THE COMPOUND // FIELD NOTES // WEEK 17
Fraction ladders, conjugates, and special triangles
Field Book
Lessons 64 to 67
Due
Test 15 due · Supplement 17
Method
Paper first, every step
The Midterm Sidequest posts to Google Classroom this week. Layered fractions unwind one rung at a time: add, divide, repeat.
What you have to be able to do
- Simplify layered complex fractions
- Rationalize two-term denominators with conjugates
- Use the 30 60 90 side ratios
The procedures
Layered complex fractionsField Book Lesson 64
- Work from the bottom rung: add, then divide, then add, then divide.
- Never skip a rung. Each pass is one clean fraction.
Advanced substitutionField Book Lesson 65
- Four equations, four variables: substitute in waves.
- Reduce to three, then two, then one, and climb back up.
Signs of fractionsField Book Lesson 66
- A negative can sit on the top, the bottom, or out front: all three name the same number.
- In a 30 60 90 triangle the sides run short, short times √3, and double the short.
ConjugatesField Book Lesson 67
- To rationalize a two-term denominator, multiply top and bottom by its conjugate.
- The product of conjugates kills the radical middle term.
Worked: 30 60 90
- The short leg of a 30 60 90 triangle is 5.
- Hypotenuse is double the short leg: 10.
- Long leg is 5√3: check it against the Pythagorean theorem.
Terms on the record
| Term | Meaning |
|---|
| conjugate | the same two terms with the middle sign flipped |
| 30 60 90 triangle | the special right triangle with sides in ratio 1, √3, 2 |
| complex fraction ladder | a stack of fractions simplified by alternating add and divide |