Lesson 11 | Quadratic Equations and Applications | 9th Grade Mathematics

Objective

Write and analyze quadratic functions for projectile motion and falling bodies applications.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F.IF.C.8.A — 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.

Interpreting Functions

F.IF.C.8.A — 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.C.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.

Interpreting Functions

F.IF.C.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.

Foundational Standards

The foundational standards covered in this lesson

A.SSE.A.1

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
A.SSE.B.3.A

Seeing Structure in Expressions

A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe how features of a quadratic function relate to a real-world context involving projectile motion.
Write a quadratic function to represent the motion path of a free-falling object using the formula $${h(t)=-16t^2+{v_0} t+h_0}$$ for customary units and $${h(t)=-4.9t^2+{v_0} t+h_0}$$ for metric units, where $$h$$ represents the height of the object, $$t$$ represents the number of seconds, $${v_0}$$ represents the initial velocity, and $$h_0 $$ represents the height of the object at $$0$$ seconds.
Compare two projectile motion situations.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students had a brief introduction to quadratic applications in Lesson 13 of Unit 7, and in Lesson 4 of this unit, students interpreted quadratic functions using the vertex form. In Lessons 11–13 of this unit, students delve into writing and analyzing applications of quadratic functions at a deeper level. In each application problem, students will use their understanding of quadratic functions to interpret key features in context. For example, students will solve for the roots and determine what the value of the roots mean in projectile motion problems, area problems, and revenue applications.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Suppose $${h(t)=-5t^2+10t+3}$$ is the height of a diver above the water (in meters), $$t$$ seconds after the diver leaves the springboard.

How high above the water is the springboard? Explain how you know.
When does the diver hit the water?
At what time on the diver's descent toward the water is the diver again at the same height as the springboard?
When does the diver reach the peak of the dive?

Guiding Questions

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References

Illustrative Mathematics Springboard Dive

Springboard Dive, accessed on Aug. 18, 2017, 2:42 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

Chris stands on the edge of a building at a height of 60 feet. and throws a ball upward with an initial velocity of 68 feet per second. The ball eventually falls all the way to the ground.

Write a function to represent the height, $$h$$, of the ball after $$t$$ seconds.
What is the maximum height reached by the ball? After how many seconds will it reach that height?
After how many seconds will the ball land on the ground after being thrown?
Sketch a graph of the motion of the ball. Label key features including the vertex, $${y-}$$intercept, and root(s).

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 4 > Topic C > Lesson 23 — Mathematic Modeling Exercise 1

Algebra I > Module 4 > Topic C > Lesson 23 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

A softball player throws a ball with an initial velocity of 26 feet per second. The ball leaves her hand at 3 feet above the ground.

At the same time, her softball coach throws a ball with an initial velocity of 32 feet per second. The ball leaves her hand at 2 feet above the ground.

Whose ball takes longer to land on the ground after being thrown? By approximately how many seconds? Give your answer to the nearest tenth of a second.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include examples similar to the Target Task where students write equations for two similar situations and answer questions to compare features
Give students an equation and have them write a context for it and questions that go along with the context

Illustrative Mathematics Throwing a Ball
EngageNY Mathematics Algebra I > Module 4 > Topic C > Lesson 23 — Exercise 1, Problem Set #1 and #4

Lesson 10

Lesson 12

Lesson Map

Topic A: Deriving the Quadratic Formula

Describe features of the vertex form of a quadratic function and write quadratic equations in vertex form from graphs.

A.SSE.B.3 F.IF.B.4 F.IF.C.8

Complete the square.

A.SSE.B.3.B

Complete the square to identify the vertex and solve for the roots of a quadratic function.

A.REI.B.4.B A.SSE.B.3.B

Solve and interpret quadratic applications using the vertex form of the equation.

A.SSE.B.3.B F.IF.C.8.A

Convert and compare quadratic functions in standard form, vertex form, and intercept form.

F.IF.B.4 F.IF.C.9

Derive the quadratic formula. Use the quadratic formula to find the roots of a quadratic function.

A.REI.B.4.A

Determine the number of real roots of a quadratic function using the discriminant of the quadratic formula.

A.REI.B.4.B F.IF.C.7.A

Graph quadratic functions from all three forms of a quadratic equation.

F.IF.C.7.A

Topic B: Transformations and Applications

Describe transformations to quadratic functions. Write equations for transformed quadratic functions.

F.BF.B.3

Graph and describe transformations to quadratic functions in mathematical and real-world situations.

F.BF.B.3

Write and analyze quadratic functions for projectile motion and falling bodies applications.

A.CED.A.2 F.IF.C.8.A F.IF.C.9

Write and analyze quadratic functions for geometric area applications.

A.CED.A.2 F.IF.C.8.A

Write and analyze quadratic functions for revenue applications.

A.CED.A.2 F.BF.A.1.B F.IF.C.8.A

Solve and identify solutions to systems of quadratic and linear equations when two solutions are present.

A.REI.C.7 A.REI.D.11

Solve and identify solutions to systems of quadratic and linear equations when two, one, or no solutions are present.

A.REI.C.7 A.REI.D.11

Quadratic Equations and Applications

Objective

Common Core Standards

Core Standards

Creating Equations

Interpreting Functions

Interpreting Functions

Foundational Standards

Seeing Structure in Expressions

Seeing Structure in Expressions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem Set

Target Task

Additional Practice

Lesson Map

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