1. Mathematical Tools UNIT 1: Measurements

2. Scientific Measurement Two types of measurement: 1. Qualitative – uses words to describe Ex: long, cold, heavy 2. Quantitative - uses numbers to describe Ex: 4 meters long, 31 ºC, 15 kilograms Scientists prefer…. QUANTITATIVE measurements  Easy to check, no personal bias  We like numbers!!!

3. How good are the measurements? Scientists use two words to describe how good the measurements are: 1. Accuracy – how close the measurement is to actual value 2. Precision – how well the measurement can be repeated

4. Accuracy & Precision…the differences Accuracy Can be true of a single measurement OR of multiple trials Precision MUST have multiple trials before anything can be said about it

5. Practice The following students measure the density of a piece of lead 3 times. The density of lead is actually 11.34 g/cm 3. Considering all the results, which person’s results were accurate? Which were precise? Were any both accurate and precise? a. Rachel: 11.32 g/cm 3, 11.35 g/cm 3, 11.33 g/cm 3 b. Daniel: 11.43 g/cm 3, 11.44 g/cm 3, 11.42 g/cm 3 c. Leah: 11.55 g/cm 3, 11.34 g/cm 3, 11.04 g/cm 3

6. Practice The following students measure the density of a piece of lead 3 times. The density of lead is actually 11.34 g/cm 3. Considering all the results, which person’s results were accurate? Which were precise? Were any both accurate and precise? a. Rachel: 11.32 g/cm 3, 11.35 g/cm 3, 11.33 g/cm 3 b. Daniel: 11.43 g/cm 3, 11.44 g/cm 3, 11.42 g/cm 3 c. Leah: 11.55 g/cm 3, 11.34 g/cm 3, 11.04 g/cm 3 A, P P neither

7. Recording Measurements A measurement describes the limitations of the measuring instrument. A measurement cannot be more exact than the instrument that was used to make the measurement. For instance, you cannot record a measurement in nanometers with a meter stick that is only calibrated in millimeters.

8. Recording Measurements one digit beyond Measuring with precision – Always estimate one digit beyond what you can read for sure from the graduations Example: 13.50 cm

9. SI Units Système Internationale d’Unités

10. Derived Units – a combination of SI units Velocity…… m/s Acceleration……m/s 2 Energy…… kg·m 2 /s 2 (Joule) Force…… kg·m/s 2 (Newton)

11. Memorize the Prefixes! PrefixSymbolConversion Factor Picop 1 pico = 1 x 10 -12 base unit Nanon1 nano = 1 x 10 -9 base unit Microμ1 micro = 1 x 10 -6 base unit Millim1 milli = 1 x 10 -3 base unit Centic1 centi = 1 x 10 -2 base unit Decid1 deci = 1 x 10 -1 base unit Kilok1 kilo = 1 x 10 3 base unit MegaM1 Mega = 1 x 10 6 base unit GigaG1 Giga = 1 x 10 9 base unit BASE Unit!

12. Conversion Factors to have…. (we will use these all year!) Length: 2.54 cm = 1 in 5280 ft = 1 mi 1.6 km = 1 mi 1600 m = 1 mi 3 ft = 1 yd Time: 1 hr = 3600 s Mass / Weight: 2.2 lb = 1 kg Volume: 1 liter = 1000 ml = 1000 cm 3 3.785 liters (l) = 1 gal

13. Scientific Notation Looks like this: 6.02 x 10 23 It is a more convenient way to work with numbers that are very large or very small Move the decimal point until only one non-zero digit remains on the left. Count the number of places you moved the decimal point and use that number as the exponent of 10. If you move the decimal place to the right, the exponent is negative; if you move the decimal place to the left, the exponent is positive. Your calculator can convert numbers to and from scientific mode. Make sure that you know how.

14. Practice Express the following quantities in scientific notation Convert these numbers to scientific notation: 1. 43,456 2. 0.0035 3. –702 4. –0.0778 5. 1000 Convert these numbers to standard form: 1. 8.87 x 10 -4 2. 3.1 x 10 2 3. 2.06 x 10 -6 4. 1 x 10 5

15. ANSWERS ANSWERS Express the following quantities in scientific notation Answers in scientific notation: 1. 4.3456 x 10 4 2. 3.5 x 10 -3 3. –7.02 x 10 2 4. –7.78 x 10 -2 5. 1 x 10 3 Answers in standard form: 1. 0.000887 2. 310 3. 0.00000206 4. 100000