Math Skills for the Laboratory

1. Math Skills for the Laboratory

3. 1000 or 103 100 or 102 10 or 101 1 or 100 0.1 or 10-1 0.01 or 10-2 0.001 or 10-3 The order of prefixes in the metric system, for every power of ten from 3 to -3, is Kilometer, Hectometer, Decameter, Meter, Decimeter, Centimeter, Millimeter You can remember it by thinking of… King Hector Distributed Money… Dollars & Cents to Millie Changing the Size of Units

4. Length, Width, and Height • Length is the (longest) distance from side to side between an object’s ends. • Width is the distance from side to side, across the object from side to side at right angles to the length. • Height is the vertical (top to bottom) distance between an object’s ends.

5. Mass • The amount of matter or stuff in something • Not to be confused with weight! • Which is the pull of the earth’s gravity on an object. • Weight can change but the amount of matter in an object stays the same.

6. A triple-beam (pan) balance or an electronic balance • is used to measure mass. • The unit we use is grams (g.). Mass

7. height length width Volume • The amount of space an object occupies. Volume of a cube or rectangular prism. • Measure the length, width, and height of the object. • Then, multiply the three measurements. • V= l x w x h • The units are cm3.

8. Volume of an Irregularly Shaped Object • A graduated cylinder is used to find the volume of an irregularly shaped object. • The unit we use is milliliters (mL).

9. How do I find the volume of an irregularly shaped object?Use the water displacement method. 1. Fill a graduated cylinder with water & record the amount. 2. Put the object in the water & record the new amount. 3. Find the difference. volume of water & object – volume of water ----------------------------------- volume of object

10. Temperature • Temperature is a measure of how fast the atoms and molecules of a substance are moving. • The faster the movement, the hotter the object’s temperature. • The slower the movement, the cooler the object’s temperature. • Temperature is measured, using a thermometer, in degrees on the Fahrenheit, Celsius, and Kelvin scales. • In science we use Celsius (°C). (Fahrenheit is sometimes used in meteorology…the study of weather.)

11. Time • Seconds (s) are used to measure the duration of an event (how long it took). • We use a stopwatch to measure time.

12. Derived measurements are made up of more than one measurement • Ex. • Gradient = • Density = Change in field value ft Distance m Mass g volume mL Derived measurements have “compound” units…more than one type of unit.

13. Density • The relationship between the mass and volume of an object. In other words, how tightly packed the matter is inside the object. • Example - a classroom with 35 students has a higher density than one with 5 students…. • Changing the size (volume) of an object does not change its density!!!!!!!!!!!!!!!!!!!! • It is still made up of the same type of matter!

14. Density and Water An object with a density less than 1.0 g/mL will float in water. The density of water is 1.0 g/mL. An object with a density greater than 1.0 g/mL will sink in water.

15. How do I find the density of an object? use these units • Density = mass grams -------------------------------------------------- volume mL or cm3 DON’T FORGET: Round to the nearest tenth.

16. Density Problems Find the density of an object whose mass is 50.0 g and volume is 25.0 mL. D = 50.0 g 25.0 mL D = 2.0 g/mL

17. The density triangle You can use the density triangle to figure out any of the variables if you already know two of them. M ÷ D V X

18. Use the density triangle • Mass = 20.0 g • Density = 0.5 g/mL • Volume = ? 40.0 mL • Volume = 10.0 cm3 • Density = 0.7 g/cm3 • Mass = ? 7.0 g

19. Percent Deviation (Error) • A way of comparing a measurement with the most commonly accepted value for that measurement. (How far off your answer is from the “correct” answer.” or “How accurate you answer is.”) % error = your result - accepted value x 100 %                           accepted value

20. Accuracy vs. Precision Precise = repeated measurements are close to each other Accurate = close to the accepted value (“bulls-eye”) Your goal is to be both accurate and precise.

21. How many cm is this pencil? With a more precise scale, how many?

22. 10  10 = 100 Scientific Notation A short-hand way of writing very large or very small numbers without writing all of the zeros. • Coefficient – any number from greater than zero and less than ten • Exponent – shows the number of times we move the decimal point (represents the # of times 10’s are multiplied together. Ex. 102 = )

23. The Distance From the Sun to the Earth 93,000,000 miles

24. Changing from Standard Notation to Scientific Notation Step 1: for a number = to or greater than 1 • Move decimal left • Leave only one number in front of decimal 93,000,000 = 9.3000000

25. Step 2 • Write number without zeros 93,000,000 = 9.3

26. 7 93,000,000 = 9.3 x 10 Step 3 • Count how many places you moved decimal • Make that your power of ten • Since you moved the decimal point to the left, your exponent will be positive.

27. 7 93,000,000 = 9.3 x 10 The power of ten is positive 7 because the decimal moved 7 places to the left. 93,000,000 ---Standard Form 9.3 x 107 ---Scientific Notation

28. Example • Given: 289,800,000 • Use: 2.898 (moved 8 places to the left) • Answer: 2.898 x 108

29. 9.85 x 107 -----> 6.41 x 1010 -----> 2.79 x 108 -----> 4.2 x 106 -----> Practice Problems Write in scientific notation. Decide the power of ten. • 98,500,000 = 9.85 x 10? • 64,100,000,000 = 6.41 x 10? • 279,000,000 = 2.79 x 10? • 4,200,000 = 4.2 x 10?

30. More Practice Problems For these, decide where the decimal will be moved. • 734,000,000 = ______ x 108 • 870,000,000,000 = ______x 1011 • 90,000,000,000 = _____ x 1010 Answers: 3) 9 x 1010 • 7.34 x 108 2)8.7 x 1011

31. Complete Practice Problems Write in scientific notation. • 50,000 • 7,200,000 • 802,000,000,000 Answers 1) 5 x 104 2) 7.2 x 106 3) 8.02 x 1011

32. The length of an E. coli bacterium 0.0000021 m

33. Changing from Standard Notation to Scientific Notation Step 1: for a number less than 1 but greater than zero*Move decimal to the right *Leave only one number in front of decimal 0.0000021 = 0000002.1

34. Step 2 • Write number without zeros 0000002.1 = 2.1

35. -6 0.0000021 = 2.1 x 10 Step 3 • Count how many places you moved decimal • Make that your power of ten • Since you moved the decimal point to the right, your exponent will be negative.

36. -6 0.0000021 = 2.1 x 10 The power of ten is negative 6 because the decimal moved 6 places to the right.

37. Example • Given: 0.000567 • Use: 5.67 (moved 4 places) • Answer: 5.67 x 10-4 Since the original number was less than 1, the exponent is negative.

38. Changing Numbers in Scientific Notation to Standard Notation • If the exponent is (+) move the decimal to the right the same number of places as the exponent. • 1.65  101 = • 1.65  103 = • If the exponent is (-) move the decimal to the left the same number of places as the exponent. • 4.6  10-2 = • 1.23  10-3 = 16.5 1650 0.046 0.00123

39. Scientific Notation to Standard Form Move the decimal to the right • 3.4 x 105 in scientific notation • 3.40000 --- move the decimal • right 5 places Move the decimal to the left -> • 3.4 x 10-5 in scientific notation • 340,000 in standard form • 00003.4 --- move the decimal • left 5 places <- • 0.00003.4 in standard form

40. 6.27 x 106 9.01 x 104 6.27 x 10-6 9.01 x 10-4 6,270,000 90,100 0.0000067 0.000901 Write in Standard Form Move the decimal to the right for + exponents or to the left for - exponents.