Polarization and Partisanship
Exploring the causes and consequences of increasing political division in the U.S.
Key Questions
- Explain why political compromise has become more difficult in recent years.
- Analyze how geographic sorting contributes to political polarization.
- Evaluate the risks to a democracy when citizens view the opposing party as an enemy.
Common Core State Standards
About This Topic
Comparing quadratic and linear growth helps students understand how different types of functions increase over time. In 9th grade, students learn that while a linear function grows at a constant rate, a quadratic function grows at an increasing rate. This is a critical Common Core standard that teaches students to recognize that quadratic growth will eventually exceed any linear growth, no matter how steep the line starts.
Students learn to use 'first and second differences' in tables to distinguish between these models. This topic comes alive when students can engage in 'growth races' or collaborative investigations where they model real-world scenarios, like comparing a flat hourly wage to a commission-based structure. Structured discussions about the 'long-term' behavior of these functions help students develop a sense of mathematical scale.
Active Learning Ideas
Inquiry Circle: The Growth Race
Groups are given two 'investment' options: one that adds $100 every year (linear) and one that adds an amount equal to the square of the year (quadratic). They must create a table and graph for both and identify the 'crossover point' where the quadratic option becomes more profitable.
Think-Pair-Share: Difference Detectives
Give students two tables of values. One student calculates the 'first differences' (the change between y-values). The other checks if those differences are constant (linear) or if the 'second differences' are constant (quadratic). They then explain their findings to each other.
Formal Debate: Which Model Fits?
Present data for a real-world scenario, like the spread of a rumor or the area of a growing garden. Students must debate whether a linear or quadratic model is a better fit, using the 'rate of change' as their primary evidence.
Watch Out for These Misconceptions
Common MisconceptionStudents often think a steep linear function will always stay ahead of a 'slow' quadratic function.
What to Teach Instead
Use 'The Growth Race' activity. Peer discussion about the 'crossover point' helps students see that because the quadratic rate is always increasing, it is mathematically guaranteed to eventually pass any straight line.
Common MisconceptionConfusing quadratic growth with exponential growth.
What to Teach Instead
Use 'Difference Detectives.' Collaborative analysis shows that quadratic growth has a constant SECOND difference, while exponential growth has a constant RATIO. This distinction is key for choosing the right model.
Suggested Methodologies
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Frequently Asked Questions
What is a 'second difference'?
How can active learning help students understand growth rates?
Why does a quadratic function eventually beat a linear one?
In what real-world scenarios do we see quadratic growth?
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