Grade 10 Mathematics Extended Global Warming – Is it Really Happening?
Guiding Question: How can I use trigonometric models to examine global warming? (AOI: Environment)
Sine and cosine functions can be used to model cyclical trends. Examples include the seasonal variation of demand for certain items, the cyclical nature of recession and prosperity, average daily temperatures, and tidal patterns. (http://people.hofstra.edu/stefan_Waner/RealWorld/trig/trigintro.html, accessed Nov 6, 2007). This assignment will consist of three parts, each marked separately. You will first be investigating trigonometric graphs and their transformations using your graphing calculator. You will then use the knowledge you gain to produce a mathematical model for real-life data. Finally, you will use trigonometric models to explore the effects of global warming.
Part 1 – Investigating Trigonometric Graphs. You will be given 80 minutes in class to complete this assignment. You will be assessed on MYP Criteria B – Investigating Patterns C – Communication in Mathematics D – Reflection in Mathematics (to be completed on moodle) You will work through the activity to investigate each family of graphs and record your results. You are then expected to write a short paragraph summarizing your results. You will need to describe the effects of A, B, C, and D in the graph of: y = A sin [B(x – C)] + D and y = A cos [B(x – C)] + D Questions to consider: Have I explained all the patterns and described them as relationships or general rules? Have I given justifications or proofs as to why I think my generalizations work? Have I presented all my information clearly and logically? Have I clearly explained my reasoning? Have I used a wide range of mathematical symbols and language?
Part 2 – Using a Trigonometric Function to Model Data: Due Monday, January 18 th . You will complete this as a homework assignment. You will be assessed on MYP Criteria A – Knowledge and Understanding C – Communication in Mathematics D – Reflection in Mathematics You will be working with the following situation: The depth of water at your favourite snorkelling spot varies from 1.2 meters to 4.4 meters, depending on the time. Last Sunday, high tide occurred at 5:00 am and the next high tide occurred at 6:30 pm. Create a model to describe the depth of water as a function of time t in hours since midnight on Sunday morning. Create a second model using data from the depth of water at your other favourite snorkelling spot, where the tide last Sunday varies from a low of 1.4 meters at 4:00 am to a high of 3 meters at noon. (http://people.hofstra.edu/stefan_Waner/RealWorld/trig/trigintro.html, accessed Nov 6, 2007) You will hand in a one to two page written report on your findings. You are expected to explain the situation and the way you came up with your models. You will be assessed on your mathematical content as well as the clarity of your explanations. Remember to write as if the reader does not know the problem. In particular, you should focus on carefully reflecting and evaluating on all your work. Questions to consider: Have I shown that I thoroughly understand the mathematics involved? Have I presented all my information clearly and logically? Have I clearly explained my reasoning and my solution to the problem? Have I used a wide range of mathematical symbols and language? Have I discussed why I chose the methods and processes I used? Have I discussed other ways in which I could have approached the problem and why I chose not to use those ways? Have I suggested improvements to my method? Have I discussed how I know that my model is correct? Have I discussed the accuracy of my model in the context of the problem? Have I discussed the significance of my findings? Have I given a detailed explanation of my findings in connection to real life? Part 3 – Using Trigonometric Models to Examine Global Warming – Due Tuesday, January 26 th . You will be assessed on MYP Criteria A – Knowledge and Understanding C – Communication D – Reflection and Evaluation
Your task is to do research on temperature trends over time and determine whether global warming is happening. You will need to find appropriate data that you can examine. You will then need to come up with models that you think fit this data. You should use technology as well as your mathematical skills to produce these models. You will then need to analyze your different models and make a judgment on the seriousness of global warning.
This is an open-ended problem with a number of different approaches, so it gives you a lot of opportunity for reflection and evaluation.
To get you started thinking: The table below gives the 24-hour Average Temperature at Dar es Salaam Airport over an entire century.
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year °C 27.5 27.6 27.2 26.2 25.1 23.9 23.3 23.5 24.0 25.1 26.2 27.3 25.6 °F 81.5 81.7 81.0 79.2 77.2 75.0 73.9 74.3 75.2 77.2 79.2 81.1 78.1
Source: DAR ES SALAAM AIRPORT data derived from GHCN 2 Beta. 607 months between 1895 and 1991 (http://www.worldclimate.com/, accessed Nov 6, 2007)
What are your variables? What is an appropriate domain and range for this data? How well does this data fit a trigonometric model? Can you come up with a trigonometric model by hand? Can you use technology to produce a better model? (Hint: How do you find a regression line of best fit on the calculator?)
Would this example give a good idea of global warming? Do you think Dar es Salaam is a good indicator? Why or why not? Is there other data which might be better? What additional data might you need to make a judgment on the effects of global warming?
Some questions to consider: Have I collected appropriate data? Have I shown that I thoroughly understand the mathematics involved? Have I been able to make appropriate conclusions in this situation based on my mathematical knowledge? Have I used different suitable ways of representing my ideas (tables, graphs, figures, formulas, etc.)? How have I used technology effectively to solve the problem? Have I used technology effectively to present my information? Have I presented all my information clearly and logically in report format? Have I clearly explained my reasoning and my solution to the problem? Have I used a wide range of mathematical symbols and language? Have I discussed why I chose the methods and processes I used? Have I discussed other ways in which I could have approached the problem and why I chose not to use those ways? Have I suggested improvements to my method when needed? Have I discussed how I know that my models are correct? Have I explained whether my results make sense in the problem situation? Have I discussed the accuracy of my models in the context of the problem? Have I discussed the significance of my findings in relation to trends in global warming? Grade 10 Mathematics Extended Global Warming – Is it Really Happening? Name ______
Criterion A: Knowledge and Understanding Level Descriptors Part Part Part 1 2 3 0 I have not met any of the descriptors 1-2 I attempt to make conclusions when solving simple problems in familiar situations. 3-4 I sometimes make appropriate conclusions when solving simple and more complex problems in familiar situations. 5-6 I generally make appropriate conclusions when solving challenging problems in a variety of familiar situations. 7-8 I consistently make appropriate conclusions when solving challenging problems in a variety of situations including unfamiliar ones. Comments:
Criterion B: Investigating Patterns Level Descriptors Part Part Part 1 2 3 0 I have not met any of the descriptors 1-2 With some help, I apply mathematical problem-solving techniques to recognize simple patterns. 3-4 I choose and then apply mathematical problem-solving techniques to recognize patterns I suggest relationships or general rules. 5-6 I choose and then apply mathematical problem-solving techniques to recognize patterns I describe the patterns as relationships or general rules I make conclusions that fit the results. 7-8 I choose and then apply mathematical problem-solving techniques to recognize patterns I describe the patterns as relationships or general rules I make conclusions that fit the results. I justify or prove my mathematical relationships or general rules. Comments: Criterion C: Communication In Mathematics Level Descriptors Part Part Part 1 2 3 0 I have not met any of the descriptors 1-2 I use basic mathematical language (notation, symbols, vocabulary) and/or ways of representing mathematical ideas (formulae, diagrams, tables, charts, graphs, models). My reasoning is difficult to follow. 3-4 I use sufficient mathematical language (notation, symbols, vocabulary) and ways of representing mathematical ideas (formulae, diagrams, tables, charts, graphs, models). My reasoning is clear but not always logical or complete. I move between different forms of representation with some success. 5-6 I use good mathematical language (notation, symbols, vocabulary) and ways of representing mathematical ideas (formulae, diagrams, tables, charts, graphs, models). My reasoning is to the point, logical and complete. I move effectively between different forms of representation. Comments:
Criterion D: Reflection in Mathematics Level Descriptors Part Part Part 1 2 3 I have not met any of the descriptors I try to explain whether my results make sense in the problem situation. I try to describe the importance of my findings in connection to real life. I correctly but briefly explain whether my results make sense in the problem situation. I describe the importance of my findings in connection to real life. I try to justify the degree of accuracy in my results, where appropriate. I critically explain whether my results make sense in the problem situation. I give a detailed explanation of the importance of my findings in connection to real life. I justify the degree of accuracy in my results, where appropriate. I suggest improvements to my method when needed. Comments: Grade 10 Extended Mathematics Investigating Trigonometric Graphs
Here is some terminology that might be helpful in your investigation. Periodic function: a function that repeats its values in regular intervals or periods. Period: how many degrees it takes to complete one cycle. Frequency: how many cycles there would be in 360°. Amplitude: how far up and how far down the graph goes from the center line. Translation: a transformation that "slides" an object in a fixed distance and in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. Stretch: Just what it sounds like!
2 center line
amplitude
amplitude
-2 y
Task 1 – Investigating y = sin x and y = cos x
Set your WINDOW as follows: Y min = -2 Sketch the graphs of y = sin x andx y = cos x Y max = 2 on the set of axes below. Y scl = 1
X min = -360 X max = 360 X scl = 90
Degrees
Find the period and amplitude of these graphs: y = sin x y = cos x
Period ______Period ______
Amplitude ______Amplitude ______
Explain how the graphs of y = sin x and y = cos x are related to each other. y
x
Task 2 – Investigating y = -sin x
Graph y = sin x and y = - sin x on the same set of axes
Degrees
What effect does the negative have on the graph?
y
Can you give a justification or proof as to why you think this is the case?
x
Task 3 – Investigating graphs in the form y = A sin x 1 Compare the graphs of y = sin x, y = 2 sin x, y = sin x on the axes below. 2
Degrees
Explore y = A sin x for other values for ‘A’. Make sure you include a good range of values. Explain what happens to y = sin x as A changes. Describe any patterns you see as relationships or general rules.
Can you give a justification or proof for your relationship or rule? y
x
Task 4 – Investigating graphs of y = sin x + D
Compare the graphs of y = sin (x), y = sin (x) + 1 and y = sin (x) – 2 on the axes below.
Degrees
Explore y = sin x + D for other values of ‘D’. Make sure you include a good range of values. Explain what happens to y = sin x + D as ‘D’ changes. Describe any patterns you see as relationships or general rules.
y
Can you give a justification or proof for your relationship or rule?
x
Task 5 – Investigating y = sin Bx 1 Compare the graphs of y = sin x, y = sin 2x, and y = sin x on the axes below. 2
Degrees
Explore y = sin Bx for other values of ‘B’. Make sure you include a good range of values. Explain what happens to y = sin Bx as ‘B’ changes. Describe any patterns you see as relationships or general rules.
Can you give a justification or proof for your relationship or rule? y
x
Task 6 – Investigating the graphs of y = sin (x – C)
Compare the graphs of y = sin x, y = sin (x -90º), y = sin (x - 45º) on the axes below.
Degrees
Explore y = sin (x – C) for other values of ‘C’. Make sure you include a good range of values. Explain what happens to y = sin (x – C) as ‘C’ changes. Describe any patterns you see as relationships or general rules.
Can you give a justification or proof for your relationship or rule?
Summary What would happen if you combined values? Describe the effects of A, B, C, and D in the graph of: y = A sin [B(x – C)] + D Make sure you include positive and negative values of A, B, C and D in your summary. Also, be sure to include justification as to why you think your generalizations work.
If Time: Investigate y = A cos [B(x – C)] + D and describe the effects of A, B, C, and D on the graph. What are the similarities and differences with the sine function? 3 y
2
Drawing Conclusions –1 NO CALCULATORS ALLOWED! x
-360 -270 -180 -90 90 180 270 360 1. Use your results from above-1 to find the period and amplitude of these graphs. Then explain how to transform-2 y = sin x to form these graphs. -3 Finally, sketch the graph. -4
-5 a) y = 2 sin x – 1 period ____ amplitude ____ Transformations: ______
y
x
b) y = - 1/2sin (x - 45º) period ____ amplitude ____ Transformations: ______
y
x
c) y = 0.5 sin 2x period ____ amplitude ____ Transformations: ______
3 y
2
1
x
-360 -270 -180 -90 90 180 270 360
-1
-2
-3
-4
-5 d) y = sin (x/2) + 2 period ____ amplitude ____ Transformations: ______3 y
2
1
x
-360 -270 -180 -90 90 180 270 360
-1
-2
-3
-4
-5 e) y = 3sin (x + 90º) – 2 period ____amplitude ____ Transformations: ______
3 y
2
1
x
-360 -270 -180 -90 90 180 270 360
-1
-2
-3
-4
-5 f) y = -sin (x - 90º) period ____amplitude ____ Transformations: ______
y
x
2. For each of these graphs determine the: a) period and amplitude and b) the sine or cosine equation of the graph 1.
y
x
Period ____ Amplitude ____
Equation of the graph ______
2.
Period ____ Amplitude ____
Equation of the graph ______ y
y
x
x
3.
4.
Period ____ Amplitude ____
Equation of the graph ______Period ____ Amplitude ____
Equation of the graph ______10 y
5
x
-360 -270 -180 -90 90 180 270 360
2 y
1 x
-360 -270 -180 -90 90 180 270 360
-1
-2
-3
-4
-5
-6
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-8