Section 3: Displacements and Vectors We have run a some relay races and represented the races as tables. Let s find a way to represent the race mathematically. Suppose Team 1 had the following relay legs and coin tosses: Leg Number Number of Cubes Coin Toss 1 5 heads 2 8 tails 3 4 heads Suppose Team 2 had the following relay legs and coin tosses: Leg Number Number of Cubes Coin Toss 1 4 tails 2 6 tails 3 6 heads How can we capture this information? One way would be to tell a story. There are many different stories that could be told about the race above. Let s see what kind of stories each team can create using the information above? 18 Racing Against Time
Team Work - People Talk Each team should write a story about what might have happened in the race described in the tables above. Describe carefully everything that may have happened in the race. Be creative, and try to describe a real life situation that might have produced the data from the race tables, using names of some of your classmates as characters in the story. Each team should choose someone to tell the team s story of the race to the class. Post the stories from each team on chart paper in the classroom. Class Work Racing Against Time 19
Another way to do it would be to represent the movements in the race with a diagram or drawing Team Work - People Drawing Each team should create below a drawing or diagram showing what might have happened in the race described in the table above. Class Work Each team should choose someone to describe the team s drawing of the race to the class. Post the drawing from each team on chart paper in the classroom next to the team s story of the race. Class Discussion Discuss as a class the following question: Which representation of the story did you like better, the story or the drawing? Why? 20 Racing Against Time
Let s look more carefully at the drawings of the race by the class and try to extract the features that are contained in the drawings. We ve talked about features many times by now. The general categories we ve used in the past are: Objects, Actions, and Relations. Each team should describe below the features that are evident in the drawings that the class made of the relay race. Remember to concentrate on features which are in the drawing of the race. Team Work Feature Category Each team should choose someone to describe the team s list of features of the race drawing to the class. Post the list of features from each team on chart paper in the classroom next to the team s drawing of the race. Class Work Racing Against Time 21
What was the list of features that the class determined were evident in the diagrams that the teams drew? Below is a one possible diagram for the race that Team 1 ran in the example above. Let s look at it together, and extract some mathematical features. We ll write the features in bold type. The first leg of the race is built by stacking as many cubes as you can. The number of cubes in a race leg determines the length of the race leg. Next a coin flip determines which direction the leg will travel. The first leg of the race also has a starting location and an ending location. The second leg of the race is built the same way. Since we are running a relay race, the second leg s starting location is the same as the first leg s ending location. Finally, the third leg s starting location is the same as the second leg s ending location. The total displacement of the team is the combination of the three legs of the race. The words that are in bold above capture some of the features that are in the diagram above. Were these some of the features your class listed? Let s concentrate on these features, and try to mathematize them, but instead of creating mathematical sentences, like we did in the Trip Line, we want to create a mathematical drawing. In the Trip Line we used the term icon for a symbolic figure that looks something like the object it represents. Mathematical drawings are also a type of icon. What s the best way to represent each of the legs of the race as an iconic drawing? We need a symbolic drawing that possesses the two main characteristics of a leg: length and direction. We ll use the term feature drawing for an iconic drawing that captures the essential features of a people drawing or diagram. 22 Racing Against Time
Each team should create a feature drawing to represent the relay race legs and expresses the features of length and direction. Below is one possible people drawing of the relay race by Team 1. Team Work Create your team s feature drawing below. Racing Against Time 23
Class Work The class should collect each team s feature drawing on a sheet (or sheets) of chart paper. Post the chart paper with the feature drawings in the classroom. Now we will show the kind of feature drawing that many mathematicians use for objects that have the same type of characteristics as the relay race legs. Here is a possible people drawing of the relay race legs of Team 1. Here s the race represented using arrows. 24 Racing Against Time
The legs of the race are represented by the arrows. Mathematicians call the arrows vectors. Vectors are objects that have two essential characteristics: a length and a direction. Recall that in both the Trip Line and Road Coloring Units, once we had new objects we needed symbols to represent the objects. We will use letters like v or w to stand for the vectors. There was another feature that reminds us of the Trip Line. In that unit, the stops on the trip came in a certain order. We used a subscript (subscript comes from Latin: sub means below, like submarine and script come from the word for write, so a subscript is just something which is written below ). The subscript just tells us which leg of the race the vector came from. Can you tell which leg of the race v 3 is? If we wrote that the last leg of a race was v 8, could you tell how many legs the race had? Below we have our previous arrow diagram labeled with our new vector variables. v 1 v 2 v 3 Racing Against Time 25
Individual Work For each of the relay race people drawings below, find the equivalent feature drawing using arrows for the relay race legs. Make sure to label each of the arrows with a vector variable that has a subscript. 1) 2) 26 Racing Against Time
3) 4) Racing Against Time 27