1、MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 1Unit 5: Day 6: Whats your Vector, Victor? MCT 4CMinds On: 15Action: 40Consolidate:20Total=75 minLearning Goal: Investigate the differences between vector and scalar quantities. Interpret information about real-world applications of
2、vectors. Discover careers that use vectors.Materials BLM 5.6.1-5.6.5 PPT 5.6.1 pencil crayons Measuring tapes Centimetre-graph paper RulersAssessmentOpportunitiesMinds On Whole Class Activity InstructionsSet context by reading BLM 5.6.1 (first half) to the classGroups of 3 Jigsaw (Home Group)Student
3、s number off 1 to 3 in their groups. Assign each number a reading from BLM 5.6.1.Groups Jigsaw (Expert Group)Students complete BLM 5.6.2 with information from their reading.Groups of 3 Jigsaw (Home Group)Share their information in round robin fashion and complete the organizerAction! Whole Class Pre
4、sentationIntroduce the idea of vector, distance and displacement with PowerPoint (PPT 5.6.1).Think/Pair/Share SummarizingStudents complete BLM 5.6.3. They then complete BLM 5.6.4 for distance and displacement.Groups of Four PresentationPairs join another pair and share their completed BLM 5.6.4 with
5、 each other and fill-in any additional informationPairs ActivityUsing BLM 5.6.5, create a map from the classroom to the cafeteria or library. Learning Skills/Teamwork/Checkbric: Assess students group work skills throughout the Action portion.Mathematical Process Focus: Reasoning and Proving, Reflect
6、ing, Connecting Students connect mathematics to a context outside mathematics. Consolidate DebriefSmall Groups DiscussionRecord the distance measurements on the blackboardDiscuss the discrepancies in the numbers if they existFor the discrepancies, how did those students get to the cafeteria?Is this
7、a vector quantity or a scalar quantity?Literacy Strategies:Structured Overview-modifiedThink, Pair, ShareFrayer ModelWord Wall:ScalarVectorDistanceDisplacementmagnitudeIf internet is available in the classroom, use www.resources.elearningontario.ca. You will want ELO10739000. This is a great animate
8、d visual of distance and displacement.Questions and PowerPoint modified from www.resources.elearningontario.ca ELO1073900ExplorationApplicationHome Activity or Further Classroom ConsolidationComplete BLM 5.6.6.MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 25.6.1: Whats your Vect
9、or, Victor?(source: www.mathscareers.org)Imagine having to describe something anything at all to another person only using a pen and paper. If theres any movement at all involved in what you have to describe, chances are youll soon find yourself drawing arrows. And what characterises an arrow? Well,
10、 its the direction its pointing in and its length. This, in a nutshell, is a vector: an object that has a direction and a magnitude. Its not hard to come up with some precise examples of the uses of vectors. The movement of a speeding car is described by its direction and its speed a direction and a
11、 magnitude, in other words a vector. To understand how a force like gravity acts on an object you need to know the direction and the intensity of the force, so again you have the two bits of information that form a vector. And when you watch the weather report youll be told which way and how strongl
12、y the wind will blow tomorrow, again a direction and a magnitude together making up a vector. Mathematics gives you a way of formalising all the information contained in the visual concept of an arrow. Using mathematical machinery, like algebra and arithmetic, you can go far beyond your doodlings an
13、d gain information that you wouldnt be able to see by simply looking at a bunch of arrows. And once you get used to thinking of vectors mathematically, you will stop seeing them simply as arrows and so be able to apply vector maths to many situations you wouldnt initially have thought of. Here is a
14、selection of examples of how vectors are used in real life that you will discover and discuss in your groups. cut here-Vectors and languageOur highly developed language is one thing that separates us from all the other animals on our planet. For this reason many people believe that language and the
15、way we use it can tell us a lot about who we, as humans, really are. The study of language called linguistics has become an important field within psychology.But there are also more practical reasons for trying to understand the way language works. Search engines and word processors work by picking
16、up on certain structures within texts to find the websites most relevant to your search and to weed out grammatical mistakes in your texts. The more they understand language, the more efficient theyll get. The same goes of course for automated speech recognition systems like those that sometimes ans
17、wer the phone when you ring up a company or information line. Both psychologists and people involved in computing want to understand the structures within language. Mathematics is a great tool for capturing structure and vectors seem to be especially useful for understanding language. Words or bits
18、of text can be represented by vectors and vector mathematics can help you see how the different components of a text interact, helping you to find structures within a text that you might not see otherwise. And much more .These are only three examples of how vectors are used. Because they are so esse
19、ntial in physics and convey visual information, many, many people, from engineers, architects and designers to meteorologists and oceanographers, use them at work every day.MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 35.6.1: Whats your Vector, Victor? (continued)(source: www.m
20、athscareers.org)Vectors in sportsIn the 1950s a group of talented Brazilian footballers invented the swerving free kick. By kicking the ball in just the right place, they managed to make it curl around the wall of defending players and, quite often, go straight into the back of the net. Some people
21、might think that such skill is pure magic, but really its just physics. When a ball is in flight its acted upon by various forces and some of these depend on the way the ball is spinning around its own axis. If you manage to give it just the right spin, the forces will interact in just the right way
22、 to deflect the ball while its flying, resulting in a curved flight path.The forces at work here can be described by vectors. Understanding their interaction requires vector maths. The footballers themselves rarely think about the mathematics of course, but today science is increasingly used to impr
23、ove the performance of athletes equipment. The exact shape of a football can have important implications on how it moves through the air and teams of scientists are employed to work out how to make the perfect football.When the equipment is more complicated than a football then the use of science is
24、 even more important. Formula One teams, for example, always employ physicists and mathematicians to help build perfect cars. Tiny differences in the shape of the car can make a difference to its speed that can determine the outcome of a race.Both the science of footballs and of race cars are really
25、 just examples of the same thing: aerodynamics, the study of how air moves. This is pure physics and since vectors can describe movement and forces, they lie at the very heart of it. cut here-Vectors and visualsVector mathematics is used extensively in computer graphics. Suppose you want to create a
26、n image on a computer screen. One way of doing this is to tell the computer the exact colour of each pixel on the screen. This requires a lot of memory and has another disadvantage: if youd like the image to move, for example to give the viewer the impression that he or she is moving around a scene,
27、 you need to constantly renew the information of the pixel colours from scratch. Its much easier to describe your set-up mathematically. Vectors are very useful here. Say, for example, that youre creating a scene lit by sunlight and ruffled by a strong wind. The sunlight and wind both come from a sp
28、ecific direction and have a certain intensity so both can be represented by vectors. Using these vectors you can create a program that calculates exactly how an object in the scene should be coloured and move to give a realistic impression of lighting and wind. Even better, you can write your progra
29、m so that the vectors representing sun and wind constantly change their direction and magnitude thus you can create gusts of wind and clouds passing overhead.MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 45.6.2: Whats your vector, Victor? SummaryVectors and Language Vectors in S
30、ports Vectors and VisualsName the career where vectors are used?How are the vectors used in this career?Name the career where vectors are used?How are the vectors used in this career?Name the career where vectors are used?How are the vectors used in this career?After listening and discussing the var
31、ious careers, why would it be necessary for you to learn about vectors?MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 55.6.3: Distance and Displacement DiscoveredComplete your questions by indicating the distance and displacement for each graphic. Work with your partner to verify
32、 your results.Partner A:_ Partner B:_Distance:_Displacement:_Distance:_Displacement:_Distance:_Displacement:_Distance:_Displacement:_Distance:_Displacement:_Distance:_Displacement:_MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 65.6.4: Frayer ModelDefinition: Facts/Characteristic
33、s:Examples: Non-examples:MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 75.6.5: Vectors in the SchoolDirections: You have been assigned to find how far it is from your math classroom to the cafeteria You will need to provide a vector diagram of the path that person would need to
34、take in order to get to the cafeteria. Remember: A person who is not familiar with the school should be able to follow your map.Drawing Vectors: The Rules: Decide on an appropriate scale for the map. (1cm on the graph paper = 1 m on the floor) Draw a compass to indicate directions NORTH, SOUTH, EAST
35、 and WEST. Draw arrow heads at the end of your vectors and make sure you have the arrows pointing in the right direction.Measurement recording sheet: You will use the length of your foot to determine the magnitude of your vectors. Length of your foot in metres:_ Make sure to use heel-to-toe when cou
36、nting your steps. Start making your map.Reflecting: How far is it to the cafeteria? Is this a vector quantity or a scalar quantity? What are your reasons for this answer?Class Recording: Record your measurement on the board. Looking at the class data, are all the measurements the same? Can you give
37、some possible explanations for why the measurements are different?MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 85.6.6: Vector AnalysisThe Grade 12 Leadership class has decided to go backpacking in Algonquin Provincial Park. There are two groups of students and each group is sta
38、rting at the same access point. Johns group starts at access point A and travels 3 km north, 2 km east, 1 km south, 4 km east and 5 km north to reach the campsite. Cathys group decides to take another trail to the campsite. Her group starts at access point A and travels 4 km east, to start. Both gro
39、ups are to meet at the same campsite for dinner. The last group to the campsite makes dinner and sets up the camp.1. Draw a vector diagram to show Johns journey. The diagram will need to have a scale and a compass to denote direction. 2. Since Cathys group started out in an easterly direction, how f
40、ar and in which direction would her group need to travel to reach the campsite? Give clear reasons to support your answer. 3. Draw a separate vector diagram to show Cathys journey to the campsite. Remember to include the scale and compass.4. Compare the distances the two groups travelled. 5. Both gr
41、oups were travelling at 3 km/hour, how long did it take each group to reach the campsite. Which group is making dinner and setting up the camp? Show your calculations to justify your reasoning. Questions modified from www.resources.elearningontario.ca ELO1073900MCT4C : Unit 5 Vectors (Draft August 2
42、007) - Last saved 27/09/2018 9Unit 5: Day 7: “Vector”y is ours!Minds On: 10Action: 50Consolidate:15Total=75 minLearning Goal: Understand the equality of vectors Represent a vector as directed line segment with directions expressed in different ways Resolving a vector into horizontal and vertical com
43、ponents in context using Pythagorean theorem where appropriateMaterials BLM 5.7.1-5.7.5 Protractors, rulers, graph paper MCT_U5L7GSP1.gsp MCT_U5L7GSP2.gsp MCT_U5L7AVI1.av (optional)AssessmentOpportunitiesMinds On Pairs ActivityDevelop the concept of equal vectors using BLM 5.7.1 and revisiting the i
44、deas of magnitude and direction from the previous lesson.Ensure that students have discerned that the conditions for equal vectors as same magnitude and direction.Action! Pairs ExplorationDistribute BLM 5.7.2 to 5.7.4. Students may wish to explore using MCT_U5L7GSP1.gsp.Note: This portion of the les
45、son could be set up as stations (though 5.7.4 requires some knowledge of direction).Learning Skills/Observation/CheckbricUse this opportunity to collect data about students independent learning skills.Mathematical Process Focus: Connecting Students will connect the mathematics to situations drawn fr
46、om other contexts.Consolidate DebriefWhole Class DiscussionHave students summarize their learning by creating a class note.Distribute BLM 5.7.5. The overview of the assignment should lead to a discussion of the concepts covered during the period. The map is available in MCT_U5L7GSP2.gsp.Students wil
47、l require two copies of the map.Concept PracticeApplicationHome Activity or Further Classroom ConsolidationComplete BLM 5.7.5.MCT4C : Unit 5 Vectors (Draft August 2007) - Last saved 27/09/2018 105.7.1: All Vectors are not Created Equal1. Complete the following statement.A vector is a quantity which has and .2. Examine the vectors in the diagram. Classify the vectors in the table below using appropriate vector notation (e.g., ).ABSame Magnitude Only Same Directio
