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Using video segments and web interactives from Get the Math, students engage in an exploration of mathematics, specifically reasoning and sense making, to solve real world problems and learn how special effects designers use math in their work. In this lesson, students focus on understanding the Big Ideas of Algebra: patterns, relationships, equivalence, and linearity; learn to use a variety of representations, including modeling with variables; build connections between numeric and algebraic expressions; and use what they have learned previously about number and operations, measurement, proportionality, and discrete mathematics as applications of algebra.  Methodology includes guided instruction, student-partner investigations, and communication of problem-solving strategies and solutions.

In the Introductory Activity, students view a video segment in which they learn how Jeremy Chernick, a designer at J & M Special Effects, uses math in his work  as he presents a mathematical challenge connected to a high-speed effect from a music video.  In Learning Activity 1, students solve the challenge that Jeremy posed in the video, which involves using algebraic concepts and reasoning to figure out the relationship between light intensity and distance from a light source in order to help fix an underexposed shot.  As students solve the problem, they have an opportunity to use an online simulation to find a solution.  Students summarize how they solved the problem, followed by a viewing of the strategies and solutions used by the Get the Math teams.  In Learning Activity 2, students try to solve additional interactive lighting (inverse relationship) challenges. In the Culminating Activity, students reflect upon and discuss their strategies and talk about the ways in which algebra can be applied in the world of special effects, lighting, and beyond.

LEARNING OBJECTIVES

Students will be able to:

• Describe scenarios that require special effects technicians to use mathematics and algebraic reasoning in lighting and high-speed photography.
• Identify a strategy and create a model for problem solving.
• Recognize, describe, and represent inverse relationships using words, tables, numerical patterns, graphs, and/or equations.
• Learn to recognize and interpret inverse relationships and exponential functions that arise in applications in terms of a context, such as light intensity.
• Compare direct and inverse variation.
• Solve real-life and mathematical problems involving the area of a circle.

[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.
• Creating Equations
  • A.CED.2, 4 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.
• 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

• Interpreting Functions
  • F.IF. 1, 2 Understand the concept of a function and use function notation.
  • F.IF.4, 5, 6 Interpret functions that arise in applications in terms of a context.
• Analyzing Functions using different representations
  • F.IF.7e. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases (i.e., exponential).
• Building Functions
  • F.BF.1 Build a function that models a relationship between two quantities.
• Build new functions from existing functions
  • F-BF 4a, 4b, 4c.  Find inverse functions:
    a.  Solve and equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
    b. Verify by composition that one function is the inverse of another.
    c. Read values of an inverse function from a graph or a table, given that the function has an inverse.

Geometry Overview

• Modeling with Geometry
  • Apply geometric concepts in modeling situations:
    G.MG.1.  Use geometric shapes, their measures, and their properties to describe objects.

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.

• The Setup (video) Optional
  An introduction to Get the Math and the professionals and student teams featured in the program.
• Math in Special Effects: Introduction (video)
  Jeremy Chernick, special effects designer, describes how he got involved in special effects, gives an introduction to the mathematics used in high speed photography, and poses a related math challenge.
• Math in Special Effects: Take the challenge (web interactive)
  In this interactive activity, users try to solve the challenge posed by Jeremy Chernick in the introductory video segment.
• Math in Special Effects: See how the teams solved the challenge (video)
  The teams use algebra to solve the Special Effects challenge in two distinct ways.
• Math in Special Effects: Try other challenges (web interactive)
  This interactive provides users additional opportunities to use key variables and different lens sizes to solve related problems.

MATERIALS/RESOURCES

For the class:

• Computer, projection screen, and speakers (for class viewing of online/downloaded video segments)
• One copy of the “Math in Special Effects: Take the challenge” answer key (DOC | PDF)
• One copy of the “Math in Special Effects: Try other challenges” answer key (DOC | PDF)

For each student:

• One copy of “Math in Special Effects: Take the challenge” handout (DOC | PDF)
• One copy of the “Math in Special Effects: Try other challenges” handout (DOC | PDF)
• One graphing calculator (optional)
• Rulers, grid paper, chart paper, whiteboards/markers, overhead transparency grids, or other materials for students to display their math strategies used to solve the challenges in the Learning Activities
• Colored sticker dots and markers of two different colors (optional)
• Computers with internet access for Learning Activities 1 and 2 (optional)

(Note: These activities can either be conducted with handouts provided in the lesson and/or by using the web interactives on the Get the Math website.)

BEFORE THE LESSON

Prior to teaching this lesson, you will need to:

• Preview all of the video segments and web interactives used in this lesson.
• Download the video clips used in the lesson to your classroom computer(s) or prepare to watch them using your classroom’s internet connection.
• Bookmark all websites you plan to use in the lesson on each computer in your classroom.  Using a social bookmarking tool (such as delicious, diigo, or portaportal) will allow you to organize all the links in a central location.
• Make one copy of the “Math in Special Effects: Take the challenge” and “Math in Special Effects: Try other challenges” handouts for each student.
• Print out one copy of the “Math in Special Effects: Take the challenge” and the “Math in Special Effects: Try other challenges” answer keys.
• Get rulers, graph paper, chart paper, grid whiteboards, overhead transparency grids, etc. for students to record their work during the learning activities.
• Get colored stickers (optional) and colored markers, for students to mark the points and construct graphs of the Special Effects data in Learning Activities 1 & 2.

THE LESSON

INTRODUCTORY ACTIVITY

1. Begin with a brief discussion about special effects and lighting.  For instance, if any of your students have used a camera or shot a video, ask them to discuss the math they have used.  Ask students where they may have seen special effects visuals and what might need to be considered in terms of lighting.
2. Explain that today’s lesson focuses on the use of math in Special Effects.  Ask students to brainstorm how they think mathematics might be used in a filming a music video. (Sample responses: calculating the time it takes to capture a shot, using measurement in terms of where to place the camera and the object being filmed, adjusting the camera for different settings.)
3. Explain that today’s lesson features video segments and web interactives from Get the Math, a program that highlights how math is used in the real world. If this is your first time using the program with this class, you may choose to play the video segment The Setup, which introduces all the professionals and student teams featured in Get the Math.
4. Introduce the video segment Math in Special Effects: Introduction by letting students know that you will now be showing them a segment from Get the Math, which features Jeremy Chernick, a Special Effects designer.  Ask students to watch for the math that he uses in his work and to write down their observations as they watch the video.
5. Play Math in Special Effects: Introduction. After showing the segment, ask students to discuss the different ways that Jeremy Chernick uses math in his work.  (Sample responses: He uses math to control the lighting while filming special effects; knowing the relationship between variables such as distance and light intensity can help Jeremy figure out how to adjust the camera settings for a better shot.)
6. Ask students to describe the challenge that Jeremy and his colleague, , Andrew Flowers, posed to the teens in the video segment. (The challenge is to figure out the relationship between distance and light intensity, and use that information to help fix a shot of an exploding flower that was underexposed.)

LEARNING ACTIVITY 1

1. Explain that the students will now have an opportunity to solve the problem, which involves using a fundamental principle of photography.  The flash power, or intensity of a light source, is often measured with a light meter in a studio, but intensity readings can vary based on the distance from the light source.  Understanding light intensity and how it changes, or varies, with distance can help the photographer to achieve the proper lighting for a given shot.
2. Ask students to think of situations in their daily life where they may need to apply the concept of variation between two sets of data, where one variable changes and another changes proportionally, either directly or indirectly.  (Sample responses: cost of downloading apps varies by the quantity selected; the time it takes to complete a chore or task varies by the number of people who are assisting; driving time varies with the speed of the car.)
3. Discuss why you would need to look for the variation between light intensity and distance in this challenge.  (Sample responses: it will help to determine how to find the light intensity for any distance to adjust the shot for the music video; a model can help identify how the two sets of data are changing in relation to each other.)
4. Review the following terminology with your students:
   • Direct variation means that the ratio between two variables remains constant.  As one variable increases, the other variable also increases.  This relationship can be represented by a function in the form y = kx where k ≠ 0.
   • Inverse variation means that the product of two variables remains constant.  As one variable increases, the other variable decreases.  This relationship can be represented by a function in the form xy = k or y = k/x where k ≠0.
   • Constant of variation for an inverse variation is k, the product of xy for an ordered pair (x, y) that satisfies the inverse relationship.
   • Aperture – An opening in the lens, or circular hole, that can vary in size.  It is adjusted to increase or decrease the amount of light.
   • Diameter of aperture (a) – The physical size of the hole measured through its center point.  Half of the diameter is the radius of the lens.
   • Area of aperture (A) – The measurement, in square mm, of the opening of the hole formed by the aperture.
   • Focal length (f) – The distance between the optical center of the lens (typically, where the aperture is located) and the image plane, when the lens is focused at infinity.
   • Image plane – The fixed area behind a camera lens – inside the camera – at which the sensor or film is located, and on which pictures are focused.
   • F-stop (s) – The camera setting that regulates how much light is allowed by changing the aperture size, which is the opening of the lens.
5. Distribute the “Math in Special Effects: Take the challenge” handout. Let your students know that it is now their turn to solve the challenge that Jeremy and Andrew posed to the teams in the video. Explain that in the activity, students should use the recorded data about light intensity and distance in order to look for patterns to solve the problem.
6. Ask students to work in pairs or small groups to complete the “Math in Special Effects: Take the challenge” handout. Use the “Math in Special Effects: Take the challenge” answer key as a guide to help students as they complete the activity. (Note: The handout is designed to be used in conjunction with the Math in Special Effects: Take the challenge activity on the website.)
   • If you have access to multiple computers, ask students to work in pairs to explore the interactive and complete the handout.
   • If you only have one computer, have students work in pairs to complete the assignment using their handouts and grid or graph paper and then ask them to report their results to the group and input their solutions into the online interactive for all to see the results.
7. Review the directions listed on the handout.
8. As students complete the challenge, encourage them to use the following 6-step mathematical modeling cycle to solve the problem:
   • Step 1: Understand the problem: Identify variables in the situation that represent essential features. (For example, light intensity, distance, and a constant of variation.)
   • Step 2: Formulate a model by creating and selecting multiple representations. (For example, students may use visual representations in sketching a graph, algebraic representations such as an equation, or an explanation/plan written in words.  Student should examine their algebraic equation to identify the type of variation that is represented by the variables.)
   • Step 3: Compute by analyzing and performing operations on relationships to draw conclusions. (For example, operations include calculating the changing light intensity.)
   • Step 4: Interpret the results in terms of the original situation. (The results of the first three steps should be examined in the context of the challenge to solve real-world applications of variations, including more than just two types: direct and inverse.  Inverse relationships may occur where a product of corresponding data values of any degree is constant and may involve exponential functions.)
   • Step 5: Ask students to validate their conclusions by comparing them with the situation, and then either improving the model or, if acceptable,
   • Step 6: Report on the conclusions and the reasoning behind them.  (This step allows students to explain their strategy and justify their choices in a specific context.)

   

   
   Ongoing Assessment: Ask students to reflect upon the following:

   • After recording the light intensity at each distance, what patterns do you notice?
   • Using the data, can you identify a recursive formula that models the relationship?
   • Using the data, can you identify an explicit formula that models the relationship?
   • Is there only one strategy for solving the challenge? (You may wish to have students solve graphically to determine the exponential change visually.)
9. After students have completed the activity, ask them to share their solutions and problem-solving strategies with the class through discussion and visual materials, such as chart graph paper, grid whiteboards, overhead transparency grids, etc.  Encourage students to discuss how their strategy helped (or didn’t help) figure out the relationship between light intensity and distance, as well as how to adjust the camera for any shot.  Ask students to discuss any difficulties they faced in completing the challenge and how they overcame those obstacles.
10. As students present their solutions, ask them to discuss the mathematics they used in solving the challenge. Ask students to describe how they selected the equation to model the relationship and any graphs used, how they calculated the values for light intensity (I), distance (d), d², and k, why rounding was important, and the strategy they would recommend to Jeremy and Andrew for finding the light intensity for any distance in a high-speed special effects shot.
11. Introduce the Math in Special Effects: See how the teams solved the challenge video segment by letting students know that they will now be seeing how the teams in the video solved the Special Effects challenge. Ask students to observe what strategies the teams used and whether they are similar to or different from the strategies presented by the class.
12. Play Math in Special Effects: See how the teams solved the challenge. After showing the video, ask students to discuss the strategies the teams used and to compare them to the strategies used by the class. How are they similar? How are they different? During the discussion, point out that the two teams in the video solved the Special Effects challenge in two distinct ways.  Discuss the strategies listed in the “Math in Special Effects: Take the challenge” answer key, as desired.

LEARNING ACTIVITY 2:

1. Go to the Math in Special Effects: Try other  challenges interactive. Explain to your students that they will use the web interactive to solve a series of problems similar to the one Jeremy Chernick presented in the video segment.  In this multi-level activity, students are challenged to learn how a camera setting called an f-stop affects the amount of light coming through the lens and use the information to improve a shot.
   Note: As in Learning Activity 1, you can conduct this activity with one computer and an LCD projector in front of the entire class or your students can work in small groups on multiple computers. This can also be assigned to students to complete as an independent project or homework using the accompanying handout as a guide.
2. Distribute the “Math in Special Effects: Try other challenges” handout. Clarify and discuss the directions.
3. Ask students to complete the handout as they explore the online challenges.
   Note: If you are using one computer, have your students work in pairs, taking turns inputting their responses into the web interactive to test their choices as they determine how focal length, aperture, and f-stop are related; how much light passes through at a given f-stop, the relationship between f-stop and area of the aperture, and how to use an understanding of f-stops to adjust the camera setting for a given shot.
4. As in Learning Activity 1, encourage your students to use the 6-step mathematical modeling cycle as they develop a strategy to solve the challenges.
5. After students have completed the activity, lead a group discussion and encourage students to share their strategies and solutions to the challenges. (Sample responses: There is a relationship between f-stops and light intensity (or area of the aperture).  “Opening up” to the next lower f-stop (for instance, going from f/1.4 to f/1) lets in twice as much light by increasing the diameter of the opening by a factor of , or about 1.414.  As diameter increases by a factor of , the area of the aperture, and therefore, light intensity, doubles.  Conversely, “closing down” to each greater f-stop (for instance, going from f/2 to f/2.8) reduces the light intensity by half.)

CULMINATING ACTIVITY

1. Assess deeper understanding: Ask your students to reflect upon and write down their thoughts about the following:
   • How did you determine an effective strategy for solving the challenges in this lesson? What are your conclusions and the reasoning behind them?
   • Compare and contrast the various algebraic and graphical representations possible for the problem. How does the approach used to solve the challenge affect the choice of representations? (Sample responses: attempting to estimate the direction and speed of the variation, or change, between two sets of data with a quick visual can be determined by using a graph or pictorial model; an approach that attempts to show a proportional relationship by quantifying the variability, or change, would be best represented with an algebraic model.)
   • Why is it useful to represent real-life situations algebraically?  (Sample responses: Using symbols, graphs, and equations can help visualize solutions when there are situations that require inverse relationships.)
   • What are some ways to represent, describe, and analyze patterns that occur in our world? (Sample responses: patterns can be represented with graphs, expressions, and equations to show and understand changes between two sets of data such as light intensity and distance.)
2. After students have written their reflections, lead a group discussion where students can discuss their responses. During the discussion, ask students to share their thoughts about how algebra can be applied to the world of Special Effects. Ask students to brainstorm other real-world situations which involve the type of math and problem solving that they used in this lesson. (Sample responses: there is an inverse relation between the temperature of the water in a lake or ocean and the depth where the measurement is taken; the speed of a bicycle gear is inversely proportional to the number of teeth in the gear.)

[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.
  • 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.

• The Setup (video) Optional
  An introduction to Get the Math and the professionals and student teams featured in the program.
• Math in Basketball: Introduction (video)
  Elton Brand, basketball player and NBA star, describes how he got involved in sports, gives an introduction to the mathematics used in maximizing a free throw shot, and poses a related math challenge.
• Math in Basketball: Take the challenge (web interactive)
  In this interactive activity, users try to solve the challenge posed by Elton Brand in the introductory video segment.
• Math in Basketball: See how the teams solved the challenge (video)
  The teams use algebra to solve the basketball challenge in two distinct ways.
• Math in Basketball: Try other basketball challenges (web interactive)
  This interactive provides users additional opportunities to use key variables and players’ individual statistics to solve related problems.

MATERIALS/RESOURCES

For the class:

• Computer, projection screen, and speakers (for class viewing of online/downloaded video segments)
• One copy of the “Math in Basketball: Take the challenge” answer key (DOC | PDF)
• One copy of the “Math in Basketball: Try other basketball challenges” answer key (DOC | PDF)

For each student:

• One copy of “Math in Basketball: Take the challenge” handout (DOC | PDF)
• One copy of the “Math in Basketball: Try other basketball challenges” handout (DOC | PDF)
• One graphing calculator (optional)
• Rulers, grid paper, chart paper, whiteboards/markers, overhead transparency grids, or other materials for students to display their math strategies used to solve the challenges in the Learning Activities
• Colored sticker dots and markers of two different colors (optional)
• Computers with internet access for Learning Activities 1 and 2 (optional)

(Note: These activities can either be conducted with handouts provided in the lesson and/or by using the web interactives on the Get the Math website.)

BEFORE THE LESSON

Prior to teaching this lesson, you will need to:

• Preview all of the video segments and web interactives used in this lesson.
• Download the video clips used in the lesson to your classroom computer(s) or prepare to watch them using your classroom’s internet connection.
• Bookmark all websites you plan to use in the lesson on each computer in your classroom.  Using a social bookmarking tool (such as delicious, diigo, or portaportal) will allow you to organize all the links in a central location.
• Make one copy of the “Math in Basketball: Take the challenge” and “Math in Basketball: Try other basketball challenges” handouts for each student.
• Print out one copy of the “Math in Basketball: Take the challenge” and the “Math in Basketball: Try other basketball challenges” answer keys.
• Get rulers, graph paper, chart paper, grid whiteboards, overhead transparency grids, etc. for students to record their work during the learning activities.
• Get colored stickers (optional) and colored markers, for students to mark the points and construct the trajectory, or path, of the basketball in Learning Activity 1 & 2.

THE LESSON

INTRODUCTORY ACTIVITY

1. Begin with a brief discussion about sports.  For instance, if any of your students play a sport, ask them to discuss the math they have used as athletes.  Ask students what sports they like to watch and how they keep track of their team’s progress.  Ask students to discuss the mathematics that players may use to track and maximize their performance.
2. Explain that today’s lesson focuses on the use of math in basketball.  Ask students to brainstorm how they think mathematics might be used in the sport. (Sample responses: knowing the rules of the game in terms of scoring, such as the shot clock timing, overtime, types and point values of shots and fouls allowed; knowing the dimensions of the court; statistical box scores, such as assists, turnovers, blocked shots, steals, field goal attempts, three-point goals and attempts, and playing time; ratios between two related statistical units, such as offensive rebounds and second-shot baskets, or two that contradict each other, such as assists and turnovers; per-minute and per-game statistics.)
3. Explain that today’s lesson features video segments and web interactives from Get the Math, a program that highlights how math is used in the real world. If this is your first time using the program with this class, you may choose to play the video segment The Setup, which introduces the professionals and student teams featured in Get the Math.
4. Introduce the video segment Math in Basketball: Introduction by letting students know that you will now be showing them a segment from Get the Math, which features Elton Brand, an NBA basketball player.  Ask students to watch for the math that he uses in his work and to write down their observations as they watch the video.
5. Play Math in Basketball: Introduction. After showing the segment, ask students to discuss the different ways that Elton Brand uses math in his work.  (Sample responses: He uses math to help improve his performance by using three key variables to influence his free throw shot; he uses acceleration of gravity or downward pull, the ball’s initial vertical velocity, and his release height to figure out the height of the basketball at any given time; he uses statistics to maximize the height of the basketball so it has the best chance of going into the basket.)
6. Ask students to describe the challenge that Elton Brand posed to the teens in the video segment. (The challenge is to use the three key variables and his stats to figure out the maximum height the ball reaches on its way into the basket in order to make a free throw shot.)

LEARNING ACTIVITY 1

1. Explain that the students will now have an opportunity to solve the problem, which involves using the Fast Break Stats for information about the three key variables (acceleration of gravity, initial vertical velocity, release height) and Elton’s stats.
2. Ask students to think of situations in their daily life where they may need to apply the concept of maximizing. (Sample response: finding the best price to charge for the school play to get the most people to attend while still making a profit.)
3. Discuss why you would need to maximize the height of the basketball trajectory. (Sample responses: to make sure it reaches the hoop; the higher the shot, the further from the basket it peaks or reaches maximum height, increasing the likelihood the player will make the shot; higher arcs require a player to have more strength and use the proper mechanics.)
4. Review the following terminology with your students:
   • Coordinates: an ordered pair of numbers that identify a point on a coordinate plane.
   • Function: a relation in which every input (x-value) has a unique output (y-value).
   • Acceleration of Gravity: causes a ball to speed up, or accelerate, when falling at a rate of -32 ft/sec².  Use only downward pull or half of -32 ft/sec², which is -16 t².
   • Initial Vertical Velocity: the angle and speed when the ball leaves the player’s hand.  Multiply by time to get the vertical distance traveled.
   • Release Height: the starting position of the ball when it leaves the player’s hand.
   • Trajectory: the path that a basketball follows through space as a function of time.
   • Maximum Height: the value in the data set where the basketball reaches its greatest vertical distance at a given time on its way into the basket.
   • Parabola: the graph of a function in the family of functions with parent function y = x².
     • The path of the ball when thrown is a trajectory represented by a parabola which can be modeled mathematically with a quadratic equation.  This equation represents the position of the path over time.
     • The height (h) of a ball, in feet, at a given time (t) is represented by the equation h(t) = -16t² + v₀t + h₀ where v₀ is the initial vertical velocity and h₀ is the initial height.
   • Vertex: the highest point of the parabola.
5. Distribute the “Math in Basketball: Take the challenge” handout. Let your students know that it is now their turn to solve the challenge that Elton Brand posed to the teams in the video. Explain that in the activity, students should use the Fast Break Facts for information about the three key variables and Elton’s stats to figure out the maximum height the ball reaches on its way into the basket when making a free throw shot.
6. Ask students to work in pairs or small groups to complete the “Math in Basketball: Take the challenge” handout. Use the “Math in Basketball: Take the challenge” answer key as a guide to help students as they complete the activity. Note: The handout can be used by itself or in conjunction with the “Math in Basketball: Take the challenge” activity on the website.
   • If you have access to multiple computers, ask students to work in pairs to explore the interactive and complete the handout.
   • If you only have one computer, have students work in pairs to complete the assignment using their handouts and grid or graph paper and then ask them to report their results to the group and input their solutions into the online interactive for all to see the results.
7. Review the directions listed on the handout.
8. As students complete the challenge, encourage them to use the following 6-step mathematical modeling cycle to solve the problem:
   • Step 1: Understand the problem: Identify variables in the situation that represent essential features. (For example, use the three key variables: acceleration of gravity, Elton’s initial vertical velocity, and his release height.)
   • Step 2: Formulate a model by creating and selecting multiple representations. (For example, students may use visual representations in sketching a graph, algebraic representations such as combining the three key variables and Elton’s stats: 24 ft/sec and a release height of 7 feet to write an equation that models the projectile motion, or an explanation/plan written in words.)
   • Step 3: Compute by analyzing and performing operations on relationships to draw conclusions. (For example, operations include calculating the values of t when the ball reaches a height of 10 feet, the value of t when the ball reaches a maximum height, and the maximum height of the basketball at this time.)
   • Step 4: Interpret the results in terms of the original situation. (The results of the first three steps should be examined in the context of the challenge to maximize the height of the basketball during the free throw shot using Elton’s release height and initial vertical velocity, as well as the acceleration of gravity.
   • Step 5: Ask students to validate their conclusions by comparing them with the situation, and then either improving the model or, if acceptable,
   • Step 6: Report on the conclusions and the reasoning behind them.  (This step allows students to explain their strategy and justify their choices in a specific context.)

   

   Ongoing Assessment: Ask students to reflect upon the following:

   • How can you combine the three key variables: acceleration of gravity, initial vertical velocity, and release height, to determine the maximum height of the basketball?
   • At what time(s) does the ball reach 10 feet?
   • At what time does the ball reach the maximum height?
   • Is there only one path or trajectory for this to occur using Elton’s stats?  How do you know? (You may wish to have students solve graphically to determine that this is the path using the given stats.)
9. After students have completed the activity, ask students to share their solutions and problem-solving strategies with the class through discussion and visual materials, such as chart graph paper, grid whiteboards, overhead transparency grids, etc.  Encourage students to discuss how their strategy helped (or didn’t help) figure out the maximum height of the path of the ball during the free throw shot.  Ask students to discuss any difficulties they faced in completing the challenge and how they overcame those obstacles.
10. As students present their solutions, ask them to discuss the mathematics they used in solving the challenge. (Sample responses: Using a graphical model by plotting (time, distance) points for the start time and release height (0, 7), and the end time and rim height (t, 10) on a coordinate graph; representing functions using a mathematical model such as a table of values; identifying variables and writing expressions and/or a quadratic equation; using the properties of the graph of the equation to  find the value of the x-coordinate of the vertex  (), then solving the equation for t to find the maximum height at this time; using a quadratic equation and solving by factoring, completing the square, or the quadratic formula.)
11. Introduce the Math in Basketball: See how the teams solved the challenge video segment by letting students know that they will now be seeing how the teams in the video solved the basketball challenge. Ask students to observe what strategies the teams used and whether they are similar to or different from the strategies presented by the class.
12. Play Math in Basketball: See how the teams solved the challenge. After showing the video, ask students to discuss the strategies the teams used and to compare them to the strategies presented by the class. How are they similar? How are they different? During the discussion, point out that the two teams in the video solved the basketball challenge in two distinct ways.  Discuss the strategies listed in the “Math in Basketball: Take the challenge” answer key, as desired.

LEARNING ACTIVITY 2:

1. Go to the Math in Basketball: Try other challenges interactive. Explain to your students that they will use the web interactive to solve a series of problems similar to the one Elton Brand presented in the video segment.  In this multi-level activity, students are challenged to use the 3 key variables, using a choice of player stats, to figure out the maximum height the ball reaches on its way into the basket to make the shot.  Choices include: Initial Vertical Velocity of 5 feet, 6 feet, or 8 feet; Release Height of 20 ft/sec, 22 ft/sec, or 24 ft/sec.  Students are encouraged to use the 3 key variables and the stats to calculate the ball’s height, h, at a given time, t, by setting up an equation to get started.  
   [Note: As in Learning Activity 1, you can conduct this activity with one computer and an LCD projector in front of the entire class or your students can work in small groups on multiple computers. This can also be assigned to students to complete as an independent project or homework using the accompanying handout as a guide.]
2. Distribute the “Math in Basketball: Try other challenges” handout. Clarify and discuss the directions.
3. Ask students to complete the handout as they explore the online challenges.
   [Note: If you are using one computer, have your students work in pairs to plot points on graph or chart paper and to write the quadratic equation using the three key variables and the player’s stats. Have students take turns inputting their responses into the web interactive to test their choices as they determine the time(s) the ball reaches 10 feet, the time when the ball reaches maximum height, and the maximum height at this time.]
4. As in Learning Activity 1, encourage your students to use the 6-step mathematical modeling cycle as they develop a strategy to solve the challenges.
5. After students have completed the activity, lead a group discussion and encourage students to share their strategies and solutions to the challenges. Ask students to discuss how they selected the equation and graphs used, and how they calculated the values for time and height using each set of player stats.

CULMINATING ACTIVITY

1. Assess deeper understanding: Ask your students to reflect upon and write down their thoughts about the following:
   • How did you determine an effective strategy for solving the challenges in this lesson? What are your conclusions and the reasoning behind them? (Sample answer: First you could find the total flight time of the ball.  Since the height of the ball is a function of the time the basketball is in the air, and the path is a trajectory or parabola, it has an axis of symmetry that passes through the vertex or highest point.  Students may use this fact to make a table of values, and since it is U-shaped between the two points it is at 10 feet, students may use the symmetry to include values to the left and right of the vertex.  A trace function or key in a graphing calculator, as well as a sketch of the graph, may be used to solve the problem.)
   • Compare and contrast the various algebraic and graphical representations possible for the problem. How does the approach used to solve the challenge affect the choice of representations? (Sample answers: If you decide to graph the points and then think of the basketball as an object that is traveling on a parabolic path, or trajectory, you would use this information to find the maximum height by finding the average between the two points it is at 10 feet; if you decide to write the equation of the function by combining the three key variables: acceleration of gravity, initial vertical velocity, and release height for Elton Brand or a given player, you could use transformations to write it in Standard Form for a quadratic equation, then find the times by using the quadratic formula or completing the square as algebraic strategies.)
   • Why is it useful to represent real-life situations algebraically?  (Sample responses: Using symbols, graphs, and equations can help visualize solutions when there are situations that require using data sets or statistics to maximum performance of an athlete.)
   • What are some ways to represent, describe, and analyze patterns that occur in our world? (Sample responses: patterns can be represented with graphs, expressions, and equations to show and understand optimization.)
2. After students have written their reflections, lead a group discussion where students can discuss their responses. During the discussion, ask students to share their thoughts about how algebra can be applied to the world of sports. Ask students to brainstorm other real-world situations which involve the type of math and problem solving that they used in this lesson. (Sample responses: sports-related problems might include “catching air” in snowboarding, throwing a baseball or football, hitting a golf ball, and shooting a model rocket to maximize the height of the ball or rocket; maximizing the area of a garden/farm given specific fencing options; modeling relationships between revenue and cost.)

• Educator’s Media Kit #1, a .zip file containing the Math in Music, Fashion, and Videogames modules (file size: 641 MB)
• Educator’s Media Kit #2, a .zip file containing the Math in Restaurants, Basketball, and Special Effects modules (file size: 665 MB)
  

After your download is complete, you will need to extract the materials from the .zip file using a program like WinZip or StuffIt Expander.

Once the files are extracted, open the file named “launch.html” as follows:

• PC users: Right-click on the file, then click on “Open with…” to select Internet Explorer, Mozilla Firefox, or another internet browser of your choosing.
• Mac users: Press “Ctrl” while clicking on the file, then click on “Open with…” to select Mozilla Firefox, Safari, or another internet browser of your choosing. (Note: if you have a mouse with more than one button, you may also simply right-click on the file instead of using Ctrl+click.)

The launch menu will connect you to all the videos, interactives, lesson plans, and handouts.

Note: You will not need to be connected to the internet, but you will need to use an internet browser program to open the files that are saved locally to your computer.  For PC users, we recommend using Internet Explorer or Mozilla Firefox. For Mac users, we recommend using Mozilla Firefox or Safari.

[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.

Statistics and Probability

• Summarize, represent, and interpret data on a single count or measurement variable.
  • S.ID.1.  Represent data with plots on the real number line (dot plots, histograms, and box plots).
  • S.ID.2  Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
  • S.ID. 3  Interpret differences in shape, center, and spread in the context of data sets, accounting for the possible effects of extreme data points (outliers).
• Summarize, represent, and interpret data on two categorical and quantitative variables.
  • S.ID.5  Recognize possible associations and trends in data.
  • S.ID.6  Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
    a)     Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
    b)     Informally assess the fit of a function by plotting and analyzing residuals.
    c)      Fit a linear function for a scatter plot that suggests a linear association.
• Interpret linear models: Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.
  • S.ID.7  Interpret the slope (rate of change) and the intercept (constant term of a linear model in the context of the data.
  • S.ID.8  compute (using technology) and interpret the correlation coefficient of a linear fit.

Algebra

• Perform arithmetic operations on polynomials.
• Create equations that describe numbers or relationships.
  • A.CED.2  Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• Understand solving equations as a process of reasoning and explain the reasoning.
  • A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution.  Construct a viable argument to justify a solution method.
• 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

• Interpret functions that arise in applications in terms of a context.
  • F.IF.4  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.  Key features include: intercepts, intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

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.

• The Setup (video) Optional
  An introduction to Get the Math and the professionals and student teams featured in the program.
• Math in Restaurants: Introduction (video)
  Sue Torres, chef and owner of a Mexican restaurant, describes how she got involved in the cooking world, gives an introduction to the mathematics used in menu costs, and poses a math challenge related to menu pricing.
• Math in Restaurants: Take the challenge (web interactive)
  In this interactive activity, users try to solve the challenge posed by Sue Torres in the introductory video segment, by examining avocado prices from the past three years to suggest a menu price for guacamole.
• Math in Restaurants: See how the teams solved the challenge (video)
  The teams solve the restaurant challenge in two distinct ways by using algebra to examine costs of avocados and suggest a menu price .
• Math in Restaurants: Try other menu pricing challenges (web interactive)
  This interactive provides users additional opportunities to set future menu prices for three dishes in Sue Torres’ restaurant, based on trends in costs over recent years.

MATERIALS/RESOURCES

For the class:

• Computer, projection screen, and speakers (for class viewing of online/downloaded video segments)
• One copy of the “Math in Restaurants: Take the challenge” answer key (DOC | PDF)
• One copy of the “Math in Restaurants: Try other challenges” answer key (DOC | PDF)

For each student:

• One copy of “Math in Restaurants: Take the challenge” handout (DOC | PDF)
• One copy of the “Math in Restaurants: Try other restaurant challenges” handout (DOC | PDF)
• One graphing calculator (Optional)
• Rulers, grid paper, chart paper, whiteboards/markers, overhead transparency grids, or other materials for students to display their math strategies when solving the challenges in the Learning Activities.
• Colored sticker dots and markers of two different colors (Optional)
• Computers with internet access for Learning Activities 1 and 2. (Optional)

(Note: These activities can either be conducted with handouts provided in the lesson and/or by using the web interactives on the Get the Math website.)

BEFORE THE LESSON

Prior to teaching this lesson, you will need to:

• Preview all of the video segments and web interactives used in this lesson.
• Download the video clips used in the lesson to your classroom computer(s) or prepare to watch them using your classroom’s internet connection.
• Bookmark all websites you plan to use in the lesson on each computer in your classroom.  Using a social bookmarking tool (such as delicious, diigo, or portaportal) will allow you to organize all the links in a central location.
• Make one copy of the “Math in Restaurants: Take the challenge” and “Math in Restaurants: Try other restaurant challenges” handouts for each student.
• Print out one copy of the “Math in Restaurants: Take the challenge” and the “Math in Restaurants: Try other restaurants challenges” answer keys.
• Get rulers, graph paper, chart paper, grid whiteboards, overhead transparency grids, etc. for students to record their work during the learning activities.
• Get colored stickers (optional) and colored markers, for students to mark the points and construct the trend lines in the scatter plots in the learning activities.

THE LESSON

INTRODUCTORY ACTIVITY

1. Ask students to discuss their favorite foods and whether they have ever prepared a meal. Ask students to brainstorm how mathematics is used in cooking and food preparation (measurements, proportions, etc.) Ask students to discuss how they use math when purchasing and selecting food items in a store or restaurant (determining the prices of items, calculating the “best buy” by comparing items of different quantities, etc.).
2. Explain that today’s lesson focuses on the use of math in restaurants.  Ask students to brainstorm how mathematics might be used in restaurants, in ways other than those mentioned in the previous discussion. (Paying employees, paying for costs of operating a business, pricing menu items, etc.)
3. Explain that today’s lesson features video segments and web interactives from Get the Math, a program that highlights how math is used in the real world. If this is your first time using the program with this class, you may choose to play the video segment The Setup, which introduces the professionals and student teams featured in Get the Math.
4. Introduce the video segment Math in Restaurants: Introduction by letting students know that you will now be showing them a segment from Get the Math, which features Sue Torres, chef and owner of Sueños, a restaurant in Chelsea, New York. Ask students to watch for the math that she uses in her work and to write down their observations as they watch the video.
5. Play Math in Restaurants: Introduction. After showing the segment, ask students to discuss the different ways that Sue Torres uses math in her work.  (Sample responses: She uses math to make recipes with many ingredients; collects data about the costs of ingredients; looks for trends in the data over time; makes predictions about prices using decimal estimations; sets the menu prices using a rule of thumb that involves using addition and multiplication of rational numbers.)
6. Ask students to describe the challenge that Sue Torres posed to the teens in the video segment. (Sue must set a menu price for guacamole for the coming year.  Avocado is the main ingredient.  The challenge is to look at avocado prices from the past 3 years to predict what avocados might cost in the next 14 months.  Then, using this prediction, , as well as Sue’s Rule of Thumb for establishing menu prices, recommend a menu price for guacamole for next year at Sueños.)

LEARNING ACTIVITY 1

1. Explain that the students will now have an opportunity to solve the problem, which will require them to graph and analyze the data, look for possible relationships, and make a prediction to determine the price of guacamole using Sue’s Rule of Thumb.
2. Ask students to think of situations in their daily life where they may need to apply the concepts of analyzing data and finding a “trend line” to make a prediction. (Sample responses: Analyzing the number of hits a baseball player makes each game or a runner’s times at a particular distance; looking at how many customers come to at a popular restaurant or store at different times of day to predict when would be the best time to eat or shop there;  looking at the number of hits your blog is getting to see if there is a trend over time; collecting and analyzing the quality and water temperature of a lake over time to see when it is safe to swim; collecting local census data over time to predict the need for a new school or additional youth programs; analyzing the amount of money you might save each year to make a prediction about how much you will have when you graduate high school.)
3. Discuss why you would need to find an average to determine the menu price of an appetizer, main course, or dessert item.  (Sample responses:  to calculate the average cost of an ingredient, or several ingredients, over time; to locate the median cost in the data set; to set one price that can remain the same over a period of time, even though the costs might be lower or higher at different times.)
4. Review the following terminology with your students:
   • Coordinates: an ordered pair of numbers that identify a point on a coordinate plane.
   • Scatter Plot: a graph that displays the relationship between two different sets of data. The values on the horizontal axis represent one data set and the values on the vertical axis represent the other data set.  The coordinates of each point represent the ordered pair of these data values.
   • Trend line:  a line on a scatter plot that shows a correlation between two sets of data.
   • Correlation: a relationship between two sets of data that can be positive, negative, or none.
   • Line of best fit: the trend line that most accurately models the relationship between the two sets of data. It has about the same number of data points above it and below it. It is used to make a prediction based on a scatter plot that appears to be linear.
   • Linear: in a straight line.
   • Extrapolation: Predicting a value outside the range of known values.
   • Interpolation: Predicting a value between two known values.
   • Slope: a ratio or rate of change.  Slope represents the change in the y-values to the change in the x-values on a coordinate graph using any two points on a line.  It is a ratio of the vertical change to the horizontal change.
   • Box and whisker plot: a display that summarizes one set of data along a number line.  It shows a 5-number summary of a data set.  The left “whisker” or segment extends from the minimum to the first quartile; the box extends from the first quartile to the third quartile, with a line segment through the median; the right “whisker” or segment extends from the third quartile to the maximum.
   • 5-number summary: five numbers in a data set that show how the data is spread.  The numbers represent the boundary points: the minimum and maximum, the median, the first quartile (median of the first half), and the third quartile (median of the second half).
   • Median: the middle value in a data set with an odd number of values that have been listed in order.  If there is an even number of values in the data set, the median is the mean of the two middle values after they have been listed in order.
   • Average: a measure of central tendency that is often displayed as the mean of the data values.  It is calculated by finding the sum of the values and dividing by the number of data values in the set.
5. Distribute the “Math in Restaurants: Take the challenge” handout. Let your students know that it is now their turn to solve the challenge that Sue Torres posed to the teams in the video.
6. Explain that in the activity, students will use the Avocado Cost Chart and scatter plot to analyze the real world data for avocados.  Students need to decide on a strategy to find a model that will show the general direction of the data.  After constructing a line that will appear to “fit” the data, called a “line of best fit” or “trend line,” they will need to make a prediction about the cost of avocados in the next 14 months.   Then, using “Sue’s Rule of Thumb,” they will make a recommendation for the menu price of guacamole.
7. Ask students to work in pairs or small groups to complete the “Math in Restaurants: Take the challenge” handout. Use the “Math in Restaurants: Take the challenge” answer key as a guide to help students as they complete the activity. Note: The handout can be used by itself or in conjunction with the “Math in Restaurants: Take the challenge” activity on the website.
   • If you have access to multiple computers, ask students to work in pairs to explore the interactive and complete the handout.
   • If you only have one computer, have students work in pairs to complete the assignment using their handouts and grid or graph paper and then ask them to report their results to the group and input their solutions into the online interactive for all to see the results.
8. Review the directions listed on the handout.
9. As students complete the challenge, encourage them to use the following 6-step mathematical modeling cycle to solve the problem:
   • Step 1: Understand the problem: Identify variables in the situation that represent essential features (For example, students may use x to represent the number of months over time and y to represent the cost of a case of avocados.)
   • Step 2: Formulate a model by creating and selecting multiple representations (For example, students may use visual representations in graphing, algebraic representations such as slope and an equation of a line of best fit, a box and whisker plot, or an explanation/plan written in words.)
   • Step 3: Compute by analyzing and performing operations on relationships to draw conclusions (For example, operations include solving for slope– the relationship between the change in y-values and the change in x-values that allows a student to conclude the rate of change for the line of best fit.  Several strategies can be used to find the lines of fit including finding the slope of the line between two representative points, then graphing the slope-intercept or point-slope form of the line of fit; finding the 5-number summary and using Q-points (the x-coordinate of the first or third quartile in the data set and the y-coordinate of the first or third quartile to form a rectangle); finding the average cost of the main ingredient and adding the additional ingredients, then calculating the menu price using a “Rule of Thumb.”)
   • Step 4: Interpret the results in terms of the original situation (The results of the first three steps should be examined in the context of the challenge to determine a menu price for the guacamole dish.)
   • Step 5: Ask students to validate their conclusions by comparing them with the situation, and then either improving the model or, if acceptable,
   • Step 6: Report on the conclusions and the reasoning behind them.  (This step allows a student to explain their strategy and justify their choices in a specific context.)

   

   Ongoing Assessment: Ask students to reflect upon the following:

   • How can you use the line of best fit to predict the cost of avocados in the future?
   • Is there only one value for the price of avocados that can be used to set the menu price?  How do you know? (You may wish to have students solve graphically to determine that there are several possibilities for the trend lines and equations, and, therefore, menu price. An extension would be to have students solve the problem using another method, such as the method of least squares, linear regression, or the median-median method.)
10. After students have completed the activity, ask students to share their solutions and problem-solving strategies with the class through discussion and visual materials, such as chart graph paper, grid whiteboards, overhead transparency grids, etc.  Encourage students to discuss how their strategy helped (or didn’t help) them predict the cost of avocados for next year and recommend a menu price for guacamole.   Ask students to discuss any difficulties they faced in completing the challenge and how they overcame those obstacles.
11. As students present their solutions, ask them to discuss the mathematics they used in solving the challenge. (Sample responses: Using coordinate graphs and scatter plots to solve problems, identifying variables and writing expressions and/or an equation of a line, finding slope or rate of change, representing a trend between two sets of data using a line of best fit, displaying data using a box and whisker plot and calculating a 5-number summary, finding the average of the cost of one ingredient and determining the menu price by multiplying this price times four.)
12. Introduce the Math in Restaurants: See how the teams solved the challenge video segment by letting students know that they will now be seeing how the teams in the video solved the restaurant challenge. Ask students to observe what strategies the teams used and whether they are similar to or different from the strategies presented by the class.
13. Play Math in Restaurants: See how the teams solved the challenge. After showing the video, ask students to discuss the strategies the teams used and to compare them to the strategies presented by the class. How are they similar? How are they different? During the discussion, point out that the two teams in the video solved the restaurant challenge in two distinct ways.  Discuss the strategies listed in the “Math in Restaurants: Take the challenge” answer key, as desired.

LEARNING ACTIVITY 2:

1. Go to the Math in Restaurants: Try other challenges interactive. Explain to your students that they will use the web interactive to solve a series of problems similar to the one Sue Torres presented in the video segment.  In this activity, students are challenged to use a Cost Chart and scatter plot to analyze the real world data for three different main ingredients: beef, shrimp, and chicken.  Students decide on a strategy to find a model that will show the general direction of the data.  After constructing a line that will appear to “fit” the data, a “line of best fit” or “trend line,” they will need to make a prediction about the cost of the main ingredient in the next 12 months.   Then, using “Sue’s Rule of Thumb,” students will make a recommendation for the price of three menu items: Shredded Beef Mini Tacos with Queso Fresco and Pico de Gallo, Shrimp Flautas with Guajillo Sauce and Guacamole, and Organic Chicken & 3-Cheese Quesadilla with Avocado Tempura and Chipotle Cream.  [Note: As in Learning Activity 1, you can conduct this activity with one computer and an LCD projector in front of the entire class or your students can work in small groups on multiple computers. This can also be assigned to students to complete as an independent project or homework using the accompanying handout as a guide.]
2. Distribute the “Math in Restaurants: Try other challenges” handout. Clarify and discuss the directions.
3. Ask students to complete the handout as they explore the online challenges. [Note: If you are using one computer, have your students work in pairs to analyze the given scatter plot (or plot the points on graph or chart paper), identify the trend line, and to write the equation of the line of best fit for the data. Have students take turns inputting their responses into the web interactive to test their choices.  Partners should complete the Cost Chart and predict the average cost of one serving of the main ingredient, using Sue’s Rule of Thumb to recommend a menu price.]
4. As in Learning Activity 1, encourage your students to use the 6-step mathematical modeling cycle as they develop a strategy to solve the challenges.
5. After students have completed the activity, lead a group discussion and encourage students to share their strategies and solutions to the challenges. Ask students to discuss how they selected the trend lines and lines of best fit, as well as the average cost per serving for the menu item.

LEARNING ACTIVITY 3 (OPTIONAL)

1. Ask students to brainstorm situations where they can apply a line of best fit (including the ways they mentioned in the introductory activity)
2. Ask students to collect data over a certain period of time. (Encourage students to go online to find existing stats, such as stats for professional sports teams and athletes.)
3. Ask students to graph this data. If it is linear, ask them to come up with a line of best fit, calculate an equation for the line, and make conclusions based on their findings. (Note: If the data is not linear, a different fit may be required.)
4. Ask students to share their findings with the class.

CULMINATING ACTIVITY

1. Assess deeper understanding: Ask your students to reflect upon and write down their thoughts about the following:
   • How did you determine an effective strategy for solving the challenges in this lesson? What are your conclusions and the reasoning behind them? (Sample answer: First you can determine whether the data is linear and has a positive or negative correlation.  Then, after choosing a method to find a line of best fit, you can use the y-coordinate representing the cost of the main ingredient to predict an average price for the next year.  Finally, you can use this price for one serving to set the menu price by using the Rule of Thumb.)
   • Compare and contrast the various algebraic and graphical representations possible for the problem. How does the approach used to solve the challenge affect the choice of representations? (Sample answers: If you decide to graph the points and then use two representative points to find the slope, you can use the point-slope form of the line algebraically to find the average cost as the median of the data in the next year.  You could use your line of best fit graphically without an algebraic equation by determining the coordinates on the graph and arrive at the same solution using visual representations.  Using a third method, you could represent the line of best fit as the diagonal in the rectangle formed by the Q-points as determined by the 5-number summaries and box and whisker plots.)
   • Why is it useful to represent real-life situations algebraically?  (Sample responses: Using symbols, graphs, and equations can help visualize solutions when there is more than one, such as different lines of fit that can be used to predict the cost of an item, as well as using different averages for each serving.)
   • What are some ways to represent, describe, and analyze patterns that occur in our world? (Sample responses: patterns can be represented with graphs, expressions, and equations to show change.)
2. After students have written their reflections, lead a group discussion where students can discuss their responses. During the discussion, ask students to share their thoughts about how the algebra concepts and problem-solving skills they used in this lesson (including recognizing trends in data and determining a line of best fit) are used in restaurants and how these concepts and skills can be applied to other real-world situations.

Using video segments and web interactives from Get the Math, students engage in an exploration of mathematics, specifically reasoning and sense making, to solve real world problems. In this lesson, students focus on understanding the Big Ideas of Algebra: patterns, relationships, equivalence, and linearity; learn to use a variety of representations, including modeling with variables; build connections between numeric and algebraic expressions; and use what they have learned previously about number and operations, measurement, statistics, as applications of algebra. Methodology includes guided instruction, student-partner investigations, and communication of problem-solving strategies and solutions.

In the Introductory Activity, students view a video segment in which they learn how Sue Torres, an accomplished chef, uses math in her work and are presented with a mathematical restaurant challenge. In Learning Activity 1, students solve the challenge that Sue posed to the teams in the video. As students solve the problem, they have an opportunity to use an online simulation to find a solution. Students summarize how they solved the problem, followed by a viewing of the strategies and solutions used by the Get the Math teams. In Learning Activity 2, students try to solve additional interactive menu pricing challenges. In the Culminating Activity, students reflect upon and discuss their strategies and talk about the ways in which algebra can be applied in the world of restaurants and beyond.

LEARNING OBJECTIVES

Students will be able to:

• Describe scenarios that require chefs to use mathematics and algebraic reasoning in creating menu pricing.
• Identify a strategy and create a model for problem solving.
• Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
• Learn to recognize trend lines and predict a line of best fit.
• Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
• Represent data with box and whisker plots.
• Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
• Create equations in two or more variables to represent relationships between quantities, including point-slope form and slope-intercept form of a line.
• Understand, explain, and use algebraic and numeric expressions and equations that are interconnected and build on one another to produce a coherent whole.