Graphing Sine and Cosine: Amplitude and Period
Students will graph sine and cosine functions, identifying and applying transformations related to amplitude and period.
Key Questions
- Analyze how changes in amplitude affect the graph of a sine or cosine wave.
- Explain the relationship between the period of a trigonometric function and its frequency.
- Construct a sine or cosine function that models a given amplitude and period.
Common Core State Standards
About This Topic
Urbanization and New Immigration covers the dramatic growth of American cities and the shift in immigration patterns during the late 19th century. This topic examines the 'New Immigrants' from Southern and Eastern Europe, the rise of nativism, and the challenging living conditions in urban tenements. Students also analyze the role of political machines like Tammany Hall in providing services to immigrants in exchange for political power.
For 11th graders, this topic is essential for understanding the origins of modern urban life and the persistent debates over immigration and assimilation. It highlights the work of muckrakers like Jacob Riis in exposing urban poverty. Students grasp these social dynamics faster through collaborative investigations into tenement life and role-playing the interactions between immigrants and political 'bosses.'
Active Learning Ideas
Inquiry Circle: Tenement Life
Small groups analyze photos from Jacob Riis's 'How the Other Half Lives' and floor plans of 'dumbbell' tenements. They must identify the health and safety hazards and propose reforms that would eventually be passed in the Progressive Era.
Role Play: The Political Machine
Students assume the roles of a neighborhood 'ward boss' and a newly arrived immigrant. They negotiate a 'deal' where the boss provides a job or coal for the winter in exchange for the immigrant's vote, discussing the ethics of this system.
Think-Pair-Share: Nativism and the Chinese Exclusion Act
Students read excerpts from anti-immigrant pamphlets of the 1880s. They work in pairs to identify the economic and racial fears that led to the first major law restricting immigration based on nationality.
Watch Out for These Misconceptions
Common MisconceptionAll immigrants were welcomed with open arms at the Statue of Liberty.
What to Teach Instead
While the statue was a symbol of hope, many immigrants faced intense discrimination, medical inspections at Ellis Island, and outright exclusion (like the Chinese). A 'station rotation' on immigrant experiences helps students see the reality of the 'golden door.'
Common MisconceptionPolitical machines were purely evil and corrupt.
What to Teach Instead
While they were certainly corrupt, they also provided vital social services that the government did not yet offer, such as help with housing and jobs. Peer-led discussion on the 'pros and cons' of the machine helps students understand why they were so popular.
Suggested Methodologies
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Frequently Asked Questions
Who were the 'New Immigrants'?
What was a 'political machine'?
How did muckrakers influence urban reform?
How can active learning help students understand urbanization?
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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