**How can a decimal be a ratio?**

- Sketch a tower that has a 2:3 height to width (h:w) ratio.
- Sketch one with a h:w ratio of 5/4.
- Sketch one with a h:w ratio of 0.4, 0.45, and 1.458.
- My tower has a h:w ratio of 1.00001. What can you tell me about my tower?

**Special (“Famous”) Right Triangles (**Chapter 10, Geometry Labs**)**

- Two of my right triangle’s sides are 5 cm and 5 cm. Use the Pythagorean Theorem to find the third side.
- Scale the “famous” half-square to find that third side more quickly.
- Two of my right triangle’s sides are 1 cm and 2 cm. Why can’t I be sure how long the third side is?

**Which is steeper? Part I**

**Meet Tangent**

- Play with the tangent button on your calculator a bunch. If you find anything cool or interesting, write it down.
- Tangent takes an angle and gives you the slope of its ramp. Make a smart guess: what’s tan(40)? tan(80)? tan(1)?
- Check out this table. What do you notice? What questions do you have?

4. Suppose we graphed this thing, slope/tangent as a function of degrees. What would that graph look like?

5. What are some possible sides for a 35 degree ramp?

**Which is Steeper? Part II**

Draw two ramps whose tangents are very close, but not quite, equal. Extra points for cleverness.

**Moving Between Degrees and Tangent**

- What’s the first slope that is unskiiable?

2. Try this Desmos activity about slanty hills.

**Solving Ratio Equations**

- Solve: x/5 = 1.3
- Solve: 5/x = 1.3
- Which of these is more difficult for you to solve?

**Scaling, Ratios, and Solving Triangles **(Geometry Labs, Chapter 11)

- Figure out as many things as you can about the triangle in this diagram.

2. This one too.

3. All of these as well.

4. Check out these 20 degree ramps.

5. How tall is the top ramp? How wide is the bottom?

6. Which way should you spin this to make the problem as nice as possible? What angle of incidence do you prefer? What’s your height, your width?

7. (Ala Geometry Labs, Chapter 11): Given one hypotenuse and one leg of a right triangle, what other parts can you find?

**Sine is a Trig Functions Also, I Guess**

- Sine!

**And then Cosine**

- Cosine!

~fin

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0: find the ratio for this building?

Love tangent first. That’s so obvious now that you’ve proposed it.

Favorite framing of trig is what you end with; one way to think of math is as the study of ‘what else do we know?’

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