THE TRANSITION LINE // FIELD NOTES // STOP 23
Triple ratios, signed sums, and exponent rules
Field Book
Lessons 85 to 88
Method
Pencil and paper. Calculators never.
Wins, losses, and the whole season ride one multiplier. Triple ratios put part, part, and total on three lines with one shared multiplier, signed numbers add by dots and moves on the line, the order of operations gets its full formal statement, and the exponent rules compress repeated bases.
What you have to be able to do
- Solve triple-ratio problems
- Add signed numbers on the number line
- State and apply the full order of operations
- Apply the product, quotient, and power rules for exponents
The procedures
Triple ratiosField Book Lesson 85
- Stack part, part, and total on three labeled lines.
- One multiplier, found from the known pair, feeds all three lines.
Signed addition on the lineField Book Lesson 86
- Place a dot over the first number.
- Move by the second number, and dot where you stop: that is the sum.
Order of operations, formalField Book Lesson 87
- Grouping symbols, then exponents and roots, then multiply-divide left to right, then add-subtract left to right.
- Inside each rank, sweep left to right.
Exponent rulesField Book Lesson 88
- Same base multiplied: add the exponents. Same base divided: subtract them.
- A power of a power multiplies the exponents.
Worked: rules only
- x5 · x3 = x8
- k9 ÷ k4 = k5, and (m2)4 = m8.
Worked: opposite directions
- -7 + 12: the signs differ, so subtract the values: 12 - 7 = 5.
- The larger value is positive, so the answer is 5.
Terms on the record
| Term | Meaning |
|---|
| power of a power | an exponential expression raised again, multiplying the exponents |
| signed number | a number carrying a positive or negative sign |