Mathematics/8/Geometry 14.1 Students prove the Pythagorean theorem. 15.1 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
Science/8/Investigation and Experimentation 9.0 Scientific progress is made by asking meaningful questions and conducting careful investigations. As a basis for understanding this concept and addressing the content in the other three strands, students should develop their own questions and perform investigations. Students will:
a. Plan and conduct a scientific investigation to test a hypothesis.
b. Evaluate the accuracy and reproducibility of data.
c. Distinguish between variable and controlled parameters in a test.
d. Recognize the slope of the linear graph as the constant in the relationship y=kx and apply this principle in interpreting graphs constructed from data.
e. Construct appropriate graphs from data and develop quantitative statements about the relationships between variables.
f. Apply simple mathematic relationships to determine a missing quantity in a mathematic expression, given the two remaining terms (including speed = distance/time, density = mass/volume, force = pressure x area, volume = area x height).
g. Distinguish between linear and nonlinear relationships on a graph of data.
To understand and apply the Pythagorean Theorem, students will work in groups to make a simulated model of two NFL players setting up a run for the goal line.
Students will view a Science of NFL Football video: The Pythagorean Theorem. To understand and apply the Pythagorean Theorem, students will work in groups to make a simulated model of two NFL players setting up a run for the goal line.
Students will be able to:
Ask scientific questions.
Explore the science and math behind using the Pythagorean Theorem in football.
Make a simulated model of two NFL players setting up a run for the goal line.
Maintain a record of their observations.
Use the record of their observations to construct reasonable explanations for questions presented to them.
Science of NFL Football video: The Pythagorean Theorem, white board and markers. For each group of students: a paper towel tube cut lengthwise in half to form ramps or two 12-inch wooden rulers with grooves to be used as ramps, two glass marbles or two steelies, a 2-foot square piece of butcher paper, a metric ruler, a protractor, and masking tape.
Anticipatory Set (Lead-in):
Tell the students, “Many years ago, a Greek philosopher by the name of Pythagoras discovered an amazing fact about triangles; if the triangle had a right angle (90°) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together! This amazing discovery is called The Pythagorean Theorem. In geometry, the Pythagorean Theorem is the relationship among the three sides of a right triangle. The theorem states that the sum of the squares of the lengths of the two legs of the triangle (sides a and b) is equal to the square of the length of the hypotenuse (c). Also, the sum of the areas of squares with side equal to the legs of the triangle (sides a and b) equals the area of the square with its side equal to the hypotenuse.”
“Why is the Pythagorean Theorem useful to know? How can we use this amazing discovery about triangles in our everyday lives? Right triangles are everywhere aren’t they? Anywhere that there is a right triangle is a place where the Pythagorean Theorem could be used. The Pythagorean Theorem has been used in many ways including: playing baseball, constructing a building, and measuring a ramp (like on a moving truck). How many of you like to watch or play football? If you do, the Pythagorean Theorem could help you understand some of what is going on. Today we are going to watch a video about the Pythagorean Theorem and make a really exciting simulated model of two NFL players setting up a run for the goal line.”
Tell the students that they are going to make a simulated model of two NFL players setting up a run for the goal line.
Tell the students that as they make this model they should keep the following question in mind: What factors affect the ability of the defensive player (LB - line backer) to catch up with the offensive player (WR - wide receiver) before the offensive player reaches the goal line? (Write this question on the board.)
Divide the class into groups of 4-5 students. Make sure that each group has a paper towel tube cut lengthwise in half to form ramps or two 12-inch wooden rulers with grooves to be used as ramps, two glass marbles or two steelies, a 2-foot square piece of butcher paper, a metric ruler, a protractor, and masking tape.
Next, each group should complete the following steps:
a. Draw a horizontal line near the top of the paper; label this “Goal Line”.
b. Draw another horizontal line parallel to the Goal Line and about 30 cm below it. Label this the “Action Line”.
c. On the left side of the Action Line, mark a spot labeled LB. On the same line, mark another spot about 30 cm to the right and label it WR.
d. Draw a line from the LB to the WR, this will form the baseline of the right triangle.
e. For the second leg of the triangle, construct a 90-degree angle from the WR to the Goal Line.
f. The third leg is the hypotenuse of the triangle which starts at the LB’s spot on the Action Line. The angle that will be formed between the baseline and the hypotenuse is called the “angle of pursuit”.
g. Use masking tape to secure one ramp at the 90-degree position on the Action Line headed towards the Goal Line. This is the place where the WR begins his run.
h. Use masking tape to secure the second ramp on the Action Line at the place where the LB’s spot is marked, about 30 cm from the WR.
i. Elevate the two ramps using textbooks, one under the WR’s ramp and two under the LB’s.
j. Put a mark the same distance from the bottom of the ramp about halfway up on both. This is the spot where you will place the marbles to begin their runs.
k. Release the two marbles on both ramps at the same time and in the same spot.
l. Observe what happens and adjust the angle and the elevation of the LB’s ramp only. The WR is a fixed position; do not change its ramp at all.
m. Release the marbles again. Observe and adjust until the objective is achieved: the LB is able to hit the WR before it reaches the Goal Line.
n. Use your protractor to measure the angle of pursuit.
o. Change the position of the LB on the Action Line and repeat the steps above. When the objective is met; use your protractor to measure the angle of pursuit.
Closure (Reflect Anticipatory Set):
One person from each group should be chosen from within the group to share with the class the answer to the original question: What factors affect the defensive player (LB - line backer) to catch up with the offensive player (WR - wide receiver) before the offensive player reaches the goal line?
Assessments & notes
Plan for Independent Practice:
Tell the students that they are to work in their same group to answer the following questions about the simulation:
How does changing the elevation change the velocity of the LB?
How does changing the angle change the velocity of the LB?
How does changing the angle change the distance that the LB runs?
Measure in centimeters the length of the two legs of the triangle and use the formula a^2 + b^2= c^2 to calculate the hypotenuse (c^2) and compare this calculation with the measured value from your paper.
Explain how the angle of pursuit affects the LB’s velocity when the LB’s position changes.
For a bigger challenge, use the physics of v = d/t to determine the velocity of each player. NOTE: the time is the same for both players. Set up a ratio and proportion equation and discuss how much faster the LB must run than the WR.
Assessment Based on Objectives:
Begin the next day’s lesson with the quiz titled, “The Pythagorean Theorem”. (See attached quiz under "Resources")
Possible Connections to Other Subjects:
Math/Art/Technology: Students could find other applications for the Pythagorean Theorem in the world around them and make a PowerPoint presentation about what they found using detailed written descriptions and illustrations.