Mathematical Toolkit Chapter 2 Pg. 14-41

2.1 The Measures of Science This should be review The metric system – used by everyone but the United States. Also called SI or Systeme de Internationle d’Unites or international systems of units and measurements Length – meter Mass – gram Time - second Temperature – kelvin Amount of substance – mole Electric current – ampere (amp) Luminous intensity - candela

Derived Units (prefixes) Kilo – 1000 base units Centi - .01 base units or 1/100th of a base unit Milli - .001 or 1/1000th of a base unit micro - .000001 or 1/1,000,000 of a base unit Nano - .000000001 or 1/1,000,000,000 of a base unit Full list is on page 18 of textbook

Scientific Notation Shorthand way to express a large number with lots of 0’s 1,000 is 103 1,000,000 is 106 1/1,000 is 10-3 Can also be used to represent numbers that are not 10 base units. 3,000 is 3 x 103 3,000,000 is 3 x 106 .003 is 3 x 10-3

Metric Conversions Does everyone remember the “stairstep”? 1000g x 1kg/1000g = 1kg 500cm x 1m/100cm = 5m Do Practice Problems 1-5 on pages 20-21 in the textbook ( 5-6 minutes) GO!!

Adding, Subtracting with Scientific Notation To Add or subtract: Convert to same exponential value then perform addition or subtraction Ex: 4 x 108 + 3 x 108 = 7 x 108 4.02 x 106 + 1.89 x 102 40,200 x 102 + 1.89 x 102 (40,200 + 1.89) = 40,201.89 x 102 1.89 is a small value so 4.02 x 106 once we move decimal points.

Multiplying and Dividing with Scientific Notation To multiple and divide: no need to convert just multiple or divide M number then add/subtract exponents to solve Ex: (4 x 103) x (5 x 1011) (4 x 5)= 20 103 + 1011 = 1014 So 20 x 1014 or 2.0 x 1015 Do Practice Problems p. 22-23 (8abcd, 9ab, 10ab)

Assignment: Section 2.1 Review Pg. 23 Questions 1-4 Use complete thoughts and grammatically correct sentences. Be worth around 20pts Due tomorrow.

2.2 Measurement Uncertainties Comparing Results Must have overlap when the standard deviation is factored in. 19.0 +/- 0.2cm and 18.8 +/- 0.3cm Overlap covers 18.8cm to 19.1cm These two measurements agree.

Accuracy and Precision Precision – describes the exactness of a measurement. Accuracy – describes how well the results agree with the standard value. Target Example:

Techniques for Good Measurement Be sure you are using the correct instrument to measure with. Use that instrument correctly Meter stick should be laid flat so that your eye is directly above the stick. Reading at an angle will cause inaccuracies in measurement due to parallax – the apparent shift in the position of an object when it is viewed at different angles. Thumb Parallax

Significant Digits…..Yay!!! Significant digits – the valid digits in a measurement. Usually limited by the instrument you use. Ex: A meter stick or ruler may only have marks showing 0.1cm so measurements can have only two digits past your decimal. The known and then the final estimated digit.

Rules for Significant Digits All non-zero digits are significant. Ex: 1.45, 55, 453.9 All final zeroes after the decimal are significant Ex: 1.20, 1.000 Zeroes between other significant digits are also significant. Ex: 405, 10,101 Zeroes used as a placeholder are not significant Ex: 0.001, 0.0005

Arithmetic with Sig. Digits Your answer can NEVER be more precise than your least precise measurement. When adding and subtracting you simply round your answer to the value with the lowest number of significant digits. Ex: 24.355 + 12.45 + 10.9 = 47.71 but using the significant digits the answer we need is 47.7 because 10.9 only has one significant digit after the decimal and 47.71 rounds down to 47.7

Arithmetic with Sig. Digits When multiplying or dividing: do the math then round again to the number with the lowest number of significant digits. Ex: 3.22 x 2.1 = 6.762, should round to 6.8 Using a Calculator: Can often display many meaningless digits. Be sure to pay attention to significant digits. I don’t want an answer given to 12 decimal places, just because you are anal retentive and tend to be too precise. I will wear out a red pen on that answer.

Section 2.2 Review Pg. 29 Questions 2.1-2.4 Due tomorrow

2.3 Visualizing Data Objectives: Be able to read and interpret graphs showing linear and nonlinear relationships between variables Use graph trends to predict outcomes Recognize quadratic and inverse relationships

Graphing Data Graphs are the easiest way to analyze data. When creating a graph, one must be sure to plot the pertinent information: independent vs dependent variables. Look at Table 2.3 on pg. 30 of the textbook: What type of experiment is shown in that table?

Graphing Data Strategy for making a Line graph ID the variables: independent is plotted on horizontal (X-axis) and dependent on the vertical (Y-axis). Establish a range for each measurement and establish your origin (0,0) Create axis labels that give an even spacing of numbers within your range. Plot the data to the best of your ability. Draw a “best fit” line or a curve through as many data points as possible Decide on a title for graph that states what is shown on it.

Linear Relationships All you need is SLOPE Linear relationships are plotted using slope intercept formula: y=mx+b Y = y coordinate M = slope B = y intercept X = x coordinate Slope is calculated using Rise/Run or change in Y divided by change in X

Linear Relationships You need any 3 points and you can solve for the remaining. To solve for B, you must assume X = 0 and you will need slope as well. Example: Two sets of data points (2,2) and (4, 6) Calculate slope, find the y-intercept, and derive the full equation for the line. 5 minutes……GO!!!

Solutions: (2,2) and (4,6) Slope = rise/ run or (6-2)/(4-2) = 4/2 or 2 Y intercept = y = 2x + b 2 = 2(2) + b 2 = 4 + b b = -2 Equation for line: y = 2x – 2 Also on line: (0, -2) (1, 0) (3, 4)……

Nonlinear Relationships Represented by parabolic (parabola) shaped line. One variable depends on the square of the other variable. Quadratic: y = ax2 + bx + c If you have a graphing calculator then use it for these, its okay to be lazy.

Nonlinear Relationships Some graphs can be represent an inverse relationship. This means when one variable is doubled the other variable goes down by a factor of 2 or ½ of original value. Y = a/x or xy = a Both formulas show the proper relationship. Graph shape called a hyperbola Practice Problem 21: pg. 36 in text….Solve it

Section 2.3 Review Pg. 36 Questions 1-4 Due tomorrow.

Chapter 2 Review Pg. 37-39 1-17 odds, 18-29 odds Due tomorrow.