133-146 SB MS3 SE U02 A11.indd Page 133 3 . -
133-146 SB MS3 TE U02 A11.indd Page 133 16/07/13 5:43 PM user-s068a wgf133-146 SB MS3 SE U02 A11.indd Page 133 3/12/13 2:37 AM dksharma vxp/103/SB00001 DEL/work/indd/TE/Math 03/Application files/TE M3 Unit 02/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .ACTIVITYExploring SlopeActivity 11High Ratio MountainLesson 11-1 Linear Equations and SlopeActivity Standards FocusThis activity deals with the connectionsbetween proportional relationships,lines and linear equations. Here studentsdevelop their understanding of slope asrate of change and as a ratio. They willgraph proportional relationships,determine slope and y-intercept fromgraphs, and interpret slope andy-intercept in the context of real-worldand mathematical problems.My NotesLearning Targets:in ybetween any two Understand the concept of slope as the ratio changechange in xpoints on a line. Graph proportional relationships; interpret the slope and the y-intercept(0, 0) of the graph. Use similar right triangles to develop an understanding of slope.SUGGESTED LEARNING STRATEGIES: Create Representations,Marking The Text, Discussion Groups, Sharing and Responding,Interactive Word WallLesson 11-1Misty Flipp worked odd jobs all summer long and saved her money to buypasses to the ski lift at the High Ratio Mountain Ski Resort. In August, Mistyresearched lift ticket prices and found several options. Since she worked sohard to earn this money, Misty carefully investigated each of her options.PLANPacing: 1–2 class periodsChunking the Lesson#1–3#4–6#7–8#9–11#13#14Check Your UnderstandingLesson Practice 11–1High Ratio MountainSki ResortStudent Lift Ticket pricesDaily Lift Ticket 3010-Day Package 80 upon purchase and 20 per day (up to 10 days)Unlimited Season Pass 390TEACHBell-Ringer ActivityGive students a few minutes to writedown all the terms and concepts thatcome to mind when they hear the wordslope. Have a few volunteers read theirlists aloud. Use any results relating to theideas of slant or steepness to introducethe concept of the slope of a line. 2014 College Board. All rights reserved. 2014 College Board. All rights reserved.1. Suppose Misty purchases a daily lift ticket each time she goes skiing.Complete the table below to determine the total cost for lift tickets.Number of DaysTotal Cost ofLift Tickets11Investigative012345603060901201501801–3 Shared Reading, CreateRepresentations, Look for a Pattern,Discussion Groups, Sharing andResponding Students will use thetable, a verbal description and anequation to represent the relationshipbetween the number of days that Mistyskis and the total cost of her ski season.These items are designed to movestudents toward expressing rate ofchange verbally.2. According to the table, what is the relationship between the cost of thelift tickets and the number of days?As the number of days increases by 1, the total cost increases by 30.Common Core State Standards for Activity 11Activity 11 Exploring Slope1338.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Comparetwo different proportional relationships represented in different ways. For example, compare adistance-time graph to a distance-time equation to determine which of two moving objects hasgreater speed.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points ona non-vertical line in the coordinate plane; derive the equation y mx for a line through theorigin and the equation y mx b for a line intercepting the vertical axis at b.Activity 11 Exploring Slope133
133-146 SB MS3 TE U02 A11.indd Page 134 08/03/14 8:56 AM user-g-w-728/103/SB00001 DEL/work/indd/TE/M01 Middile School/Math 03/Application files/TE M3 .133-146 SB MS3 SE U02 A11.indd Page 134 06/03/14 8:36 AM user-g-w-728Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedMy Notes3. Let d represent the number of days for which Misty bought lift ticketsand C represent Misty’s total cost. Write an equation that can be used todetermine the total cost of lift tickets if Misty skis for d days.C 30d4. Model with mathematics. Plot the data from the table on the graphbelow. The data points appear to be linear. What do you think thismeans?7–8 Look for a Pattern, DiscussionGroups, Sharing and RespondingThese items have students think aboutthe rate of change through horizontaland vertical movements on the graph.That leads directly into the concept offinding a slope graphically by askingstudents to write the rate of change as aratio comparing vertical change tohorizontal change.Sample answer: Linear means that the points could be connected tocreate a line.y275Total Cost of Lift Tickets250225200G175F150E125D10075C5025BA1234567 8Days9x10 11 12 13 145. Label the leftmost point on the graph point A. Label the next 6 points,from left to right, points B, C, D, E, F, and G. See graph above.MATH TIPVertical change is the number ofspaces moved up or down on agraph. “Up” movement isrepresented by a positive number.“Down” is a negative number.Horizontal change is the numberof spaces moved right or left ona graph. Movement to the right isindicated by a positive number.Movement to the left is indicatedby a negative number.6. Reason quantitatively. According to the graph, what happensto the total cost of lift tickets as the number of days increases? Justifyyour answer.As the number of days increases by 1, the total cost increases by 30.For each space you move right, you move 30 spaces up.7. Describe the movement, on the graph, from one point to another.A to B:Vertical Change 30Horizontal Change1B to C:Vertical Change 30Horizontal Change1C to D:Vertical Change 30Horizontal Change1D to E:Vertical Change 30Horizontal Change1E to F:Vertical Change 30Horizontal Change1Vertical Change 30Horizontal Change1F to G:134SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 EquationsMINI-LESSON: RatiosFor students who are struggling with ratios, a mini-lesson is available toprovide examples and practice. In this lesson, students are given real-worldsituations that can be represented by ratios and asked to provide theratios, as well as to express them in different ways: verbally, in fractionalnotation, and with a semi-colon.See SpringBoard’s eBook Teacher Resources for a student page for thismini-lesson.134SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 Equations 2014 College Board. All rights reserved.4–6 Activating Prior Knowledge,Create Representations, Think-PairShare, Sharing and RespondingStudents will use the graph and a verbaldescription to represent the relationshipbetween the number of days that Mistyskis and the total cost of her ski season.These items are also designed to movestudents toward expressing rate ofchange verbally. 2014 College Board. All rights reserved.ACTIVITY 11 Continued/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .
133-146 SB MS3 TE U02 A11.indd Page 135 16/07/13 5:50 PM user-s068a wgf133-146 SB MS3 SE U02 A11.indd Page 135 3/12/13 2:37 AM dksharma vxp/103/SB00001 DEL/work/indd/TE/Math 03/Application files/TE M3 Unit 02/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeActivity 11continued8. a. The movements you traced in Item 7 can be written as ratios. Writeratios in the form vertical change to describe the movement from:horizontal changeA to B: 30B to C: 30C to D: 30D to E: 301111b. Vertical change can also be described as the change in y. Similarly,the horizontal change is often referred to as the change in x.Therefore, the ratio vertical change can also be written ashorizontal changechange in y. Determine the change in y and change in x from A to Cchange in xchange in y.in Item 4. Write the ratio aschange in xMy NotesREadiNg aNdWRitiNg MatHWhen writing a ratio, you can alsorepresent the relationship byseparating each quantity with acolon. For example, the ratio 1:4 isread “one to four.”ACTIVITY 11 Continued7–8 (continued) Students use thechange in yratiograph to calculatechange in xbetween a series of points and discoverthat the ratio that represents the rate ofchange is a constant, includingcalculations for which the change in yand change in x are negative. Make surethat the students properly distinguishthe direction of the change in y andchange in x in each case, rather thansimply determining the magnitude ofthe change.A is (0, 0) and C is (2, 60).vertical changechange in y0 to 60 60 horizontal change change in x0 to 22Continue to use the data from Item 4. Determine the change in y andchange in x for each movement described below. Then write thechange in yratio.change in xc. From B to E:903e. From B to A:1204f. From E to B: 90 3 2014 College Board. All rights reserved. 2014 College Board. All rights reserved. 30 1d. From A to E:Activity 11 Exploring Slope135Activity 11 Exploring Slope135
133-146 SB MS3 TE U02 A11.indd Page 136 8/10/13 11:51 PM deepaksharma pjh/103/SB00001 DEL/work/indd/TE/Algebra 01/Application files/TE A1 BM133-146 SB MS3 SE U02 A11.indd Page 136 8/1/13 8:38 PM deepaksharma 000/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeACTIVITY 119–11 Shared Reading, Marking theText, Discussion Groups, Look for aPattern, Sharing and RespondingThese items, are designed to facilitatestudent understanding of the fact thatthe rate of change here is constant andto aid students in relating the rate ofchange to Misty’s ski scenario. Studentsshould see that the ratios they write areequivalent, although not all in lowestterms, and understand that the sign ofthe ratio is not dependent on directionsince the division of two negativenumbers yields a positive quotient.continued12 Shared Reading, Marking theText, Activating Prior Knowledge,Think-Pair-Share Students use whatthey know about scale factors andsimilar triangles to further develop theirunderstanding of a constant rate ofchange. By using the proportional sidesin similar triangles they can see that thechange in yratiobetween any twochange in xpoints on a line is constant.MATH TIPMy Notes9. Describe the similarities and differences in the ratios written in Item 8.How are the ratios related?The ratios are all equivalent to 30 , although they are not all in lowest1terms.10. Make sense of problems. What are the units of the ratios created inItem 8? Explain how the ratios and units relate to Misty’s situation.Dollars each day. The ratios show the 30 increase in price for eachday that she skis.The ratio 30 represents the 30 that is multiplied by the number of1days in the equation.change in ybetween any two points on a line is constant.change in xUse the diagram below and what you know about similar triangles tochange in yratios are equivalent for the movementsexplain why thechange in xchange in xdescribed.12. The ratio105This activity discusses slope as aconstant rate of change. Students canuse this concept to determine slopegiven a graph, a table or two points.The slope formula itself is confusingto some students.From W to Z: 610V3W63change in y 5 10change in xThey are equivalent because the triangles are similar and thecorresponding sides are being compared.Having a strong grounding in theconcept of slope as a rate of changewill make the transition to theformulaic understanding easier forstudents and be beneficial in higherlevel mathematics courses such asAlgebra II, Precalculus and Calculus.1361366SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 EquationsSpringBoard Mathematics Course 3/PreAlgebra, Unit 2 Equations 2014 College Board. All rights reserved.From W to V: 3TEACHER to TEACHERZ5 2014 College Board. All rights reserved.In similar triangles, correspondingangles are congruent andcorresponding sides are inproportion.11. How do the ratios relate to the equation you wrote in Item 3?change in yACTIVITY 11 Continued
133-146 SB MS3 TE U02 A11.indd Page 137 08/03/14 8:56 AM user-g-w-728133-146 SB MS3 SE U02 A11.indd Page 137 06/03/14 8:37 AM user-g-w-728/103/SB00001 DEL/work/indd/TE/M01 Middile School/Math 03/Application files/TE M3 ./103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedchange in ybetween any twoThe slope of a line is determined by the ratiochange in xpoints that lie on the line. The slope is the constant rate of change of a line. It is also sometimescalled the average rate of change. All linear relationships have a constant rate of change. The slope of a line is what determines how steep or flat the line is. The y-intercept of a line is the point at which the line crosses they-axis, (0, y).My NotesMATH TERMSSlope is the ratio of verticalchange to horizontal change, orchange in y .change in x13. Draw a line through the points you graphed in Item 4. Use the graph todetermine the slope and y-intercept of the line. How do the slope andy-intercept of this line relate to the equation you wrote in Item 3?Slope: 30 ; y-intercept: (0, 0). The slope is the coefficient of d in the1equation. (0, 0) represents the cost of tickets for 0 days.14. Complete the table to show the data points you graphed in Item 4. Usechange in ythe table to indicate the ratioand to determine the slope ofchange in xthe line. 2014 College Board. All rights reserved. 2014 College Board. All rights reserved.Number of Days0030260390412051506180change in y:change in x30301The slope of a line, change in y , ischange in x yalso expressed symbolically as x. is the Greek letter delta, and inmathematics it means “change in.”Total Cost ofLift Tickets1change in y:READING MATHchange in x:slope:ACTIVITY 11 ContinuedDeveloping Math LanguageThis lesson contains several vocabularyterms. The word slope is introducedhere and related words linear andy-intercept are reviewed Time shouldbe spent to read closely, mark the textand develop the relationships betweenthese terms. Attention should also beydrawn to the notation.xAs you guide students through theirlearning of these essential mathematicalterms, explain meanings in languagethat is accessible for your students.Whenever possible, provide concreteexamples to help students gainunderstanding. Encourage students tomake notes about new terms and theirunderstanding of what they mean, andabout how to use them to describeprecise mathematical concepts andprocesses.TEACHER to TEACHERAdd words like the terms introducedabove to your classroom Word Wallregularly. Include math terms,academic vocabulary, and otherwords that students use regularly intheir group or class discussions. Toremind students to refer to the WordWall, ask them to choose words toadd. Another way to reinforcelanguage acquisition is to have eachstudent choose a word from theWord Wall and then pair-share fora few minutes to discuss meaningand use.1301Activity 11 Exploring Slope13713 Create Representations, Look fora Pattern, Discussion Groups,Sharing and Responding Studentsidentify the slope and y-intercept of thelinear model they create and thenexplore the relationships between thevalues they find on the graph and theterms of the equation they created torepresent Misty’s ski season. The ideahere is that students will begin to makeconnections between the differentrepresentations of the data.14 Create Representations, Look fora Pattern, Discussion Groups, Sharingand Responding Students return towhat they know about the ratiochange in yand use the rate of changechange in xidentified in a table to determine theslope of the line in question. Again, theidea here is that students begin to makeconnections between differentrepresentations of the data and the slope.Activity 11 Exploring Slope137
133-146 SB MS3 TE U02 A11.indd Page 138 08/03/14 8:56 AM user-g-w-728/103/SB00001 DEL/work/indd/TE/M01 Middile School/Math 03/Application files/TE M3 .133-146 SB MS3 SE U02 A11.indd Page 138 06/03/14 9:05 AM user-g-w-728Debrief this section of the activity byasking students to describe thechange in yrelationship between slope, change in xyand. How would they use each toxdetermine the rate of change of a linearmodel from a graph? From a table?Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedMy NotesCheck Your Understanding15. Find the slope and the y-intercept for each of the following. Rememberchange in y.to use the ratiochange in xAnswers15. a. slope: 3; y-intercept: (0, 0)b. slope: 2 ; y-intercept: (0, 1)1c. slope: 2.5 ; y-intercept: (0, 0)1d. slope: 2 ; y-intercept: (0, 2)1e. Yes. Sample explanation:From W, extend the change-in-ysegment to 9 units and extendthe change-in-x segment to 15units to locate point P and createanother right triangle. The ratioof the change in y to the changein x is 9 or 3 , the same as the15 5ratios for W and V and for Wand Z, so P must be on the sameline.change in y 2500 1000 150016. 5 23change in xJohn’s rate is 500 feet per ��1–123x–1–2–2–3–4CONNECT TO SPORTSLongboards are larger than themore trick-oriented skateboards.Longboards are heavier andsturdier than skateboards. Somepeople even use them instead ofbicycles.c.xy012402.5510d.xy 1013420 4e. Look back at the figure for Item 12. Would a point P that is 9 units upfrom point W and 15 units to the right be on the line that containspoints W, V, and Z? Use similar triangles to explain your answer.16. John is longboarding at a constant rate down the road. If 2 minutesafter he leaves his house he is 1,000 feet away and at 5 minutes he is2,500 feet from his house, what would his average rate of change be?138SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 EquationsMINI-LESSON: Finding Slope Given a Table or a GraphSome students may need further practice in determining slope fromgraphs and/or tables representing linear relationships. For that purpose amini-lesson has been provided, containing a problem of each kind as anexample to show the student. The instruction for the problems suggestsworking with them in such a way as to emphasize the fact that sloperemains constant in a linear relationship.See SpringBoard’s eBook Teacher Resources for a student page for thismini-lesson.138SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 Equations 2014 College Board. All rights reserved.Check Your Understanding 2014 College Board. All rights reserved.ACTIVITY 11 Continued/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .
133-146 SB MS3 TE U02 A11.indd Page 139 27/03/14 11:23 AM user-g-w-728133-146 SB MS3 SE U02 A11.indd Page 139 3/12/13 2:37 AM dksharma vxp/103/SB00001 DEL/work/indd/TE/M01 Middile School/Math 03/Application files/TE M3 ./103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeActivity 11continuedThe Tran family is driving across the country. They drive 400 miles each day.Use the table below to answer Items 17–20.DaySee the Activity Practice for additionalproblems for this lesson. You may assignthe problems here or use them as aculmination for the activity.Total Miles Driven140028003LESSON 11-1 PRACTICE417.51400280018. Draw a graph for the data in the table. Be sure to title the graph andlabel the axes. Draw a line through the points.3120019. Write an equation that can be used to determine the total miles, M,driven over d days.4160052000ADAPTSome students may need more explicitinstruction on how slope and y-interceptare determined from the variousrepresentations of linear relationships(equations, tables and graphs) and whatthey mean. Such students may benefitby being guided through a problemwhich moves students back and forthbetween the different representations ofsome linear relationship, focusing onslope and y-intercept at each step of theway and allows them to makeconnections.The graph below shows the money a student earns as she tutors. Use thegraph to answer Items 21–24.Money Earned Tutoringy 350Money Earned 300 250 200 150 100 501234Weeks Tutoring567x21. What is the slope of the line?22. What is the y-intercept of the line?23. Write an equation that can be used to determine how much money, D,the student has earned after w weeks.24. attend to precision. Calculate how much money the student willhave earned after 52 weeks.18. Miles Travelled by the Tran FamilyyMiles Travelled bythe Tran Family2400Total Miles Driven 2014 College Board. All rights reserved.Days Total Miles Driven17. Complete the table.20. Find the slope and the y-intercept of the line you created, using thegraph you drew or the equation you wrote. Explain what each representsfor the Tran family’s situation. 2014 College Board. All rights reserved.ASSESSStudents’ answers to lesson practiceproblems will provide you with aformative assessment of theirunderstanding of the lesson conceptsand their ability to apply their learning.My NotesLESSON 11-1 PRACTICEACTIVITY 11 Continued200016001200800400123 4Days56Activity 11 Exploring Slope19. M 400d20. The slope is 400 . The y-intercept is1(0, 0). The slope represents howmany miles the family drives eachday. The y-intercept is how manymiles they had driven at the verystart of the trip, or 0.21. 50122. (0, 0)23. D 50w24. 50 52 weeks 2600139Give students a verbal description of alinear relationship situation, and thendemonstrate how to represent thesituation using an equation. Work withthe students to identify the slope andy-intercept in the equation. Studentsmay then create a table of values, againhighlighting the slope and y-intercept,and relate the meaning of the slope andy-intercept to the context of theproblem. Finally, they should identifythe slope and y-intercept on a graph ofthe relationship, while also reiteratingthe meaning of the terms in the contextof the problem. It may be helpful tocreate a mat that shows the threedifferent representations, along with theslope and y-intercept highlighted ineach.xActivity 11 Exploring Slope139
133-146 SB MS3 SE U02 A11.indd Page 133 3/12/13 2:37 AM dksharma vxp/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Exploring SlopeActivity 11High Ratio MountainLesson 11-1 Linear Equations and SlopeMy NotesLearning Targets:in ybetween any two Understand the concept of slope as the ratio changechange in xpoints on a line. Graph proportional relationships; interpret the slope and the y-intercept(0, 0) of the graph. Use similar right triangles to develop an understanding of slope.SUGGESTED LEARNING STRATEGIES: Create Representations,Marking The Text, Discussion Groups, Sharing and Responding,Interactive Word WallMisty Flipp worked odd jobs all summer long and saved her money to buypasses to the ski lift at the High Ratio Mountain Ski Resort. In August, Mistyresearched lift ticket prices and found several options. Since she worked sohard to earn this money, Misty carefully investigated each of her options.High Ratio MountainSki ResortStudent Lift Ticket pricesDaily Lift Ticket 3010-Day Package 80 upon purchase and 20 per day (up to 10 days)Unlimited Season Pass 390 2014 College Board. All rights reserved.1. Suppose Misty purchases a daily lift ticket each time she goes skiing.Complete the table below to determine the total cost for lift tickets.Number of DaysTotal Cost ofLift Tickets01234562. According to the table, what is the relationship between the cost of thelift tickets and the number of days?Activity 11 Exploring Slope133
133-146 SB MS3 SE U02 A11.indd Page 134 06/03/14 8:36 AM user-g-w-728/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedMy Notes3. Let d represent the number of days for which Misty bought lift ticketsand C represent Misty’s total cost. Write an equation that can be used todetermine the total cost of lift tickets if Misty skis for d days.4. Model with mathematics. Plot the data from the table on the graphbelow. The data points appear to be linear. What do you think thismeans?y275Total Cost of Lift Tickets250225200175150125100755025234567 8Days910 11 12 13 14x5. Label the leftmost point on the graph point A. Label the next 6 points,from left to right, points B, C, D, E, F, and G.MATH TIPVertical change is the number ofspaces moved up or down on agraph. “Up” movement isrepresented by a positive number.“Down” is a negative number.Horizontal change is the numberof spaces moved right or left ona graph. Movement to the right isindicated by a positive number.Movement to the left is indicatedby a negative number.6. Reason quantitatively. According to the graph, what happensto the total cost of lift tickets as the number of days increases? Justifyyour answer.7. Describe the movement, on the graph, from one point to another.A to B:Vertical ChangeHorizontal ChangeB to C: Vertical ChangeHorizontal ChangeC to D: Vertical ChangeHorizontal ChangeD to E: Vertical ChangeHorizontal ChangeE to F: Vertical ChangeHorizontal ChangeF to G: Vertical ChangeHorizontal Change134 SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 Equations 2014 College Board. All rights reserved.1
133-146 SB MS3 SE U02 A11.indd Page 135 3/12/13 2:37 AM dksharma vxp/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeActivity 11continued8. a. The movements you traced in Item 7 can be written as ratios. Writeratios in the form vertical change to describe the movement from:horizontal changeMy NotesA to B:B to C:Reading andWriting MathC to D:D to E:When writing a ratio, you can alsorepresent the relationship byseparating each quantity with acolon. For example, the ratio 1:4 isread “one to four.”b. Vertical change can also be described as the change in y. Similarly,the horizontal change is often referred to as the change in x.Therefore, the ratio vertical change can also be written ashorizontal changechange in y. Determine the change in y and change in x from A to Cchange in xchange in y.in Item 4. Write the ratio aschange in xContinue to use the data from Item 4. Determine the change in y andchange in x for each movement described below. Then write thechange in yratio.change in xd. From A to E:e. From B to A:f. From E to B: 2014 College Board. All rights reserved.c. From B to E:Activity 11 Exploring Slope135
133-146 SB MS3 SE U02 A11.indd Page 136 8/1/13 8:38 PM deepaksharma 000/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedMy Notes9. Describe the similarities and differences in the ratios written in Item 8.How are the ratios related?10. Make sense of problems. What are the units of the ratios created inItem 8? Explain how the ratios and units relate to Misty’s situation.MATH TIPIn similar triangles, correspondingangles are congruent andcorresponding sides are inproportion.11. How do the ratios relate to the equation you wrote in Item 3?change in ybetween any two points on a line is constant.change in xUse the diagram below and what you know about similar triangles tochange in yratios are equivalent for the movementsexplain why thechange in xchange in xdescribed.12. The ratioFrom W to V:Z56V3W63change in y 5 10change in x 2014 College Board. All rights reserved.From W to Z:change in y10136SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 Equations
133-146 SB MS3 SE U02 A11.indd Page 137 06/03/14 8:37 AM user-g-w-728/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedchange in ybetween any twoThe slope of a line is determined by the ratiochange in xpoints that lie on the line. The slope is the constant rate of change of a line. It is also sometimescalled the average rate of change. All linear relationships have a constant rate of change. The slope of a line is what determines how steep or flat the line is. The y-intercept of a line is the point at which the line crosses they-axis, (0, y).My NotesMATH TERMSSlope is the ratio of verticalchange to horizontal change, orchange in y .change in x13. Draw a line through the points you graphed in Item 4. Use the graph todetermine the slope and y-intercept of the line. How do the slope andy-intercept of this line relate to the equation you wrote in Item 3?READING MATH14. Complete the table to show the data points you graphed in Item 4. Usechange in ythe table to indicate the ratioand to determine the slope ofchange in xthe line.Number of DaysThe slope of a line, change in y , ischange in x yalso expressed symbolically as x. is the Greek letter delta, and inmathematics it means “change in.”Total Cost ofLift Tickets0123 2014 College Board. All rights reserved.456change in y:change in x:change in y:change in xslope:Activity 11 Exploring Slope137
133-146 SB MS3 SE U02 A11.indd Page 138 06/03/14 9:05 AM user-g-w-728/103/SB00001 DEL/work/indd/SE/M01 Middile School/Math 03/Application files/SE M3 .Lesson 11-1Linear Equations and SlopeACTIVITY 11continuedMy NotesCheck Your Understanding15. Find the slope and the y-intercept for each of the following. Rememberchange in y.to use the ratiochange in 123x–1–2–2–3–4CONNECT TO SPORTSLongboards are larger than themore trick-oriented skateboards.Longboards are heavier andsturdier than skateboards. Somepeople even use them instead ofbicycles.c.xy012402.5510d.xy 1013420 416. John is longboarding at a constant rate down the road. If 2 minutesafter he leaves his house he is 1,000 feet away and at 5 minutes he is2,500 feet from his house, what would his average rate of change be?138SpringBoard Mathematics Course 3/PreAlgebra, Unit 2 Equations 2014 College Board. All rights reserved.e. Look back at the figure for Item 12. Would a point P that is 9 units upfrom point W and 15 units to the right be on the line that conta
As the number of days increases by 1, total cost 30. Activity 11 Exploring Slope 133 . and C represent Misty’s total cost. Write an equation that can be used to . represents the 30
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3 300 500 133.35 (5.25) 47.75 (1.88) 146.05 (5.75) 12.70 (0.50) cg-3150 3 550 750 — — — — cg-3175 3-1/2 8 4 133.35 (5.25) 22.35 (0.88) 133.35 (5.25) 9.65 (0.38) cg-3604 3-1/2 4 2/0 133.35 (5.25) 30.23 (1.19) 143.00 (5.63) 9.65 (0.38) cg-3622 3-1/2 2/0 250 133.35 (5.25) 36.58 (1.44) 149.35 (5.88) 9.65 (0.38) cg-3625 3-1/2 300 500 146.05 .
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MS3 Survival Guide 2016-2017 Your treasure map to surviving the 3rd year of Medical School By the TAs of the Med
It is possible to power data logger directly from dc voltage source to input for AC/DC adapter. Types MS2 and MS3 works with power voltage from 9 to 30V DC, MS4 . If the RS module is located at channel 6th position, then link RS485 connected to channel 6. serves for channels 6 to 16. Physical positions of input modules for channels 7 to 16
Part 2 Formulation of Strategy 129 7 Strategic intent 131 About this chapter 133 Introduction 133 Mission 133 Statement of strategic intent 138 Nature of support for the mission statement 138 Goals and objectives 140 Hierarchy of objectives 142 Long-term versus short-term goals 143 The balanced scorecard 144 Gap analysis 146 Summary 146 vi Contents
C11.4. Information for Analysis 132 C12. CHAPTER 12 - AE BUILDING DESIGN AND LAYOUT 133 C12.1. General 133 C12.2. Building Exteriors 133 C12.3. Interiors, Walls, Roofs, and Ceilings 133 C12.4. Floors and Work Surfaces 133 C12.5. Substantial Dividing Walls 134 C12.6. Exits and Doors 134 C12.7. Emergency Egress 134 C12.8. Passageways 134 C12.9.
ZO-104: Practical Zoology Uttarakhand Open University Page 6 Habit and Habitat: Amoeba proteus is widely distributed. It is commonly found on the bottom mud or on underside of aquatic vegetation in fresh water ponds, lakes, springs, pools and slow running streams. It is .
