Most of the actual investigation and experimentation will be done during class time. For homework, you will be asked to document the results of your experiments in the form of a lab report. Make sure that you keep a notebook for your classroom experiments so you can still reference them when you go home to write up your report!
Guidelines for writing
Lab reports consist of at least three sections, outlined below. In the Introduction, the textbook describes one possible layout for your write-ups. You may follow their advice, or alternatively, mine here:
Section 1: Begin by writing an introduction explaining the topic of the module. Assume that your audience is somebody who is familiar with mathematical language, but not necessarily with this particular topic. Be sure to include motivation for studying this topic, either from the text or from your own thoughts. Write careful definitions for any of the fundamental concepts used in the module (again, remember your audience). Finally, preview some of the questions you will be addressing in the upcoming sections.
Section 2: In this section, describe the technical results of your investigations. Results come in three types: examples and counterexamples, conjectures supported by experimental evidence, and theorems supported by formal proof. Provide context for each of your results, giving thought to the order in which you give them and the transitions between them. Try to state each result as a mathematical statement using language which is as precise as possible. Sometimes, you have more results than you have space to write up, and in that case choose the most important ones. Be sure to include at least one formal proof in this section.
Section 3: In the last section, summarize the state of your progress on this line of research from a very high level. Always include material for further research—it is impossible to answer everything in one go! What were the questions you were unable to answer, and why are they important to you? Did you make any further conjectures that you are currently unable to support with solid evidence or proof?
Your labs will be assessed on the following four criteria:
Exposition: The quality and effectiveness of your writing, not including proofs. Includes introduction and motivation, presentation of results, style, structure, and clarity throughout.
Proofs: The quality and clarity of your technical mathematical writing, that is, the statements of results and definitions and the proofs of results.
Depth: The strength and completeness of your experimental and theoretical results, scientifically speaking.
Synthesis: How well you conveyed understanding of the results and goals of the lab as a whole. This can be done in many ways, including: summarizing results, motivating and asking questions, making conjectures, and drawing connections between the results, evidence, and conjectures.