Algebra Learning Objective: We will solve quadratic projectile application problems by factoring. Standards for Mathematical Practice (SMPs) SMP1 Make sense of problems SMP2 Reason abstractly and quantitatively SMP3 Construct arguments and critique the reasoning of others SMP4 Model with math SMP5 Use appropriate tools SMP6 Attend to precision SMP7 Make use of structure SMP8 Look for regularity in repeated reasoning Focus Standard: A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to solve quadratic projectile motion applications. Support Standard: SMP 1, SMP 3, SMP 6 Prerequisite(s): Manipulative(s): Special Instructions: This lesson is a conceptual introduction to zeros of a quadratic function. Students eventually learn the physics involving in projectile motion using quadratic function as a general method of finding zeros of any quadratic function. Common Core Shifts in Mathematics Shift 1 Focus Shift 2 Coherence Shift 3 Fluency Shift 4 Deep Understanding Shift 5 Application Shift 6 Dual Intensity Vocabulary Zero of a Function, Root of an Equation, Binomial, & Trinomial Academic: Content: Zeros of a function Support:

Standards for Mathematical Practice that we will focus on today are: Connection: Standard 1.1   Focus on Quadratic Functions: Studying the properties of parabola (U-shape curves). This lesson, which will be taught over the next two day, ties into the real world applications of parabola with introduction to zeros of a quadratic function. Students will eventually learn the physics involving in projectile motion, gravitational acceleration, and trajectories. Standards for Mathematical Practice that we will focus on today are: Make sense of problems Model with math Reason abstractly and quantitatively Use appropriate tools in solving & factoring problem

5-3 Learning Objective We shall solve quadratic projectile application problems by factoring. 1 2 3 CFU What are we going to learn today? CFU What does Solve & Projectile mean? What is an Quadratic Equation? Factor each expression. Activate Prior Knowledge Students, you already know how to factor quadratic expressions & equations. Now, we will factor to solve and find zero in a quadratic function. Make Connection The process of working through details of a problem to reach a solution or answer. Second degree(exponent) equations. An object thrown into space with force.

Function is Zero Height(h) Parabola=–16t2 + 32t Seconds(t) Connection When a soccer ball is kicked into the air, how long will the ball take to hit the ground? The height h in feet of the ball after t seconds can be modeled by the projectile quadratic function h(t) = –16t2 + 32t. In this situation, the value of the function represents the height of the soccer ball. When the ball hits the ground, the value of the function is zero. CFU A-What is “Zero of a function”? B-What is “Symmetric” mean? Parabola=–16t2 + 32t Height(h) Function is Zero Initial Speed After t sec: Height=0 Max Height Seconds(t) A zero of a function is a value of the input x that makes the output f(x) equal zero. The zeros of a function are the x-intercepts. These zeros are always symmetric about the axis of symmetry. equal distance

Concept Development Zero of a Function:- Root of an Equation:- the x-intercepts of the quadratic function. On your whiteboard, draw the x- and y-axes. Draw the graph of a function with no, one, or two zeros. Explain. In your own words, what are the zeros of a function? The zeros of a function are ______________. CFU Binomial:- a Polynomial with two terms. 81 - Trinomial:- a Polynomial with three terms. 2x Academic Vocabulary Zero of a Function Root of an Equation Binomial Trinomial

Skill Development/Guided Practice We shall solve quadratic projectile application problems by factoring. You can also find zeros by using algebra. For example, to find the zeros of f(x)= 2d 2 + 11d + 5, you can set the function equal to zero. The solutions to the related equation represent the zeros of the function. 2 d CFU A-How do you use X-Box Method? B-What is “Zero Product Property”? . + or or

We shall solve quadratic projectile application problems by factoring. Skill Development/Guided Practice We shall solve quadratic projectile application problems by factoring. Find the roots (Zeros) of the following Quadratic Functions by factoring.

We shall solve quadratic projectile application problems by factoring. Skill Development/Guided Practice We shall solve quadratic projectile application problems by factoring. Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile. The general function that approximates the height(h) in feet of a projectile on Earth after(t)seconds is given. _ *Note that this model has limitations because it does not account for air resistance, wind, and other real-world factors. CFU A-What is V0 and h0 stand for? B-What is Vertex & zero represents?

We shall solve quadratic projectile application problems by factoring. Skill Development/Guided Practice We shall solve quadratic projectile application problems by factoring. Check It Out! A football is kicked from ground level with an initial vertical velocity of 48 ft/s. How long is the ball in the air? h(t) = –16t2 + v0t + h0 Write the general projectile function. h(t) = –16t2 + 48t + 0 Substitute 48 for v0 and 0 for h0. The ball will hit the ground when its height is zero. –16t2 + 48t = 0 Set h(t) equal to 0. –16t(t – 3) = 0 Factor: The GCF is –16t. –16t = 0 or (t – 3) = 0 Apply the Zero Product Property. t = 0 or t = 3 Solve each equation. The football will hit the ground after 3 seconds. Notice that the height is also zero when t = 0, the instant that the football is hit. CFU A-What is the General Projectile function? B-How did we find the time in air?

Skill Development/Guided Practice We shall solve quadratic projectile application problems by factoring. A rocket is launched from ground level with an initial vertical velocity of 176ft/s. After how many seconds will the rocket hit the ground? A cannon ball is fired with an initial vertical velocity of 27ft/s from 10ft above the ground level. After how many seconds will the cannonball hit the ground? Vertical Velocity of 176ft/s Vertical Velocity of 27ft/s From Ground level 10 ft. from Ground level - t 2 16t (11 - t) = 0 (16t+2)(-t+2)=0 32t -160 16t 16t = 0 or 11 - t = 0 16t+5=0 or -t+2=0 -5 32 5 -5t t = 2 27 t = 0 or t = 11 After 2 seconds the cannonball will hit the ground. After 11 seconds the rocket will hit the ground. CFU How did you solve the problem? CFU How did you solve the problem?

Relevance Understanding projectile motion is important to many engineering designs. Any engineered design that includes a projectile, an object in motion close to the Earth's surface subject to gravitational acceleration, requires an understanding of the physics involved in projectile motion. This includes machines such as motocross bikes made for launching off jumps to weapons such as missiles, turrets and high-powered cannons. NASA engineers apply projectile motion concepts as they predict meteorite paths that may enter the Earth's atmosphere or disrupt satellite transmissions. The combination of a physical understanding of projectile motion and the mathematical ability to solve equations enables engineers to predict the projectile trajectories. 0 ft 400 ft Using quadratic models, we can estimate the distance of a homerun. Does anyone else have another reason why it is relevant to find the zeros of a quadratic function? (Pair-Share) Why is it relevant to find the zeros of a quadratic function? You may give one of my reasons or one of your own. Which reason is more relevant to you? Why? CFU

a. Zeros Projectile motion Quadratic Time in air Distance The zeros of a function are when f(x) equal 0. A quadratic function can have no, 1, or 2 zeros. The zeros of a quadratic function can be found by factoring the quadratic expression. Word Bank Skill Closure Evaluate the function by setting equal to zero and factoring. Find the zeros of quadratic functions. 1 Access Common Core a. Word Bank Zeros Projectile motion Quadratic Time in air Distance Summary Closure What did you learn today about finding the zeros of quadratic functions? (Pair-Share) Use words from the word bank.

Independent Practice/Periodic Reviews

EDI – Cognitive, Teaching, and English Learner Strategies Learning Objective: We will find the zeros of quadratic functions. Cognitive Strategies Teaching Strategies Elaboration Demonstration Language Strategies Targeted Vocabulary Academic Content zeros of a function Support Vocabulary Strategy Multiple-Meaning Synonym Definition zeros of a function Homophone Internal Context Clue Listen, Speak Similar Sounds Read Tracked Reading Write Writing Content Access Strategies Comprehensible Input Cognates Contextual Clues Graphic Organizer Contextualized Definitions Pictures

We shall solve quadratic projectile application problems by factoring. Summary Closure What did you learn today about finding the zeros of quadratic functions? (Pair-Share) Use words from the word bank. Word Bank Zeros Projectile motion Quadratic Time in air Distance