Notes
Units and conversions
- A quantity is a number with a unit. Lengths use units like m, area uses $\text{m}^2$, and volume uses $\text{m}^3$.
- Use dimensional analysis: multiply by conversion factors written as fractions so that unwanted units cancel.
- Example: $\displaystyle 5 \text{ m/s} = 5 \cdot \frac{3600 \text{ s}}{1 \text{ h}} \cdot \frac{1 \text{ km}}{1000 \text{ m}} = 18 \text{ km/h}$.
Expressions
- Follow the order of operations: parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right).
- Properties (commutative, associative, distributive, identity, inverse) justify rewriting expressions.
- Combine like terms only. Distribute a negative sign to every term in the parentheses.
Exponents
- $x^a \cdot x^b = x^{a+b}$, $\displaystyle \frac{x^a}{x^b} = x^{a-b}$, $(x^a)^b = x^{ab}$, $x^0 = 1$, $\displaystyle x^{-n} = \frac{1}{x^n}$.
- Multiply coefficients separately: $(3x^2)(2x^4) = 6x^6$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$400 \div 80 = 5$ m/s, and $5 \cdot 3.6 = 18$ km/h.
Parentheses: $4$. Exponent: $16$. Multiply: $32$. Add: $35$.
Only the grouping changes, not the order.
$4x + 8 - 3x + 3 = x + 11$. Subtracting $-1$ times $-3$ gives $+3$.
Multiply coefficients: $3 \cdot 2 = 6$. Add exponents: $2 + 4 = 6$.
$2 \cdot 1440 = 2880$ mL, and $2880 \div 1000 = 2.88$ L.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A delivery driver averages 45 miles per hour. Her van uses 1 gallon of fuel for every 15 miles, and fuel costs $4 per gallon.
(a) How many gallons of fuel does the van use per hour? Show the units.
Show the answer and scoring guide
$\displaystyle \frac{45 \text{ miles}}{1 \text{ h}} \cdot \frac{1 \text{ gal}}{15 \text{ miles}} = 3$ gallons per hour.
Check off the points you earned:
(b) What is the fuel cost per hour of driving?
Show the answer and scoring guide
$\displaystyle \frac{3 \text{ gal}}{1 \text{ h}} \cdot \frac{$4}{1 \text{ gal}} = $12$ per hour.
Check off the points you earned:
(c) Write an expression for the fuel cost after $h$ hours, then find the cost of a 6.5-hour shift.
Show the answer and scoring guide
The cost is $12h$ dollars. For $h = 6.5$: $12(6.5) = 78$, so the cost is $78.
Check off the points you earned: