THE COMPOUND // FIELD NOTES // WEEK 08
Dividing fractions, opposite motion, and clean reasoning
Field Book
Lessons 28 to 31
Due
Test 6 due ยท Supplement 8
Method
Paper first, every step
Escape 2 posts this week. Lesson 30 is the reasoning lesson: the Examiner grades arrows, not vibes.
What you have to be able to do
- Divide rational expressions by invert and multiply
- Set up opposite-direction motion with a distance total
- Test conclusions with arrow diagrams and use perpendicular slopes
The procedures
Divide fractionsField Book Lesson 28
- Invert the divisor and multiply.
- Factor and cancel before multiplying anything out.
Motion in opposite directionsField Book Lesson 29
- Two travelers apart: D1 + D2 equals the total distance.
- Write each distance as rate times time and solve.
Deductive reasoningField Book Lesson 30
- Draw the arrow from statement to conclusion and ask if it only runs one way.
- Euclid's vertical angle argument is the model: every step earns its arrow.
Perpendicular slopesField Book Lesson 31
- Perpendicular lines have slopes that are negative reciprocals.
- A remote interior pair of a triangle sums to the exterior angle.
Worked: closing the gap
- Two riders 300 miles apart head toward each other at 40 and 60 mph.
- 40t + 60t = 300.
- 100t = 300, so t = 3 hours.
Terms on the record
| Term | Meaning |
|---|
| reciprocal | the flip of a fraction, so their product is 1 |
| deductive reasoning | drawing a conclusion that must follow from the given statements |
| negative reciprocal | the opposite of the flip, the slope relation of perpendicular lines |