# LESSON TITLE: Math in Basketball (by Deborah L. Ives, Ed.D) GRADE LEVEL/COURSE: Grades 7-12 Algebra. TIME ALLOTMENT: Two 45-minute class periods

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9 LEARNING STANDARDS & SAMPLE END-OF-COURSE (EOC) QUESTIONS Sample Related End-of-Course (EOC) Questions (available for download at These sample questions, selected from Common Core Resources, cover the same algebraic concepts explored in this lesson. Common Core State Standards 2010 [Note: You may also wish to view Pathways 1 and 2 for Algebra I connections in the CCSS] Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Algebra Overview Seeing Structure in Expressions A.SSE.1a, 1b, 2 Interpret the structure of expressions. A.SSE.3a, 3b Write expressions in equivalent forms to solve problems. Arithmetic with Polynomials and Rational Functions A.APR.1 Perform arithmetic operations on polynomials. A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Creating Equations A.CED.1, 2 Create equations that describe numbers or relationships. Reasoning with Equations and Inequalities A.REI.1 Understand solving equations as a process of reasoning and explain the reasoning. A.REI.4. Solve quadratic equations in one variable. A.REI.4b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Represent and solve equations and inequalities graphically. A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Functions Overview (Quadratics) Interpreting Functions F.IF. 1, 2 Understand the concept of a function and use function notation. 9

10 F.IF.4, 5, 6 Interpret functions that arise in applications in terms of a context. Analyzing Functions using different representations F.IF. 7a Graph linear and quadratic functions and show intercepts, maxima, and minima F.IF.8a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions F.BF.1 Build a function that models a relationship between two quantities. Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice. 10

11 Name: Date: Math in Basketball: Take the Challenge Student Handout When NBA player Elton Brand steps to the free throw line, a number of key variables can influence his shot. Your challenge is to use the 3 key variables and Elton s stats to figure out the maximum height the ball reaches on its way into the basket to make the shot. (This activity can also be completed online. Go to click on The Challenges, then scroll down and click on Math in Basketball: Take the Challenge. ) FAST BREAK FACTS: KNOW THE STATS 1. Identify what you already know. Look at the Fast Break Facts (following the last question in this handout) for information about the 3 key variables and Elton s stats. The Acceleration of Gravity: Elton s Initial Vertical Velocity : Elton s Release Height: Combine these 3 key variables used to calculate the ball s height, h, at a given time, t, by setting up an equation to get started. h(t) = AT WHAT TIME(S) DOES THE BALL REACH 10 FEET? 2. Plan it out. What strategy will you use? Select one or more representations, such as your equation or a graph (found on the last page), to calculate the value(s) of t when the ball reaches a height of 10 feet.

14 Math in Basketball: Take the challenge Student Handout FAST BREAK FACTS THE 3 KEY VARIABLES The Acceleration of Gravity which causes a ball to speed up, or accelerate, when falling at a rate of -32ft/sec 2. Use only downward pull or half of -32ft/sec 2, which is -16t 2. Initial Upward Velocity (v 0 ) - the angle and speed when it leaves the player s hand. Multiply by time to get the vertical distance traveled. Release Height (h 0 ) - the starting position of the ball. ELTON BRAND STATS Elton s Average Initial Upward Velocity: Elton s Average Release Height: STANDARD COURT MEASUREMENTS Height of the basketball hoop off the floor: Distance from the free throw line to backboard Diameter of hoop/rim 24 ft/sec 7.0 ft 10 ft 15 ft 18 in GRAPH YOUR DATA Height (in feet) Time (in seconds)

15 Name: Date: Math in Basketball: Try other basketball challenges Student Handout In this challenge, you get to choose a new set of player stats, then use the 3 key variables to figure out the maximum height the ball reaches during a free throw shot. (This activity can also be completed online. Go to click on The Challenges, then scroll down and click on Math in Basketball: Try other challenges. ) FAST BREAK FACTS: KNOW THE STATS 1. Identify what you already know. Look at the Fast Break Facts on the last page of this handout for information about the 3 key variables and select a player s stats from the choices below: The Acceleration of Gravity: Initial Vertical Velocity (Select one): 20 ft/sec 22 ft/sec 24 ft/sec Release Height (Select one): 5 ft 6 ft 8 ft Combine these 3 key variables used to calculate the ball s height, h, at a given time, t, by setting up an equation to get started. h(t) = AT WHAT TIME(S) DOES THE BALL REACH 10 FEET? 2. Plan it out. What strategy will you use? Select one or more representations, such as your equation or a graph (found on the last page), to calculate the value(s) of t when the ball reaches a height of 10 feet.

18 Math in Basketball: Try other challenges Student Handout FAST BREAK FACTS THE 3 KEY VARIABLES The Acceleration of Gravity which causes a ball to speed up, or accelerate, when falling at a rate of -32ft/sec 2. Use only downward pull or half of -32ft/sec 2, which is -16t 2. Initial Upward Velocity (v 0 ) - the angle and speed when it leaves the player s hand. Multiply by time to get the vertical distance traveled. Release Height (h 0 ) - the starting position of the ball. PLAYER S STATS (Select one of each) Initial Upward Velocity: 20 ft/sec 22 ft/sec 24 ft/sec Release Height: 5 ft 6 ft 8 ft STANDARD COURT MEASUREMENTS Height of the basketball hoop off the floor: Distance from the free throw line to backboard Diameter of hoop/rim 10 ft 15 ft 18 in GRAPH YOUR DATA Height (in feet) Time (in seconds)

19 Math in Basketball: Take the Challenge ANSWER KEY When NBA player Elton Brand steps to the free throw line, a number of key variables can influence his shot. Your challenge is to use the 3 key variables and Elton s stats to figure out the maximum height the ball reaches on its way into the basket to make the shot. (This activity can also be completed online. Go to click on The Challenges, then scroll down and click on Math in Basketball: Take the Challenge. ) FAST BREAK FACTS: KNOW THE STATS 1. Identify what you already know. Look at the Fast Break Facts (following the last question in this handout) for information about the 3 key variables and Elton s stats. The Acceleration of Gravity: -16t 2 Elton s Initial Vertical Velocity : 24 ft/sec Elton s Release Height: 7 feet Combine these 3 key variables used to calculate the ball s height, h, at a given time, t, by setting up an equation to get started. h(t) = -16t t + 7 AT WHAT TIME(S) DOES THE BALL REACH 10 FEET? 2. Plan it out. What strategy will you use? Select one or more representations, such as your equation or a graph (found on the last page), to calculate the value(s) of t when the ball reaches a height of 10 feet. Strategy A: The height (h) of the ball, in feet, at a given time (t) is represented by the equation: h(t) = -16t t + 7 where 24 is the initial vertical velocity and 7 is the release height The value for t at 10 feet would occur at two points in time, one on the way up, the other at the hoop. Substitute 10 feet for h(t): 10 = -16t t + 7 Write in standard form: 0 = at 2 + bt + c by subtracting 10 from each side: 0 = -16t t + (-3); where a = -16, b = 24, and c = -3 Solve algebraically using the quadratic formula, t =, or complete the square to find two values of t.

22 Math in Basketball: Take the challenge Answer Key FAST BREAK FACTS THE 3 KEY VARIABLES The Acceleration of Gravity which causes a ball to speed up, or accelerate, when falling at a rate of -32ft/sec 2. Use only downward pull or half of -32ft/sec 2, which is -16t 2. Initial Upward Velocity (v 0 ) - the angle and speed when it leaves the player s hand. Multiply by time to get the vertical distance traveled. Release Height (h 0 ) - the starting position of the ball. ELTON BRAND STATS Elton s Average Initial Upward Velocity: Elton s Average Release Height: STANDARD COURT MEASUREMENTS Height of the basketball hoop off the floor: Distance from the free throw line to backboard Diameter of hoop/rim 24 ft/sec 7.0 ft 10 ft 15 ft 18 in GRAPH YOUR DATA (0.75, 16) Height (feet) (0, 7) (0.14, 10) (1.36, 10) Time (seconds)

23 Math in Basketball: Try other basketball challenges ANSWER KEY In this challenge, you get to choose a new set of player stats, then use the 3 key variables to figure out the maximum height the ball reaches during a free throw shot. (This activity can also be completed online. Go to click on The Challenges, then scroll down and click on Math in Basketball: Try other challenges. ) FAST BREAK FACTS: KNOW THE STATS 1. Identify what you already know. Look at the Fast Break Facts on the last page of this handout for information about the 3 key variables and select a player s stats from the choices below: The Acceleration of Gravity: Initial Vertical Velocity (Select one): 20 ft/sec 22 ft/sec 24 ft/sec Release Height (Select one): 5 ft 6 ft 8 ft Combine these 3 key variables used to calculate the ball s height, h, at a given time, t, by setting up an equation to get started. h(t) = The answer will vary depending on the stats chosen by the student, but will follow this model: h(t) = -16t 2 + v 0 t + h 0 (Note: Initial Vertical Velocity = v 0 ; Release Height = h 0 ) For example, if the student selected Initial Vertical Velocity = 5 ft and Release Height = 20 ft/sec, the correct equation would be h(t) = -16t t + 5. AT WHAT TIME(S) DOES THE BALL REACH 10 FEET? 2. Plan it out. What strategy will you use? Select one or more representations, such as your equation or a graph (found on the last page), to calculate the value(s) of t when the ball reaches a height of 10 feet.

25 Math in Basketball: Try other challenges Answer Key WHAT IS THE MAXIMUM HEIGHT OF THE BASKETBALL? 6. Plan it out. What strategy will you use? Select one or more representations, such as your equation or a graph (found on the last page), to calculate the maximum height the ball will reach on its way to the basket. Using the value of t, or time, when the ball reaches its maximum height, you can substitute that value into the equation you set up for h(t) to find the height of the ball at that time, or use graphical representation. 7. Solve your problem. Show all your steps. You may use the graph on the last page of this handout or show your work in the space below. See below for all solutions. ALL FINAL SOLUTIONS: [h 0 = Release Height and v 0 = Initial Vertical Velocity] h 0 = 5 and v 0 = 20 ft/sec h(t) = -16t t + 5 At what times (in sec) does the ball reach 10 feet? t = 0.35 t = 0.90 At what time does the ball reach its maximum height? t = 0.63 (or 5/8 sec) Maximum height = feet h 0 = 5 and v 0 = 22 ft/sec h(t) = -16t t + 5 At what times (in sec) does the ball reach 10 feet? t = 0.29 t = 1.09 At what time does the ball reach its maximum height? t = 0.69 (or 11/16 sec) Maximum height = feet h 0 = 5 and v 0 = 24 ft/sec h(t) = -16t t + 5 At what times (in sec) does the ball reach 10 feet? t = 0.25 t = 1.25 At what time does the ball reach its maximum height? t = 0.75 (or ¾ sec) Maximum height = 14 feet h 0 = 6 and v 0 = 20 ft/sec h(t) = -16t t + 6 At what times (in sec) does the ball reach 10 feet? t = 0.25 t = 1.0 At what time does the ball reach its maximum height? t = (or 5/8 sec) Maximum height = feet h 0 = 6 and v 0 = 22 ft/sec h(t) = -16t t + 6 At what times (in sec) does the ball reach 10 feet? t = 0.22 t = 1.16 At what time does the ball reach its maximum height? t = (or 11/16 sec) Maximum height = feet

26 Math in Basketball: Try other challenges Answer Key h 0 = 6 and v 0 = 24 ft/sec h(t) = -16t t + 6 At what times (in sec) does the ball reach 10 feet? t = 0.19 t = 1.31 At what time does the ball reach its maximum height? t = 0.75 (or ¾ sec) Maximum height = 15 feet h 0 = 8 and v 0 = 20 ft/sec h(t) = -16t t + 8 At what times (in sec) does the ball reach 10 feet? t = 0.11 t = 1.14 At what time does the ball reach its maximum height? t = 0.63 (or 5/8 sec) Maximum height = feet h 0 = 8 and v 0 = 22 ft/sec h(t) = -16t t + 8 At what times (in sec) does the ball reach 10 feet? t = 0.10 t = 1.28 At what time does the ball reach its maximum height? t = (or 11/16 sec) Maximum height = feet h 0 = 8 and v 0 = 24 ft/sec h(t) = -16t t + 8 At what times (in sec) does the ball reach 10 feet? t = 0.09 t = 1.41 At what time does the ball reach its maximum height? t = 0.75 (or ¾ sec) Maximum height = 17 feet FAST BREAK FACTS THE 3 KEY VARIABLES The Acceleration of Gravity which causes a ball to speed up, or accelerate, when falling at a rate of -32ft/sec 2. Use only downward pull or half of -32ft/sec 2, which is -16t 2. Initial Upward Velocity (v 0 ) - the angle and speed when it leaves the player s hand. Multiply by time to get the vertical distance traveled. Release Height (h 0 ) - the starting position of the ball. PLAYER S STATS (Select one of each) Initial Upward Velocity: 20 ft/sec 22 ft/sec 24 ft/sec Release Height: 5 ft 6 ft 8 ft STANDARD COURT MEASUREMENTS Height of the basketball hoop off the floor: Distance from the free throw line to backboard Diameter of hoop/rim 10 ft 15 ft 18 in

27 Math in Basketball: Try other challenges Answer Key GRAPH YOUR DATA Release height = 5 ft, Initial Vertical Velocity = 20 ft/sec Release height = 5 ft, Initial Vertical Velocity = 22 ft/sec

28 Math in Basketball: Try other challenges Answer Key Release height = 5 ft, Initial Vertical Velocity = 24 ft/sec Release height = 6 ft, Initial Vertical Velocity = 20 ft/sec

29 Math in Basketball: Try other challenges Answer Key Release height = 6 ft, Initial Vertical Velocity = 22 ft/sec Release height = 6 ft, Initial Vertical Velocity = 24 ft/sec

30 Math in Basketball: Try other challenges Release height = 8 ft, Initial Vertical Velocity = 20 ft/sec Answer Key Release height = 8 ft, Initial Vertical Velocity = 22 ft/sec

31 Math in Basketball: Try other challenges Answer Key Release height = 8 ft, Initial Vertical Velocity = 24 ft/sec

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