Graphs of Sine and Cosine Functions
Analyzing the characteristics (amplitude, period, phase shift, vertical shift) of sinusoidal graphs.
Key Questions
- Explain how changes in amplitude and period affect the visual representation of sine and cosine waves.
- Differentiate between phase shift and vertical shift in their impact on the graph.
- Construct the equation of a sinusoidal function given its graph or key characteristics.
Common Core State Standards
About This Topic
This topic explores the 'Second Founding' of America through the 14th Amendment and the doctrine of Selective Incorporation. Students learn how the Supreme Court used the Due Process Clause to gradually apply the Bill of Rights to state governments, case by case. They also examine the Equal Protection Clause as the constitutional engine for modern civil rights movements.
For 12th graders, this is the most important legal concept for understanding why state laws must respect federal rights. It connects the 1791 Bill of Rights to the 1868 14th Amendment and the 20th-century rights revolution. This topic comes alive when students can physically model the patterns of legal expansion by 'incorporating' different amendments through a historical timeline activity.
Active Learning Ideas
Inquiry Circle: Incorporation Timeline
In small groups, students are assigned a specific amendment (e.g., 2nd, 4th, 8th). They must find the landmark Supreme Court case that 'incorporated' it and explain how the law changed in their state after that ruling.
Formal Debate: Total vs. Selective Incorporation
Students debate whether the Supreme Court should have 'totally' incorporated the Bill of Rights all at once in 1868, or if the 'selective' (case-by-case) approach was a better way to respect federalism.
Gallery Walk: The 14th Amendment's 'Children'
Display various modern rights (e.g., marriage equality, desegregation, birthright citizenship). Students rotate to identify how each 'child' was born from either the 'Due Process' or 'Equal Protection' clause of the 14th Amendment.
Watch Out for These Misconceptions
Common MisconceptionThe Bill of Rights has always applied to the states.
What to Teach Instead
Originally, it only applied to the *federal* government (Barron v. Baltimore). Peer-led 'Before and After' comparisons of state power help students realize that for most of US history, states could legally restrict speech or religion.
Common MisconceptionThe 14th Amendment only applies to formerly enslaved people.
What to Teach Instead
While that was the original intent, the wording 'any person' has allowed the Court to protect immigrants, corporations, and all citizens. Peer discussion about the 'universal' language of the amendment helps clarify its broad reach.
Suggested Methodologies
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Frequently Asked Questions
What is 'Selective Incorporation'?
Which rights are still NOT incorporated?
How can active learning help students understand the 14th Amendment?
What is the 'Equal Protection Clause'?
Planning templates for Mathematics
5E Model
The 5E Model structures lessons through five phases (Engage, Explore, Explain, Elaborate, and Evaluate), guiding students from curiosity to deep understanding through inquiry-based learning.
unit plannerMath Unit
Plan a multi-week math unit with conceptual coherence: from building number sense and procedural fluency to applying skills in context and developing mathematical reasoning across a connected sequence of lessons.
rubricMath Rubric
Build a math rubric that assesses problem-solving, mathematical reasoning, and communication alongside procedural accuracy, giving students feedback on how they think, not just whether they got the right answer.
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