Common Core Geometry Unpacked and Backwards

The Common Core State Standards are designed to get students ready for college and careers by helping them be critical thinkers.  Unpacking the standards set by the Common Core allows a teacher to develop a lesson plan that engages students to learn at a higher level by relating curriculum to real life application.  Backwards planning is a process where a teacher can create a learning experience to meet specific learning goals staring from the end result.  The teacher starts by identifying the objectives and learning standards required for students to meet or determining the end result of the lesson.  A list of skills and concepts that students need to master the specific learning objective follows.  Next a final test or assessment is designed that will determine students learned the objectives of the standard.  A series of lessons and projects are then designed to progressively move students to the complete understanding of the objective.  The teacher then decides how to assess the students at each step of the learning process for the unit.  Assessment can be in many forms, from a quiz, project or even verbal questions during class, allowing the teacher to determine if students are learning the material needed to be successful.  Finally the teacher can review and reflect on the lesson plan and make any changes or additions. 

Currently I am teaching a first and second semester online Geometry course through Oaks Christian Online School, but not a teacher in the traditional classroom.  The online course corresponds to the in class curriculum at Oaks Christian High School located in Westlake Village, CA and common core state standards are used.  I chose CCSS.Math.Content.HSG.SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. High school students taking Geometry range from 9^th grade to 11^th grade.  I choose this standard because of the applications of right triangles, trigonometry and the best known and most useful theorem in mathematics, the Pythagorean Theorem. 

The following are goals or proficiencies I want students to achieve by the end of the unit: 

1.       Algebra review for this standard: Finding the square root and rationalizing the denominator.

2.       Understand and use mathematical vocabulary associated with this standard: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent, Opposite, Adjacent, Hypotenuse. 

3.       State and Apply the Pythagorean Theorem and the Converse of the Pythagorean Theorem.

4.       Prove the Pythagorean Theorem.

5.       Know and apply the basic trigonometric functions defined by the angles of a right triangle.

6.       Use trigonometric functions to solve for an unknown length of a side of a right triangle.

 

 

I will use the following assessments to know students are meeting and understanding the standard:

1.     Assigned daily homework to reinforce concepts learned.  Students will have time to ask homework questions and receive feedback from homework assignment. Ability to call on a student who understood a problem to explain how they arrived at the solution. 

2.     Two to three quizzes prior to the test to assess student’s readiness and understanding of material.  All quiz grades below 70% can retake prior to the test for additional credit and support.

3.     Verbal questions throughout the class time, engaging the class to participate in the learning experience.

Learning experiences or activities I will use to help students meet the standard are as follows:

Student will design their own right triangle word problem and present to the class.  I will encourage students to design the problems as it relates to their own life.  For example:  I play golf and I could design a right triangle to find out how far I need to hit the golf ball or at what angle to the flag from a position on the golf course.   Students will submit their word problem with diagram using paper, poster board, or digital presentation; answer key provided on separate sheet of paper.

Words-Symbols-Models Project:  Students will write in words the trigonometric functions with specific ratio associated with that function, followed by the symbol represented by the specific function anda model to represent the function.  Students can present their work in google slides, Prezi, or presentation method of choice.  I would choose the top three presentations to present to the class and winning designs would receive a homework pass or first release from class for the week.

Students will work in groups of 2-3 on an applied word problem of my choice (involving sailing, gardening, painting, geography) using a right triangle and the Pythagorean theorem to determine the missing distance, find the Perimeter of the figure, Area of the figure, and cost analysis of project in word problem.  This will be an in class project.  Each group will make a drawing/ diagram labeling important parts.  On a separate page the group will present all math used for the drawing/diagram and in finding the cost of the perimeter/area of the project. 

 

Consequences in the Classroom

After the class has acknowledge the rules and procedures the next step is to announce the consequences of following and not following these agreed upon rules.  Consequences should be both positive and negative and established in the first week of class and addressed routinely and frequently over the class year.  Without consequences rules and procedures are meaningless and have no standing with the students.  Consequences must be used in an appropriate way with a mindset of enhancing the learning environment and maintaining and managing a successful classroom. 

Example of positive consequences:  Algebra 2 class of twenty students - Using the "I do it / We do it / You do it together / You do it alone" method, after  I do the example problems on the smart board the students are to work on three specific problems as a class for the We Do It part of the lesson.  Raising hands and participating quietly is crucial for a positive and productive learning environment for all students.  This part of the lesson requires me to remain in the front of the room and guide the students through the problems while having students interact to solve the problems.  Each student that raises their hand I will reply with verbal comments like: "Thank you for raising your hand and following the class rules, what is your question?", "Excellent choice being quiet and waiting to be called on 'students name' what do you think the next step is in solving this problem?".  Using non verbal responses by smiling at students working quietly and giving them a thumbs up when they are sitting quietly waiting for the class to finish as a whole is another way to respond to students in a positive and appreciative way.  For students that demonstrate an excellence of upholding the rules and procedures of the classroom I will send a note home to the parent(s) or guardian acknowledging the students excellent behavior and diligent work in the classroom. 

Example of negative consequences:  Algebra 2 class of twenty students - Using the "I do it / We do it / You do it together / You do it alone" method, while I am demonstrating the "I do it" part of the lesson a student is talking and disrupting the learning - first the student is met with concerned eye contact.  If the behavior continues I will use the students name remaining calm and positive with comments like: "Student name please respect the classroom rules and hold your conversation or comments for after class."; "Verbal warning Students name"; "Just a reminder to the class that being silent is necessary for all students learning, right Students name!".  I would monitor this student during the class time by keeping a close proximity having watchful eye (withitness).  If a student is unable to contain and control breaking the rules and procedures of the class I would initiate a three strike rule: three verbal warnings results in parent/guardian phone call or email. 

A coogle diagram of positive and negative consequences follows:

https://drive.google.com/open?id=100lwQqw-bu68gz4yBmyr6l8tkNxwcdNK

https://coggle.it/diagram/WgUvluojMQABDQqi/f57975feaf72fa3ecc475c43ee482a13341461d584cc4df65dc4a5ad91492f3a

Teaching Learning Situations

Analysis of three teaching-learning situations


I watched and read about three different teaching techniques:

In the project based learning video Roller Coaster Physics, a 5th/6th grade teacher developed a physics class by having students design a model roller coaster that must be both safe and fun. The teacher incorporates science, technology, engineering and math in the roller coaster project. The project has students learning and applying physics concepts like potential energy, kinetic energy and Newton’s three laws of motion. The roller coaster physics project is detailed and requires students to learn and perform at a high academic level.

The teacher has a clear and concise outline for the project that she follows from the lesson plan in Glog. The outline shows the project objectives, specific terminology students are to use, concepts to be learned and demonstrated, an action plan for the project with past examples, rubric and participation rules. For a project based learning activity to be successful strong and comprehensive outline is necessary.

The Roller Coaster Physics project begins with the class being grouped into teams of four to five. Small groups allow students to build and improve their collaboration and communication skills. The entire group participates in the building and designing of the roller coaster but each student has a specific role or job they are responsible for. These roles include group organizer, measurer, treasurer, record keeper. The teacher sets a budget for the project and has additional materials the groups can purchase. The idea of a set budget helps the students apply the learning to real life and ties the math into the physics part of the project.

A technique called chiming was used where students gathered at a table and a representative from each group discussed issues with their group’s project. Students would then provide suggestions and brainstorm about how to solve each issue encountered building on communication skills, while using specific physics terminology as required by the teacher for a high level of learning and communicating. The idea of chiming builds students’ knowledge of how to solve a problem based on the physics and allows the teacher to assess each student individually and each group.  

The project continues to build on the high academic expectations when each group chose who will sketch the project to be used in a computer simulator. Another student in the group is responsible for marking and labeling the sketch, for example the rise and run of the initial loop, to be entered into the simulator. The results allow students to make adjustments and improve on the model design. This project based learning has hands on combined with technical learning to push students to learn on a higher level and making learning fun and exciting.

The Roller Coaster Physics project is so engaging and fun for the students that excellent student behavior is expected. As students are having fun while learning there should be no negative behaviors and as students are working with their peers a healthy and happy atmosphere should dominate a productive and engaging classroom. Norms and procedures were not discussed in the video, but from how the project was conducted the students had an awareness of what was expected. I felt that the norms and procedures for the class was understood by all the students prior to the project base learning or there would have been more challenges for the teacher to deal with.



In the video 3rd Grade Chinese Math Class, the teacher uses repetition in the form of song or with a rhythmic pattern to teach the third graders by memorization. The teacher used the white board after the class finished with the song and demonstrated a problem. In the video students raised their hands to solve the problem. This is what I could understand from the video as the language was Chinese and no subscripts were provided.

The article “What makes Chinese math lessons so good?” explains that repetition is the main way to efficient and effective learning. “The goal of math education in China is to develop conceptual and procedural knowledge through rigid practice” according to the article (2014). I remember learning my math facts in elementary school in a similar methodology. High academic expectations are achieved by the students through the method of repetition and practice. The students are engaged as a whole when repeating the song and communicating as a group and focused on the teacher.  

Behavior expectations are strict in the Chinese culture and students must be quiet and raise their hands to participate in the call. Norms and procedures are based on the customs of the culture and not discussed in the video (to my knowledge) or in the article. The overall class, in my opinion, is rigid and militant compared to the project based learning class but statistics show that this method of teaching develops a strong base for the concepts of mathematics.


Whole Brain Teaching (WBT) is a technique that uses hand gestures, call and response, and fun sounds to teach lessons to students. The video of a 9th grade geography class used these techniques for the lesson. The WBT technique had the whole class, including the teacher, waiving their hands, shaking their heads and acting in a bizarre but the class in completely engaged and responding to the direction of the teacher. In the video the teacher makes hand gestures to teach a lesson about longitude and latitude and location. The students mimic the hand motions. The class is structured around the WBT and in terms of norms and procedures the students are responding to specific calls and hand motions from the teacher. The rules would seem to be in the form of the specific gesture the teacher initiates.   

The behavior expectations are that every student is participating in the motions and sounds. I would expect to have this kind of participation that much time was invested in the norms and procedures and expectations of the students. The video continues to demonstrate how students learn to speed read. The students where in pairs and making motions and reading to each other. This activity is useful to develop reading and vocabulary practicing the hundred most common phonic words in the English language. I feel the Whole Brain Teaching technique would be a technique that requires a lot of practice and patient.  

After watching and analyzing these three techniques the project based learning technique would be my choice to implement. As a secondary mathematics teacher the Whole Brain Teaching technique would be difficult to implement. I feel the WBT would be a technique used primary education due to the hand motions, calls and responses – something I do not envision high school aged students participating in. I feel I could use the Chinese method of repetition by song or rhyme in a classroom of Algebra to Calculus students, but only for a small portion. Currently a song for the quadratic formula is used to help students memorize the formula they need to use in solving quadratic equations. The project based learning technique would require a great deal of planning and cooperation from the whole mathematics department. I feel that the project based learning technique would be most engaging for the age group I work with. Working in teams and applying mathematics to real world situations and creating a learning environment with high expectations where the students are collaborating and communicating is the ideal class that I would want to provide for my students.



References

The Teaching Channel. (2012). Roller Coaster Physics STEM in Action. Retrieved  https://www.teachingchannel.org/videos/teaching-stem-strategies

Chen, C. (2011). Third Grade Chinese Math Class. Retrieved https://www.youtube.com/watch?v=h7LseF6Db5g

Glogster EDU. (2012). Roller Coaster Physics. Retrieved http://edu.glogster.com/glog/roller-coaster-lab/1gku0vrn4cn

Wei, C. (2014). The Conversation Explainer. What Makes Chinese Math Lessons So Good? Retrieved http://theconversation.com/explainer-what-makes-chinese-maths-lessons-so-good-24380

Mackens, R. (2011). Whole Brain Teaching-Richwood High-The Basics. Retrieved https://www.youtube.com/watch?v=8iXTtR7lfWU&feature=youtu.be

My name is Marlene Peters and I teach secondary mathematics.  I was born and grew up in Homestead Florida.  My parents have been married fifty years and I fabulous younger sister.  I had a modest upbringing and have a small family dynamic.  I am a good mix of French, Irish, German and some Native American Indian but on paper I am Caucasian.  I have felt fortunate to be exposed to many different cultures growing up in South Florida.  I met my husband in South Florida and in our first year of marriage we moved to California because of an amazing job opportunity.  I have two children, a boy and girl, both of whom I am so proud of. 

The first week of school is the most important week to me.  During this week I will set the tone for the year.  I want students to feel welcomed and cared for when they enter the classroom.  Yes the class will be challenging, but I want them to know we will meet these challenges as a team and I will support them to the finish line. 

In the first week I will review classroom policies.  These policies are given out by the school and will discuss matters of bullying, social media, cell phone use and plagiarism.  I want the students to know these are rules that are for their protection and guidance. 

During the first week I will tell them I am the math teacher who embraces technology and looks forward not only using the calculator but applying the calculator and other technologies during the year. The class will be working on four group projects over the course of the year and small groups during class time. 

First homework assignment is to make a short video or a presentation of their choice answering the following questions:

What is your name?
What is your favorite subject?  Least favorite subject?
What is your favorite food? Favorite candy or treat?
Do you have any allergies?
Do you have a pet? What kind/how many?
What is your favorite song or band? 
What do you like to do in your free time?
Do you work/ play a sport(s)? 
Do you speak any other language(s)?
What is the farthest place from school you have traveled to and why?
How can I help you be successful in this class?
Anything else you would like me to know about yourself?
Tell me about your family:)

I believe this assignment will be engaging, fun and show the students I am interested in who they are inside and outside the classroom.  I want my students to know I care.  As the teacher I would also share a video I will make from questions the class will write down or email to me.  This allows the students to know me on a personal and professional level.  Of course I will not answer all of the questions and will have a few of my own, but this setting opens an interaction between the students and myself and sets a tone that the class is going to be more than boring math.  Questions I will address in my video for the students:

What is my name?
Discuss my family - married for twenty five years, two children one boy and one girl.
Share my dog Stanley
Wear my hockey jersey and say "Go Hawks"
Show my set of golf clubs
Share my rose garden
Share where I went to school

Now that the foundation is set, developing a positive environment in the classroom starts with the teacher - me.  Modeling behavior I want to see from my students is the best way to ensure a positive environment.  Enthusiasm is contagious and inspiring; this is how I want to engage my students daily.  Taking an "I don't know" response to "how can you find out?" moment is my method to taking the student to a positive "I can do this" attitude. 

Class time is precious but taking in and processing complicated information can be draining.  To maintain a classroom environment that is positive and energetic I would take the much needed brain break.  Brain breaks can be daily, twice daily or as needed.  Taking a brain break can be as simple as having the class just stand up and try to touch the ceiling and then reaching down to the group to fine motor skill exercises.  I believe having a period of time where as a community the class is doing something other than math reenergizes the class. 

The first week video assignment is useful in understanding the various cultures in my classroom.
Multicultural education is crucial for students to be successful in life.  Learning and understanding other cultures provides students with social and collaboration skills used in the real life situations.
Providing open communication with students and understanding that everyone is unique and special makes for a successful multicultural classroom.  Providing resources and examples from around the world, and demonstrating that mathematics is a subject that has been influence by men and women from all backgrounds.  I think it is important to embrace students cultures as each culture has a particular method of communicating and learning.  As a teacher it is my responsibility to embrace all the students needs and provide support during the learning process.

Making the students feel cared for and safe in the classroom means a bully free classroom.  Bullying is not acceptable in High School and most students (if not all) have heard about what to do if they witness bullying in school.  As a teacher I need to pay attention to any changes in a students participation, attitude and attendance as these are signs of a troubled student.  Open and honest communication that is caring and direct is the best way to address possible bullying situations.  Gentle reminders throughout the year will keep the topic open and refresh students awareness. 

Utilizing all the above actions will result in a classroom that is positive, welcoming, embraces all cultures and free from bullying.  Listening to students is key in understanding what is important and motivating in their environment.  For example if a majority of the students are discussing the National Debt, I have the ability to use this emotional situation and incorporate it into the classroom in some part of a lesson or just take time to converse about how the topic makes them feel.  Discussing topics of concern or major interest is how a teacher can address the emotional and social needs of their students. 

The classroom environment has a multitude of variables that the teacher has the ability to shape and model.  Variable like a welcome and caring environment, use of technology, multicultural classroom, bully free zone and addressing social and emotional concerns of students, all make up a dynamic and fascinating space where not only mathematics but real life takes place.