Creating Opportunities for Mathematical Discourse

• Content objectives – Answer a chosen (or self-written) question using graph theory
• Literacy/language objectives – Communicate informally within groups to answer chosen question; communicate in writing to convey understanding to others; communicate formally by explaining solution in a video
• Engagement/interaction objectives – Collaborate with peers and exercise choice

Common Core State Standards for Mathematics

• CCSS.MATH.PRACTICE.MP1
  Make sense of problems and persevere in solving them.
• CCSS.MATH.PRACTICE.MP3
  Construct viable arguments and critique the reasoning of others.
• CCSS.MATH.PRACTICE.MP7
  Look for and make use of structure.

Common Core State Standards for English Language Arts

• CCSS.ELA-LITERACY.CCRA.W.2
  Write informative/explanatory texts to examine and convey complex ideas and information clearly and accurately through the effective selection, organization, and analysis of content.
• CCSS.ELA-LITERACY.SL.11-12.1
  Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 11–12 topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively.
• CCSS.ELA-LITERACY.SL.11-12.4
  Present information, findings, and supporting evidence, conveying a clear and distinct perspective, such that listeners can follow the line of reasoning, alternative or opposing perspectives are addressed, and the organization, development, substance, and style are appropriate to purpose, audience, and a range of formal and informal tasks.

The Unit
This 20-day unit on graph theory was the last unit of the school year. Graph theory is the study of points and lines; it is a relatively new branch of math that many people do not encounter until college. During the unit, students learned about Eulerian paths and circuits as well as Hamilton paths and circuits. In the featured lesson, which was the final assessment for the graph theory unit, students applied graph theory knowledge to explore a question of their choosing and created a video to explain their solution.

Before the Video
Before this lesson, students progressed through learning about graph theory by investigating the school basement (which is an Eulerian path); exploring the Konigsberg bridge problem; and sorting cards into graphs with a Eulerian circuit, Eulerian path, or neither. Most students also determined rules about the degree of the vertices in a graph and whether a graph would have a circuit or path.  A few students also looked at Hamiltonian paths and circuits or applied Eulerian circuits to snow plowing routes.

During the Video
During this lesson, Ms. Langer explained to students that they would be working in small groups to find a solution to a question and create a poster and a video of their explanation. Students had to first form groups and discuss the six possible questions to answer. They then selected one question to solve and began their projects by sorting cards into Eulerian paths, circuits, or neither. Ms. Langer scaffolded their work by asking questions and giving prompts to push their thinking. Students talked with each other or a student teacher to discuss their ideas and clarify their explanations. Tools such as different colors for tracing, or a plastic sleeve that allowed them to trace the path on the graph with a dry erase marker, helped students work through challenges.

After the Video
The next day, students focused their attention on explaining their project so that other people could understand it. They created a poster to help articulate their question and solution and create clear visuals. They then created a video presentation to formally teach an audience about their question and answer.

Teacher Prep
Before the lesson, Ms. Langer wrote up the questions that students could choose from, prepared sort cards of various types of graphs, and determined the criteria by which to evaluate students.

Prior Knowledge
To participate in this lesson, students needed prior knowledge about graph theory. They needed to understand Eulerian circuits and paths as well as have had practice with problem solving in groups and supporting each other’s learning.

Differentiated Instruction
Ms. Langer provides various tools and supports to reach students with different learning styles and abilities. For example, depending on how a student processes information, students might find it helpful to use multiple colors on graphs to keep different solutions separate. In addition, plastic sheet protectors allow students to trace (and erase) on a graph to help visualize the path. Some students benefit from working with bigger graphs. She also provides a problem-solving rubric with sentence starters for the poster and criteria for success. Students who finished their project early could solve a second question. Ms. Langer had students working in groups; however, one student preferred to work alone. A student teacher checked in with him to make sure that he still was able to discuss ideas with someone.

Group Interaction
Students worked in small groups to choose and solve problems and create a formal video presentation. Ms. Langer utilizes group work throughout the year and generally does not answer questions; she encourages students to talk to each other and does not answer a question unless the whole class is stuck. She tries to design curriculum with challenging problems so that a student will not automatically know the answer and take control of the group.

Because this project took place over two days, students could spend time first working and talking together to really understand the question and the answer. They then made the poster in groups to make sure that they had all the information for the explanation and examples clearly described before making the video.

• Sort cards with various graphs of Eulerian circuits, paths, or neither
• Sheet protector and dry erase markers
• Graph Theory Open Questions handout

Formative Assessment
During the lesson, Ms. Langer listened for students sharing ideas and making progress toward the solution. She listened for who was participating and how, whether they were accurate, and what they were saying if they were not accurate. Before starting their poster, Ms. Langer had each group explain their answer to her and made sure that all group members participated in the explanation.

Student Self-Assessment
At the beginning of the lesson, students were given criteria for success for math work and for the video presentation by which to do their self-assessment.

Summative Assessment
Ms. Langer assessed the poster and final video for a grade.

Impact of Assessment
If a significant number of students get stuck on one aspect of the lesson, Ms. Langer incorporates a discussion about it into the beginning of the next class so that she can address the problem before moving on.

Ongoing Assessment
Ms. Langer keeps a spreadsheet record of which problems each student has completed. The problems are linked to various standards that will be covered within the unit. This allows her to track where each student is with regard to her unit goals. In addition, she often uses exit tickets to help assess which students may need more help.