Mathematics: Reasoning about Large Systems

Objectives

• D05.S1.12.01 Apply the ideas of limits and continuity informally to describe change, approximation, and the behavior of quantities.
• D05.S1.12.02 Reason quantitatively about large systems such as populations, energy use, or carbon, using orders of magnitude and estimation, and critique the numbers used in public claims.
• D05.S2.12.01 Model systems of relationships with equations or matrices and interpret the meaning of a solution in a real context.
• D05.S2.12.02 Describe an algorithm as a step-by-step procedure for a real problem and analyze whether it works correctly and fairly.
• D05.S3.12.01 Use coordinates and vectors to analyze geometric situations and prove results.
• D05.S3.12.02 Apply geometric reasoning to real-world design, such as architecture, mapping, or the layout of shared spaces, considering access and sustainability.
• D05.S4.12.01 Use inferential reasoning, including margin of error, confidence, and significance, to evaluate claims made from data.
• D05.S4.12.02 Detect and critique misleading statistics, data visualizations, and algorithmic bias in media and public life, and defend a conclusion with honest data.

Essential questions

• How do limits and continuity describe change and approximation, and how do I reason quantitatively about large systems and critique the numbers in public claims?
• How do I model systems of relationships and analyze algorithms for correctness and fairness?
• How do I use inferential reasoning — and detect misleading statistics and algorithmic bias — to reach and defend conclusions with honest data?

Big ideas

• Limits and continuity describe change, approximation, and the behavior of quantities, and quantitative reasoning about large systems uses orders of magnitude and estimation to test public claims.
• Systems of relationships can be modeled with equations or matrices, and algorithms can be analyzed — for whether they work correctly and fairly.
• Inferential reasoning — margin of error, confidence, significance — plus a critical eye for misleading statistics and algorithmic bias, lets me defend conclusions with honest data.

Lessons

1. 1 The Idea of Getting Closer 55 minutes
2. 2 Approximation and the Shape of Change 55 minutes
3. 3 Orders of Magnitude: Thinking in Powers of Ten 55 minutes
4. 4 Critiquing the Numbers in Public Claims 55 minutes
5. 5 Systems of Relationships as Equations 55 minutes
6. 6 Matrices as Many Equations at Once 55 minutes
7. 7 Algorithms: Step-by-Step Procedures 55 minutes
8. 8 Are Algorithms Fair? 55 minutes
9. 9 Coordinates and Vectors: Position and Direction 55 minutes
10. 10 Proving with Vectors 55 minutes
11. 11 Geometry in Real Design: Architecture and Mapping 55 minutes
12. 12 Designing Shared Spaces: Access and Sustainability 55 minutes
13. 13 Inference from Samples: Margin of Error and Confidence 55 minutes
14. 14 Detecting Misleading Statistics and Defending with Honest Data 55 minutes

Assessment plan

Limits-and-continuity work; a quantitative large-systems critique of a public claim; an equations-and-matrices system model with a real-context solution; an algorithm correctness-and-fairness analysis; coordinate-vector proofs; a real-world geometric design with access and sustainability notes; an inferential-reasoning evaluation; and a misleading-data and algorithmic-bias critique with an honest-data defense, gathered as capstone portfolio artifacts.