STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part I. 4 th Nine Weeks,

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1 STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part I 4 th Nine Weeks,

2 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 found at the end of the document. problems are 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

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4 Equations of Lines and Transformational Graphing A.6 The student will graph linear equations and linear inequalities in two variables including Use the parent function y=x and describe transformations defined by changes in the slope or y-intercept Use transformational graphing to investigate effects of changes in equation parameters on the graph of the equation b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line. If you are asked to translate (or shift) a line, just move two points on the line the specified units and draw a new line. Remember, the new line should be parallel to the original line. This means the slope should remain the same. Only the y-intercept will change. Example 1: Shift the line down 5 units and right 2 units. Choose two points on the line. From each point, count down 5 units and right 2 units. Plot a new point. Then, draw a line through the two new points. The equation of the translated line is. Notice that the slopes of the lines are the same. Example 2: The function. Choose two points on the function f(x), which is. From each point, count up 3 units. Plot a new point. Then, draw a line through the two new points. The equation of the translated line is. Notice that the slopes of the lines are the same. is displayed on the graph below. Graph the function 4

5 Example 1: What is the equation of the line with a slope of and a y-intercept of? Example 2: What is the equation of the line that passes through and? First find the slope: Now you can use the slope and either of the points to solve for b. Now that you have and you can write an equation in slope intercept form. Graphs of y = equations Graphs of x = equations y = 4 x = -2 Each coordinate on the line has a y-coordinate of 4. Each coordinate on the line has an x-coordinate of -2. (-5, 4), (-2, 4), (0, 4), (1, 4), (3, 4) (-2, 6), (-2, 3), (-2, 0), (-2,-1), (-2, -4) 5

6 Equations of Lines & Transformational Graphing 1. What is the equation of a line whose y-intercept is and slope is? 2. What is the equation of the line that passes through and whose slope is? 3. What is the equation of the line that passes through (3, 4) and has a slope of -2? 4. What is the equation of the line that passes through (3, -2) and (9, -4)? 5. The function is displayed on the graph below. What is the equation of the function? 6. What is the equation of the line graphed below? 6

7 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. 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. 7

8 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. There is also a way to calculate a line of best fit using your calculator. 8

9 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 Year Average cost for one gallon Start by entering this data into the list in your calculator 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). 9

10 Example 3: What is the equation of the curve of best fit for the data below? Time Height (sec) (feet) 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 (months) Length (inches) 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) Height (ft)

11 Direct and Inverse Variation A.8 The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically. If the ratio between two variables is a constant, then a direct variation exists. A direct variation can be written in the form, where is the constant of variation. If the product of two variables is a constant, then an inverse variation exists. An inverse variation can be written in the form or. Example 1: Determine if each relation is a direct variation, inverse variation, or neither. x y x y First check the ratios: Does the ratio? NO! Therefore this is NOT a direct variation! Next check the products: Does? NO! Therefore this is NOT an inverse variation! First check the ratios: Does the ratio? YES! Therefore this IS a direct variation! Notice that we did not use the ordered pair to check the ratios. It is impossible to divide by zero, therefore we used the other points. x y First check the ratios: Does the ratio? NO! Therefore this is NOT a direct variation! Next check the products: Does? Does this also equal? YES! Therefore this IS an inverse variation! To write an equation of a direct variation, use a given point (other than plug into, to solve for. To write an equation of an inverse variation, use a given point to plug into, to solve for. ) to 11

12 Example 2: Suppose varies directly with, and when. What direct variation equation relates and? Start with. We are given a value for and, so plug those in and solve for. This is the constant of variation! This is the direct variation equation! Once you have a direct variation equation, you can use that equation to determine other values. Example 3: The distance that you jog,, varies directly with the amount of time you jog,. If you can jog 9 miles in 1.5 hours, how long will it take you to jog 4 miles? Jogging varies directly with time Now we need to solve for k in order to write a direct variation equation. Use the values that are related to one another. This is the constant of variation! This is the direct variation equation! Now, we can use this equation to solve for the time it takes to jog 4 miles. We are given that you jog 4 miles. This will be plugged in for j. Then, solve for t. Therefore, it would take you of an hour to jog 4 miles. Scan this QR code to go to a video tutorial on direct and inverse variations. 12

13 Direct and Inverse Variation 1. Determine if each of the following relations are a direct variation, an inverse variation, or neither. A x y x y Suppose varies directly with, and when. What direct variation equation relates and? 3. Suppose varies directly with, and when. What will be when? 4. The amount of money spent at the mall varies directly with the amount of time you spend shopping. If you spend $90 when you are in the mall for 2.5 hours, how much time did you spend in the mall when you spent $340? 5. Jason s earnings ( ) during his summer job is directly proportional to the amount of hours he worked ( ). When,, what is the constant of variation? 6. The time it takes to complete a job ( ) is inversely proportional to the amount of workers assigned to the job ( ). What value would accurately represent this relationship? Time (t) B Amount of Workers (w) ?

14 Graphing Inequalities A.6 The student will graph linear equations and linear inequalities in two variables including Graph linear inequalities in two variables, including those that arise from a variety of real-world situations A linear inequality can be formed by replacing the equal sign in any linear equation with an inequality symbol. The solutions for a linear inequality are any ordered pairs that make it a true statement. Example 1: Identify which ordered pairs are a solution of. This is true, therefore is a solution. This is false, therefore is NOT a solution. This is true, therefore is a solution. As you can see from the last example, linear inequalities will have more than one solution. In fact, they will have infinitely many solutions. The graph of a linear inequality will indicate all of the solutions, and it is called a half-plane, and is bounded by a boundary line. All of the points on one side of this boundary are solutions, while all of the points on the other side of the boundary are not solutions. You graph a linear inequality the same way that you graph a linear equation. The line that you graph will either be dashed or solid depending on the inequality symbol. Dashed lines are used for < and >. This indicates that the points on the line are not part of the solution set. Solid lines are used for the solution set. and. This indicates that the points on the line are part of To determine which half-plane to shade in, you can select one point that is not on the graph and determine if it is a solution or not. If it is a solution, shade that side of the boundary. If it is not a solution, shade the other side. The point (0, 0) is often an easy point to check with. 14

15 Example 2: Graph First, determine if you will be using a dashed or solid line (dashed in this case because you have < ) Then, graph the equation of the line by plotting the y-intercept and counting the slope as rise over run. Once you have a couple of points graphed, connect them with a dashed line. Finally, figure out which half-plane to color in. Select a point to see if it works. This is false. Therefore, is NOT a solution. So, we will shade the other side. All of the points on the shaded side will satisfy the inequality. Scan this QR code to go to a video tutorial on graphing linear inequalities. Example 3: Graph First, determine if you will be using a dashed or solid line (solid in this case because you have greater than or = equal to. 15

16 Before we graph this equation, we should put it in slope intercept form! Remember that if you multiply or divide by a negative number you will have to switch the inequality symbol s direction. You divided by here! Don t forget to switch the sign! Now we can put a point at the y-intercept, and count the slope as rise over run. Finally, figure out which half-plane to color in. Select a point to see if it works. You can plug this point into the original equation or the transformed equation. This is true. Therefore, is a solution. So, we will shade that side. All of the points on the shaded side will satisfy the inequality. 16

17 Graphing Linear Inequalities 1. Is a solution to the inequality? 2. Is a solution to the inequality? 4. Graph. 5. Graph. 6. Write the inequality that is graphed here 17

18 Answers to the problems: Equations of Lines and Transformational Graphing Graphing Linear Inequalities cont Scatterplots 1. 9 month old snake Direct and Inverse Variation 1. A ; B Graphing Linear Inequalities 1. No 2. Yes 3. Yes 6. 18