The paths to "Eureka" moments: Teaching Mathematics in Secondary Education

Posts tagged ‘Algebra’

H5- Honor student potential for roles in the greater society

H5- Honor student potential for roles in the greater society

Teacher-candidates prepare students to be responsible citizens for an environmentally sustainable, globally interconnected, and diverse society.

This means that teachers should teach in such a way so as to guide students to be responsible people in diverse society. In the math classroom, this includes thinking critically so as to solve real-world problems (such as environmental sustainability).

As a math teacher, there has been numerous times in which my classes have engaged in problem-solving exercises and activities. As a warm up before starting a unit on algebraic functions, I used a youtube video about the Enigma Machine. This machine was used in World War II to encode and decode messages by the Germans. The video, along with explaining the mathematics surrounding the machine, provided historical background and global facts surrounding the machine and its uses. The clip also explained how different countries had attempted to crack the code so as to intercept messages.

Four-rotor German Enigma cypher machine, 1939-1945.

This short film allowed students to think of mathematical topics as globally interconnected- a universal language. Following the films, students worked collaboratively to solve coded problems (functions with inputs and outputs) and determine the code (function rule). In working collaboratively, diverse students gained insight from multiple perspectives and were encouraged in their mutual respect for the skills of others (responsible citizenship).

Through using the video as inspiration and encouraging group work, I learned the power of collaboration among students and effective ways of motivating student learning. In the future, I would like to use this video as an inspiration for students to write more about what they learned and express their understanding in written reflection.


Technology in Math Class: Function Machines and Jeopardy!

P4-Practice the integration of appropriate technology with instruction


Teacher-candidates use technology that is effectively integrated to create technology proficient learners.

As a student teacher, I regularly utilize technology as a catalyst for learning and as a way of enhancing my instruction. In this way, students are given the opportunity to see and use technology as a learning tool.

In my 8th grade algebra class, we have begun a unit on functions. As a way of introducing the “big picture” concept of what functions are and how they work, I decided to show a short Youtube video created by several middle school math teachers.


This video is quirky and engaging visually (there are no words spoken). Explanations and definitions are given in written text along with humorous and cliché sayings. The end of the clip provides an opportunity for students to interact by guessing specific outputs and function rules. In this section of the video, I paused the video and asked for student guesses.

After the first introduction day, and in the second lesson, I used the game of Jeopardy to motivate student learning and provide an interactive way of reviewing key concepts associated with functions. I used a Powerpoint template to customize each question to fit our exact content and new vocabulary.

Jeopardy Functions!

Students were asked to form teams at their tables and then were given a question to answer in 1 minute. If the table answered incorrectly, then the question was bumped over to the next table. Because students had to determine answers to each question (just in case the first group got it wrong), all students participated in every question.

In both examples, technology was used as a way of enhancing the learning experience of students by providing opportunities for students to actively engage with the content.

In creating these learning experiences, I learned how technology can be used to instruct the class as a whole and yet engage individual students. This allowed me to gain experience in presenting math content using different methodologies other than direct instruction.

Through these two lessons, students were able to have new content presented in a lively and interactive way. The visuals and humor presented in the video and the group competition in the Jeopardy game alleviated some of the fear associated with new mathematical content and instead created a culture associating learning with fun.

As a teacher, one of my goals is to identify students’ fears and insecurities related to math content and provide transformative approaches to teaching so as to create a safe learning community. I plan to do this by using technology as a tool for instruction and student engagement, in this way, creating technology proficient learners.

Student Diversity Honored in Design of Application Activity

P-1 Practice intentional inquiry and planning for instruction.

Teacher-candidates plan and/or adapt standards-based curricula that are personalized to the diverse needs of each student.

As a student teacher, I create curricula based on standards and the readiness levels of my students.

Savings Problem

savings accountIn my 8th grade honors classes with Algebra students, we continued to work on recursive sequences using contextual problems to relate the concept to real life scenarios. I created a savings account activity and word problem where students were asked to determine how long it would take them to save $500 under different circumstances. Previous to this lesson, we had worked on recursive sequences for about 5 class periods and I had determined student readiness levels through formative assessments. In the activity, students were separated into 6 different groups and asked to solve one of two word problems. The two word problems were formulated for different readiness levels. The first problem was for those who were not quite proficient in finding explicit formulas and the second problem was for those who were proficient.

This activity was based on the standard:

A1.7.C Express arithmetic and geometric sequences in both explicit and recursive forms, translate between the two forms, explain how rate of change is represented in each form, and use the forms to find specific terms in the sequence.

Students were asked in both problems to find the explicit formula of the savings scenario they were given and then find the term when the balance reached $500. Those who were not proficient received additional aid in interpreting the problem as well as steps to finding the explicit formula. The sequence they were given was arithmetic and thus was less challenging mathematically than the second geometric sequence. In this way, students in the first readiness group were able to focus more on interpretation and application of their skills rather than the more complex mathematics operations.

Creating this activity provided me the experience of using formative assessment and knowledge of students’ readiness levels to plan for instruction which challenges students appropriately. I was able to isolate specific learning objectives and provide the opportunity for diverse students to be individually and collaboratively successful.

This activity allowed students to build on their prior understanding of recursive sequences through contextual examples. Additionally, students were able to collaborate in their groups, creating a learning community, and using their peers as resources for their academic growth.

Finally, this activity is exemplary of future lessons that I will plan to challenge diverse students. While this lesson was generated to meet student readiness levels, in the future, I will use similar strategies to fit diversity in learning styles, learning profiles, and multicultural backgrounds.

The Classroom as an Environment for Learning: A Holistic Approach

H-3 Honor the classroom/school community as a milieu for learning. When the classroom and school community are respected as places of learning, student success is promoted and learning is uninhibited by unnecessary restrictions.

In the lesson plan presented below, students are encouraged to work collaboratively within groups of 3-6 students to solve for recursive equations given a sequence. Each group was given a different sequence and asked to solve using dry erase markers directly on their tables before then copying their work into math journals. In this way, students teach each other by discussing and writing their work for each other to see. Having completed their assigned problem, groups were then asked to review the work of at least two other groups and writing the others’ results in their personal math journals. Finally, one group was selected to demonstrate their work and rational in front of the class.

Recursive Review Lesson

IMG_20130117_113030Students were uninhibited by the tables as work spaces since they were able to use as much space as needed. Students were also able to quickly erase mistakes and make corrections. When the groups were satisfied with their own work, they were asked to leave it on the tables for other groups to review.  In reviewing the work of others, students were able to look to the entire class of peers as a resource for learning. They could learn from the work of others, ask questions, and then take notes in their personal math journals.


Finally, one group of students was asked to come forward and present their work in front of the class. Though their work was observed by several groups, the specific sequence they were given was at a level of greater difficulty and importance. In this way, all students were able to benefit from a more instructional time of learning (peer to peer instruction). The teacher, instead of directly instructing, led the presenting group through a series of questions about how they had solved their problem and why they were able to perform the operations they did. Not only did the students presenting have the opportunity to grow in communication and further understanding of their own work, but the entire class benefited from their work without having to struggle through it individually.


Through this lesson, I learned the power of peer instruction and collaboration. The teacher need not be the only source of learning. When students are in an environment which promotes peer communication and creativity in displaying their learning, they are able to thrive and grow as learners.

In the future, table tops will be continued to be used as resources for displayed work and examples. Additionally, I plan to continue selecting students to present in front of the class to demonstrate their work. I will especially look to encourage less confident students to do so (in content they display proficiency). I believe this to be one way to promote a healthy learning community where all students are valued.

Showcase Lesson Plan – Functions as Literal Equations

As a result of the conclusion to my first quarter in the Education Program at Seattle Pacific University, I have completed my first showcase lesson plan!

Please view it, and give me your feedback!

Attached are four documents which make up a showcase lesson plan centered around a lesson titled: Functions as Literal Equations.

The documents are as follows:

1) Unit Plan Alg1

2) csc function lesson

3) Functions as a Machine of Literal Equations Rational

4) Function Lesson