Expressions&Equations+(Salem+Hills+CA,+refers+to+A.SSE.3)

SWBAT: use the concepts of factoring and completing the square to create equivalent expressions for quadratics in order to find critical information regarding the quadratic function.
 * __Unit Goal__**


 * __Content Standards__**
 * A.SSE.3 ** Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 
 * a. ** Factor a quadratic expression to reveal the zeros of the function it defines.
 * b. ** Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

1) Make sense of problems and persevere in solving them. 2) Reason abstractly and quantitatively. 3) Construct viable arguments and critique the reasoning of others 4) Model with mathematics 5) Use appropriate tools strategically 6) Attend to precision 7) Look for and make use of structure 8) Look for and express regularity in repeated reasoning
 * __Integrate the Practice Standards throughout the learning cycles:__**

Launch: These represent the path of a football thrown during the last play at the homecoming game. Are any of the expressions given equivalent? Show your work and explain your reasoning why or why not.
 * __Develop:__**
 * __Activity 1:__**

(a) 1 3/16 + 18//t// – 16//t//2 (b) –16(//t// – 19/16 )(//t// + 1/16 ) (c) 16t2 + 18t – 19/16 (d) –16(//t// – 9/16 )2 + 100/16 (e) 19/16 + 18t – 16t2 (f) 1/16 (19 – 16//t//)(16//t// + 1)

Explore: Students work to prove if any are equivalent Discuss: Bring class back together to share which expressions are indeed equivalent

Actvity 2 Problem 1: The shape of the Gateway arch in St Louis, Missouri, is a catenary curve, which closely resembles a parabola. The function y = -2/315x2 + 4x models the shape of the arch, where y is the height in feet and x is the horizontal distance from the base of the left side of the arch in feet.

According to the model, what is the maximum height of the arch?

Problem 2: A model for the height of an arrow shot into the air is y = -16t2 + 72t + 5, where x is time and y is height. What can you learn by finding the graph’s intercept(s) with the x-axis?

Activity 3: The activity from this NCTM page was used for activity 1 and activity 3. []
 * __Solidify__**

As shown in the graph, the height of a thrown horseshoe depends on the time that has elapsed since its release. (Note that this graph of the horseshoe’s height is parabolic, but it is not the same as the graph of the horseshoe’s flight path.) The height of the horseshoe (measured in feet) as a function of time (measured in seconds and represented by the variable //t//) from the instant of release is 1 3/16 + 18//t// – 16//t//2. (use the picture from the horseshoe activity listed above) What information can you conclude about the horseshoe’s flight from other equivalent expressions? Explain your answers.

Activity 4: Problem 1: The Brick Bakery sells more bagels when it reduces its prices, but then its profit changes. The function y = -1000x2 + 1100x – 2.5 models the bakery’s daily profit in dollars, from selling bagels, where x is the price of a bagel in dollars. The bakery wants to maximize the profit. What will the daily profit be for selling the bagels at $0.40 each ; for $0.85 each? What price should the bakery charge to maximize its profit from bagels? What is the maximum profit? Explain your reasoning and show your work.
 * __Practice__**

Problem 2: The profit P from handmade sweaters depends on the price s at which each sweater is sold. The function P = -(s – 20)(s – 100) models the monthly profit from sweaters for one custom tailor. What will be the maximum profit for the sweaters?

Launch Explore Discuss