Model 1: SI Prefix Multipliers
| SI Prefixes | ||||
| Prefix | Symbol | Multiplier | ||
| exa | E | 1,000,000,000,000,000,000 | 1018 | |
| peta | P | 1,000,000,000,000,000 | 1015 | |
| tera | T | 1,000,000,000,000 | 1012 | |
| giga | G | 1,000,000,000 | 109 | |
| mega | M | 1,000,000 | 106 | |
| kilo | k | 1000 | 103 | |
| 1 | 100 | |||
| deci | d | 0.1 | 10–1 | |
| centi | c | 0.01 | 10–2 | |
| milli | m | 0.001 | 10–3 | |
| micro | µ | 0.000001 | 10–6 | |
| nano | n | 0.000000001 | 10–9 | |
| pico | p | 0.000000000001 | 10–12 | |
| femto | f | 0.000000000000001 | 10–15 | |
| atto | a | 0.000000000000000001 | 10–18 | |
1 kilometer = 1000 meters = 103 meters
1 millimeter = 0.001 meters = 10–3 meters
Critical Thinking Questions:
- How many meters are in one kilometer?
- How many meters are in one millimeter?
- How many millimeters are in one meter?
- How many millimeters are in one kilometer?
- What is the exponential factor (10x) that describes the difference between a kilometer and a meter?
- What is the exponential factor that describes the difference between a meter and a millimeter?
- What is the exponential factor that describes the difference between a kilometer and a millimeter?
- How is your answer for problem #5 related to your answer to problem #1?
- Using your calculator, enter your answer to problem #5 and multiply that by your answer to problem #6. How is this answer related to your answer to Problem #7?
- Using grammatically correct English sentences, explain how to convert from one prefix multiplier to another.
Exercises:
- Use the best prefix multiplier to express each measurement without any exponents: (i.e. 5.6 km)
- 5.6 x 10–6 g
- 2.1 x 10–3 L
- 5.0 x 104 m
- 4.9 x 10–5 s
- Convert the following measurements to scientific notation without any prefix multipliers: (i.e. 5.6 x 103 m)
- 1.1 ns
- 2.7 µm
- 105 kg
- 25 mL
Model 2: Derived Units
A derived unit is formed from a combination of other units.
| Derived Unit | Source | Unit |
| Area | length (m) x width (m) | m2 |
| Volume | length (m) x width (m) x height (m) | m3 |
| Velocity | distance (m) / time (s) | m/s |
| Density | mass (g) / volume (mL) | g/mL |
Critical Thinking Questions:
- How many one centimeter squares are contained in a larger square that is 10 cm long and 10 cm wide?
- What is the area of a square that is 10 cm long and 10 cm wide? (make sure to include the proper units)
- What is the area of a square that is 0.1 m long and 0.1 m wide?
- How are the two squares in problems 14 & 15 related? How are the areas of the two squares related?
- By what exponential factor do the two areas in Problem #16 differ? How does that compare to the difference between the exponential factors of a centimeter and a meter?
- How many one centimeter cubes are contained in a larger cube that is 10 cm long, 10 cm wide, and 10 cm high?
- What is the volume of a cube that is 10 cm long, 10 cm wide, and 10 cm high? (make sure to include the proper units)
- What is the volume of a cube that is 0.1 m long, 0.1 m wide, and 0.1 m high?
- How are the two cubes in problems 19 & 20 related? How are the volumes of the two cube related?
- By what exponential factor do the two volumes in Problem #21 differ? How does that compare to the difference between the exponential factors of a centimeter and a meter?
- Using grammatically correct English sentences, explain how to convert from one prefix multiplier to another in a derived unit.
- If you had 1.00 mL of water, and that volume of water had a mass of 1.00 gram, calculate the density of water.
- If you had 1000 mL of water, and that volume of water had a mass of 1000 gram, calculate the density of water.
- How is the density of water related to the overall quantity of water?
- Write a mathematical (i.e. algebraic) expression to describe density.
- Algebraically rearrange your answer to problem #28, so that the mass term is all by itself on one side of the equation.
- Algebraically rearrange your answer to problem #28, so that volume is all by itself on one side of the equation.
Exercises:
- What would be the mass of 738 mL of water?
- Acetone (nail polish remover) has a density of 0.7857 g / mL.
- Calculate the mass of 39.84 mL of acetone
- Calculate the volume occupied by 469 g of acetone.
Model 3: Dimensional Analysis
Earlier we learned about metric prefixes, and learned how to convert between different metric prefixes. We will now expand upon that to convert between different units. Most problems follow the following scheme:
- Start with your measurement
- Insert conversion factors so that units to be cancelled are on opposite sides
- Cancel units
- Multiply across the top, divide by the bottom
- Check that the units of your answer are appropriate
Critical Thinking Questions.
- How many meters are in a kilometer?
- Starting with a distance of 96 kilometers, use your conversion factor from Problem #32 to convert this distance to meters.
- Using grammatically correct English sentences, explain how to insert the conversion factor into your calculations so that the units cancelled?
Exercises.
- Convert 1.76 yards to centimeters. (1.094 yard = 1 m)
- Convert the density of water from problem 25 to units of kg / L.
- Take a look back at problem 4, and use the dimensional analysis technique to convert 1 kilometer to millimeters. Verify that you get the same answer.
- Take a look back at your answers to Problem 17 and 22, and use the dimensional analysis technique to convert the Area from cm2 to m2, and the Volume from cm3 to m3.
Challenge Problems:
- A Cessna 172 takes on 145 L of aviation gasoline (avgas). Avgas has a density of 0.821 g / mL. Calculate the mass of fuel (in kg) onboard the Cessna.
- Find the density of a metal cylinder with a mass of 8.3 g and a length of 1.94 cm, and a radius of 0.55 cm.
- The world record in the 100 m dash is 9.58 s. Calculate the velocity of this runner in miles per hour. (1 mile = 1609.3 m)
- The Toyota Prius, a hybrid electric vehicle, has an EPA gas mileage rating of 57 miles per gallon in the city. How many kilometers can the Prius travel on 15 L of gasoline?
