Author: Justin Field, Chesapeake High School, Baltimore County Public Schools

STEM-Centric Unit Quadratic Functions Author: Justin Field, Chesapeake High School, Baltimore County Public Schools Background Information Subject: Algebra I Identify the course the unit will be implemented in.

Grade Band: 9-12 Identify the appropriate grade band for the lesson. Duration: Two 90 minute class periods Identify the time frame for the unit. This lesson is an interactive introduction to quadratic functions. Students will conduct Overview: experiments in order to determine a quadratic regression equation of launched objects at a Provide a concise summary of what students will learn in given angle of elevation. A STEM Specialist will help students interpret their results and the lesson. It explains the unit’s focus, connection to engage students in hands-on learning experiences that demonstrates how quadratic content, and real world connection. functions are used by STEM professionals. Students will begin to explore real-life applications for finding the vertex of a quadratic function as well as its zeros. A quadratic function is a function of the second degree [i.e., a function of the form f(x) = ax2 + bx + c]; in a rectangular coordinate system. The graph of a quadratic function is a parabola. Quadratic functions have numerous real-world applications. This link >> 101 uses of quadratic Background Information: functions provides a basic idea of concepts that involve quadratics. Throughout the lesson, Identify information or resources that will help teachers understand and facilitate the lesson. students will be responsible for identifying the vertex, zeros, and shape of the quadratic function and how it applies to a given real world situation. Another helpful resource to find out more information about quadratics can be found at the Algebra I Open Course Professional Development Lessons found here >> Quadratic Lesson Plan see page 14 The STEM Specialist can:  help students interpret the results of their experiments.  engage students in hands-on learning experiences that explain what forces cause a quadratic STEM Specialist Connection: regression to be appropriate for predictive purposes. Describe how a STEM Specialist may be used to enhance the learning experience. STEM Specialist may be found at  engage students in hands-on learning experiences that demonstrate how quadratic functions http://www.thestemnet.com/ are used in their work including examples of the vertex, zeros, and axis of symmetry of a quadratic function being used as a direct application to solve a problem. This will help the students make sense of their data and serve as a motivation for the students to learn how to calculate the zeros and vertex of a quadratic function.

of 19 STEM-Centric Unit Quadratic Functions Background Information Enduring Understanding: Identify discrete facts or skills to focus on larger concepts,  Mathematical models are used to develop solutions to real-world problems. principles, or processes. They are transferable - applicable  Quadratic functions can be used to model motion. to new situations within or beyond the subject. Essential Questions: Identify several open-ended questions to provoke inquiry 1. How can quadratic functions be used to model real-world problems and solutions? about the core ideas for the lesson. They are grade-level appropriate questions that prompt intellectual 2. How does STEM professional use quadratic functions? exploration of a topic. Student Outcomes: Identify the transferable knowledge and skills that Students will be able to: students should understand and be able to do when the 1. graph functions expressed symbolically and show key features of the graph by hand. lesson is completed. Outcomes must align with but not limited to Maryland State Curriculum and/or national 2. graph quadratic functions and show intercepts, maxima, and minima. standards. Audience: Students will work in collaborative teams to collect data on the ☒Peers Product, Process, Action, Performance, etc.: ☒ Identify what students will produce to demonstrate projectile motion of a launched object to gain understanding on how a Experts / that they have met the challenge, learned content, quadratic function is described algebraically and what it means. Practitioners and employed 21st century skills. Additionally, Students will have the goal of being able to describe what a vertex is, ☒Teacher(s) identify the audience they will present what they what zeros or roots are, and how they play a role in the importance of a ☐ have produced to. School quadratic function. Community ☐Other______Domain: Quadratic Functions and Modeling Cluster Statement: Analyze functions using different representations. Standards Addressed in the Unit: Identify the Maryland State Curriculum Standards Standard: F.IF.7 Graph functions expressed symbolically and show key features of addressed in the unit. the graph, by hand in simple cases and using technology for more complicated cases.★ a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

of 19 STEM-Centric Unit Quadratic Functions Background Information Equipment:  rubber bands  ruler  protractor  measuring tape  goggles  TI-83+ or 84+ graphing calculator  target with bull’s-eye  action figure  computer with internet access  projector Suggested Materials and Resources: Identify materials needed to complete the unit. This Websites* includes but is not limited to websites, equipment,  101 uses of quadratic functions PowerPoints, rubrics, worksheets, and answer keys.  Engagement Activity  Shape Shifter  Multimedia Applet: Quadratic sliders * Throughout the lesson, students are linked to online resources in order to conduct research. The sites have been chosen for their content and grade-level appropriateness. Teachers should preview all websites before introducing the activities to students and adhere to their school system’s policy for internet use. People, Facilities:  STEM Specialist  Students will need a safe location to launch rubber bands. Materials (rubrics, worksheets, PowerPoints, answer keys, etc.):  Quadratic Functions Student Note Sheet Day 1  Quadratic Functions Student Note Sheet Day 2

of 19 STEM-Centric Unit Quadratic Functions Lesson 1 of 2 Duration: 90 minutes

Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. Materials: ☒ Engagement ☐Make sense of problems and  Motivation Video: Demonstration and online activities - Mentos persevere in solving them.  Motivation Activity: Shape Shifter. Students will need a computer with ☐Exploration internet access to complete this activity. ☒ Reason abstractly and  Quadratic Functions Student Note Sheet Day 1 ☐Explanation quantitatively. Preparation: ☐ ☐Extension Construct viable arguments  The instructor should be sure that the motivation video plays on a and critique the reasoning of others. ☐Evaluation computer/through a projector and be visible for the whole class.  Students will begin class with a warm-up activity from previously learned material and begin a quadratics unit with the above resources. ☐Model with mathematics.  Each student will need one copy of the Quadratic Functions Student Note ☐Use appropriate tools Sheet Day 1 strategically.

Facilitation of Learning Experience: ☐Attend to precision. 1. Provide each student with a copy of Quadratic Functions Student Note Sheet Day 1. Ask students to draw on their note sheet the path of a bouncing basketball, ☒ Look for and make use of water from a water fountain, and the outline of a satellite dish. Ask students what structure. all these have in common? Accept all answers. 2. Have a cone shaped object that you have pre-sliced at an angle to form a ☐ Look for and express regularity parabolic cross section. Display the cross-sectional shape for the students and ask in repeated reasoning. the similarities between the paths they drew and the cross section. Tell them that the mathematical term for the curve or arc is called a parabola and

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. algebraically it is the shape of a quadratic function. 3. Projectile Motion: Show the demonstration video (Demonstration and online activities - Mentos) and have the students volunteer answers to the questions beneath the video. 4. Curve fitting: Have the class participate in the shape shifter activity (Shape Shifter) by letting the class work together or having students work in groups to match each curve to its parabolic equation. Ask questions such as “how is the number in the ‘a’ position affecting the parabola, the ‘b’ position, the ‘c’ position? 5. Parabolic brainstorm: ask students to summarize the events above by brainstorming other objects, motions, or concepts they think could be described by a parabola and explain in their own words why they think so.

Transition: We are going to work in collaborative groups to collect real data on the projectile motion of a rubber band to gain understanding on how a quadratic function is described algebraically and what it means. We will have the goal of being able to describe what a vertex is, what zeros or roots are, and how they play a role in the importance of a quadratic function.

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. Activity inspired by page 10-17 of this project based learning lesson ☐Engagement ☐Make sense of problems and persevere in solving them. Materials: ☒Exploration  rubber bands ☐Reason abstractly and  ruler ☐Explanation quantitatively.  protractor ☐ ☐Extension  measuring tape Construct viable arguments  goggles and critique the reasoning of others. ☐Evaluation  TI-83+ or 84+ graphing calculator  target with bull’s-eye ☒Model with mathematics. Preparation: ☒ Place students into groups of four with each student having one of the following Use appropriate tools strategically. roles: spotter, recorder, holder, and launcher. Each group will have a station where they will perform the rubber band experiment and record their data on the ☐Attend to precision. Quadratic Functions Student Note Sheet Day 1. Have students read through directions and have a student volunteer paraphrase what each job will do during ☐Look for and make use of the experiment. structure.

Facilitation of Learning Experience: ☐ Look for and express regularity Phase 1. Experiment 1: Parabolic Motion in repeated reasoning. Holder and the Launcher will work together. The holder will keep the ruler level at about waist height. The launcher will place one end of the rubber band on the end of the ruler and pull back the elastic to measure the starting length at rest. For all the trials, the launcher will stretch the rubber band 5 cm beyond the starting point and release. The ruler will be held at the various angles of elevation measured

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. using a protractor. Spotter will measure the horizontal flight distance. Recorder will record the results in the table below.

Phase 2. The group will now use the TI Graphing calculators in order to create a quadratic regression equation to model the data and graph the function on their note-sheet. Regression steps 1. Press STAT, Enter on your calculator to bring up list editor.

2. Enter angles of elevation as ‘x’ values by entering them in L1 and enter horizontal

distances in L2 3. Press STAT, CALC, QuadReg, Enter, in order to calculate the quadratic equation. The calculator will give a screen that says y=ax2+bx+c and then values for a, b, and c. Use this to write the quadratic regression equation for your data.

Phase 3: On the students note-sheet they should create a table of estimated values by substituting in the given angles of elevation from the experiment as ‘x’ values into their regression equation. They should then graph the results with angle of elevation on the x axis and horizontal distance on the y axis.

Transition: Each group will trade regression equation/graphs with another group and be given a “small object to act as a target such as a model, lego person, or action figure.”

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. Materials: ☐Engagement ☐Make sense of problems and  graphing calculator persevere in solving them.  quadratic regression graph ☐Exploration  action figure ☐Reason abstractly and  ☐Explanation ruler quantitatively.  protractor ☒ ☒Extension  measuring tape Construct viable arguments  rubber bands and critique the reasoning of others. ☐Evaluation  safety goggles Preparation: ☐Model with mathematics. Each group will trade work with another group and use the regression equation in order to calculate the angle of elevation necessary to hit a target. ☐Use appropriate tools strategically. Facilitation of Learning Experience: Experiment 2: Ballistics training ☒Attend to precision. Directions: 1. Each group member will have the same role as in the first experiment, but this ☐Look for and make use of time place the target at a horizontal distance within range of the rubber band. The structure. first attempt being a short distance, second being a medium distance, and the third ☐ being near the full range of the rubber band. Use the graph given to you in order to Look for and express regularity in repeated reasoning. choose the angle of elevation necessary to hit the target. 2. Calculate the error by subtracting the target distance from the actual distance traveled.

Transition: Have groups summarize what we have explored today and how quadratic functions

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. are related to projectile motion and other concepts. Materials: ☐Engagement ☐Make sense of problems and Quadratic Functions Student Note Sheet Day 1 persevere in solving them. ☐Exploration Preparation: ☒Reason abstractly and Students should return to the formation they were in during class discussion for ☒Explanation quantitatively. independent work. ☐ ☐Extension Construct viable arguments Closure: and critique the reasoning of Today we looked at several examples of quadratic functions and created data to others. ☐Evaluation model projectile motion. Explain in your own words a falling object or a parabolic design that you might consider researching. Why do you think it can be modeled ☐Model with mathematics. by a quadratic function? ☐Use appropriate tools Inform students that a STEM Specialist will visit the class tomorrow to help with strategically. the interpretation of today’s rubber band launch. The STEM Specialist will also ☐ discuss how quadratic functions are applied in the workplace. Attend to precision. ☐Look for and make use of structure.

☐ Look for and express regularity in repeated reasoning.

of 19 STEM-Centric Unit Quadratic Functions Supporting Information Struggling Learners  Group students based upon ability, learning style, or other appropriate criteria, so all students can equally contribute to group work.  Questions asked during class discussion are open ended, perhaps provide struggling students a specific example to write about.  Specific deadlines for work completion would be important to establish with the teams, so class time is effectively used.  Provide resources to define and/or pronounce difficult vocabulary.  Break work into chunks for teams, so they are able to achieve small goals and meet all expectations.  Provide additional time for work completion or assign some parts for homework. English Language Learners Interventions/Enrichments  Strategies to help English Language Learners are similar to those listed above. Identify interventions and enrichments for  Provide resources to define and/or pronounce difficult vocabulary. A native diverse learners. language dictionary may also be beneficial.  Use visuals (pictures displayed on a document camera or PowerPoint presentation), when appropriate.  Read directions and documents aloud to students, when appropriate. Gifted and Talented  Ask students to research further a particular situation that appears to have a quadratic relationship. Find an example of a quadratic application where the ‘a’ in ax2 is positive and one where it is negative.  The instructor will foster independent thinking and collaboration between the partners. No one student should take over the work for the partnership.  Higher level thinking questions should be asked throughout the lesson with the expectation of responses that are thoughtful and elaborate.  Encourage students to develop discussion questions for the STEM Specialist. Lesson 2 of 2 Duration: 90 Minutes

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. Materials: ☐Engagement ☐Make sense of problems and  Computer persevere in solving them.  Projector ☐Exploration  Quadratic Functions Student Note Sheet Day 2 ☐Reason abstractly and ☒Explanation quantitatively. Preparation: ☒ ☐Extension  Contact the STEM Specialist in advance to co-plan the lesson and explain Construct viable arguments his/her role in facilitating instruction. Provide the STEM Specialist a and critique the reasoning of others. ☐Evaluation description of the ability level of the students and the prior knowledge your students may have of quadratic functions. Discuss available technology and classroom set-up with the Specialist. Prepare a list of questions to help guide ☒Model with mathematics. the learning experience with the STEM Specialist or have students prepare ☐ some questions in advance. Use appropriate tools strategically.  Students should be organized in seating for optimal interaction with the STEM Specialist. ☐Attend to precision.  Provide each student with a copy of Quadratic Functions Student Note Sheet Day 2. ☐Look for and make use of structure. Facilitation of Learning Experience: 1. Interpretation of rubber band launch: The STEM Specialist will work with ☐ Look for and express students to interpret the results of the horizontal distance of a rubber band at a regularity in repeated reasoning. given angle of elevation. The STEM Specialist will be responsible for explaining what forces cause a quadratic regression to be appropriate for predictive purposes. 2. Student paraphrase: Students will paraphrase on their note-sheet why the angle of elevation and the horizontal distance of the rubber band formed a quadratic relationship; students will volunteer to share what they wrote as a summary.

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. 3. Quadratic Field-work Applications: The STEM Specialist will engage students in hands-on learning experiences that demonstrate how quadratic functions are used in their work including examples of the vertex, zeros, and axis of symmetry of a quadratic function being used as a direct application to solve a problem. This will help the students make sense of their data and serve as a motivation for the students to learn how to calculate the zeros and vertex of a quadratic function. 4. My Parabola: Students will use their imagination to sketch the graph of a quadratic function based on the STEM Specialist presentation, they will label the vertex, zeros, and axis of symmetry and describe what they represent in terms of the situation they are imagining (for example: my parabola represents the flight of a cannonball, the zeros represent when the cannonball is on the ground, the vertex represents its highest point, the axis of symmetry separates the graph into where the cannonball is rising in the air verses when it is falling.) Students will share their graphs with the class.

Transition: We will now go to the computer lab and explore how the different algebraic components of a quadratic function change its position in shape when it is given in ‘vertex form’ and when it is given in ‘standard form.’

Materials: ☐Engagement ☐Make sense of problems and  Computers with internet access persevere in solving them.  Quadratic Functions Student Note Sheet Day 2. ☒Exploration ☐Reason abstractly and Preparation:

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. ☐Explanation Students will transition to the computer lab or laptops within the classroom. They quantitatively. can use Edmodo or another platform to guide them to the correct link in order to ☐Extension complete the next portion of the lesson ☐Construct viable arguments and critique the reasoning of others. ☐Evaluation Facilitation of Learning Experience: Phase 2 Quadratic Sliders: Students will use the following directions to guide them ☐Model with mathematics. through the activity within the following link: Quadratic Sliders 1. Click on the link: Quadratic Sliders ☒Use appropriate tools 2. Follow the directions below the graph under the heading “The simplest Case strategically. Y=constant, (y=c)” 2 Sketch the graph of y=0x +0x+12 on your note-sheet, in the standard form for a ☐Attend to precision. quadratic equation y=ax2+bx+c, what does it appear that ‘c’ does to the graph? 3. Follow the directions underneath the heading ‘Linear Equations. (y=bx) ☐Look for and make use of 4. Sketch the graph of three different b values with at least one being negative structure. when a and c are zero, what does it appear that b represents? What happens when a c value other than zero is added to the equation? ☐ Look for and express 5. Follow the directions underneath the heading “The squared term. (y=ax2)” regularity in repeated reasoning.

Transition: Let’s log off the computers, summarize our findings, and apply what we learned about quadratic functions to two different situations.

Materials: ☐Engagement ☐Make sense of problems and Quadratic Functions Student Note Sheet Day 2. persevere in solving them. ☐Exploration Preparation: ☒Reason abstractly and Students can either come back from computer lab or get into a

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. ☐Explanation discussion/independent practice formation. Each student will answer the questions quantitatively. below independently. This will serve as a formative assessment for the lesson. ☐Extension ☐Construct viable arguments Facilitation of Learning Experience: and critique the reasoning of others. ☒Evaluation Phase 3: Summarize Sketch a graph and write one sentence in order to answer the following questions ☐Model with mathematics. based on the “Quadratic Slider” exploration. a. How is a quadratic function when ‘a’ is positive different from one where ☐Use appropriate tools ‘a’ is negative? strategically. b. How is a quadratic function when ‘b’ is positive different from one where ☐Attend to precision. ‘b’ is negative? c. How is a quadratic function when ‘c’ is positive different from one where ☐Look for and make use of ‘c’ is negative? structure.

Use your answers from above and apply your understanding of the rubber ☐Look for and express band experiment to answer the following regularity in repeated reasoning. d. The path of a baseball is modeled by the quadratic function where y=height and x=time, ? i. Explain why the sign on each coefficient makes sense. ii. What does the vertex represent in this situation? iii. What do the zeros represent in this situation? e. A ball rolling down a ramp can be modeled by the quadratic function where y=speed and x=time, i. Explain why the sign on each coefficient makes sense.

of 19 STEM-Centric Unit Quadratic Functions Learning Experience 5E Component Identify the 5E component Standards for Mathematical addressed for the learning Details experience. The 5E model is Practice not linear. ii. Sketch a possible behavior of this graph in the first quadrant, is either the vertex or either zero located there? Why or why not? Closure: Monitor students’ progress and provide assistance as needed. At the end of class, collect student note sheets from day 1 and day 2 for grading.

of 19 STEM-Centric Unit Quadratic Functions Supporting Information Struggling Learners  Have students work in pairs to complete website exploration.  Provide students with the actual graph of the summary questions and ask them to describe its shape, vertex, and zeros.  Group students based upon ability, learning style, or other appropriate criteria, so all students can equally contribute to group work.  Specific deadlines for work completion would be important to establish with the teams, so class time is effectively used.  Provide resources to define and/or pronounce difficult vocabulary.  Provide additional time for work completion or assign some parts for homework. English Language Learners  Strategies to help English Language Learners are similar to those listed above.  Create a video tutorial for how to operate the website using a program such as Interventions/Enrichments jing. Identify interventions and enrichments for  Provide resources to define and/or pronounce difficult vocabulary. A native diverse learners. language dictionary may also be beneficial.  Use visuals (pictures displayed on a document camera or PowerPoint presentation), when appropriate.  Read directions and documents aloud to students, when appropriate. Gifted and Talented  Have students complete ‘vertex form’ of the quadratic sliders activity as well. Ask students to describe a situation that can be modeled by a quadratic function when ‘a’ is negative, when ‘a’ is positive and what the vertex and zeros represent in each case.  The instructor will foster independent thinking and collaboration between the partners. No one student should take over the work for the partnership.  Higher level thinking questions should be asked throughout the lesson with the expectation of responses that are thoughtful and elaborate.

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