Algebra I Notes Unit One: Real Number System

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1 Syllabus Objectives: 1.1 The student will organize statistical data through the use of matrices (with and without technology). 1.2 The student will perform addition, subtraction, and scalar multiplication on matrices. Teacher Note: A review of operations with real numbers should be incorporated throughout the first unit. Matrix: a rectangular arrangement of numbers in horizontal rows and vertical columns Entry (or element): each number in a matrix Order of a Matrix: number of rows number of columns = by Square Matrix: a matrix with the same number of rows as columns Ex: Ex: a. Find the order of the matrix. R R C1 C2 C This matrix has 2 rows and 3 columns. Therefore, its order is 2 3. (Read two by three.) b. Find the entry in the first row, second column in the matrix above The entry is 1. Matrices are used to organize data. Ex: Write a matrix to organize the following information regarding school lunch. A high school offers 5 entrees, 3 desserts, and 5 drinks. A middle school offers 3 entrees, 2 desserts, and 4 drinks. Ent Des Dri HS MS Question: What is another way to write this matrix? Adding and Subtracting Matrices: in order to add or subtract two matrices, they must have the same order. To add or subtract matrices, add or subtract the corresponding entries. Page 1 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

2 Ex: Add the matrices , so we can find the sum. Note: Both matrices have the same order ( ) Note: Corresponding entries are the same color = Ex: Find the difference of the matrices. Note: Both matrices has the same order ( ) , so we can find the difference. It is helpful to rewrite the subtraction problem as addition (add the opposite) Now add the corresponding entries: = Scalar: a real number Multiplying by a Scalar: multiply every entry in the matrix by the scalar Ex: Find the product Multiply every entry by the scalar: = Page 2 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

3 Using Matrices on the Graphing Calculator Ex: Find the sum and difference of the matrices on the graphing calculator , Use the MATRIX screen, then the EDIT menu to enter the matrices. Note that you must enter the order of the matrix (4 3. Keystrokes for Matrix A are below. Then enter the entries. ) To enter the second matrix, edit matrix B. Operations with the matrices can be calculated on the home screen. Below are the keystrokes for finding the sum: To view the rest of the matrix, use the right arrow to scroll over. Below are the keystrokes for finding the difference: You Try: Simplify by performing the operations QOD: Why must the orders of two matrices be the same to find the sum or difference? Page 3 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

4 Syllabus Objective: 1.4 The student will determine the probability of an event with and without replacement using sample spaces. Probability of an Event: a number between 0 and 1, inclusive, that is a measure of the likelihood that the event will occur Note: An impossible event has a probability of 0, and a certain event has a probability of 1. Outcomes: the possible results of an experiment Event: a collection of outcomes Favorable Outcomes: the outcomes of an event that you wish to occur Calculating Theoretical Probability: P = # of Favorable Outcomes Total # of Outcomes Ex: What is the probability that you will roll an even number in one roll of a fair number cube? There are 3 even numbers on a number cube (2,4,and 6). These are the favorable outcomes. There are 6 numbers that are equally likely to roll on a number cube (1,2,3,4,5 and 6). These are the total outcomes. Therefore, the theoretical probability that you will roll an even number in one roll is Ex: Two coins are tossed. What is the probability that both are tails? List all of the possible outcomes when tossing two coins: HH, HT, TH, TT One of the outcomes is two tails. This is the favorable outcome. There are 4 possible outcomes that are equally likely. 3 1 P = = 6 2 Therefore, the theoretical probability that you will toss both tails is P = 1 4 Experimental Probability: the probability based on repetitions of an actual experiment Ex: Toss three coins 10 times and record the number of heads for each of the 10 tosses. Combine your results with the rest of the class. Find the experimental probability of getting three heads when three coins are tossed. Graphing Calculator: You can simulate a coin toss on the graphing calculator. Use the following keystrokes to begin the Probability Simulator: Page 4 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

5 After collecting the data, compare the experimental probability with the theoretical probability. List all of the possible outcomes: HHH, HHT, HTT, HTH, THH, THT, TTH, TTT One of the outcomes is three heads. This is the favorable outcome. Therefore, the theoretical probability of obtaining three heads is P = 1 8 Probability of Two Events: calculate the product of the theoretical probabilities Ex: Find the probability of obtaining a head and rolling a 5 when tossing a coin and rolling a die. Probability of tossing a head: 1 2 Probability of rolling a 5: 1 6 Probability of tossing a head and rolling a 5: 1 1 = Note: You may also write out all of the possible outcomes. Probability With and Without Replacement Ex: A drawer has 6 blue socks, 4 red socks, and 10 white socks. Find the probability of randomly picking a blue sock and then a red sock out of the drawer if you replace the first sock after it is picked. Probability of picking a blue sock: 6 3 = Probability of picking a red sock: = 20 5 Probability of picking a blue sock and then a red sock (with replacement): 3 1 = Ex: A drawer has 6 blue socks, 4 red socks, and 10 white socks. Find the probability of randomly picking a blue sock and then a red sock out of the drawer if you do not replace the first sock after it is picked. Probability of picking a blue sock: 6 3 = Probability of picking a red sock (if the first sock, which is blue, is NOT replaced): 4 19 Probability of picking a blue sock and then a red sock (without replacement): 3 4 = 3 2 = Page 5 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

6 You Try: Find the probability of randomly choosing an M from the letters in the word MATHEMATICS. QOD: The probability of one event is 2 3. The probability of a second event is 3. Which event is more 8 likely to occur? Page 6 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

7 Syllabus Objective: 1.3 The student will collect, organize, display, and analyze data using graphical representations including box and whisker plots. Measures of Central Tendency: a number used to represent a typical value in a set of data. There are three commonly used measures of central tendency: Mean (average): the sum of the data values divided by the number of data values Median: the middle number when the data values are written in order Note: If there are two numbers in the middle, the average of these two numbers is the median. Mode: the data value(s) that occurs most often Note: A set of data may have more than one mode or no mode. Ex: Find the mean, median, and mode of the set of data. 45,1,52, 42,10, 40,50, 40,7,52 Mean: = 339 = Median: Put values in order 1,7,10,40,40,42,45,50,52,52 40,4There are two numbers the middle 1,7,10, 40,, 45,50,52,52 the average of the two middle numbers. 2Find = 82 = Mode: There are two values that occur the most, 40 and 52. Stem-and-Leaf Plot: an arrangement of digits used to display numerical data in order Ex: Create a stem-and-leaf plot for the data in the example above. Use the digits in the tens place for the stems and the digits in the ones place for the leaves. Note: We will include all of the tens places from 0 to Key: 52 represents the number 52. Box-and-Whisker Plot: a data display that divides a set of data into four parts Median (Second Quartile): the middle number of a set of data that is written in order First Quartile: the median of the lower half of the data Third Quartile: the median of the upper half of the data Page 7 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

8 Range: the difference between the largest and smallest data values Interquartile Range (IQR): the difference between the Third and First Quartiles Ex: Use the set of data 10, 20, 5, 34, 9, 25, 28, 15, 16, 12, 13, 7 to draw a box-and-whisker plot. Then find the range and IQR. Step One: Put the data values in order. 5, 7, 9, 11, 12, 13, 15, 16, 21, 25, 28, 34 Step Two: Find the three quartiles Second Quartile (median): Q2 = = = First Quartile: 5,7,9,11,12,13, 15,16,21,25,28, Median of lower half: Q1 = = = Third Quartile: 5,7,9,11,12,13, 15,16,21,25,28, Median of upper half: Q3 = = = Step Three: Draw the box-and-whisker plot above a number line. Label the number line according to the data values. Note: The whiskers connect the box to the smallest and largest numbers in the data set. The number line above begins at 0 and ends at 35 with a scale of 5. Range: 34 5 = 29 IQR: = 13 Ex: Use the box-and-whisker plot above to answer the following questions: 1. What percent of the data is located within the box? Solution: Each piece of the box is 25% of the data, which means 50% of the data lie within the box. 2. What fraction of the data are less than 23? Solution: 23 is the third quartile, so 3 quarters of the data lie below it. Therefore, the fraction is What percent of the data are greater than 10? Solution: 10 is the first quartile, which means 25% (a quarter) of the data are less than 10. Therefore, 75% of the data are greater than 10. Page 8 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7

9 You Try: Use the data in the stem-and-leaf plot to create a box-and-whisker plot. Find the measures of central tendency, the range, and the interquartile range. You may use your calculator for computation Key: 16 4 = 164 QOD: Describe a set of data that would have no mode. Page 9 of 9 McDougal Littell: , 2.7, 2.8, 6.6, 6.7