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Transformations and Congruence · Weeks 10-18

Proving Triangle Congruence

Students will apply SSS, SAS, ASA, AAS, and HL congruence postulates to write formal proofs.

Key Questions

  1. Construct a two-column proof to demonstrate triangle congruence using a specific criterion.
  2. Differentiate between the conditions required for each triangle congruence postulate.
  3. Evaluate which congruence criterion is most appropriate for a given set of information.

Common Core State Standards

CCSS.Math.Content.HSG.CO.B.8
Grade: 10th Grade
Subject: Mathematics
Unit: Transformations and Congruence
Period: Weeks 10-18

About This Topic

Imperialism in China during the 19th century was characterized not by direct colonial rule, but by 'Spheres of Influence' and 'Unequal Treaties.' This topic covers the Opium Wars, where Britain used military force to protect its illegal drug trade, and the subsequent carving up of China by various powers. Students also examine internal responses, such as the devastating Taiping Rebellion and the anti-foreign Boxer Rebellion.

For 10th graders, this topic explains the 'Century of Humiliation' that still informs Chinese foreign policy today. It highlights the role of economic interests in driving international conflict and the impact of the US 'Open Door Policy.' This topic particularly benefits from hands-on, student-centered approaches like a mock negotiation of an 'Unequal Treaty,' where students see the power imbalance firsthand.

Active Learning Ideas

Watch Out for These Misconceptions

Common MisconceptionChina was colonized by the British just like India was.

What to Teach Instead

China remained technically independent but lost control over its economy and key ports (Spheres of Influence). A map activity showing 'treaty ports' versus 'colonies' helps students understand this distinction.

Common MisconceptionThe Opium Wars were about the British defending 'free trade.'

What to Teach Instead

While the British used that rhetoric, they were specifically forcing the sale of an addictive drug that was illegal in China and Britain. Peer analysis of Lin Zexu's letter to Queen Victoria helps students see the moral argument.

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Frequently Asked Questions

What were the Opium Wars?
Two wars in the mid-19th century between China and Western powers (primarily Britain). They were sparked by the Chinese government's attempt to stop the British from smuggling opium into the country, which was causing widespread addiction and economic drain.
What are 'Spheres of Influence'?
These were areas in China where a foreign nation (like France, Germany, or Japan) had exclusive trading rights and economic privileges, even though the territory still officially belonged to the Chinese government.
What was the Taiping Rebellion?
A massive civil war in China (1850-1864) led by Hong Xiuquan, who claimed to be the brother of Jesus. It was a response to economic hardship and the perceived weakness of the Qing Dynasty, resulting in an estimated 20-30 million deaths.
How can active learning help students understand imperialism in China?
By role-playing the negotiation of an 'Unequal Treaty,' students experience the frustration of losing sovereignty. This active approach helps them understand why these events are still remembered as a 'Century of Humiliation' in China, making the historical grievances more tangible and less abstract.

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