Alex and Morgan were asked to graph the equation y = 2x + 1

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1 Which is better? Ale and Morgan were asked to graph the equation = Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged them into the original equation to find corresponding - values. Net, I plotted m points.. I looked at the equation and saw that it was in slopeintercept form. So I could see that the slope, m, was 2 and the -intercept, b, was at (0, 1). Net, I plotted the - intercept. Then I found. other points b using the slope. Since the slope is 2, this means that the rise is up 2 and the run is to the right 1. Finall, I connected m points and now I have m line. Here is m final graph. * Describe Ale s wa to a new student in our class. * Describe Morgan s wa to a new student in our class. * What are some similarities and differences between Ale s and Morgan s was?* Wh did Ale choose the -values he chose? * Even though Ale and Morgan did different steps, wh did the get the same answer? * Which wa is easier, Ale s wa or Morgan s wa? Wh? 4.1.1

2 Which is better? Ale and Morgan were asked to graph the equation = Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged them into the original equation to find corresponding - values. Net, I plotted m points. Finall, I connected m points and now I have m line. When graphing a linear equation, making a table of values and plotting the slope and -intercept both give ou the same graph because both methods allow ou to find. several points that fall on the line. There is more than one wa to graph a line. Before ou start graphing, ou can look at the problem first and tr to see which wa will be easier. I looked at the equation and saw that it was in slopeintercept form. So I could see that the slope, m, was 2 and the -intercept, b, was at (0, 1). Net, I plotted the - intercept. Then I found. other points b using the slope. Since the slope is 2, this means that the rise is up 2 and the run is to the right 1. Here is m final graph. * Describe Ale s wa to a new student in our class. * Describe Morgan s wa to a new student in our class. * What are some similarities and differences between Ale s and Morgan s was? * Wh did Ale choose the -values he chose? * Which wa is easier, Ale s wa or Morgan s wa? Wh? 4.1.1

3 Student Worksheet Describe Ale s wa to a new student in our class. 2 Describe Morgan s wa to a new student in our class. 3 What are some similarities and differences between Ale s and Morgan s was? 4 Wh did Ale choose the -values he chose? 5 Even though Ale and Morgan did different steps, wh did the get the same answer? 6 Which wa is easier, Ale s wa or Morgan s wa? Wh?

4 Which is better? Ale and Morgan were asked to graph the equation using a table of values. Ale s choose tpical values wa Morgan s choose values more carefull wa First I chose some values. As I usuall do when I make a table of values, I picked to be 0, 1, 2, 3, and First I chose some - values. I chose multiples of For each value in the table, I plugged it into the equation to find the corresponding value /3 2 14/ For each value in the table, I plugged it into the equation to find the corresponding value. Then I plotted each ordered pair and connected the dots to give a graph of this line. 4 16/ Then I plotted each ordered pair and connected the dots to give a graph of this line. * How did Ale graph the equation? How did Morgan graph the equation? * What are some similarities and differences between Ale s and Morgan s was? Wh did Morgan chose to use onl multiples of 3 for? * Whose wa is easier, Ale s or Morgan s? Wh? 4.1.2

5 Which is better? Ale and Morgan were asked to graph the equation using a table of values. Ale s choose tpical values wa Morgan s choose values more carefull wa First I chose some values. As I usuall do when I make a table of values, I picked to be 0, 1, 2, 3, and 4. For each value in the table, I plugged it into the equation to find the corresponding value. Then I plotted each ordered pair and connected the dots to give a graph of this line When creating a table of values 3 to graph an equation, it is 6 helpful to choose values that 9 will generate whole 12 number values for (rather than simpl choosing values without considering the specific equation to be graphed). 1 13/3 2 14/3 There is more 3 5 than one wa 9 to / pick points for graphing a line. Before ou start picking points, ou can look at the problem first and tr to pick points in the easiest wa. First I chose some - values. I chose multiples of 3. For each value in the table, I plugged it into the equation to find the corresponding value. Then I plotted each ordered pair and connected the dots to give a graph of this line. * How did Ale graph the equation? How did Morgan graph the equation? * What are some similarities and differences between Ale s and Morgan s was? Wh did Morgan chose to use onl multiples of 3 for? * Whose wa is easier, Ale s or Morgan s? Wh? 4.1.2

6 1a Student Worksheet How did Ale graph the equation? 1b How did Morgan graph the equation? 2 What are some similarities and differences between Ale s and Morgan s was? 3 Wh did Morgan choose to use onl multiples of 3 for? 4 Whose wa is easier, Ale s or Morgan s? Wh? 5 If the problem were changed to = 2 + 4, and ou were tring to graph it using Morgan s wa, then 7 what values would ou choose?

7 Wh does it work? Ale and Morgan were asked to find the slope of the line passing through (3,4) and (2,-1) Ale s graph wa Morgan s formula wa First I drew a graph and plotted the two given points. m = First I wrote out the formula for slope. Starting from the bottom point, I counted the number of units up and to the right necessar to get to the other point. I went up 5 and to the right 1, so the slope is 5/1, which is 5. m = 5 1 = 5 m = m = 5 1 Then I substituted in (3, 4) for ( 1, 1 ) and (2, -1) for ( 2, 2 ). Then I simplified the numerator and denominator. m = 5 Then I divided to get the slope. * Describe Ale's wa to a new student in our class. * Describe Morgan's wa to a new student in our class. * Ale counted the spaces between the two points, beginning at the point (2, -1). Would Ale have gotten the same answer b starting from the other point, (3, 4)? * How are Ale's and Morgan's was similar? * How are Ale's and Morgan's was different? * If the two points in the problem were changed to (-20, 6) and (14, 35), would Ale's wa or Morgan's wa be better? * Even though Ale and Morgan did different first steps, wh did the both get the same answer? 4.2.1

8 Wh does it work? Ale and Morgan were asked to find the slope of the line passing through (3,4) and (2,-1) Ale s graph wa Morgan s formula wa First I drew a graph and plotted the two given points. m = First I wrote out the formula for slope. Starting from the bottom point, I counted the number of units up and to the right necessar to get to the other point. I went up 5 and to the right 1, so the slope is 5/1, which is 5. m = 5 1 = 5 To find the slope of a line passing through two given points, ou can m = 1 plot 4 the points and get the 2 slope 3 from the graph or use the slope formula, and ou will get the same answer. m = 5 There is more than one wa to find the slope of a line; before ou start, ou can look at the problem first and tr to see which wa will be easier. 1 m = 5 Then I substituted in (3, 4) for ( 1, 1 ) and (2, -1) for ( 2, 2 ). Then I simplified the numerator and denominator. Then I divided to get the slope. * Describe Ale's wa to a new student in our class. * Describe Morgan's wa to a new student in our class. * Ale counted the spaces between the two points, beginning at the point (2, -1). Would Ale have gotten the same answer b starting from the other point, (3, 4)? * How are Ale's and Morgan's was similar? * How are Ale's and Morgan's was different? * If the two points in the problem were changed to (-20, 6) and (14, 35), would Ale's wa or Morgan's wa be better? * Even though Ale and Morgan did different first steps, wh did the both get the same answer? 4.2.1

9 Student Worksheet Describe Ale's wa to a new student in our class. 2 Describe Morgan's wa to a new student in our class. 3 Ale counted the spaces between the two points, beginning at the point (2, -1). Would Ale have gotten the same answer b starting from the other point, (3, 4)? 4 How are Ale's and Morgan's was similar? 5 How are Ale's and Morgan's was different? 6 If the two points in the problem were changed to (-20, 6) and (14, 35), would Ale's wa or Morgan's wa be better? 7 Even though Ale and Morgan did different first steps, wh did the both get the same answer?

10 Wh does it work? Ale and Morgan were asked to graph the equation 4 3 = 12 b plotting the - and -intercepts. Ale s find the intercepts b setting and equal to zero wa Morgan s find the intercepts b covering terms up wa First I found the - intercept. I found it b setting equal to zero and solving the equation. 4 3 = 12 When = 0 : 4(0) 3 = 12 = 4 (0, 4) 4 3 = 12 3 = 12 = 4 (0, 4) First I found the - intercept. I covered up the term and solved the equation that was still showing, in m head, to get the intercept of 4. Then I found the - intercept. I found it b setting equal to zero and solving the equation. 4 3 = 12 When = 0 : 4 3(0) = 12 = 3 ( 3, 0) 4 3 = 12 4 = 12 = 3 ( 3, 0) Then I found the - intercept. I covered up the term and solved the equation that was still showing, in m head, to get the intercept of -3. Then I plotted the - and -intercepts and connected the dots to give a graph of this line. Then I plotted the - and -intercepts and connected the dots to give a graph of this line. * How did Ale find the intercepts? How did Morgan find the intercepts? * What are some similarities and differences between Ale s and Morgan s was? * Wh did Ale and Morgan get the same intercepts, even though their was are different? * On a timed test, whose wa would ou use, Ale's or Morgan's? 4.3.1

11 Wh does it work? Ale and Morgan were asked to graph the equation 4 3 = 12 b plotting the - and -intercepts. Ale s find the intercepts b setting and equal to zero wa Morgan s find the intercepts b covering terms up wa First I found the - intercept. I found it b setting equal to zero and solving the equation. Then I found the - intercept. I found it b setting equal to zero and solving the equation. Then I plotted the - and -intercepts and connected the dots to give a graph of this line. 4 3 = 12 When = 0 : 4(0) 3 = 12 = 4 (0, 4) 4 3 = 12 When = 0 : 4 3(0) = = = 12 = 4 He Morgan, (0, what 4) did comparing these two eamples 4 help 3 = 12 us to see? When ou are = 3finding the intercepts ( 3, of an 0) equation in standard form, ou can use the shortcut of covering up the terms rather than writing out the step where ou substitute 0 in for and. Covering up a term is a shortcut for substituting 0 for the variable. = 12 = 3 ( 3, 0) First I found the - intercept. I covered up the term and solved the equation that was still showing, in m head, to get the intercept of 4. Then I found the - intercept. I covered up the term and solved the equation that was still showing, in m head, to get the intercept of -3. Then I plotted the - and -intercepts and connected the dots to give a graph of this line. * How did Ale find the intercepts? How did Morgan find the intercepts? * What are some similarities and differences between Ale s and Morgan s was? * Wh did Ale and Morgan get the same intercepts, even though their was are different? * On a timed test, whose wa would ou use, Ale's or Morgan's? 4.3.1

12 1a Student Worksheet How did Ale find the intercepts? 1b How did Morgan find the intercepts? 2 What are some similarities and differences between Ale s and Morgan s was? 3 Wh did Ale and Morgan get the same intercepts, even though their was are different? 4 On a timed test, whose wa would ou use, Ale's or Morgan's? Wh?

13 Wh does it work? Ale and Morgan were asked to write the equation of the line in point-slope form that passes through (3,7) and (8,22). Ale s use the first point wa Morgan s use the second point wa First I wrote the point-slope form of a line. 1 = m( 1 ) 1 = m( 1 ) First I wrote the point-slope form of a line. Net, I found the slope using the slope formula. I used (3,7) as point #1 and (8,22) as point #2. m = = 3 m = = 3 Net, I found the slope using the slope formula. I used (3,7) as point #1 and (8,22) as point #2. Then I plugged in the coordinates of point #1 to get m equation. 7 = 3( 3) 22 = 3( 8) Then I plugged in the coordinates of point #2 to get m equation. Finall, I wanted to check m answer with Morgan so I put it in slope-intercept form. 7 = 3( 3) 7 = 3 9 = = 3( 8) 22 = 3 24 = 3 2 Finall, I wanted to check m answer with Ale so I put it in slopeintercept form. * Describe Ale's wa to a new student in our class. Describe Morgan's wa to a new student in our class. * How do ou know whether Ale s or Morgan s wa is correct? * Describe how Ale s and Morgan s was are similar and different. * Even though Ale and Morgan did different steps, wh did the get the same answer? * Would Ale and Morgan have gotten the same slope value if the had used (8,22) as point #1 and (3,7) as point #2? 4.4.1

14 Wh does it work? Ale and Morgan were asked to write the equation of the line in point-slope form that passes through (3,7) and (8,22). Ale s use the first point wa Morgan s use the second point wa First I wrote the point-slope form of a line. 1 = m( 1 ) 1 = m( 1 ) First I wrote the point-slope form of a line. Net, I found the slope using the slope formula. I used (3,7) as point #1 and (8,22) as point #2. Then I plugged in the coordinates of point #1 to get m equation. Finall, I wanted to check m answer with Morgan so I put it in slope-intercept form. m = 2 1 m = When writing 8the 3equation of 15 5 = 3 a line in point-slope 15 5 = 3 form, ou can choose to substitute in either of the given points for. Either wa, the equation 7 = 3( will give 3) ou the 22 same = 3(line. 8) 7 = 3( 3) 22 = 3( 8) There is more than one wa to 7 = = 3 24 find the equation of a line. Before ou start, = 3 ou 2can look at the = 3 2 problem first and tr to see which wa will be easier. Net, I found the slope using the slope formula. I used (3,7) as point #1 and (8,22) as point #2. Then I plugged in the coordinates of point #2 to get m equation. Finall, I wanted to check m answer with Ale so I put it in slopeintercept form. * Describe Ale's wa to a new student in our class. Describe Morgan's wa to a new student in our class. * How do ou know whether Ale s or Morgan s wa is correct? * Describe how Ale s and Morgan s was are similar and different. * Even though Ale and Morgan did different steps, wh did the get the same answer? * Would Ale and Morgan have gotten the same slope value if the had used (8,22) as point #1 and (3,7) as point #2? 4.4.1

15 Student Worksheet Describe Ale's wa to a new student in our class. 2 Describe Morgan's wa to a new student in our class. 3 How do ou know whether Ale s or Morgan s wa is correct? 4 Describe how Ale s and Morgan s was are similar and different. 5 Even though Ale and Morgan did different steps, wh did the get the same answer? 6 Would Ale and Morgan have gotten the same slope value if the had used (8,22) as point #1 and (3,7) as point #2?

16 Which is better? Ale and Morgan were asked to graph 1 = 4( 2). Ale s convert to slope-intercept form wa Morgan s use point-slope form wa First, I noticed that this line is given in pointslope form. Before graphing, I wanted to convert it to slope-intercept form. I added 1 on either side. 1 = 4( 2) = 4( 2) = m( 1 ) 1 = 4( 2) First I wrote the general equation for a line in point-slope form. I compared that to the equation given in the problem. Then I distributed the 4. = B looking at m equation, I can find one ordered pair that falls on the line. It is (2, 1). So I plotted this point. I simplified m epression. Now the equation is in slope-intercept form. To graph it, first I plotted the -intercept, which is at (0, -7). Then I looked at m equation and found the slope of the line, which is 4. I used the slope to plot several more points that fall on the line. I started at m point (0, -7) and counted up 4, right 1 to plot several more points. = = 4( 2) Then I looked. at m equation again, and I found the slope of the line, which is 4. I used the slope to plot several more points that fall on the line. I started at m point (2, 1) and counted up 4, right 1, and also down 4, left 1 to plot several more points. Then I connected the dots to make a line. This is m graph! Then I connected the dots to get the graph of m line. * How did Ale graph the equation? * Wh did Ale begin b solving for? * How did Morgan graph the equation? * Describe some similarities and differences between Ale s and Morgan s was. * Did Ale and Morgan get the same answer? How do ou know? * Whose wa is easier, Ale s or Morgan s? * The slope-intercept form of an equation and the point-slope form of an equation both tell ou the location of one point on the line. What part of the equation in slope-intercept form tells ou the location of a point? What part of the equation in point-slope form tells ou the location of a point? 4.4.2

17 Which is better? Ale and Morgan were asked to graph 1 = 4( 2). Ale s convert to slope-intercept form wa Morgan s use point-slope form wa First, I noticed that this line is given in pointslope form. Before graphing, I wanted to convert it to slope-intercept form. I added 1 on either side. Then I distributed the 4. I simplified m epression. Now the equation is in slope-intercept form. Then I connected the dots to get the graph of m line. 1 = 4( 2) = 4( 2) + 1 = = = m( 1 ) 1 = 4( 2) When ou have to graph an equation given in point-slope form, ou don t have to convert it to slope-intercept 1 = 4( 2) form. It ma be easier to graph the line directl from the point-slope form. To graph it, first I plotted the -intercept, which is at (0, -7). Then I looked at m equation and found the slope of the line, which is 4. I used the slope to plot several more points that There is more than one wa to fall on the line. I started at m point (0, -7) and counted up 4, right 1 to can look at the problem first and plot several more points. graph a line. Before ou start, ou tr to see which wa will be easier. First I wrote the general equation for a line in point-slope form. I compared that to the equation given in the problem. B looking at m equation, I can find one ordered pair that falls on the line. It is (2, 1). So I plotted this point. Then I looked. at m equation again, and I found the slope of the line, which is 4. I used the slope to plot several more points that fall on the line. I started at m point (2, 1) and counted up 4, right 1, and also down 4, left 1 to plot several more points. Then I connected the dots to make a line. This is m graph! * How did Ale graph the equation? * Wh did Ale begin b solving for? * How did Morgan graph the equation? * Describe some similarities and differences between Ale s and Morgan s was. * Did Ale and Morgan get the same answer? How do ou know? * Whose wa is easier, Ale s or Morgan s? * The slope-intercept form of an equation and the point-slope form of an equation both tell ou the location of one point on the line. What part of the equation in slope-intercept form tells ou the location of a point? What part of the equation in point-slope form tells ou the location of a point? 4.4.2

18 Student Worksheet a How did Ale graph the equation? 1b How did Morgan graph the equation? 2 Wh did Ale begin b solving for? 3 Describe some similarities and differences between Ale s and Morgan s was. 4 Did Ale and Morgan get the same answer? How do ou know? 5 Whose wa is easier, Ale s or Morgan s? 6 The slope-intercept form of an equation and the point-slope form of an equation both tell ou the location of one point on the line. What part of the equation in slope-intercept form tells ou the location of a point?

19 Which is better? Ale and Morgan were asked to find the -intercept of the line connecting the two points (-3, 1) and (-4, -1). Ale s graphing wa Morgan s algebraic wa First I drew a graph and plotted the two given points. Then I looked at the points and found the slope of the line. To get from one point to the net, I went up two and to the right one. I repeated this pattern to draw more points on the line until it crossed the -ais to get the -intercept, which is (0,7). = m + b m = ( 3) 2 1 = 2 = 2 + b 1 = 2( 4) + b 1 = 8 + b First I wrote out the slope-intercept form of the equation of a line. In this equation, b is the -intercept. Then I used the slope formula to find the slope. It is 2. Then I substituted the slope for m in the equation. Then I substituted in one of the given points, (-4, -1), for and in the equation. Then I eliminated the parentheses. 7 = b Then I solved for b to get the answer, which means that the -intercept is at (0,7). * Wh did Ale plot the two points as a first step? * Wh did Morgan write the equation for a line in slope-intercept form as a first step? * What are some similarities and differences between Ale's and Morgan's was? * Even though Ale and Morgan did different steps, wh did the both get the same answer? * If the points given were changed to (520, 657) and (-3000, -25), would Ale s wa or Morgan s wa be better? * If the points given were changed to (0.56, 29) and (-2/3, ), would Ale s wa or Morgan s wa be better? 4.5.1

20 Which is better? Ale and Morgan were asked to find the -intercept of the line connecting the two points (-3, 1) and (-4, -1). Ale s graphing wa Morgan s algebraic wa First I drew a graph and plotted the two given points. Then I looked at the points and found the slope of the line. To get from one point to the net, I went up two and to the right one. I repeated this pattern to draw more points on the line until it crossed the -ais to get the -intercept, which is (0,7). = m + b m = ( 3) 2 1 = 2 When ou are given two points, ou can determine the - intercept of the line = that 2 + passes b through both points either (1) graphicall or (2) algebraicall. For 1 = 2( 4) + b some problems, the algebraic method ma be easier than the graphing method, such as when 1 = 8 + b the numbers given in the problem are large or mess. Before ou start, ou can look at the problem first and tr to see which wa will be easier. 7 = b First I wrote out the slope-intercept form of the equation of a line. In this equation, b is the -intercept. Then I used the slope formula to find the slope. It is 2. Then I substituted the slope for m in the equation. Then I substituted in one of the given points, (-4, -1), for and in the equation. Then I eliminated the parentheses. Then I solved for b to get the answer, which means that the -intercept is at (0,7). * Wh did Ale plot the two points as a first step? * Wh did Morgan write the equation for a line in slope-intercept form as a first step? * What are some similarities and differences between Ale's and Morgan's was? * Even though Ale and Morgan did different steps, wh did the both get the same answer? * If the points given were changed to (520, 657) and (-3000, -25), would Ale s wa or Morgan s wa be better? * If the points given were changed to (0.56, 29) and (-2/3, ), would Ale s wa or Morgan s wa be better? 4.5.1

21 1a Student Worksheet Wh did Ale plot the two points as a first step? 1b Wh did Morgan write the equation for a line in slope-intercept form as a first step? 2 What are some similarities and differences between Ale's and Morgan's was? 3 Even though Ale and Morgan did different steps, wh did the both get the same answer? 4 If the points given were changed to (520, 657) and (-3000, -25), would Ale s wa or Morgan s wa be better? 5 If the points given were changed to (0.56, 29) and (-2/3, ), would Ale s wa or Morgan s wa be better?

22 Which is correct? Ale and Morgan were asked to graph the equation = 3 2 Ale s slope and -intercept wa Morgan s slope and -intercept wa = 3 2 = 3 2 First I graphed the -2, which is the -intercept. First I graphed the 3, which is the -intercept. Then I used the slope to find more points. Since the slope is 3, I went up three units and to the right 1 unit to get two more points, then connected them to get the line. Then I used the slope to find more points. Since the slope is -2, I went down two units and to the right 1 unit to get two more points, then connected them to get the line. * Describe Ale's wa to a new student in our class. * Describe two was that Ale's and Morgan's was are similar. * Describe two was that Ale's and Morgan's was are different. * Which answer is correct, Ale's or Morgan's? How do ou know? * Can ou give a general rule that describes what ou have learned from comparing Ale's and Morgan's was of solving this tpe of problem? 4.5.2

23 Which is correct? Ale and Morgan were asked to graph the equation = 3 2 Ale s slope and -intercept wa Morgan s slope and -intercept wa = 3 2 = 3 2 First I graphed the -2, which is the -intercept. First I graphed the 3, which is the -intercept. He Ale, what did we learn from comparing these right and wrong was? Then I used the slope to find more points. Since the slope is 3, I went up three units and to the right 1 unit to get two more points, then connected them to get the line. In the slope-intercept form of the equation = m + b, m represents the slope and b represents the -intercept. Be careful that ou don t confuse these! Then I used the slope to find more points. Since the slope is -2, I went down two units and to the right 1 unit to get two more points, then connected them to get the line. * Describe Ale's wa to a new student in our class. * Describe two was that Ale's and Morgan's was are similar. * Describe two was that Ale's and Morgan's was are different. * Which answer is correct, Ale's or Morgan's? How do ou know? * Can ou give a general rule that describes what ou have learned from comparing Ale's and Morgan's was of solving this tpe of problem? 4.5.2

24 Student Worksheet Describe Ale's wa to a new student in our class. 2 Describe two was that Ale's and Morgan's was are similar. 3 Describe two was that Ale's and Morgan's was are different. 4 Which answer is correct, Ale's or Morgan's? How do ou know? 5 Can ou give a general rule that describes what ou have learned from comparing Ale's and Morgan's was of solving this tpe of problem?

25 Which is correct? Ale and Morgan were asked to graph the equation = Ale s plot run over rise wa Morgan s plot rise over run wa = = First I plotted the -intercept, which is at (0,1). First I plotted the -intercept, which is at (0,1). Then I plotted two points using the slope of the equation. The slope is -2/5, so I went down five and to the right two, to get to the net point. To find another point, I went up five and left two. Then I drew a line through the three points to finish graphing the line.. Then I plotted two points using the slope of the equation. The slope is -2/5, so went down two and right five, to get to the net point. To find another point, I went up two and left five. Then I drew a line through the three points to finish graphing the line. * How did Ale graph this line? How did Morgan graph this line? * Describe two was that Ale's and Morgan's was are similar. * Describe two was that Ale's and Morgan's was are different. * Which line is correctl graphed, Ale's or Morgan's? How do ou know? * In thinking about the similarities and differences between Ale's and Morgan's was, what conclusions can ou draw about how to solve this tpe of problem? 4.5.3

26 Which is correct? Ale and Morgan were asked to graph the equation = Ale s plot run over rise wa Morgan s plot rise over run wa = = First I plotted the -intercept, which is at (0,1). He Morgan, what did we learn from comparing these right and wrong was? First I plotted the -intercept, which is at (0,1). Then I plotted two points using the slope of the equation. The slope is -2/5, so I went down five and to the right two, to get to the net point. To find another point, I went up five and left two. Then I drew a line through the three points to finish graphing the line. The slope is ratio of the vertical distance between an two points on a line, to the horizontal distance between those two points. It is the change in divided b the change in. When we sa that slope means rise over run, the rise is the change in and can be found in the numerator of the slope, and the run is the change in and can be found in the denominator of the slope.. Then I plotted two points using the slope of the equation. The slope is -2/5, so went down two and right five, to get to the net point. To find another point, I went up two and left five. Then I drew a line through the three points to finish graphing the line. * How did Ale graph this line? How did Morgan graph this line? * Describe two was that Ale's and Morgan's was are similar. * Describe two was that Ale's and Morgan's was are different. * Which line is correctl graphed, Ale's or Morgan's? How do ou know? * In thinking about the similarities and differences between Ale's and Morgan's was, what conclusions can ou draw about how to solve this tpe of problem? 4.5.3

27 1a Student Worksheet How did Ale graph this line? 1b How did Morgan graph this line? 2 Describe two was that Ale's and Morgan's was are similar. 3 Describe two was that Ale's and Morgan's was are different. 4 Which line is correctl graphed, Ale's or Morgan's? How do ou know? 5 In thinking about the similarities and differences between Ale's and Morgan's was, what conclusions can ou draw about how to solve this tpe of problem?

28 How do the differ? Ale was asked to graph the equation =, and Morgan was asked to graph the equation. = 3 Ale s graph = wa Morgan s graph = 3 wa = = 3 I rewrote the equation in = m + b form. = m + b = = m + b = I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted up 1, right 1 and down 1, left 1 to plot other points on the line. I connected the points to draw the graph of the line. I graphed the - intercept, (0,0) and counted up 3, right 1 and down 3, left 1 to plot other points on the line. I connected the points to draw the graph of the line. * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * If m were changed to -3, how would this affect the graph of the line? What about if m were 1 changed to? 3 * How does changing the value of m affect the graph of a line? 4.5.4

29 How do the differ? Ale was asked to graph the equation =, and Morgan was asked to graph the equation. = 3 Ale s graph = wa Morgan s graph = 3 wa = = 3 I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted up 1, right 1 and down 1, left 1 to plot other points on the line. I connected the points to draw the graph of the line. = m + b = In the slope-intercept form of a line ( = m + b), the coefficient of, which is m, indicates the slope. Changing the value of m changes the slope, or the steepness, of the line. = m + b He Morgan, what did we = learn from comparing these two was? I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted up 3, right 1 and down 3, left 1 to plot other points on the line. I connected the points to draw the graph of the line. * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * If m were changed to -3, how would this affect the graph of the line? What about if m were 1 changed to? 3 * How does changing the value of m affect the graph of a line? 4.5.4

30 Student Worksheet a How did Ale graph the line given b his equation? 1b How did Morgan graph the line given b her equation? 2 Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? 3 What are some similarities and differences between Ale s and Morgan s problems? 4 What are some similarities and differences between Ale s and Morgan s graphs? 5 If m were changed to -3, how would this affect the graph of the line? What about if m were changed to 1 3? 6 How does changing the value of m affect the graph of a line?

31 How do the differ? Ale was asked to graph the equation = 2, and Morgan was asked to graph the equation. = 2 Ale s graph = 2 wa Morgan s graph = -2 wa = 2 = 2 I rewrote the equation in = m + b form. = m + b = = m + b = I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. I graphed the - intercept, (0,0) and counted down 2, right 1 and up 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * How does changing the sign of m affect the graph of a line? 4.5.5

32 How do the differ? Ale was asked to graph the equation = 2, and Morgan was asked to graph the equation. = 2 Ale s graph = 2 wa Morgan s graph = -2 wa = 2 = 2 I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. = m + b = = m + b In the slope-intercept = form of a line ( = m + b), the coefficient of, which is m, indicates the slope. Changing the sign of m changes the slope, or the steepness, of the line. When a line has a positive slope, its height increases from left to right. When a line has a negative slope, its height decreases from left to right. I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted down 2, left 1 and up 2, right 1 to plot other points on the line. I connected the points to draw the graph of the line. * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * How does changing the sign of m affect the graph of a line? 4.5.5

33 Student Worksheet a How did Ale graph the line given b his equation? 1b How did Morgan graph the line given b her equation? 2 Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? 3 What are some similarities and differences between Ale s and Morgan s problems? 4 What are some similarities and differences between Ale s and Morgan s graphs? 5 How does changing the sign of m affect the graph of a line?

34 How do the differ? Ale was asked to graph the equation = 2, and Morgan was asked to graph the equation. = Ale s graph = 2 wa Morgan s graph = wa = 2 = First I made a table of values. First I made a table of values. I graphed the - intercept, (0,0) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. I graphed the - intercept, (0,3) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * How does changing the value of b affect the graph of a line? 4.5.6

35 How do the differ? Ale was asked to graph the equation = 2, and Morgan was asked to graph the equation. = Ale s graph = 2 wa Morgan s graph = wa = 2 = I rewrote the equation in = m + b form. I graphed the - intercept, (0,0) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. = m + b = Changing the value of b in the equation of a line shifts (or translates) the graph up or down verticall b units. = m + b In the slope-intercept form of a = line ( = m + b), b is the - coordinate of the point where the line intersects the -ais. I considered the equation in = m + b form. I graphed the - intercept, (0,5) and counted up 2, right 1 and down 2, left 1 to plot other points on the line. I connected the points to draw the graph of the line. * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * How does changing the value of b affect the graph of a line? 4.5.6

36 Student Worksheet a How did Ale graph the line given b his equation? 1b How did Morgan graph the line given b her equation? 2 Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? 3 What are some similarities and differences between Ale s and Morgan s problems? 4 What are some similarities and differences between Ale s and Morgan s graphs? 5 How does changing the value of b affect the graph of a line?

37 How do the differ? Ale was asked to find the -intercept of the line in the graph below, and Morgan was asked to find the -intercept of the line in the graph below. Ale s find the -intercept wa Morgan s find the -intercept wa The -intercept is the point where the line intersects the - ais. The -intercept is the point where the line intersects the - ais. It is at (3,0). (3, 0) (0, 2) It is at (0,-2). * How did Ale find the -intercept of the line? How did Morgan find the -intercept of the line? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between the ordered pairs that Ale and Morgan generated? * What is an -intercept? What is a -intercept? 4.5.7

38 How do the differ? Ale was asked to find the -intercept of the line in the graph below, and Morgan was asked to find the -intercept of the line in the graph below. Ale s find the -intercept wa Morgan s find the -intercept wa The -intercept is the point where the line intersects the - ais. He Ale, what did we learn from comparing these two was? The -intercept is the point where the line intersects the - ais. It is at (3,0). (3, 0) (0, 2) The -intercept is the point where the line intersects the -ais. The -intercept is the point where the line intersects the -ais. It is at (0,-2). * How did Ale find the -intercept of the line? How did Morgan find the -intercept of the line? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between the ordered pairs that Ale and Morgan generated? * What is an -intercept? What is a -intercept? 4.5.7

39 Student Worksheet a How did Ale find the -intercept of the line? 1b How did Morgan find the -intercept of the line? 2 What are some similarities and differences between Ale s and Morgan s problems? 3 What are some similarities and differences between the ordered pairs that Ale and Morgan generated? 4 What is an -intercept? 5 What is a -intercept?

40 Which is better? Ale and Morgan were asked to graph the equation 3 2 = 6 Ale s - and -intercepts wa Morgan s slope-intercept wa 3 2 = = 6 First I found the -intercept, b plugging in 0 for in the equation and solving for. 3 2(0) = 6 3 = 6 = 2 -intercept is (2, 0) 2 = = First I put the equation in slope-intercept form b solving for. Then I found the -intercept, b plugging in 0 for in the equation and solving for. 3(0) 2 = 6 2 = 6 = 3 -intercept is (0, 3) Then I graphed the equation using the intercept (0, -3) and the slope of 3/2. Then I plotted the two intercepts and connected them to get the line. Then I connected the points to get the line. * How did Ale graph the line? * Wh did Morgan solve the equation for as a first step? * Name a step that Morgan did that Ale did not. Wh did Ale not do that step? * What are some similarities and differences between Ale's and Morgan's was? * If the original problem were = 3, would Ale's wa or Morgan's wa be better?

41 Which is better? Ale and Morgan were asked to graph the equation 3 2 = 6 Ale s - and -intercepts wa Morgan s slope-intercept wa 3 2 = = 6 First I found the -intercept, b plugging in 0 for in the equation and solving for. 3 2(0) = 6 3 = 6 = 2 -intercept is (2, 0) 2 = = First I put the equation in slope-intercept form b solving for. Then I found the -intercept, b plugging in 0 for in the equation and solving for. 3(0) 2 = 6 2 = 6 = 3 -intercept is (0, 3) When ou have to graph an equation given in standard form, ou don t have to automaticall convert it to slope-intercept form. It ma be easier to graph the line directl from the standard form, b substituting 0 for and for to find the intercepts. Then I graphed the equation using the intercept (0, -3) and the slope of 3/2. Then I plotted the two intercepts and connected them to get the line. There is more than one wa to graph a line. Before ou start, ou can look at the problem first and tr to see which wa will be easier. Then I connected the points to get the line. * How did Ale graph the line? * Wh did Morgan solve the equation for as a first step? * Name a step that Morgan did that Ale did not. Wh did Ale not do that step? * What are some similarities and differences between Ale's and Morgan's was? * If the original problem were = 3, would Ale's wa or Morgan's wa be better?

42 Student Worksheet How did Ale graph the line? 2 Wh did Morgan solve the equation for as a first step? 3 Name a step that Morgan did that Ale did not. Wh did Ale not do that step? 4 What are some similarities and differences between Ale's and Morgan's was? 5 If the original problem were would Ale's wa or Morgan's wa be better?

43 Wh does it work? Ale and Morgan were asked to determine the slope of the horizontal line below from its graph. Ale s use the slope formula wa Morgan s inspect the patterns in a T-table wa First, I read an two points off the line. ( 5, 8) and (3, 8) First, I read several points off the line and put them into a T - table. Then I substituted these two ordered pairs into the slope formula, which is m = m = I looked at the table and noticed that the values never change, even when the values change. I simplified, to get that the slope of this line is zero. m = 0 8 = 0 The slope is 0. Since slope measures the change in values for each change in values, I know that the slope is zero. * How did Ale find the slope? How did Morgan find the slope? * What are some similarities and differences between Ale's and Morgan's was? * Which wa is easier, Ale s wa or Morgan s wa? Wh? * What are some advantages of Ale s wa? Of Morgan s wa? 4.7.1

44 Wh does it work? Ale and Morgan were asked to determine the slope of the horizontal line below from its graph. Ale s use the slope formula wa Morgan s inspect the patterns in a T-table wa First, I read an two points off the line. ( 5, 8) and (3, 8) He Morgan, what did comparing these two eamples help us to see? First, I read several points off the line and put them into a T - table. Then I substituted these two ordered pairs into the slope formula, which is m = I simplified, to get that the slope of this line is zero.. These eamples m = 8 8 helped us see wh the slope of a horizontal line is 0. For 5 a horizontal 3 line, all of the points on the line have the same value for the coordinate, so the difference between the coordinates of an two points m = 0 on the line will be 0 and thus = the 0 slope will The be slope is I looked at the table and noticed that the values never change, even when the values change. Since slope measures the change in values for each change in values, I know that the slope is zero. * How did Ale find the slope? How did Morgan find the slope? * What are some similarities and differences between Ale's and Morgan's was? * Which wa is easier, Ale s wa or Morgan s wa? Wh? * What are some advantages of Ale s wa? Of Morgan s wa? 4.7.1

45 1a Student Worksheet How did Ale find the slope? 1b How did Morgan find the slope? 2 What are some similarities and differences between Ale's and Morgan's was? 3 Which wa is easier, Ale s wa or Morgan s wa? Wh? 4 What are some advantages of Ale s wa? Of Morgan s wa?

46 Wh does it work? Ale and Morgan were asked to determine the slope of the line below from its graph. Ale s use the slope formula wa Morgan s inspect the patterns in a T-table wa First, I read an two points off the line. (7, 4) and (7, 1) First, I read several points off the line and put them into a T - table. Then I substituted these two ordered pairs into the slope formula, which is m = m = 4 ( 1) 7 7 I looked at the table and noticed that the values never change, even when the values change. I simplified. Since an number divided b zero is undefined, I found that this line has no slope. m = 5 0 This line has no slope. Since slope measures the change in values for each change in values, and since the values don t change, I know that this line has no slope. * How did Ale find the slope? How did Morgan find the slope? * What are some similarities and differences between Ale's and Morgan's was? * Which wa is easier, Ale s wa or Morgan s wa? Wh? * What are some advantages of Ale s wa? Of Morgan s wa? 4.8.1

47 Wh does it work? Ale and Morgan were asked to determine the slope of the line below from its graph. Ale s use the slope formula wa Morgan s inspect the patterns in a T-table wa First, I read an two points off the line. (7, 4) and (7, 1) What did comparing these two eamples help us to see? First, I read several points off the line and put them into a T - table. Then I substituted these two ordered pairs into the slope formula, which is m = I simplified. Since an number divided b zero is undefined, I found that this line has no slope.. These m eamples = 4 ( 1) help us see wh the slope of 7a vertical 7 line is undefined. For a vertical line, all of the points on the line have the same value for the coordinate, so the difference between the coordinates of an two points on the line will m = be 5 0. Division This b 0 line is has no slope. undefined, so thus 0 the slope of an vertical line will be undefined. I looked at the table and noticed that the values never change, even when the values change. Since slope measures the change in values for each change in values, and since the values don t change, I know that this line has no slope. * How did Ale find the slope? How did Morgan find the slope? * What are some similarities and differences between Ale's and Morgan's was? * Which wa is easier, Ale s wa or Morgan s wa? Wh? * What are some advantages of Ale s wa? Of Morgan s wa? 4.8.1

48 Student Worksheet a How did Ale find the slope? 1b How did Morgan find the slope? 2 What are some similarities and differences between Ale's and Morgan's was? 3 Which wa is easier, Ale s wa or Morgan s wa? Wh? 4 What are some advantages of Ale s wa? Of Morgan s wa?

49 How do the differ? Ale and Morgan were given the set of ordered pairs {(-3, 6), (2,5), (3,1), (2, 4)}, and asked to determine if the relation is a function. Ale s make a table of values wa Morgan s graph and use the vertical line test wa I made a table of values. I graphed the ordered pairs on a coordinate plane. I saw that the element 2 in the domain is paired with both a 5 and a 4 in the range.. Then I tried using the vertical line test. I found that I could draw a vertical line that intersected m graphed points more than once. Because of this, I believe that this relation is not a function. Not a function Not a function Because of this, I believe that this relation is not a function. * How did Ale complete the problem? * How did Morgan complete the problem? * Describe a wa in which Ale s and Morgan s was are similar. * Describe a wa in which Ale s and Morgan s was are different. * Wh does the vertical line test tell us the same thing as the table of values? * Can ou think of another wa of determining whether this relation is a function, besides Ale s wa and Morgan s wa? * If the problem were changed so ou were instead asked to determine whether = was a function, would ou use Ale s wa or Morgan s wa? Wh? 4.9.1

50 How do the differ? Ale and Morgan were given the set of ordered pairs {(-3, 6), (2,5), (3,1), (2, 4)}, and asked to determine if the relation is a function. Ale s make a table of values wa Morgan s graph and use the vertical line test wa I made a table of values. I graphed the ordered pairs on a coordinate plane. I saw that the element 2 in the domain is paired with both a 5 and a 4 in the range. Because of this, I believe that this relation is not a function. He Morgan, what did comparing these two eamples help us to see? These eamples help us see wh the vertical line test works. If ou can draw a vertical line through the graph that intersects the graph more than once, Not this a function shows that there are Not a function two points on the graph that have the same value but different values, so the graph is not a function.. Then I tried using the vertical line test. I found that I could draw a vertical line that intersected m graphed points more than once. Because of this, I believe that this relation is not a function. * How did Ale complete the problem? * How did Morgan complete the problem? * Describe a wa in which Ale s and Morgan s was are similar. * Describe a wa in which Ale s and Morgan s was are different. * Wh does the vertical line test tell us the same thing as the table of values? * Can ou think of another wa of determining whether this relation is a function, besides Ale s wa and Morgan s wa? * If the problem were changed so ou were instead asked to determine whether = was a function, would ou use Ale s wa or Morgan s wa? Wh? 4.9.1

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

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