Jazz It Up With Journals!
Jazz It Up With Journals! Leslie Banta Asst. Prof. of Mathematics Mendocino College [email protected] Explain in your own words your understanding of slope. How does this relate to the slope formula? After you have answered the first part of the prompt, read page 163 in your book. Does reading this section change how you think of slope? If yes, in what way(s)? If no, why not? In Algebra, we look at the slope (rate of change) of a line – the slope is constant throughout the line. In Calculus, we study instantaneous rate of change where we look at the slope of a curve at a single point and then throughout the entire curve. We do this using a line that is tangent to the curve – a line that touches the curve in exactly one point, skimming along the edge of the curve.
Cut out the image, below left, and tape it into your journal. Draw several tangent lines to different points along the curve (similar to the drawing on the right, but you don’t have to calculate the slope). Describe what happens with the slopes of the tangent lines as x increases (that is, as you move from left to right along the curve).
Explain the changes that would occur to the line y = x for each of the following transformations. Try to make your explanations as simple, clear and precise as you can. Could I picture the new graph from your description? You do not have to graph the lines (but you can if it helps you to picture it better). DO NOT tell me how to graph the line… tell me how the line was transformed (changed). “Mathy” words like slope, point, intercept, etc. are not allowed. Pretend you are describing the changes to a 3rd grade student who simply want to “see” the difference. • y=x+2 • y = x –3 • y = 2x • y = (1/2) x • y = –x • y = 3x – 5 • y = –2x + 1 Consider the curve y = x2. Explain the changes that would occur to the curve y = x2 for each of the following: • y = x2 + 2 • y = 2x2 These parabolas behave “nicely”. That is, their vertex is on the y-axis, but this is not always the case. When we shift a parabola horizontally, we often write the equation in vertex form so that we can easily identify the vertex. The vertex form of an equation for a parabola is y = a ( x − h ) + k . Without looking up any information online or in a book, how do you think the variables a, h, and k might change the graph of y = x2? (It’s okay if your guess is wrong – the point is for you to think about it using knowledge you already have). Now, look up some information about the topic here (read critically and completely): http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php How does what this site tells you about a, h, and k relate to what you know about the graphs of sinusoidal functions? (This is about making connections – try to communicate clearly.) 2
Graph y = x2 and y = ( x − 2) + 3 on the same graph. Be sure to include an appropriate scale. (You may use graph paper and tape it in your journal.) Briefly explain how the parent graph is changed to obtain the second graph. 2
Consider the curve y = sin(x)…
Linear Equations • Explain in your own words the process you would use to solve the following equation______________. What is your result when you follow this process? • The equation y − y1 = m( x − x1 ) is the point-slope form of a line. Explain in your own words why this is called the point-slope form. Explain how you would graph a line from an equation in this form (for example, y − 2 = 5( x + 1) ). • Explain how the process of graphing a line from an equation in slope-intercept form and graphing a line from an equation in point-slope form are similar. How are they different? General Topics • Explain your understanding of the order of operations (PEMDAS/GEMDAS). • Write an application problem that uses ___________. Write your solution to the problem, explaining each step. How could you check to see if your answer is reasonable? • Explain in your own words how you could find the length of the hypotenuse of a right triangle if you knew the length of the other two sides. • What strategy did you use to solve_______________? How did you check to see if your answer was reasonable? • What is something that you would like to learn about math? Why does this topic interest you? Higher Math • Consider two methods for finding the volume of a solid of revolution: the disk/washer method and the shell method. How do you decide which method to use? Does it make a difference which method you choose? Are there situations where one method is better than the other? Create examples that demonstrate your reasoning. • Why should you learn the techniques of integration such as integration by parts, trigonometric substitution, and partial fractions? Computers and some graphing calculators can do integration analytically much more quickly than you will ever be able to do by hand. Should these techniques be dropped from the course completely? What emphasis do you feel should be placed on pencil and paper integration in general? Just how much is necessary or beneficial to learn of these skills? Justify your answers to these questions. • What are the most important ideas that you learned in this unit? Include how_______. Explain why_______. In your own words, define___________. • From what you have read in this section, what do you consider to be the main idea? What does_________? How can you ___________? Explain_________. • What have you learned about the relationship between trigonometric and circular functions? • Explain how you can find the particular equation of a circular function if you are given its graph.
Cut out the image, below left, and tape it into your journal. Draw several tangent lines to different points along the curve (similar to the drawing on the right, but you don’t have to calculate the slope). Describe what happens with the slopes of the tangent lines as x increases (that is, as you move from left to right along the curve).
Explain the changes that would occur to the line y = x for each of the following transformations. Try to make your explanations as simple, clear and precise as you can. Could I picture the new graph from your description? You do not have to graph the lines (but you can if it helps you to picture it better). DO NOT tell me how to graph the line… tell me how the line was transformed (changed). “Mathy” words like slope, point, intercept, etc. are not allowed. Pretend you are describing the changes to a 3rd grade student who simply want to “see” the difference. • y=x+2 • y = x –3 • y = 2x • y = (1/2) x • y = –x • y = 3x – 5 • y = –2x + 1 Consider the curve y = x2. Explain the changes that would occur to the curve y = x2 for each of the following: • y = x2 + 2 • y = 2x2 These parabolas behave “nicely”. That is, their vertex is on the y-axis, but this is not always the case. When we shift a parabola horizontally, we often write the equation in vertex form so that we can easily identify the vertex. The vertex form of an equation for a parabola is y = a ( x − h ) + k . Without looking up any information online or in a book, how do you think the variables a, h, and k might change the graph of y = x2? (It’s okay if your guess is wrong – the point is for you to think about it using knowledge you already have). Now, look up some information about the topic here (read critically and completely): http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php How does what this site tells you about a, h, and k relate to what you know about the graphs of sinusoidal functions? (This is about making connections – try to communicate clearly.) 2
Graph y = x2 and y = ( x − 2) + 3 on the same graph. Be sure to include an appropriate scale. (You may use graph paper and tape it in your journal.) Briefly explain how the parent graph is changed to obtain the second graph. 2
Consider the curve y = sin(x)…
Linear Equations • Explain in your own words the process you would use to solve the following equation______________. What is your result when you follow this process? • The equation y − y1 = m( x − x1 ) is the point-slope form of a line. Explain in your own words why this is called the point-slope form. Explain how you would graph a line from an equation in this form (for example, y − 2 = 5( x + 1) ). • Explain how the process of graphing a line from an equation in slope-intercept form and graphing a line from an equation in point-slope form are similar. How are they different? General Topics • Explain your understanding of the order of operations (PEMDAS/GEMDAS). • Write an application problem that uses ___________. Write your solution to the problem, explaining each step. How could you check to see if your answer is reasonable? • Explain in your own words how you could find the length of the hypotenuse of a right triangle if you knew the length of the other two sides. • What strategy did you use to solve_______________? How did you check to see if your answer was reasonable? • What is something that you would like to learn about math? Why does this topic interest you? Higher Math • Consider two methods for finding the volume of a solid of revolution: the disk/washer method and the shell method. How do you decide which method to use? Does it make a difference which method you choose? Are there situations where one method is better than the other? Create examples that demonstrate your reasoning. • Why should you learn the techniques of integration such as integration by parts, trigonometric substitution, and partial fractions? Computers and some graphing calculators can do integration analytically much more quickly than you will ever be able to do by hand. Should these techniques be dropped from the course completely? What emphasis do you feel should be placed on pencil and paper integration in general? Just how much is necessary or beneficial to learn of these skills? Justify your answers to these questions. • What are the most important ideas that you learned in this unit? Include how_______. Explain why_______. In your own words, define___________. • From what you have read in this section, what do you consider to be the main idea? What does_________? How can you ___________? Explain_________. • What have you learned about the relationship between trigonometric and circular functions? • Explain how you can find the particular equation of a circular function if you are given its graph.
