Web viewThe 5E model is not linear. Details. Standards for Mathematical Practice ☐Engagement...

STEM-Centric UnitQuadratic Functions

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

Background InformationSubject:Identify the course the unit will be implemented in. Algebra I

Grade Band:Identify the appropriate grade band for the lesson. 9-12

Duration:Identify the time frame for the unit. Two 90 minute class periods

Overview:Provide a concise summary of what students will learn in the lesson. It explains the unit’s focus, connection to content, and real world connection.

This lesson is an interactive introduction to quadratic functions. Students will conduct experiments in order to determine a quadratic regression equation of launched objects at a given angle of elevation. A STEM Specialist will help students interpret their results and engage students in hands-on learning experiences that demonstrates how quadratic 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.

Background Information:Identify information or resources that will help teachers understand and facilitate the lesson.

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 functions provides a basic idea of concepts that involve quadratics. Throughout 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

STEM Specialist Connection:Describe how a STEM Specialist may be used to enhance the learning experience. STEM Specialist may be found at http://www.thestemnet.com/

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 regression to be appropriate for predictive purposes. 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

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http://www.thestemnet.com/

https://dl.dropboxusercontent.com/u/72483584/MBRT%20Lessons/Algebra%20I_PD_U10_InstrGuide_v1.1.pdf

http://plus.maths.org/content/os/issue30/features/quadratic/index


Background Informationfor the students to learn how to calculate the zeros and vertex of a quadratic function.

Enduring Understanding:Identify discrete facts or skills to focus on larger concepts, principles, or processes. They are transferable - applicable to new situations within or beyond the subject.

Mathematical models are used to develop solutions to real-world problems. Quadratic functions can be used to model motion.

Essential Questions:Identify several open-ended questions to provoke inquiry about the core ideas for the lesson. They are grade-level appropriate questions that prompt intellectual exploration of a topic.

1. How can quadratic functions be used to model real-world problems and solutions?

2. How does STEM professional use quadratic functions?

Student Outcomes:Identify the transferable knowledge and skills that students should understand and be able to do when the lesson is completed. Outcomes must align with but not limited to Maryland State Curriculum and/or national standards.

Students will be able to:1. graph functions expressed symbolically and show key features of the graph by

hand.2. graph quadratic functions and show intercepts, maxima, and minima.

Product, Process, Action, Performance, etc.:Identify what students will produce to demonstrate that they have met the challenge, learned content, and employed 21st century skills. Additionally, identify the audience they will present what they have produced to.

Students will work in collaborative teams to collect data on the projectile motion of a launched object to gain understanding on how a quadratic function is described algebraically and what it means. Students 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.

Audience:☒Peers☒Experts / Practitioners☒Teacher(s)☐School

Community☐Other______

Standards Addressed in the Unit:Identify the Maryland State Curriculum Standards addressed in the unit.

Domain: Quadratic Functions and Modeling

Cluster Statement: Analyze functions using different representations.

Standard: F.IF.7 Graph functions expressed symbolically and show key features of 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.Suggested Materials and Resources:Identify materials needed to complete the unit. This includes but is not limited to websites, equipment,

Equipment: rubber bands ruler

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Background Information

PowerPoints, rubrics, worksheets, and answer keys.

protractor measuring tape goggles TI-83+ or 84+ graphing calculator target with bull’s-eye action figure computer with internet access projector

Websites* 101 uses of quadratic functions 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

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http://www.mathopenref.com/quadraticexplorer.html

http://www.montereyinstitute.org/courses/Algebra1/U10GAME_RESOURCE/index.html

http://www.projectsharetexas.org/resource/analyzing-graphs-quadratic-functions

http://plus.maths.org/content/os/issue30/features/quadratic/index


Lesson 1 of 2Duration: 90 minutes

Learning Experience5E Component

Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details Standards for Mathematical Practice

☒ Engagement

☐Exploration

☐Explanation

☐Extension

☐Evaluation

Materials: Motivation Video: Demonstration and online activities - Mentos Motivation Activity: Shape Shifter. Students will need a computer with

internet access to complete this activity. Quadratic Functions Student Note Sheet Day 1

Preparation: The instructor should be sure that the motivation video plays on a

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.

Each student will need one copy of the Quadratic Functions Student Note Sheet Day 1

Facilitation of Learning Experience: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, water from a water fountain, and the outline of a satellite dish. Ask students what 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 parabolic cross section. Display the cross-sectional shape for the students and ask the similarities between the paths they drew and the cross section.

☐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.

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Tell them that the mathematical term for the curve or arc is called a parabola and 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.

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☐Engagement

☒Exploration

☐Explanation

☐Extension

☐Evaluation

Activity inspired by page 10-17 of this project based learning lesson

Materials: rubber bands ruler protractor measuring tape goggles TI-83+ or 84+ graphing calculator target with bull’s-eye

Preparation:Place students into groups of four with each student having one of the following 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 Quadratic Functions Student Note Sheet Day 1. Have students read through directions and have a student volunteer paraphrase what each job will do during the experiment.

Facilitation of Learning Experience:Phase 1. Experiment 1: Parabolic MotionHolder 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 using a protractor.Spotter will measure the horizontal flight distance.Recorder will record the results in the table below.


☐Reason abstractly and quantitatively.


☒Model with mathematics.

☒Use appropriate tools strategically.


☐Look for and make use of structure.


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https://dl.dropboxusercontent.com/u/72483584/MBRT%20Lessons/Algebra%20I_PD_U10_InstrGuide_v1.1.pdf





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 steps1. 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.”

☐Engagement Materials: graphing calculator quadratic regression graph

☐Make sense of problems and

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☐Exploration

☐Explanation

☒Extension

☐Evaluation

action figure ruler protractor measuring tape rubber bands safety goggles

Preparation: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.

Facilitation of Learning Experience:Experiment 2: Ballistics trainingDirections:1. Each group member will have the same role as in the first experiment, but this time place the target at a horizontal distance within range of the rubber band. The 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 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 are related to projectile motion and other concepts.

persevere in solving them.


☒Construct viable arguments and critique the reasoning of others.



☒Attend to precision.



☐Engagement Materials:Quadratic Functions Student Note Sheet Day 1


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☐Exploration

☒Explanation

☐Extension

☐Evaluation

Preparation:Students should return to the formation they were in during class discussion for independent work.

Closure:Today we looked at several examples of quadratic functions and created data to 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 by a quadratic function?

Inform students that a STEM Specialist will visit the class tomorrow to help with the interpretation of today’s rubber band launch. The STEM Specialist will also discuss how quadratic functions are applied in the workplace.

☒Reason abstractly and quantitatively.







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Supporting Information

Interventions/EnrichmentsIdentify interventions and enrichments for diverse learners.

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

Strategies to help English Language Learners are similar to those listed above.

Provide resources to define and/or pronounce difficult vocabulary. A native 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

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Duration: 90 Minutes




☐Engagement

☐Exploration

☒Explanation

☐Extension

☐Evaluation

Materials: Computer Projector Quadratic Functions Student Note Sheet Day 2

Preparation: Contact the STEM Specialist in advance to co-plan the lesson and

explain his/her role in facilitating instruction. Provide the STEM Specialist a 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 the learning experience with the STEM Specialist or have students prepare some questions in advance.

Students should be organized in seating for optimal interaction with the STEM Specialist.

Provide each student with a copy of Quadratic Functions Student Note Sheet Day 2.

Facilitation of Learning Experience:1. Interpretation of rubber band launch: The STEM Specialist will work with students to interpret the results of the horizontal distance of a rubber band at a 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



☒Construct viable arguments and critique the reasoning of others.

☒Model with mathematics.





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a quadratic relationship; students will volunteer to share what they wrote as a summary.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.’

☐Engagement Materials: Computers with internet access Quadratic Functions Student Note Sheet Day 2 .

☐Make sense of problems and persevere in solving

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☒Exploration

☐Explanation

☐Extension

☐Evaluation

Preparation:Students will transition to the computer lab or laptops within the classroom. They can use Edmodo or another platform to guide them to the correct link in order to complete the next portion of the lesson

Facilitation of Learning Experience:Phase 2 Quadratic Sliders: Students will use the following directions to guide them through the activity within the following link: Quadratic Sliders1. Click on the link: Quadratic Sliders2. Follow the directions below the graph under the heading “The simplest Case Y=constant, (y=c)”Sketch the graph of y=0x2+0x+12 on your note-sheet, in the standard form for a 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)4. Sketch the graph of three different b values with at least one being negative 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?5. Follow the directions underneath the heading “The squared term. (y=ax2)”

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

them.




☒Use appropriate tools strategically.




☐Engagement

☐Exploration

Materials:Quadratic Functions Student Note Sheet Day 2.

Preparation:Students can either come back from computer lab or get into a


☒Reason abstractly and

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☐Explanation

☐Extension

☒Evaluation

discussion/independent practice formation. Each student will answer the questions below independently. This will serve as a formative assessment for the lesson.

Facilitation of Learning Experience:Phase 3: SummarizeSketch a graph and write one sentence in order to answer the following questions based on the “Quadratic Slider” exploration.

a. How is a quadratic function when ‘a’ is positive different from one where ‘a’ is negative?

b. How is a quadratic function when ‘b’ is positive different from one where ‘b’ is negative?

c. How is a quadratic function when ‘c’ is positive different from one where ‘c’ is negative?

Use your answers from above and apply your understanding of the rubber band experiment to answer the following

d. The path of a baseball is modeled by the quadratic function where

y=height and x=time, y=−16 x2+64 x+8?

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, y=0.5 x2+6 x+16

quantitatively.






☐Look for and express regularity in repeated reasoning.

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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.

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Supporting Information

Interventions/EnrichmentsIdentify interventions and enrichments for diverse learners.

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 jing.

Provide resources to define and/or pronounce difficult vocabulary. A native 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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Quadratic Functions Introduction (Day 1)

Name:

Sketch the graph of the following situations

Cross Section DisplaySatellite Dish OutlineWater in a water fountainBouncing a basketball

Define Parabola: ________________________________________________________________________________________.

Shape ShifterList the objects used in the ‘shape shifter’ activity below

Parabolic BrainstormUsing the information gathered so far, explain another situation that may involve a quadratic function and why you think so. Draw a sample graph of the situation labeling the key components.

Graph Explanation

Rubber Band Launch1. Circle your role: holder, launcher, recorder, or spotter.Materials: rubber bands, ruler, measuring tape, protractor, TI 84 Calculator. 2. The 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 using a protractor. The spotter will measure the horizontal flight distance and the recorder will record the results in the table below. Then perform the quadratic regression by using the listed directions.

Angle of Elevation (In Degrees)

Trial 1 Trial 2 Trial 3

Average Horizontal Distance

Traveled (In cm.)

0102030405060708090

Quadratic Regression Directions1. Press STAT, ENTER to access list editor.2. Enter Angle of Elevations in L1 and Average Horizontal Distances in L2

3. Press STAT, CALC, QuadReg, ENTER4. The screen will read y=ax2+bx+c followed by values for a, b, and c. Use these values to write the quadratic equation that best describes your data.5. Use the table on the back of the note-sheet to graph the quadratic function.

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Rubber Band Launch Graph

Angle of Elevation (In Degrees)

Quadratic Regression Equation

y=______________________

0102030405060708090

Directions: Use the table and graph to create your quadratic function, make sure your graph has all the necessary components.

Ballistics TrainingDirections: Each group member will have the same role as in the first experiment, but this time place the target at a horizontal distance within range of the rubber band. The first attempt will be a short distance, the 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 choose the angle of elevation necessary to hit the target.

Target distance from launcher

Angle of elevation chosen Distance of rubber band launch Error (Distance of launch – Target placement)

SummaryDirections: In your own words, how are quadratic functions related to projectile motion (the rubber band launch). What did the vertex (top of the parabola) and the zeros (x-intercepts) of the graph represent in the experiment?

For further reviewDirections: Today we looked at several examples of quadratic functions and created data to 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 by a quadratic function?

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Quadratic Functions Introduction (Day 2)

Name:

STEM Specialist Learning ExperienceDirections: Respond to the following prompts below1. Paraphrase why the angle of elevation and the horizontal distance of the rubber band formed a quadratic relationship based on the explanation given by the STEM Specialist.

2. Explain what the vertex, axis of symmetry, and zeros represent in terms of the situation(s) given by the STEM Specialist.

Parabolic Brainstorm IIUsing the information provided by the STEM Specialist, explain another situation that may involve a quadratic function and why you think so. Label the vertex, axis of symmetry, and zeros and explain what they mean in your situation

Graph Explanation

Computer Lab: “Quadratic Sliders”Directions: Log into Edmodo and access the link for today’s activity called “quadratic sliders.” 1. Follow the directions below the graph under the heading “The simplest Case Y=constant, (y=c)”2. Sketch the graph of y=0x2+0x+12 below, in the standard form for a quadratic equation y=ax2+bx+c, then pick two other values for c and graph those as well. Respond to the prompt to the right.

3. Follow the directions underneath the heading ‘Linear Equations. (y=bx)4. Sketch the graph of three different b values with at least one being negative when ‘a’ and ‘c’ are both zero. 5. Follow the directions underneath the heading “The squared term. (y=ax2)”

Standard Form: y=ax2+bx+c, c-the constant termc appears to have the following effect on the graph…

b= b=b=

Standard Form: y=ax2+bx+c , b-the linear termb appears to have the following effect on the graph…

c= c=c=

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Computer Lab: “Quadratic Sliders” ContinuedDirections: Log into Edmodo and access the link for today’s activity called “quadratic sliders.” 5. Follow the directions underneath the heading “The squared term. (y=ax2)” then graph of three different a values with at least one being negative when ‘b’ and ‘c’ are both zero.

a= a=a=

Standard Form: y=ax2+bx+c , a-the quadratic termb appears to have the following effect on the graph…

SummarizeDirections: Answer the following questions in your own words to prepare for a class discussion on your findings.a. How is a quadratic function when ‘a’ is positive different from one where ‘a’ is negative?

b. How is a quadratic function when ‘b’ is positive different from one where ‘b’ is negative?

c. How is a quadratic function when ‘c’ is positive different from one where ‘c’ is negative?

Projectiles in Standard FormDirections: Complete the prompts below to demonstrate your understanding of quadratic functions in standard form.

d. The path of a baseball is modeled by the quadratic function where y=height and x=time

y=−16x2+64 x+8 ?i. Explain why the sign on each coefficient makes sense.

ii. Sketch a possible graph of the situation, 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

y=0.5 x2+6 x+16


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?

Web viewThe 5E model is not linear. Details. Standards for Mathematical Practice ☐Engagement...

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