STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I 4 th Nine Weeks, 2016-2017 1
OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents.. Each nine weeks Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A section has also been developed to provide students with the opportunity to solve similar problems and check their answers. The answers to the are found at the end of the document. problems The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall textbook series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. Algebra I Blueprint Summary Table Reporting Categories No. of Items SOL Expressions & Operations 12 A.1 A.2a c A.3 Equations & Inequalities 18 A.4a f A.5a d A.6a b Functions & Statistics 20 A.7a f A.8 A.9 A.10 A.11 Total Number of Operational Items 50 Field-Test Items* 10 Total Number of Items 60 * These field-test items will not be used to compute the students scores on the test. It is the Mathematics Instructors desire that students and parents will use this document as a tool toward the students success on the end-of-year assessment. 2
3
Rational Expressions A.2 The student will perform operations on polynomials, including b) dividing polynomials When dividing a polynomial by a monomial, first determine if there are any common factors that can be divided out. Example 1: Simplify The GCF is. If you remove this from both the numerator and the denominator you can divide it out, because Sometimes, you can simplify things further if you break the problem up into smaller pieces. This works when all of the terms do not share the same GCF. As an example, instead of writing, you could write, then simplify. Example 2: Simplify Simplify each fraction separately. When dividing a polynomial by another polynomial, first factor anything that can be factored, then simplify by dividing out common factors. Example 3: Simplify Factor the numerator! Find the greatest common factor in each row and each column. These will give you your two binomials! So, we are looking for factors of 12 that add up to! Put the now factored trinomial back into the original problem and divide out factors that are in both the numerator and denominator. 4
Rational Expressions 1. 2. 2 4 14a b 18a b ab 6x 2 2 y 6x y 2 2 6x y 3 3 3 4. 5. 2 4ab c 2 16ab 12a b 3 2x 2 13x 15 2x 3 8c Scan this QR code to go to a video tutorial on dividing polynomials. 3. 15y 3 6y 12 3y 6. 2x 2 11x 21 4x 28 Scatterplots A.11 The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions. A scatterplot is a graph made of ordered pairs relating two sets of data. Scatterplots can show trends in data. Positive Correlation Negative Correlation No Correlation As the x-value increases, the y-value also increases. As the x-value increases, the y-value decreases. The x and y-values do not appear to have any relation. When a scatterplot has a positive or negative correlation, a trend line can be drawn to help predict other values on the line. The trend line will be the line that best fits the given data. Trend lines are shown here for a positive and negative correlation. 5
You could estimate this trend line s equation, called a line of fit, by selecting two points that lie on or very near the trend line, finding a slope and calculating a y-intercept. Example 1: The scatterplot below compares hours watching television and GPA. Estimate the equation of the line of fit. Then, use this equation to determine the GPA that you could expect if you watched 17 hours of TV each week. First select two points that are either on the trend line or very near it, such as: (2.8, 15) and (3.2, 13) Use these points to find the slope of the line. You can then use the slope and one point to solve for b, and write the line of fit. To answer the second part of the question, we will use the equation that we just found. We are given the hours watching TV and are asked to find a GPA. TV hours represent the y-value, so we will plug 17 in for y and then solve for x. This will give us our estimated GPA. So, to answer the question, an estimated trend line would be, and if you watched 17 hours of TV a week, your estimated GPA would be 2.4. Scan this QR code to go to a video tutorial on scatterplots and line of fit. Scan this QR code to go to a video tutorial on line of best fit. 6
There is also a way to calculate a line of best fit using your calculator. Example 2: The cost of a gallon of gas for the past 6 years is given. Write an equation for the line of best fit, and then use this equation to predict gas prices in 2017. Year 2006 2007 2008 2009 2010 2011 Average cost for one gallon 2.59 3.16 3.29 3.25 3.51 3.56 Start by entering this data into the list in your calculator. 2006 can be year 1, 2007 can be year 2, etc. STAT ENTER Then hit the stat button again, scroll over to calc, and select number 4 (LinReg) Press enter twice and your results window will show. The line of best fit is To answer the second part of the question, we first need to determine what number the year 2017 would be associated with. Since 2011 was year 6, 2017 would be year 12. To predict the gas price in 2017, we will plug 12 in for x in our line of best fit. Sometimes a curve (parabola) will fit the data better than a straight line will. You could quickly determine which is more appropriate by typing your data points into your calculator and graphing the points. If the points look like the picture below, a curve may be a better fit for the data. To find the curve of best fit, you follow the same procedure as above, but you select number 5 (QuadReg) instead of 4 (LinReg). Example 3: What is the equation of the curve of best fit for the data below? 7
Time Height (sec) (feet) 0 5 1 12 2 21 3 19 4 14 The question told us that we will find a CURVE of fit, so we know we should attempt QuadReg. Start by entering this data into the list in your calculator. Time will be your L1 (x) values and Height will be L2 (y). STAT ENTER Then, hit the stat button again, scroll over to calc, and select number 5 (QuadReg). Press enter twice and your results window will show: The curve of best fit is This graph shows the 5 data points and the curve of best fit. Scan this QR code to go to a video tutorial on curve of best fit. Scatterplots 1. Using the data in the table, find theline of best fit. Then, use this information to predict how long the average 9-month-old rattlesnake is. Age 1 3 5 7 10 12 15 24 (months) Length (inches) 8 16 25 36 49 50 64 71 2. Use the data in the table to find the quadratic curve of best fit. Then, use this information to determine at what time the ball will reach its maximum height. Time (sec) 0 1 2 3 4 5 Height (ft) 4 8 17 21 20 14 Box and Whisker Plots 8
A.10 The student will compare and contrast multiple univariate data sets, using box-andwhisker plots. A box and whisker plot is a way to summarize a set of data by using five key values (the median, upper and lower quartiles, and upper and lower extremes). Box and whisker plots make it easy for us to see how data is distributed within a set. The data will be separated into quartiles, and one quarter of the data points are included in each quartile. Example 1: The box and whisker plot below shows how many questions students answered correctly on their last Geography quiz. What is the inner quartile range? If 48 students took the quiz, how many answered 6 or more questions correctly? The inner quartile range can be calculated by subtracting the lower quartile from the upper quartile. The inner quartile range is 10. To answer the second part of the question we have to remember that a box and whisker plot separates the data into quartiles. Thus one quarter of the data points are in each quartile. If 48 students took the quiz, one quarter of them would be 12 students ( ). If we are only considering students that answered 6 or more correctly, we will look at all of the areas from 6 to 23. Therefore we would add 12+12+12 = 36 students. 36 students answered 6 or more questions correctly 9
To create a box and whisker plot: Arrange your data in ascending order Find the median Find the median of each half (quartiles) Check for outliers (LQ (1.5 x IQR) or UQ + (1.5 x IQR)) Example 2: Create a box and whisker plot to organize the weights of the Washington Redskins offensive lineman: The median is 310.5. The lower quartile is 302, and the upper quartile is 319.5. To check for outliers we first need to find the IQR. Then we will multiply this by 1.5. Upper outliers would have to be above UQ We don t have any data points higher than this, so no upper outliers. Lower outliers would have to be below LQ We don t have any data points below this, so no lower outliers You will put a point at the lower extreme (292), the upper extreme (329), the lower quartile (302), the upper quartile (319.5), and the median (310.5). Connect the quartiles to create a box, and extend whiskers to the extremes. Scan this QR code to go to a video tutorial on box and whisker plots. 10
You can also use your calculator to create a box and whisker plot. Example 3: Use your calculator to create a box and whisker plot for the data below, which shows the bowling scores of a team. Begin by entering the data into L1. Then, press 2 ND and then y = to bring up the STAT PLOT menu. Press ENTER to select plot 1. Make sure that On is highlighted, then highlight the box and whisker plot with outliers. Also make sure that the Xlist is L1, which is where our data is entered. Go to the ZOOM menu and select ZoomStat and press ENTER. A box and whisker plot will be graphed. If you press TRACE you will be able to determine each of the quartile values and any outliers. You can see two outliers on this graph, 205 and 290. Box and Whisker Plots Use the box and whisker plot showing students 1 st nine weeks science grades to answer questions 1 4. 1. What is the IQR for this data? 2. What value would constitute an upper outlier? Lower outlier? 3. What percentage of students scored below 66%? 4. If this plot was created with data from 600 students, how many students scored between 66 and 85? 5. Sketch a box and whisker plot for the data below. Be sure to check for outliers! 48 52 39 41 48 50 48 30 51 46 11
You may be asked to compare box-and-whisker plots or analyze the plots with the addition or removal of data. You will most likely be informed that each value included in the box-and-whisker plot is different, which is extremely important. This takes away the uncertainty of repeat values, which could keep a boundary in the same place even with addition or removal of data. Remember that each quartile represents 25% of the data. Determine how many values are contained in each quartile, make a note of each boundary, and visualize how the quartiles of data will shift based on the introduction or removal of values. Box and Whisker Plots 6. A salesman recorded the amount of cars he sold each day for seven days. He sold a different number of cars each day. The box-and-whisker plot summarizes his data. The salesman only sold 6 cars on the eighth day. He redraws the box-andwhisker plot to include his data. Which statements comparing the new box-andwhisker plot to the original box-and-whisker plot are true? (Select all that apply.) A. The median decreased. B. The lower extreme decreased. C. The upper extreme decreased. D. The value of the interquartile range decreased. 7. The number of math problems given during each lesson by two different teachers is summarized in these box-and-whisker plots. Teacher A Teacher B Teacher A gave a different amount of problems during each of the 9 lessons. Teacher B gave a different amount of problems during each of the 8 lessons. What is the total amount of lessons that had at least 10 problems? 12
Statistics A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores. Standard Deviation and Variance The standard deviation of a data set tells us how spread out the data is. If the data is very spread out, the standard deviation will be higher than if the data is all clumped together. The variance is another measure of how spread out the data is. Standard deviation is represented by σ (lowercase Greek letter sigma). The variance is just the standard deviation squared, σ². The formula for both of these measures is on the Algebra I formula sheet, but there is a way to calculate these values in the graphing calculator. Example 1: The height in inches of the Washington Wizards starting lineup is shown below. Find the standard deviation and the variance of the data, round your answer to the nearest hundredth. 75, 80, 76, 79, 81 Start by entering the data into L1 in your STAT menu. Then go to STAT, scroll over to CALC, and select 1: 1-Var Stats When you press ENTER twice, your calculator will display the single variable statistics. KEY MEAN SUM of the DATA SUM Squared Sample Standard Deviation Population Standard Deviation Sample Size We want to use the standard deviation that is represented by σ. Therefore, our standard deviation is 2.32.The variance is just the standard deviation squared which = (2.32)² = 5.38. Standard Deviation = 2.32 inches Variance = 5.38 inches 13
Example 2: Using the data from Example 1, how many of the starting lineups heights are within one standard deviation of the mean? The heights were 75, 80, 76, 79, 81 This question is referring to players who are both one standard deviation above the mean, and one standard deviation below the mean. According to the calculator stats, the mean was 78.2 inches and the standard deviation was 2.32 inches. 78.2 + 2.32 = 80.52 inches 78.2 2.32 = 75.88 inches There is one player (81 ) who is taller than one standard deviation above the mean and one player (75 ) who is shorter than one standard deviation below the mean. This means that 3 players (80, 76, and 79 ) are all within one standard deviation of the mean. Example 3: How short would a player have to be to be 2.5 standard deviations below the mean? First we need to calculate how many inches is 2.5 standard deviations. We can do this by multiplying the standard deviation by 2.5. We can then subtract 5.8 from the mean of 78.2. A player would have to be 72.4 tall to be 2.5 standard deviations below the mean. Scan this QR code to go to a video tutorial on standard deviation. Statistics Use the speeds of the top 10 fastest roller coasters, to answer the questions below. 128, 120, 107, 100, 100, 95, 93, 85, 85, 82 1. What is the standard deviation? (round to nearest hundredth) 2. What is the variance? (round to nearest hundredth) 3. How many coasters are within 1.25 standard deviations of the mean? 4. How fast would a coaster have to be going to be 3 standard deviations above the mean? 14
Z-Scores A z-score tells us how many standard deviations a specific data point is from the mean. Z-scores can be positive or negative. If a z-score is positive, it indicates that the data point is that many standard deviations above the mean. If a z-score is negative, it indicates that the data point is that many standard deviations below the mean. If a data point has a z-score of zero, then that data point is the same as the mean of the data. (It did not deviate from the mean.) The formula for calculating a z-score is on the Algebra I formula sheet. (The variables are defined for you on the formula sheet.) Example 4: A class history midterm grades are shown below. What is the z-score for a score of 78? Grades: 81, 62, 90, 77, 82, 86, 98, 100, 90, 75, 83, 88, 79, 76, 85 First calculate the 1-Var Stats in your calculator. We need the Mean and Standard Deviation. We want to know the z-score for a 78. This shows that a score of 78 is a little more than ½ standard deviation below the mean. Scan this QR code to go to a video tutorial on z-scores. 15
Example 5: Jessie s teacher wouldn t tell her the actual score that she received; only that she had a z-score of 1.26. Determine Jessie s score. (Refer to the data above.) Now, we are given the z-score and asked to find the data point. Plug everything that you know into the formula, and then solve the equation for the missing piece. We know the z-score, mean and standard deviation. Now we just need to solve for X! Jessie s score was a 95% Example 6: Another class took the exam and had the same class average but a standard deviation of 3.60. If Jessie had been in this class (and still had a z-score of 1.26), would her score be lower or higher? Explain. We could calculate the answer to this problem, but this isn t necessary. The class average is exactly the same, but the standard deviation is lower. Jessie still scored 1.26 standard deviations above the mean, but now those standard deviations are smaller, therefore her grade will be lower. We know the z-score, mean and standard deviation. Now we just need to solve for X! Jessie s score was a 88% 16
Mean Absolute Deviation The mean absolute deviation of a set of data is the average distance between each data point and the mean. To find the mean absolute deviation: First, find the mean Next, find the distance between each data point and the mean Finally, find the average (mean) of these distances. Example 6: The horsepower of the top 6 sports cars is shown below. Calculate the mean absolute deviation for the data. 691, 662, 651, 638, 631, 621 First, find the mean of the data. The mean of this data is 649 horsepower. Next, find the distance between each data point and the mean. (Remember, distances are always positive!). 691 649 = 42 662-649 = 13 651 649 = 2 649 638 = 11 649 631 = 18 649 621 = 28 Finally, find the average (mean) of these distances. The mean absolute deviation of the data is 19. The average distance between each data point and the mean is 19 horsepower. Scan this QR code to go to a video tutorial on mean absolute deviation. Statistics The heights of the tallest 7 men ever recorded are shown below (in inches). Use these to answer the questions. 107, 105, 103.5, 99, 99, 99, 98 5. Calculate the mean absolute deviation. 6. What is the z-score for 99 inches? 7. What is the z-score for 107 inches? 8. The tallest woman ever confirmed would have had a z-score of -1.13. How tall was she? 17
Answers to the problems: Rational Expressions 1. 2. 3. 4. 5. 6. Box and Whisker Plots 1. 19 2. Upper outliers above 113.5 Lower outliers below 37.5 3. 25% 4. 300 students 5. Scatterplots 1. 9 month old snake 2. Ball reaches maximum height at 6. A & B 7. 11 lessons Statistics 1. 14.42 mph 2. 207.94 3. 8 4. 142.76 mph 5. 3.14 inches 6. -0.75 7. 1.65 8. 97.74 inches 18