Arrow Flight and the Quadratic Equation Brent Sauve Hannahville Indian School

Transcription

1 Introduction By Jennifer (McLeod) Anziano Arrow Flight and the Quadratic Equation Brent Sauve Hannahville Indian School Wild game was an important part of my families food. We had very little money, and were lucky that my Dad was an avid hunter. I am the oldest of seven children, and I was taught how to hunt at a very early age. I accompanied my Dad even before Kindergarten, riding on his shoulders and he trudged through the swamps hunting for ducks. By the time I was 10, I was hunting deer on my own, using a 50# lemonwood long-bow. It took lots of practice before I was strong enough to pull the bow, and lots more practice before I could even hit the target. Over time, my brothers and sisters also learned to shoot a bow, and backyard competitions were an every night event. We became good shots, and had to make up new rules because we were splitting too many arrows. Then my Dad decided to make it interesting. First, he made us increase the distance to the target. Little by little, he increased the distance, and initially we would watch our arrows fall short of the target. Eventually, even though we didn t know the math, we had to calculate how much to raise the front of the arrow to compensate for it s drop due to the increased distance (therefore giving gravity more time to act.) We got to the point where we were shooting our arrows from across the street to hit the target in our backyard. This created quite an attraction in our neighborhood. People would come to watch us shoot (and to stop any traffic coming up the road during our competition). They would cheer and laugh, depending on whether we hit the target, split someone else s arrow, or miss the target completely. Of course my Dad could win whenever he wanted to, but once in a while he would miss so that one of us could win (At the time, we honestly believed we won!).

2 When we got to the point where distance wasn t a big enough challenge anymore, Dad made it more interesting by making us shoot the arrows that weren t straight. This proved to be a bit more dangerous, but we got pretty good at figuring out which way the arrow would go. We had to quit though, when too many arrows were ending up in our neighbors yards. My Dad passed away, and the backyard competitions ended. But I ll never forget the joy we shared when one of us could finally pull the bow, actually hit the target, and especially when we brought home food. Born to Laurence and Carleen McLeod, 1955 in Pontiac, Michigan. She is a member of the Sault Ste Marie Tribe of Chippewa Indians, Eagle Clan, first degree Mide. Jennifer is married to George Anziano (Little Traverse Bay Band of Odawa Indians), and she is an instructor at the Hannahville Indian School. Together, George and Jennifer have 11 children and 10 grandchildren. Jennifer and George live in a log cabin in Wilson, MI with their two malamute dogs, a wild mustang horse and a brand new kitten (who already hunts rabbits!). Jennifer and George follow a traditional life, and also continue to hunt deer, geese, moose, bear and elk for food. The bow and arrow was the weapon of choice for many different cultures throughout history. Native Americans used the bow for defense and for the taking of game animals for food. Many aspects of the flight of the arrow can be described using mathematics. When shot, the general flight of an arrow is a parabolic arc. Of course, any parabola can be described using a quadratic equation, ax + bx + c = 0. This lesson helps to define for students how a, b, and c affect the graph of the quadratic equation, and should be considered as an introduction into graphing of quadratics. The National Council of Teachers of Mathematics (NCTM) standards that are best addressed include the Algebra Standard of the Content Standards, and the Connections Standard of the Processes Standards. Quadratic equations and the graphing of them definitely fits into the Algebra Standard. Connections between the cultural heritage of Native Americans and the mathematics defining the flight of an arrow also are apparent. By using a culturally important object, such as the bow and arrow, the mathematics that sometimes can seem so abstract can take on important meaning and give ownership of the ideas of quadratics to our students. This ownership of a mathematical idea also helps fulfill the concept of the NCTM connections.

3 Quadratic Equations/Expressions The flight of an arrow can be described with a quadratic expression. The basic quadratic expression takes the form ax + bx + c. We will explore the quadratic expression/equation by looking at the flight of an arrow. The graph of a basic quadratic equation is shown below and doesn t look much like path of an arrow. f ( x) = x + x + 1 Where a = 1, b = 1, and c = 1. If we let the y-axis be the midpoint between were we are shooting from and where the arrow hits, we can see that the graph is not centered on the y-axis. It is slightly to the left. This is caused by the bx term of our quadratic equation. Below is the same graph without the bx term (actually when b=0). = x + 1. This centers our graph on the y-axis, but it still doesn t look like the flight of an arrow. We can turn the graph upside down by letting a = -1. = x + 1 This looks a little better, but if the tick marks on the graph represent a yard each, our arrow only went about two yards. While being accurate at two yards is important, we probably will have to shoot a little farther to bring game home. Lets look at the quadratic equation a little closer.

4 Obviously, in our quadratic equation f ( x) = ax + bx + c the coefficient of x term, the a in the equation, must be negative to get the graph to look like the flight of an arrow. The coefficient a really could indicate the forces acting on the arrow. These forces include the weight of the arrow, initial arrow speed, and wind resistance among others. If we look at changing the a coefficient to a -.01 we will get a graph that more closely resembles the flight of an arrow. At this point we don t need the second term of the equation (b = 0). =.01x + 1. This looks much more like the graph of an arrow in flight. In fact, if the distance between each grid line is equal to one yard, our arrow rose 1 yard while it flew 0 yards. This is much more realistic than the other graphs we made. If we are to interpret the above graph as shooting an arrow, then the arrow was shot from the ground, flew twenty yards, and then hit the ground again. Since most of the things we will be shooting at will be above the ground, and since when we shoot we are not shooting from ground level we should raise the graph up a little. By changing the c term of our quadratic equation ( the overall graph. f ( x) = ax + bx + c ) we can raise or lower =.01x +. Again this appears to have helped our graph represent the flight of a real arrow. The arrow was fired from a yard above the ground, flew twenty yards, and hit an object a yard above the ground. During its flight the arrow rose one yard before coming back down. If our arrows are heavier, or if we shoot into a strong wind, or if our arrows have bad fletching, the path that our arrow flew would change. Any of the above problems would greatly affect our shot.

5 Here is the flight of an arrow if we only change the a part of the first term of our equation, perhaps by shooting an arrow that was a little heavier. Here is f ( x) =.0x + This shows that the arrow would still only rise one yard above the line of sight. Obviously, changing the a of the quadratic equation f ( x) = ax + c from -.01 to -.0 had a great affect on our graph. To show that we are not shooting off the ground again we will have to move the graph to the left a little. To do so we have to reinsert the bx part of the quadratic equation. =.0x + This shows that the added weight of our arrow would have made us miss whatever we were shooting at. To hit our target we ll have to aim higher, thus taking into account the added weight of the arrow. =.0x + 3. By adding one to the third term it will raise our aim and we should hit what we aimed at. Here is f ( x) =.0x + 3 Again we will have to adjust our graph to the right so that we are shooting from 1 yard above the ground. We can do this by setting our b value to zero. =.0x + 3

6 Here we aimed two yards above the target to get our arrow to hit were we wanted it to. The added weight of the arrow changed our aim point. Lets compare two graphs below. A B Graph A, represented by the equation f ( x) =.01x +, was when we shot from a yard above the ground, and hit a target twenty yards away and a yard above the ground. Graph B represented by the equation f ( x) =.0x + was when our arrow had a little more weight. Remember the added weight was represented by changing -.01 to -.0. the number is more negative (smaller) because the added weight of the arrow would adversely affect the flight of the arrow. The -.1x in graph B s equation simply moved the graph to the left so that the arrow was shot from a yard above the ground. The arrow in graph B again rose 1 yard during its flight but this time it hit the ground at 18 yards, missing what we were shooting at. Equipment: TI 83 Graphing calculators. Above is the windows screen and the Y= screen from the TI 83 calculators used in this lesson. Students should use a new plot each time they put a new equation in the calculator. In this way different graphs can be quickly compared or gone back to. All equations with a darkened in = will be displayed on the calculator. The above Y = screen would show only the f ( x) =.0x +. As an example, below we can compare the graphs of ( ).01 f x = x + 1and of f ( x) =.0x + graphs and discuss how changing certain numbers in the quadratic equation changed the graph of the equation Activities:

7 Have students throw a ball, but have a rod or a stick set at a height that the students must throw under. Raise the rod and compare the distance thrown with the original distance when the rod was at a lower height. Have students see how high they can throw the ball. How far in horizontal distance did the ball go? As the students try to throw the ball higher they should notice that the horizontal distance decreases. Go back into the classroom and illustrate how the quadratic equation would change as students tried to throw for horizontal distance and as they tried to throw for vertical height. Extensions: An entire unit could easily be based around this one basic lesson. Depending on the class and the students, there are a number of extensions that could be included into this lesson to have it last a number of days. These extensions include: -Have students use different numbers in place of the a, the b, and the c in the quadratic equation f ( x) = ax + bx + c. Students should gain a deeper understanding of how small changes to parts of the equation dramatically change the graph (and thus the equation). -Discuss with students why it might not be a good idea to have the y axis for the midpoint for our graphs. Question: Can we have a negative distance? Have students adjust the quadratic equations so that the arrow is shot from the y-axis. -Solving of systems of equations. Each different flight of the arrow is a different quadratic equation. By solving a pair of equations students could see where the arrows would cross (the intersection of the graphs). Solving of systems using graphing and by algebraic means would be a great extension to this lesson. Resources: University of Chicago School of Mathematics Project; Algebra. Chapter 1, pages Scott, Foresman, Copyright TI-83 Plus Calculators. Numerous outside readings involving the use of bows and arrows and the cultural/historic use of the bow. Name 1). Change this equation, f ( x) = 3x + x + 1, so that the graph is shifted 3 places down.

8 ). What would need to be changed so that f ( x) = x x + 3 was shifted to the right 4 places. 3). Which graph is shown to the right? 1 a. f ( x) = x 3 b. f ( x) = 3x 1 c. f ( x) = x 3 d. f ( x) = 3x 4). In the quadratic equation if a is negative the parabola will open. In the quadratic equation if a is postive the parabola will open. 5). Sketch the graph of this equation: f ( x) = x + x 3 6). Match each equation with the appropriate graph.

9 a. f ( x) = x + x + 1 b. f ( x) = x + x 1 c. f ( x) = x + x + 1 d. f ( x) = x + x + 4 I II III IV