Mini-lessons

Goal: Practice explaining the "why" of mathematics.

Common student questions

These are examples of questions that have come up in various classroom over the years. Why does …

1. \( \sqrt{a+b} \neq \sqrt{a} + \sqrt{b} \)
2. \(x^0 = 1\) for \( x \neq 0 \). What does \( 0^0 \) equal?
3. \( (-1)(-1) = 1 \)
4. \( \frac{1}{x} \div \frac{1}{y} = \frac{1}{x} \cdot \frac{y}{1} \)
5. \(x + x^2 \neq x^3 \)
6. \((a + b)^2 \neq a^2 + b^2 \)
7. \(\dfrac{a + b}{a} \neq 1 + b\), when \( a \neq 0 \)
8. \((x)(x) \neq 2x \)
9. \(\dfrac{ 4(x+y) }{ 4x+y } \neq 1 \)
10. \(\log(a \cdot b) = \log(a) + \log(b) \), but \( \log(a + b) \neq \log(a) \cdot \log(b) \)
11. \(x^n \cdot x^m = x^{n+m} \) and \( \frac{x^n}{x^m} = x^{n-m} \)?
12. \(\frac{1}{0} \neq \frac{0}{1} \neq \frac{0}{0} \)
13. \(\frac{1+x}{2} \neq 1 + x/2 \) (notable because of calculator input)
14. \( \cot( \frac{\pi}{2} ) = 0 \) when \( \cot(x) = \frac{1}{\tan(x)} \)and \( \tan(\frac{\pi}{2}) = \text{DNE} \)
15. \( .\bar 9 = 1 \)
16. the quadratic equation work?
17. SSA not give congruent (or similar) triangles?
18. the Pythagorean theorem only work for right triangles?
19. an inequality change signs with multiplying or dividing by -1?
    (e.g., \( -x \ge 1 \implies x \le -1 \) )
20. When does a square root need a \( \pm \)? When does it not?
21. Is 1 a prime number? Why or why not?
22. \( \frac{a}{b} \div \frac{c}{d} = \frac{a \div c}{b \div d}\)

Possible resources

• common errors in writing mathematics
• Math mistakes (various)
• Khan academy is rich resource for teachers (Pershan)
• patterns that do not continue

Submission

Prepare a mini-lesson to teach the class about your chosen classroom question. Mini-lessons should be 10 minutes or less and should include more than one explanation (remember that students believe something incorrect so need extra convincing to believe what is correct).

Feel free to use handouts, activities, slideshows, manipulatives, technology, etc. Videos are not recommended because you aren't teaching during the video time.

For planning, there are 9 students in class + 1 instructor.

Scoring guide

Practice assignments are designed for formative feedback. Expect a lot of notes. I recommend using these notes to make edits to your practice assignment so you start your career with an improved version.

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