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Information Regarding This Resource The following resource will help to unify the concepts of building and interpreting functions, logarithms, trigonometry, conics, complex number systems, vectors and matrices by allowing students the opportunity to analyze data in a real-world scenario. Students will also collaborate, create and reflect on their learning. The Format of This Resource This resource is organized into flexible components that can be utilized by educators, parents or students in its entirety or can be fragmented based on desired knowledge. Each text box contains the process skills students will use in the lesson to explore the mathematical content within the Pre-Calculus Curriculum. The focus of each lesson is highlighted for easy reference. The lessons have been designed to allow multiple entry points to accommodate for different levels of understanding. All lessons have been designed for mathematical computation and graphing without requiring a calculator. Throughout this resource, students are asked to justify or explain their answers, thought process or understanding. The intent is for students to reflect on their mathematical thoughts. Students should keep in mind that justifications or explanations can take multiple forms, including, but not limited to, diagrams, graphs, text, or pictures. These are not meant to be right or wrong, rather a means of making learning visible.Connection of Standards: Process Standard(s): Students will show their understanding of polynomial identities and the Binomial Theorem by making sense of problems, persevering, reasoning and making sense of relationships, and using critical thinking skills to justify their mathematical reasoning.Content Standard(s): PC.AAPR.4 Prove polynomial identities and use them to describe numerical relationships. 23495720090Frances was asked to write a polynomial identity. She wrote the below equation on her paper.(x-2)4=x4-2x3+4x2-6x+8Use the Binomial Theorem to prove whether Frances’ identity is accurate.If the two expressions are identities, write another identity, using the Binomial Theorem, for a 5th degree binomial.If the two expressions are not identities, justify algebraically using the Binomial Theorem and write the correct equivalent expression.00Frances was asked to write a polynomial identity. She wrote the below equation on her paper.(x-2)4=x4-2x3+4x2-6x+8Use the Binomial Theorem to prove whether Frances’ identity is accurate.If the two expressions are identities, write another identity, using the Binomial Theorem, for a 5th degree binomial.If the two expressions are not identities, justify algebraically using the Binomial Theorem and write the correct equivalent expression.PC.AAPR.5 Apply the Binomial Theorem to expand powers of binomials, including those with one and with two variables. Use the Binomial Theorem to factor squares, cubes, and fourth powers of binomials. Connection of Standards: Process Standard(s): Students will show their understanding of applying the Division and Remainder Theorems by making sense of problems, persevering, reasoning and making sense of relationships, and using critical thinking skills to justify their mathematical reasoning.Content Standard(s): PC.AAPR.2 Know and apply the Division Theorem and the Remainder Theorem for polynomials. -6350224790Tamika and Sam are in the same pre-calculus class. Their teacher writes fx=12x3+13x2-27x-10 on the board and asks them to solve the equation. Sam decides to use his calculator to graph the function to find the x-intercepts. He can find that one zero is x=-2, but cannot figure out how to find the other two roots, as they are not integers. Tamika and Sam decide to work together to find the other two roots.How can Sam and Tamika use the solution x=-2 to find the other two roots?How do Sam and Tamika know if the roots are real or imaginary when looking at the graph?Find all solutions for the polynomial function.Explain how the Rational Root Theorem could have been used to refine the possibilities for finding the remaining roots.Explain how using division and the Remainder Theorem could have been used to find the remaining two roots.Explain what method you used.Algebraically justify that you have correct solutions. 00Tamika and Sam are in the same pre-calculus class. Their teacher writes fx=12x3+13x2-27x-10 on the board and asks them to solve the equation. Sam decides to use his calculator to graph the function to find the x-intercepts. He can find that one zero is x=-2, but cannot figure out how to find the other two roots, as they are not integers. Tamika and Sam decide to work together to find the other two roots.How can Sam and Tamika use the solution x=-2 to find the other two roots?How do Sam and Tamika know if the roots are real or imaginary when looking at the graph?Find all solutions for the polynomial function.Explain how the Rational Root Theorem could have been used to refine the possibilities for finding the remaining roots.Explain how using division and the Remainder Theorem could have been used to find the remaining two roots.Explain what method you used.Algebraically justify that you have correct solutions. Connection of Standards: Process Standard(s): Students will show their understanding of linear and quadratic systems by making sense of problems, persevering, reasoning and making sense of relationships, using critical thinking skills to justify their mathematical reasoning, and connecting ideas to real world situations through modeling.Content Standard(s): 29845722630Write the quadratic and linear system that corresponds with the below graph.Solve the system using any method.Justify that your solutions are correct.Explain a real-world problem that would require a system containing at least one quadratic function to be solved.00Write the quadratic and linear system that corresponds with the below graph.Solve the system using any method.Justify that your solutions are correct.Explain a real-world problem that would require a system containing at least one quadratic function to be solved.PC.AREI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. Connection of Standards: Process Standard(s): Students will show their understanding of writing and modeling polynomial functions and analyzing the graphs by making sense of problems, persevering, reasoning and making sense of relationships, using critical thinking skills to justify their mathematical reasoning, and connecting ideas to real world situations through modeling.Content Standard(s): PC.ASE.1 Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. PC.ASE.2 Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. PC.FBF.3 Describe the effect of the transformations ?(?), ?(?)+?, ?(? +?), and combinations of such transformations on the graph of ? = ?(?) for any real number ?. Find the value of ? given the graphs and write the equation of a transformed parent function given its graph.260351300292The path of the first 2 seconds of a roller coaster is shown below and can be represented by the function ht=(t+1)(t-2)2(t-4)2, where t is the amount of time in seconds and h(t) is the height in feet.How high was the platform when the roller coaster started?This model is only good for a portion of the roller coaster’s path. Determine the interval of time for which the model is realistic. Explain your reasoning.Sketch the roller coaster’s path from 0 to 5 seconds.What do the zeros represent in this real-world model? (Think in terms of an actual roller coaster and the above diagram and not x-intercepts.)State any extrema (maxima or minima) and intervals of increase and decrease for your graph.Explain what the above key features represent in terms of the roller coaster. (Use the terms seconds and feet in your explanation.)Determine how fast, in ft/sec the roller coaster was going between 1 and 4 seconds. Is this rate realistic? Justify why it is or why it is not realistic.Extension: Write a function that could model another portion of this roller coaster. Justify your model.00The path of the first 2 seconds of a roller coaster is shown below and can be represented by the function ht=(t+1)(t-2)2(t-4)2, where t is the amount of time in seconds and h(t) is the height in feet.How high was the platform when the roller coaster started?This model is only good for a portion of the roller coaster’s path. Determine the interval of time for which the model is realistic. Explain your reasoning.Sketch the roller coaster’s path from 0 to 5 seconds.What do the zeros represent in this real-world model? (Think in terms of an actual roller coaster and the above diagram and not x-intercepts.)State any extrema (maxima or minima) and intervals of increase and decrease for your graph.Explain what the above key features represent in terms of the roller coaster. (Use the terms seconds and feet in your explanation.)Determine how fast, in ft/sec the roller coaster was going between 1 and 4 seconds. Is this rate realistic? Justify why it is or why it is not realistic.Extension: Write a function that could model another portion of this roller coaster. Justify your model.PC.FIF.4 Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. PC.FIF.6 Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. Connection of Standards: Process Standard(s): Students will show their understanding of the relationship between inverse functions and composition of functions by making sense of problems, persevering, reasoning and making sense of relationships, and using critical thinking skills to justify their mathematical reasoning.Content Standard(s): PC.FBF.4 Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse functions if and only if f(x) = y and g(y) = x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain. a. Use composition to verify one function is an inverse of another. b. If a function has an inverse, find values of the inverse function from a graph or table. 32385688975Justify that fx=(x-3)2+1 and f-1x=x-1+3 are inverse functions.State the domain and range for fx=(x-3)2+1.Domain ________________________Range ____________________________State the domain and range for f-1x=x-1+3.Domain ________________________Range ____________________________What do you notice about the domain and range of inverse functions?Find (f°f-1)(x). (Also written as f(f-1x)Find (f-1°f)(x). Use the above compositions and the below statement to justify the functions are inverse functions.If f°gx and (g°f)(x) are both equivalent to x, then the two functions are inverse functions.Extension: Find the inverse of x=2log3 x . Use compositions to justify they are inverse functions. 00Justify that fx=(x-3)2+1 and f-1x=x-1+3 are inverse functions.State the domain and range for fx=(x-3)2+1.Domain ________________________Range ____________________________State the domain and range for f-1x=x-1+3.Domain ________________________Range ____________________________What do you notice about the domain and range of inverse functions?Find (f°f-1)(x). (Also written as f(f-1x)Find (f-1°f)(x). Use the above compositions and the below statement to justify the functions are inverse functions.If f°gx and (g°f)(x) are both equivalent to x, then the two functions are inverse functions.Extension: Find the inverse of fx=2log3 x . Use compositions to justify they are inverse functions. PC.FBF.5 Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents.Connection of Standards: Process Standard(s): Students will show their understanding of the unit circle and the relationship between radians and degrees by making sense of problems, persevering, reasoning and making sense of relationships, using critical thinking skills to justify their mathematical reasoning, and connecting ideas to real world situations through modeling.Content Standard(s): 26670412115Introduction to the Unit CircleDraw a circle on paper, making sure you can identify the center of the circle. Use a straightedge to draw a radius of the circle (from the center of the circle to a point on the circle).Cut a string the length of the radius.Starting at a given point on the circle, use the string to determine how many times the radius length string wraps around the circumference of the circle exactly one time. (See Diagram A at the end of this document.)Take pictures of each movement of the string to show the number of radius length strings it would take to wrap around the circle exactly one time.You should have determined that the string wraps around the circumference of the circle a little more than 6 times. As a matter of fact, it wraps around exactly 2π times.The formula for the circumference of a circle is C = 2πr. How does the above activity help you understand the formula for circumference and how it was derived?Will a radius length string always wrap around the circumference of a circle 2π times or is it dependent on the size of the circle? Test your hypothesis to justify your answer.Since the angle of one full rotation around a circle is 360° and the distance around a circle (circumference) is 2πr, we can now break a circle into degrees and radians. A radian is the SI unit for measuring angles. Because the unit circle used in trigonometry has a radius of 1 unit, the radian measure of one full rotation around the unit circle is equivalent to 2π. Can you write a conversion factor that will easily allow you to transfer between degree measure and radians?Break your circle into the below degree measures. Now find the equivalent radians for each and record these measures on your diagram. (See Diagram B at the end of this document as an example.)360° = ________ radians9. 30°= ________ radians180° = ________ radians10. 60°= ________ radians90° = ________ radians11. 120°= ________ radians45° = ________ radians12. 150°= ________ radians135°= ________ radians13. 210°= ________ radians270°= ________ radians14. 240°= ________ radians225°= ________ radians15. 300°= ________ radians315°= ________ radians16. 330°= ________ radians00Introduction to the Unit CircleDraw a circle on paper, making sure you can identify the center of the circle. Use a straightedge to draw a radius of the circle (from the center of the circle to a point on the circle).Cut a string the length of the radius.Starting at a given point on the circle, use the string to determine how many times the radius length string wraps around the circumference of the circle exactly one time. (See Diagram A at the end of this document.)Take pictures of each movement of the string to show the number of radius length strings it would take to wrap around the circle exactly one time.You should have determined that the string wraps around the circumference of the circle a little more than 6 times. As a matter of fact, it wraps around exactly 2π times.The formula for the circumference of a circle is C = 2πr. How does the above activity help you understand the formula for circumference and how it was derived?Will a radius length string always wrap around the circumference of a circle 2π times or is it dependent on the size of the circle? Test your hypothesis to justify your answer.Since the angle of one full rotation around a circle is 360° and the distance around a circle (circumference) is 2πr, we can now break a circle into degrees and radians. A radian is the SI unit for measuring angles. Because the unit circle used in trigonometry has a radius of 1 unit, the radian measure of one full rotation around the unit circle is equivalent to 2π. Can you write a conversion factor that will easily allow you to transfer between degree measure and radians?Break your circle into the below degree measures. Now find the equivalent radians for each and record these measures on your diagram. (See Diagram B at the end of this document as an example.)360° = ________ radians9. 30°= ________ radians180° = ________ radians10. 60°= ________ radians90° = ________ radians11. 120°= ________ radians45° = ________ radians12. 150°= ________ radians135°= ________ radians13. 210°= ________ radians270°= ________ radians14. 240°= ________ radians225°= ________ radians15. 300°= ________ radians315°= ________ radians16. 330°= ________ radiansPC.FT.1 Understand that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle. Reflection:Collaborate with someone in your family, a friend, or a neighbor. Ask them to look over your mathematical reasoning and ask you at least 5 guiding questions. Document the 5 questions they asked. Document the answers you gave to the 5 questions.Reflect on your work. Where did you struggle? Where did you triumph? What do you still wonder? Look over the content you covered, the processes that guided you through your discoveries, and think about your collaboration. Write down your thoughts and allow your reflection to move you forward in your mathematical thinking. 22914225158105Diagram B00Diagram B3931632027604003657601950915131532912211522915201074469158261621018500 ................
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