assessment.  I loved that the students did not have to work the problem using on method, however, they did have to show their work and reasoning.  As a grader, you could not assume anything.  If a student did not show how he or she arrived at an answer, they did not receive credit for several of the categories above.  If it was necessary to perform a process more than once, like finding the slope of a line, the student would have to calculate the slope using the same method each time, while showing their work, to receive credit for the repeated reasoning practice.

Overall the grading of the assessments reminded me of the Advanced Placement Calculus training I went through and how that test is graded.  Students have to justify arguments and show their work.  Answers are no longer enough and your method for finding the answer must be mathematically sound.  The sixth grade teacher told us that she spend a class period with her students looking over the sample tests and the scoring rubrics and discussing why some students didn’t receive full credit.  One of the benefits for students was that they discovered they could work the same problem using different methods and still receive full credit as long as they showed and explained their work.  

I would strongly encourage you to contact your state to gain access to any practice assessments and grading rubrics released.  In TN, the items are password protected or I would share them.  (I like my job and want to keep it 🙂  I plan on encouraging our Algebra 1 teachers to begin implementing the 8th grade assessments we have access to online.  Overall, I learned a lot from the training and I have high hopes for the future math students.  It will take a few years to see the results, but I believe mathematics education is finally moving in the right direction in the United States.

Roller Coaster Fun

August 28, 2012 2 comments

The STEM academy at my school is starting a Science Olympiad team this year.  To recruit members, the engineering teacher and I collaborated for a roller coaster project during class.  We took our students to the auditorium and put them in groups of three or four.  We gave them a handout explaining the details of the activity.  The students were asked to create a mock roller coaster using the supplies they were given.  They were told the roller coaster would be evaluated using the following equation:

The students had to first discuss how each of the variables in the equation would affect the final score.  The goal was to achieve the highest score, while creating an aesthetically pleasing roller coaster.  After the student finished the roller coaster, they were asked to create their own equations that would give them a higher score.  They had to defend why they weighted each item as they did and why they put it in the numerator or denominator. 

We supplied each group with a piece of foam tube for track, four notecards, four straws, a styrofoam cup, and a roll of tape, and a razor blade.  The students were allowed to use items in the room as supports, but not as actual parts of the roller coaster.  The passengers were marbles of varying weights.  The students had a great time and it was amazing to see the differences in  each roller coaster.  I definitely see expansion ideas for this project.  Next time, the engineering teacher and I will create a store for the supplies and make the students purchase their supplies with a limited budget.

Calculator Boot Camp

August 6, 2012 19 comments

I based this idea on something I stole from samjshah.  Sam does an algebra boot camp for his calculus students during the first week of school.  If you have taught calculus before, you know that the calculus is easy, but most students struggle with the algebra involved in it.  I transferred  this idea to my Algebra 2 students.

The Algebra 2 standards in TN are overwhelming.  After teaching it for 2 years, I  found that most of my time was spent teaching students how to use the graphing calculator.  My students do not use a graphing calculator in Algebra 1.  This is their first exposure to this type of technology.  This year, instead of repeating the same steps everyday and getting frustrated, I decided to do a calculator boot camp the first week.

On the second day of school, I give the students this lab on how to graph a function and change the window.  It will also address how to utilize the table feature of the TI-84.  On day two, students will work on finding zeros on the TI-84 and finding maximums and minimums on a graph.  On the third day of the boot camp, students will learn how to input data into a table and find the measures of central tendencies.    My plan is to give the students an assessment at the end of boot camp to test their calculator skills.  My goal is that students have the basic skills to navigate the calculator and to give them the confidence to explore the calculator on their own.  I will post on my success with this after the first week of school.

Pacing using Backwards Design and State Tests

June 19, 2012 5 comments

Every summer I try to evaluate the classes I taught and how I taught them.  I’m sure that most teachers do this.  It doesn’t hurt that in Tennessee teachers receive their TVAAS evaluation scores in June, which provides us with an opportunity to reflect on how our students performed on the End of Course state tests.  I was pleased with my students’ performance this year, but I know that I should never get comfortable.  Next year, I will be teaching a grade level Algebra 2 for the first time and the End of Course for this class is a beast!  My main concern for this group of students is pacing.  I never want a student to walk into an End of Course feeling unprepared because of something I did or didn’t teach. 

The first thing I did before I taught Algebra 2 was look at the End of Course test issued by Tennessee.  If you have an End of Course test for your state, you should start there.  I learned this many years ago in College when the buzzwords of the time were ‘backwards design’.  I now know they aren’t just buzzwords, but a great way to plan anything.  When I met with a personal trainer several years ago for the first time, his first question was, “What is your final goal?”  This goal dictated my workouts, schedule, and diet.  As teachers I feel like we should ask ourselves, “What is our final goal for our students?”  My answer is different for each class.  For several classes, my answer is that I want them to hate math less when they leave my class.  For my honors classes, it is to be prepared for the next level.  For my End of Course classes, it is to pass the state test.  I hate to reduce my class to a single test at the end of the year, but my job depends on it.

The first thing I did to help students pass the test is analyze the test.  I looked at each test and recorded the number of times the state asked questions on certain topics.  Pearson stresses that Tennessee provides them with a blueprint for what questions to select and how many and that the sample test is a good model of this. 

In Algebra 2, the Tennessee End of Course has 9 questions on Statistics.  I used to squeeze this section in at the end of my class if I had time, but with the state putting so much emphasis on this topic on the test, I felt it was necessary for me to place importance on it in my class.  The Algebra 2 End of Course has one question on Conic Sections.  I love Conic Sections!  I still teach them, but I do not spend the two weeks on them that I used to spend on them.  We now focus two to three days on them in Algebra 2.  The students who really need them will see them again in Pre Calculus.

Here is my analysis of the Tennessee Algebra 1 EOC test.

Here is my analysis of the Tennessee Algebra 2 EOC test.

After I prioritized my list of topics or standards, I try to fit them into a semester long calendar.  I created this one page document that lays out my entire semester in one glance.  It allows me to start my planning for the entire year and make sure everything ‘fits’ in by the End of Course test.  The worst feeling is to be two weeks from the test and realize you have 4 chapters left to cover.  I fill in the boxes on the page in pencil with the standard or lesson I plan on teaching that day.  It is an evolving document, so I erase and change often.  This paper stays in the front of my lesson plan book all semester.  I try to always keep my eye on the big picture or goal for my class. 

I know that some teachers would argue that ‘covering’ the material is not sufficient enough.  I agree.  I try to not just cover the material, but look at it in as much depth as possible on a limited time schedule.  I do not teach all of the standards equally, because not all of the standards are weighted equally on the test.  Yes, there are some things that get left out.  As a professional, it is my job to make those decisions. 

I wish someone would have walked me through this pacing process as a student teacher.  Instead, I was handed a pacing guide and said this is what you will teach and how long you will teach it.  I never had input nor did I give it much thought.  I now make my own pacing guides.  I know I may lose this freedom soon.  I fear the days of standardized learning are coming, but until then, I will continue to pace my students in way so they can be successful.

Categories: Algebra 1, Algebra 2

SBG and High Stakes Testing

April 29, 2012 1 comment

It’s that time of year again.  Time for end of year, high stakes testing.  With the introduction of Race to the Top, Tennessee’s high stakes testing got even tougher.  This year, the end of year tests count for 15% of a student’s grade in elementary, 20% in middle school, and 25% in high school.  For teachers, it is 35% of their professional evaluation.  High stakes…

While other teachers are frantically trying to cram an entire semester of material into a week of review, I am using SBG to determine what I review.  At the beginning of the year, I break my semester into bight size chunks or Standards.  I then teach and measure learning on those standards.  If a student doesn’t show mastery, they are provided several opportunities to demonstrate mastery.  The learning is in the hands of the learner and not the teacher. 

The best part about this system is the end of the year review.  I simply open my digital gradebook and look at the class averages on each standard.  The standards that are the lowest for the class are the focus of my review.  Not only does this provide much-needed review for the students, it gives them a chance to raise their grade at the end.  There is no extra credit or extra project, just measuring learning a student didn’t master the first time.  Of course, this makes me a hero in the eyes of the students, because all they see is that I allow them to make mistakes and retest.  Let me be honest, it is very selfish as well. 

Did I mention the state test is 35% of my evaluation?  I want my students to perform well.  I know their scores reflect on me and what happens in my classroom.  If a student learns how to solve a system of three equations in three variables a week before the high stakes test and not in March when I taught it, I’m ok with that.  That is one more standard he or she has  mastered.  This is one more question correct on the test. 

I do not teach to the test, but I make sure my kids have every chance to  be successful on it.  Standards based grading provides them with the motivation to do this.  It isn’t perfect.  I hate the testing as much as the next teacher, but it is here and I don’t see it going away anytime soon.  So, until there is a perfect system I will use every tool to help my students master the material and do well on the test.  Standards based grading is one tool that I feel does this.

Categories: Algebra 1, Algebra 2, SBG