# Writing to Deepen Mathematical Understanding

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Laura Mourino assesses formative understanding by having her 10th graders write letters answering four key questions about a unit circle.

Teacher: Laura Mourino

School: Harvest Collegiate High School, New York, NY

Discipline: Mathematics (Algebra 11)

Lesson Topic: Unit circle

Lesson Month: March

Number of Students: 26

### Featured Lesson’s Student Goals:

• Content objectives – Analyze a visual representation of a unit circle and make connections to various algebraic and geometric concepts, including the Pythagorean theorem, coordinate geometry, locus, transformations, and trigonometric functions
• Literacy/language objectives – Write for understanding using appropriate language; validate responses; provide sufficient quantitative analysis; pose questions; restate analytical concepts in familiar language; develop a word bank by using the chart paper
• Engagement/interaction objectives – Share observations about the visual representation of the unit circle

Common Core State Standards for Mathematics

• CCSS.Math.Content.HSF.IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

### Instruction Details:

The Unit
This 10-day unit—Unveiling Trigonometric Functions Through the Unit (and Non-Unit) Circle—was taught about three-quarters through the school year. The essential question for the unit was, Are the origins of trigonometry algebraic or geometric? The two-day lesson focused on introducing the unit circle through an inquiry approach while reinforcing students’ algebraic and geometric skills.

Before the Video
The previous lesson in this unit revisited and explored various interconnected algebraic and geometric concepts, including the Pythagorean theorem and the equation of a circle on a coordinate plane. On the first day of this two-day lesson, students were introduced to unit circles with a sketching activity and practice work using trig functions.

During the Video
On the second day of this two-day lesson, Ms. Mourino gave students a diagram to observe, analyze, and write about in a letter format. When writing the letter, students were asked follow MLA format and respond to four requests: provide the reader with numerous mathematical statements and/or quantitative analysis to describe the diagram; pose as many appropriate diagram-related questions as possible; reflect and share personal reactions and/or concerns about the diagram; and make connections or predictions about how the class might use the diagram in the future. (Honors students were also asked to compare and contrast the use of trigonometry on a coordinate plane vs. non-coordinate plane.) For each of these requirements, Ms. Mourino stressed the use of appropriate language, including validated responses and providing sufficient quantitative analysis.

Students had 15 minutes to complete the task—10 minutes of independent work followed by 5 minutes of collaborative work in table groups during which time students shared their letters, received feedback, and implemented edits. Then, based on their individual and/or group findings, students added contributions to collaborative charts at different stations around the room with the topics of Statements, Reflections, Questions, Connections, and Quantitative Analysis. Each table group shared-out major findings/conclusions to the class, during which Ms. Mourino elicited appropriate mathematical language. Ms. Mourino then led the class in a summary of the activity and asked students to share comments, critiques, and concerns. The exit slip included a two-minute summary and reflection.

After the Video
Ms. Mourino posted the chart papers throughout the room and revisited them with students regularly. In subsequent lessons, students referred to their “posed questions” from the writing assignment and checked them off as they were answered in class. The same writing assignment was given again at a later date as a graded quiz.

Teacher Prep
Ms. Mourino prepared a lesson plan and made sure she had enough handouts and materials for students. She consulted with her ELA colleagues to determine how to integrate and reinforce the concepts that they were teaching. Ms. Mourino prepared her lessons for the year using “backward design.” She started with final projects and assessments, identified the key skill sets students would need for them, developed a curriculum map, and adjusted and realigned the map to students’ needs as the semester progressed.

Prior Knowledge
To participate in this lesson, students needed to know the Pythagorean theorem and the equation of a circle.

Differentiated Instruction
Ms. Mourino modeled the activity for students and asked for clarifying questions on the task. She provided students with the choice of writing, typing, or scribing their responses and gave students a choice of entry point when responding to the four writing requirements. She played music as students worked (as a soothing mechanism and noise controller) and made sure that seating arrangements at tables were skill appropriate while also being sensitive to socialization needs. Ms. Mourino encouraged the use of colored pencils and markers to outline diagrams and visually reinforce key words and ideas. She provided adequate time countdowns for students with time management challenges. The chart portion of the activity provided a kinesthetic opportunity as students moved around the room from station to station.

Group Interaction
Students worked in table groups to collaborate on written letters and to add conclusions and findings to class charts. Throughout the year, Ms. Mourino changed seating groups at random (but with sensitivity to students’ skill levels and social needs). She offered students the opportunity to switch roles with her (students as the teacher and the teacher as the student) to help students better understand the challenges of teaching, but also to give Ms. Mourino an opportunity to see how students viewed her. With a goal of mutual respect in mind, she used this information to modify her teaching when needed.

### Resources and Tools

• Pre-Diagnostic: Unit Circle handout
• Chrome books
• Chart paper
• Colored markers
• Music

### Assessment:

Formative Assessment
The main writing activity served as a pre-diagnostic assessment to see what students understood about unit circles from past lessons. Ms. Mourino asked some students to read aloud to see who needed support and determine which new vocabulary to introduce. As she walked around the room, she observed students as they engaged in the task. She asked questions to assess and scaffold, giving prompts when needed.

Summative Assessment
The pre-diagnostic activity was given to students again after the lesson and graded as a quiz. The summative assessments for the unit included a standardized test (in SAT format), a written description of the unit circle making full connections with algebraic and geometric concepts, and an essay question about when two people on two Ferris wheels with different radii face each other.

Impact of Assessment
Ms. Mourino’s students drove the path of the curriculum. She adapted lesson planning and instruction continuously to meet their needs. The next time she teaches this lesson, she plans to extend the lesson to three days in order to offer more reflection time, increase dialogue between students, help students make deeper connections to transformations, and offer more opportunities for collaborations between multiple table groups.