Algebra Content Academy

Algebra Content Academy
Statistics SOL A.9, A.10, A.11 SOL AII.2, AII.9, AII.10, AII.11, AII.12 February 18, 2015 & March 4, 2015

Reporting Category: Functions and Statistics
Algebra 1 Algebra 2 Look for trends!

Curriculum framework What are the verbs?

Henrico Curriculum Guide

Algebra 1 Pacing SOL A.9 – 1st Nine Weeks SOL A.10, A.11

Algebra 2 Pacing SOL AII.2, AII.9, AII.10, AII.11
SOL AII.12 – 1st Nine Weeks

2014 Algebra 1 SPBQ data – Statistics

2014 Algebra 2 SPBQ data – Statistics

Algebra 1 Formulas What formulas do students need to know related to Statistics? What formulas are provided on the formula sheet related to Statistics?

Algebra 2 Formulas What formulas do students need to know related to Statistics? What formulas are provided on the formula sheet related to Statistics?

Standard deviation & Normal Curve
Statistics SOL A.9 SOL AII.11

Vertical Articulation

Performance Analysis Comparison - SOL 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 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 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. 2013 A.9 - The next standard being highlighted is SOL A.9. This standard reads: 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. In particular, students need practice calculating and interpreting standard deviation and z-scores. A.10 - The next standard being highlighted is SOL A.10. This standard reads: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. Students have difficulty with many aspects of this standard. A.11 - The next standard being highlighted is SOL A.11. In particular, students had inconsistent performance making predications using the curve of best fit. 2014 A.9 - The next standard highlighted is SOL A.9. This standard reads: 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. In particular, students need practice calculating and interpreting standard deviation. A.10 – The next standard highlighted is SOL A.10. This standard reads: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. A.11 - The next standard highlighted is SOL A.11. In particular, student performance was inconsistent when making predictions using the curve of best fit.

Performance Analysis Comparison – SOL AII.11
The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

Breakout Sessions Work problems from A.9 and AII.11 in small groups.
Make sure you are able to do the problems on the calculator and know the keystrokes.

Standard Deviation and z-score
Algebra 1 A.9 Standard Deviation and z-score

2012 - Suggested Practice for SOL A.9
Students need additional practice finding values within a given standard deviation of the mean. A teacher gave a quiz. The following stem-and-leaf plot shows the scores of the students in her class. TI-84 Need to Know Stat, Calc, 1-Var Stats Quiz Scores The mean score of this data set is 70 and the standard deviation (rounded to the nearest tenth) is 15.8. Which scores are within one standard deviation of the mean? STEM LEAF 4 5 0 0 6 0 0 0 7 8 9 10 Any student who scored a 60, 70 or 80 scored within one standard deviation of the mean. Key: 6|0 equals 60 Common Errors? Misconceptions?

Students need additional practice identifying an interval in which an element lies. A data set has a mean of 16.5 and a standard deviation of 3. The element x has a z-score of 1.5. In which interval does the element lie? 10.5 ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < 25.5 Common Errors? Misconceptions?

Students need additional practice finding an element of a data set given the mean, standard deviation, and z-score. A data set has a mean of 34 and a standard deviation of An element in the data set has a z-score of -1.2. Without doing a calculation, state whether this element is less than, equal to, or greater than 34. Determine the element of the data set. Less than 34 (the mean) because the z-score is negative. −1.2= 34−𝑥 4.5 The element is 28.6 Common Errors? Misconceptions?

Students need additional practice performing calculations with statistical information. A data set has a mean of 55 and a standard deviation of The z-score for a data point is What is the data point? A data set has a standard deviation of 3. The element 16 is an element of a data set, with a z-score of 2.4. What is the mean of the data set? 50.8 For SOL A.9 students need additional practice performing calculations with statistical information. The answers to the two examples are shown on the screen. 8.8 Common Errors? Misconceptions?

Students need additional practice performing calculations with statistical information. The number of minutes book club students read on Monday night is displayed by the graph. The mean number of minutes for this data set is 21.18, and the standard deviation of the data set is 6.5. The z-score for the data point representing the number of minutes Tim read is In which interval does this data point lie? Here is another example for SOL A.9. This example asks students to identify an interval in which a data point lies. Intervals can be presented graphically, as in this example, or algebraically. The answer to this question is shown on the screen. The interval 25 to 30 minutes. Common Errors? Misconceptions?

Students need additional practice solving problems involving standard deviation. A data set is shown. If the standard deviation of the data set is approximately 1.25, how many of these elements are within one standard deviation of the mean? For SOL A.9, students need additional practice performing calculations with statistical information. For the data set shown, the mean is In order to determine how many elements are within one standard deviation of the mean, subtract the standard deviation from the mean, and add the standard deviation to the mean to find the interval in which values within one standard deviation lie. For the example on the screen, any data elements that are greater than or equal to 2.95 and less than or equal to 5.45 are within one standard deviation of the mean. The answer is shown on the screen. Common Errors? Misconceptions?

Students need additional practice solving problems involving standard deviation. A data set has a mean of 45. An element of this data set has a value of 50 and a z-score of What is the standard deviation for this data set, rounded to the nearest hundredth? Use two of the three numbers shown in the list to complete this sentence. A data set could have a variance of and a standard deviation of 81 9 For SOL A.9, students need additional practice performing calculations with statistical information and understanding the relationship between variance and standard deviation. The answers to the two examples are shown on the screen. As an extension to example #2, ask students to give other numbers that would accurately complete the sentence. Students should realize that the standard deviation of a data set is the square root of the variance. Common Errors? Misconceptions?

Algebra 2 A.11 Normal Distribution

2012 - Suggested Practice for SOL AII.11
Students need additional practice using mean and standard deviation to find the area under a normal curve and apply properties of the normal distribution to solve problems. The running times for a group of 200 runners to complete a one mile run are normally distributed with a mean of 6.5 minutes and a standard deviation of 1.5 minutes. Approximately how many of the runners have a time greater than 8 minutes? 32 What percentage, rounded to the nearest tenth, of these runners can complete this run in less than minutes? 2.3% Casio Need to Know TI-84 Need to Know 2nd, Vars, normalcdf( Common Errors? Misconceptions?

Students need additional practice using properties of the normal distribution curve to find the probability of an event, the percent of data that falls within a specified interval, and the number of expected values that fall within a specified interval. A population of adult males had their heights measured. The heights were normally distributed. Approximately what percentage of the heights, rounded to the nearest whole number, are within one standard deviation of the mean? 34 % c. 68% b. 95% d % For this standard, students need additional practice using properties of the normal distribution curve to find the probability of an event, the percent of data that falls within a specified interval, and the number of expected values that fall within a specified interval. Students should be familiar with using the empirical rule to determine the approximate percentage of data that falls within one, two and three standard deviations of the mean. Students should also know that the total area under the curve is one. An example and its answer are shown on the screen. Common Errors? Misconceptions?

At a company, the data set containing the ages of applicants for a particular job was normally distributed. The mean age of the applicants was 30 years old, and the standard deviation of the data set was 3.5 years. Which is closest to the percent of applicants that were 21 years old or younger? a % b % c % 9.0 % The next few slides show examples of how this standard may be assessed. Here is another example that asks students to find the probability of an event by performing calculations, rather than using the empirical rule as in the previous example. The answer to this question is shown on the screen. Casio Need to Know TI-84 Need to Know 2nd, Vars, normalcdf( For additional assistance: See Technical Assistance Document for AII.11 Common Errors? Misconceptions?

A normally distributed data set of 500 values has a mean of 35 and a standard deviation of 7. Which is closest to the probability that a value in the data set will fall between 42 and 46? a b c. 10 d. 50 TI-84 Need to Know 2nd, Vars, normalcdf( This example asks students to find the probability that a value in the data set will fall between two given values. The answer is shown on the screen. For additional assistance: See Technical Assistance Document for AII.11 Common Errors? Misconceptions?

A normally distributed data set of 600 values has a mean of 18.5 and a standard deviation of 3.25. 1. What is the approximate number of values in the data set expected to be 22 or greater? 2. What is the approximate number of values in the data set expected to be 16 or fewer? 3. Which is closest to the expected number of values in the data set that lie between 21 and 27? a b c d. 467 Acceptable answers: 84 or 85 Acceptable answers: 132 or 133 These last examples for standard A2.11 ask for the approximate number of values in the data set that will be at or above, at or below or between two numbers in a data set. The answers to the questions are shown on the screen. Common Errors? Misconceptions?

Students need additional practice in recognizing the properties of a normal distribution. Which description of a normal distribution is most likely NOT true? Approximately 99.7% of the data will fall within three standard deviations of the mean. Approximately 95% of the data will fall within two standard deviations of the mean. Approximately 68% of the data will fall within one standard deviation of the mean. Approximately 34% of the data will fall within one standard deviation of the mean. An example highlighting the properties of a normal distribution is shown on the screen. For additional assistance with the content of SOL A2.11, or to find suggestions on types of questions to present to students, please refer to the Technical Assistance Document located on the Virginia Department of Education Web site by clicking on the link shown on the screen. This document gives specific instructions on how to solve problems that are similar to the one shown on this screen and the ones that follow. The answer to this question is shown on the screen. For additional assistance: See Technical Assistance Document for AII.11 Common Errors? Misconceptions?

Students need additional practice using properties of the normal distribution curve to find the probability of an event, the percent of data that falls within a specified interval, and the number of expected values that fall within a specified interval. a. A normally distributed data set has a mean of 0 and a standard deviation of What percent of the data would be expected to be between -1.5 and 1.5? b. The scores of a college history test were normally distributed with a mean of 75 and a standard deviation of 6. What is the probability of a student’s score being an 80 or lower? 95% For this standard, students need additional practice using properties of the normal distribution curve to find the probability of an event, the percent of data that falls within a specified interval, and the number of expected values that fall within a specified interval. Students would benefit from a discussion on various methods of solving these problems. Again, if additional assistance is needed on the instruction of this standard, please refer to the VDOE Technical Assistance document. Two examples and answers are shown on the screen. 80% Common Errors? Misconceptions?

Bayside Elementary School is visiting a local amusement park. One of the amusement park’s attractions requires that children must be at least 44 inches tall to ride. The heights of children at Bayside Elementary are normally distributed with a mean of 43 inches and a standard deviation of 3.4 inches. What is the probability rounded to the nearest tenth that a child selected at random does NOT meet the height requirement for the amusement park attraction? Approximately 61.6% Here is another example that asks students to find the probability of an event by performing calculations. In this example, students must find the probability that a child will NOT meet the height requirement. A common error is finding the probability for children that WILL meet the height requirement and not subtract this value from 1. The answer is shown on the screen. Common Errors? Misconceptions?

Statistics SOL A.11 SOL AII.9
Curve of Best Fit Statistics SOL A.11 SOL AII.9

Vertical Articulation

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. 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. 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. 2013 A.9 - The next standard being highlighted is SOL A.9. This standard reads: 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. In particular, students need practice calculating and interpreting standard deviation and z-scores. A.10 - The next standard being highlighted is SOL A.10. This standard reads: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. Students have difficulty with many aspects of this standard. A.11 - The next standard being highlighted is SOL A.11. In particular, students had inconsistent performance making predications using the curve of best fit. 2014 A.9 - The next standard highlighted is SOL A.9. This standard reads: 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. In particular, students need practice calculating and interpreting standard deviation. A.10 – The next standard highlighted is SOL A.10. This standard reads: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. A.11 - The next standard highlighted is SOL A.11. In particular, student performance was inconsistent when making predictions using the curve of best fit.

The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential and logarithmic functions The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions. Common Errors? Misconceptions?

Breakout Sessions Work problems from A.11 and AII.9 in small groups.
Make sure you are able to do the problems on the calculator and know the keystrokes.

Algebra 1 A.11 Curve of Best Fit

Students need additional practice making predictions using the linear or quadratic curve of best fit. Determine the quadratic curve of best fit for the data. Then estimate what the value of y will be when x = -4. TI-84 Need to Know Stat, Edit, Graph, Zoom, ZoonStat, Stat, Calc, QuadReg Common Errors? Misconceptions?

Students need additional practice making predictions using the linear or quadratic curve of best fit. This set of ordered pairs shows a relationship between x and y. What is the equation for the quadratic curve of best fit for this set of data? Predict the value of y when x = 8. For SOL A.11 students need additional practice making predictions using the linear or quadratic curve of best fit. Students performed better when they had to predict from a graph or find the equation of a line of best fit when the data points were presented on a graph. When data was presented in set form or in a table, students had more difficulty finding the curve of best fit and making a predication. The answers to the questions are shown on the screen. 200 Common Errors? Misconceptions?

This table shows the value, v, of an account at the end of m months. There was an initial deposit of $50 and no other deposits were made. If the value of the account continues to increase in the same way, predict the value of the account at the end of 13 months. Use the quadratic curve of best fit to make the prediction. m, time in months v, value in dollars 50 1 129 3 299 5 485 7 687 9 905 Here is another example that asks students to predict a value using the quadratic curve of best fit. Similar to the last example, students should determine the quadratic curve of best fit using the graphing calculator, and then use that curve to predict the value of the account at the end of thirteen months. The answer to the question is shown on the screen. $1,389.00 Common Errors? Misconceptions?

a. 437 acres b. 441 acres c. 447 acres d. 463 acres
Suggested Practice for SOL A.11 The data in the table shows the average United States farm size, in acres, for the years Average Farm Size Using the line of best fit for the data shown in the table, what is the best prediction of the average farm size in the year 2014? a acres b acres c acres d acres Year Average Acres Per Farm 2000 434 2001 437 2002 436 2003 441 2004 443 2005 444 2006 446 2007 449 Here is one last example for SOL A.11. The answer is shown on the screen. Common Errors? Misconceptions?

Students need additional practice determining the linear or quadratic curve of best fit. This set of ordered pairs shows a relationship between x and y. Which equation best represents this relationship? a. b. c. d. Extension: Using the curve of best fit, what is the value of y, rounded to the nearest whole number, when the value of x is 8? For SOL A.11, students need additional practice determining the linear or quadratic curve of best fit for a set of data. Students should be able to determine whether the curve of best fit is linear or quadratic, and then choose the equation which best models the data. The answer to the question is shown on the screen. As an extension, have students predict the value of y, rounded to the nearest whole number, when the value of x is 8. The answer is shown on the screen. Answers will vary depending on how the numbers in the curve of best fit are rounded. Using the equation in option d, y = 186. Common Errors? Misconceptions?

Students need additional practice determining the linear or quadratic curve of best fit and making predictions. This set of ordered pairs shows a relationship between x and y. Using the line of best fit, which is closest to the output when the input is 5? a. b. c. d. For SOL A.11, students need additional practice making predictions using the curve of best fit. Students should be familiar with different ways to describe the x- and y-values, including input, output, independent variable, and dependent variable. The answer to the question is shown on the screen. Common Errors? Misconceptions?

Which equation best models the relationship shown on the grid? a. b. c. d. Here is another example that asks students to select the curve of best fit given data on a grid. Students should recognize the shape as quadratic and be able to eliminate choices a and b. They should also recognize that since the curve opens downward, the coefficient on the first term will be negative. This eliminates option c and leaves the student with the correct answer, which is option d. Alternately, a less efficient strategy for this particular example is to enter the points into the graphing calculator, and then find the quadratic curve of best fit. Common Errors? Misconceptions?

Algebra 2 AII.9 Curve of Best Fit

Students need additional practice identifying the equation for the curve of best fit and making predictions using the curve of best fit. A data set is displayed in this table. Using the exponential curve of best, what is the value of y, rounded to the nearest hundredth, when x = 5? TI-84 Need to Know Stat, Calc, Stat Plot, ZoomStat, Graph x -3 -2 -1 y 3.375 2.25 1.5 1 Common Errors? Misconceptions?

Students need additional practice finding the exponential curve of best fit for a set of data and making predictions using this curve. The table provides the value of an account over time that earned annual compound interest. There was an initial deposit of $1,500 into the account, and no other deposits were made. Assuming the account continues to grow in the same way, use the exponential curve of best fit to find the value of the account at the end of 40 years, rounded to the nearest dollar. Value of Account Over Time 5 10 15 20 25 30 1,500.00 1,914.42 2,443.34 3,118.39 3,979.95 5,079.53 6,482.91 Time in years, x Value in dollars For SOL A2.9, students need additional practice finding the exponential curve of best fit for a set of data and making predictions using this curve. When the data points are graphed, students do well on choosing the equation of the curve of best fit or predicting a value. However, students struggle with this skill when they have to determine the curve of best fit and/or make a prediction when the data is represented in a table or in a set. An example that represents this skill, and its answer, are shown on the screen. $10,560 Common Errors? Misconceptions?

A Exponential B Linear C Logarithmic D Quadratic
Suggested Practice for SOL AII.9 Students need additional practice finding the curve of best fit for a set of data and making predictions using this curve. Which type of equation would best model the data in this table? A Exponential B Linear C Logarithmic D Quadratic Using the equation of best fit from the data in the table, what would be the value of y if x = 300? x y 30 2 60 4 90 8 120 16 150 32 180 64 For SOL A2.9, students need additional practice finding the curve of best fit for a set of data and also making predictions using this curve. When the data points are graphed, students do well on choosing the equation of the curve of best fit or predicting a value. However, students struggle with this skill when they have to determine the curve of best fit and/or make a prediction when the data is represented in a table or in a set of values. The answers to the two questions are shown on the screen. 1024 Common Errors? Misconceptions?

Miscellaneous Content
Statistics SOL A.10 SOL AII.2, AII.10, AII.12

Breakout Sessions Work problems from A.10 and AII.2, AII.10, AII.12 in small groups. Make sure you are able to do the problems on the calculator and know the keystrokes.

Box-and-Whisker Plots
Algebra 1 A.10 Box-and-Whisker Plots

The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. TI-84 Need to Know Stat, Stat Plot, ZoomStat, Graph 2013 A.9 - The next standard being highlighted is SOL A.9. This standard reads: 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. In particular, students need practice calculating and interpreting standard deviation and z-scores. A.10 - The next standard being highlighted is SOL A.10. This standard reads: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. Students have difficulty with many aspects of this standard. A.11 - The next standard being highlighted is SOL A.11. In particular, students had inconsistent performance making predications using the curve of best fit. 2014 A.9 - The next standard highlighted is SOL A.9. This standard reads: 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. In particular, students need practice calculating and interpreting standard deviation. A.10 – The next standard highlighted is SOL A.10. This standard reads: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots. A.11 - The next standard highlighted is SOL A.11. In particular, student performance was inconsistent when making predictions using the curve of best fit.

Students need additional practice analyzing changes to a data set when a data point is added or removed, and analyzing two box-and-whisker plots to draw a conclusion about the distribution of the data. This box-and-whisker plot represents nine pieces of data. No number is repeated. The number 23 is removed from the data set and a new box-and-whisker plot is drawn. Compared to the values in the original box-and-whisker plot, describe the changes to each of these values (increases, decreases or stays the same): the lower extreme the lower quartile the median the upper quartile the upper extreme It stays the same. Need to see it? Sample Data Sets Before: 10, 12, 14, 15, 18, 20, 21, 22, 23 After: 10, 12, 14, 15, 18, 20, 21, 22 It stays the same. It decreases. It decreases. It decreases. Common Errors? Misconceptions?

Each of these box-and-whisker plots represents a data set with 10 distinct elements. Which statements about these plots appear to be true? There are more elements in the lower quartile of plot B than plot A, because the left whisker of plot B is longer than the left whisker of plot A. Since both data sets have 10 distinct elements, the box of plot A and the box of plot B contain the same number of elements. There are fewer elements in the upper quartile of plot B than plot A, because the right whisker of plot B is shorter than the right whisker of plot A. The interquartile range of plot A is greater than the interquartile range of plot B. The range of both plots are equal. Common Errors? Misconceptions?

Students need additional practice interpreting data plotted in box-and-whisker plots. Each of these box-and-whisker plots contain 15 unique elements. Write a statement comparing the range of both plots. Which box-and-whisker plot has the greater interquartile range? Which data set has more elements with a value of 11 or greater? The value of the range for both plots is equal to 17. For SOL A.10 students need additional practice interpreting data plotted in box-and-whisker plots. Students struggle with vocabulary, such as range and interquartile range. They have difficulty calculating and comparing the range and interquartile range of two or more plots. Students also need practice comparing plots that have a specified number of elements. For instance, the third question requires students to sketch out Plots A and B and determine how many elements of each plot meet a given criteria. The answers to the questions are shown on the screen. Plot A. The value of the interquartile range for Plot A is 11, and the value of the interquartile range for Plot B is 10. Plot B. There are 12 elements in Plot B with a value of 11 and above and 8 elements in Plot A with a value of 11 and above. Common Errors? Misconceptions?

Plot A represents the total number of songs downloaded by each of 15 students in Mr. Archer’s class during October. Each student in Mr. Archer’s class downloaded a different number of songs from the others. Plot B represents the total number of songs downloaded by each of 20 students in Mrs. Baker’s class during October. Each student In Mrs. Baker’s class downloaded a different number of songs from the others. During the month of October, what is the difference between the number of students who downloaded more than 6 songs in Mrs. Baker’s class and the number of students who downloaded more than 6 songs in Mr. Archer’s class? Here is another example for SOL A.10. This question requires students to determine how many data points in each plot meet a certain criterion. The answer is shown on the screen. The difference is 4. There were 15 students who downloaded more than 6 songs in Mrs. Baker’s class, and 11 students who downloaded more than 6 songs in Mr. Archer’s class. Common Errors? Misconceptions?

This box-and-whisker plot summarizes the number of pieces of pizza each of ten volunteers served at a concession stand one night. Another volunteer served 16 pieces of pizza that night, and 16 is added to the original data set. A new box-and-whisker plot is drawn. Which two statements comparing the new box-and-whisker plot to the original box-and-whisker plot must be true? The interquartile range of the box-and-whisker plot increases. The range of the box-and-whisker plot increases. The value of the upper extreme increases. The value of the median increases. Here is an example that requires students to determine the affect on a box and whisker plot when an element is added to the data set. Paying attention to the location of the new element within the original data set helps students determine whether or not certain conclusions can be drawn. It is very important for students to realize that some statements cannot be proven true using the box and whisker plot, since the elements themselves are not given. Students must draw conclusions based only on the information that is given. For instance, the statement, “The interquartile range of the box-and-whisker plot increases,” may be true or false, depending on the data set. Likewise the location of the median of the data set will change, but the value of the median may or may not increase. The only two statements that will always be true are the ones indicated on the screen. Some statements cannot be proven true using the box and whisker plot, since the elements themselves are not given. Common Errors? Misconceptions?

Students need additional practice identifying and comparing the ranges, interquartile ranges, and medians of box-and-whisker plots. Which two plots appear to have the same value for the range? Which two plots appear to have the same value for the interquartile range? Which two plots appear to have the same value for the median? For this standard, students need additional practice identifying and comparing basic values associated with box-and-whisker plots: the range, interquartile range, and median. The range is the difference between the upper and lower extremes. The interquartile range is the difference between the upper quartile (Q3) and the lower quartile (Q1). The median value (Q2) can be found by locating the vertical line in the box, between Q1 and Q3. The answers to the questions are shown on the screen. The next slide shows these same box-and-whisker plots with a different question format. Plots A and C Plots A and B Plots B and C Common Errors? Misconceptions?

Which statement appears to be true regarding the box-and-whisker plots shown? The interquartile range of the data for plot A is greater than the interquartile range of the data for plot B. The upper extreme of the data for plot A is greater than the upper extreme of the data for plot C. The range of the data in plot A is the same as the range of the data in plot C. The median of the data in plot A is greater than the median of the data in plot B. This question asks students to identify the true statement about these box-and-whisker plots. Again, students are comparing the basic values associated with the plots: range, interquartile range, and median. This item also has a statement about the upper extreme that needs to be evaluated. The answer to the question is shown on the screen. Common Errors? Misconceptions?

Algebra 2 AII.2 Sequences and Series

2012 – None The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include and an The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include and an.

Students need additional practice finding the nth term of a sequence when a written description of the sequence is given. What is the seventh term of the geometric sequence with a first term of 729 and a common ratio of ? For SOL A2.2, students need additional practice finding the nth term of a sequence when a written description of the sequence is given. Students performed better on items when the geometric or arithmetic sequence was written out numerically. Students may have had a more difficult time selecting the correct formula from the Algebra II formula sheet when the question was presented in the format of this example. The answer to the example is provided on the screen. Common Errors? Misconceptions?

Students need additional practice finding the sum of a geometric series, particularly when the common ratio is negative. Find the sum of this series. For SOL A2.2, students also need additional practice finding the sum of a geometric series, particularly when the common ratio is negative. Students did well on multiple choice and fill-in-the-blank items when the ratio was positive. When the ratio of the series was negative, the most common student error was subtracting the ratio from one incorrectly in the denominator. The answer to the example is shown on the screen. Common Errors? Misconceptions?

Students need additional practice finding the sum of a geometric series, particularly when the common ratio is negative. Find the sum of the infinite geometric series: For SOL A2.2, students need additional practice finding the sum of a geometric series, particularly when the common ratio is negative. Example a: When finding the sum of an infinite geometric series for which the common ratio is negative, the most common student errors were subtracting the ratio from one incorrectly when simplifying 1-r in the denominator or incorrectly applying the order of operations when calculating the answer. As an extension, students would benefit from seeing an infinite geometric series presented using the summation sign as shown in example b. In this example, students need to expand the series to determine the common ratio, then find the sum of the series. The answers to example a and the extension, example b, are shown on the screen. Common Errors? Misconceptions?

Algebra 2 AII.10 Variations

The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.

Students need additional practice identifying a variation equation that models a situation, and solving problems involving variation. Common Errors? Misconceptions?

Which equations represent this situation? A car's stopping distance, d, varies directly with the speed it travels, s, and inversely with the friction value of the road surface, f. Common Errors? Misconceptions?

The amount of time required to stack boxes varies directly with the number of boxes and inversely with the number of people who are stacking them. If 2 people can stack 60 boxes in 10 minutes, how many minutes will be required for 6 people to stack 120 boxes? k = (𝑻𝒊𝒎𝒆 ×𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑷𝒆𝒐𝒑𝒍𝒆) (𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑩𝒐𝒙𝒆𝒔) 6 minutes and 40 seconds Common Errors? Misconceptions?

Students need additional practice finding the constant of proportionality and solving real-world problems involving a combination of direct and inverse variations. If y varies inversely with the square of x, what is the constant of proportionality when y = 10 and x = 5? b. Body mass index (BMI) is directly proportional to a person’s weight in pounds and inversely proportional to the square of a person’s height in inches. A person with a BMI of has a weight of 135 pounds and a height of 63 inches. Rounded to the nearest hundredth, what is the BMI of a person with a weight of 145 pounds and a height of 65 inches? 250 For SOL A2.10, students need additional practice finding the constant of proportionality and solving real-world problems involving a combination of direct and inverse variations. Two examples to practice these skills, and their answers, are shown on the screen. Answers may vary depending on how the constant was rounded. BMI should be approximately Common Errors? Misconceptions?

Students need additional practice finding the constant of proportionality involving a combination of direct and inverse variations. If y varies directly with the square of x and inversely with the cube root of t, what is the constant of proportionality if x = 4, y = 3, and t = 8 ? 𝒚= 𝒌 𝒙 𝟐 𝟑 𝒕 𝟑= 𝒌 (𝟒) 𝟐 𝟑 𝟖 𝟑 𝟖 =𝒌 For SOL A2.10, students need additional practice finding the constant of proportionality involving a combination of direct and inverse variations. An equation that can be written for this problem is shown on the screen. To find the constant of proportionality, substitute the values from the problem into the equation. The answer is shown on the screen. Common Errors? Misconceptions?

Students need additional practice finding the constant of proportionality and solving real- world problems involving a combination of direct and inverse variations. Assume that wind resistance varies jointly as an object’s surface area and velocity. If a ball with a surface area of 25 square feet traveling at a velocity of 40 miles per hour experiences a wind resistance of 225 Newtons, what velocity must a ball with 40 square feet of surface area have in order to experience a wind resistance of 270 Newtons? Where: R = wind resistance (Newtons) A = surface area (square feet) V = velocity (miles per hour) 𝑹=𝒌𝑨𝑽 Students must calculate the constant of proportionality to find the answer. The constant of proportionality (k) = For SOL A2.10, students need additional practice finding the constant of proportionality and solving real-world problems involving a combination of direct and inverse variations. An equation that students can write from this real-world example is shown on the screen. Students should then substitute into this equation to find the constant of proportionality. In this example, the constant of proportionality is Students can then use this value to find the answer. 𝟐𝟕𝟎=𝟎.𝟐𝟐𝟓 𝟒𝟎 𝑽 𝟑𝟎 miles per hour =𝑽 Common Errors? Misconceptions?

Permutations and Combinations
Algebra 2 AII.12 Permutations and Combinations

The student will compute and distinguish between permutations and combinations and use technology for applications – None None

Decide whether each of these can be answered using a permutation or a combination and then determine the answer. Twenty horses competed in a race. In how many ways could the horses have finished in first place through third place? 6,840 A 10 person student council will be selected from 18 students at a school. How many possibilities are there for this student council? 43,758 TI-84 Need to Know Math, Prob, nPr or nCr 2012 Common Errors? Misconceptions?

Final Thoughts Statistics SOL A.9, A.10, A.11
SOL AII.2, AII.9, AII.10, AII.11, AII.12 February 18, 2015 & March 4, 2015

Instructional Resources
ExamView Banks NextLesson.org HCPS Math Website - VDOE Enhanced Scope and Sequence Skills - JMU Pivotal Items ExploreLearning Teaching Strategies Student Engagement Activities

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