Exploring the Parabolic Boom in Fathom



Modeling Canada’s Youth Cohorts using E-STAT Data & Fathom

Testing Terrific Trigonometry Tools

Originators: Joel Yan, Statistics Canada, mdm4u@statcan.ca

Heather Curl, Sarnia Collegiate, curlhe@

Jennifer Brown, St. Michael Catholic High School, Kemptville, jenn.brown@cdsbeo.on.ca

Introduction

In this activity, students will learn that trigonometric forms can be used to model real world data on population. Data on specific age groups (cohorts) over time in Canada are extracted from E-STAT and imported into Fathom. Within Fathom, students then use sliders to model a trigonometric relation and adjust the parameters to maximize the visual fit of the sinusoidal curve to specific age cohorts in Canada. Sliders help students gain an understanding of the values of the different parameters in the general form of the sine curve, y = a sin (kx + d) + c. In this activity we use the following form for the sine curve that is used in several common textbooks:

y = amplitude sin (k*(x – x0) - phase shift) + displacement

Furthermore we will use k = =2*pi/b where b is the wavelength or period for the sine curve.

Note: x0 is the base year for our population data, which is 1970 in this case.

Overall Expectations (from the Ontario grade 11 curriculum, MCR3U)

• Identify and represent sinusoidal functions and solve problems involving sinusoidal functions, including those arising from real-world applications (page 40).

• Demonstrate an understanding of periodic relationships and the sine function and make connections between the numeric, graphical and algebraic representations of sine functions.

Specific Expectations

• Investigating the relationships between the graphs and the equations of sinusoidal functions:

o Determine through investigations using graphing software the effect of simple transformations (e.g. translations, stretches) on the graphs and equations of y = sin x and y = cos x

o Determine the amplitude, period, phase shift shift, domain and range of sinusoidal functions whose equations are given in the form y = a sin (kx + d) + c

• Solving problems involving sinusoidal functions:

o Students will collect data that can be modelled as a sinusoidal function… from secondary sources (e.g. websites such as Statistics Canada, E-STAT)… (Pg. 41, MCR3U)

▪ Students will identify sinusoidal functions… Sample problem: Using data from Statistics Canada, investigate to determine if there was a period of time over which changes in the population of Canadians aged 20-24 could be modelled using a sinusoidal function. (Pg. 41, MCR3U)

o Students will collect data that can be modelled as a sine function… from secondary sources (e.g., websites such as Statistics Canada, E-STAT)… (Pg. 51, MCF3M)

o Explain the relationship between the properties of a sinusoidal function and the parameters of its equation within the context of an application and over a restricted domain

o Pose and solve problems … drawn from a variety of applications and communicate the solutions with clarity and justification

Prior Knowledge

• General use of the world wide web

• Basic use of E-STAT

• Basics of Fathom

Classroom Instruction

Review the parametric form for a sine curve including the following terms that will be used later for the slider names in Fathom: displacement, amplitude, period, and phase.

You may ask the class to work through all the steps on the student worksheet, or, to save time, you may wish to start with the Fathom collection that was already created for this task. This dataset is available on the internet on our site in 2 forms: a simplified version with no sliders at , and a close to a final version with sliders already defined at in the E-STAT Folder under the heading Grade 11 in the file labeled Fathom dataset for Testing Terrific Trig Tools lesson.

Graphs and Answers to Questions

Q1. What is the interpretation of the parameter 'displacement' for this curve?

Answer: The vertical displacement about which the sine curve oscillates.

Q2. What is the interpretation of the parameter 'amplitude' for this curve?

Answer: The amplitude (i.e. maximum extent) of the sine curve above and below the central value (i.e. the displacement)

Q3. What is the interpretation of the parameter ‘period’ for this curve?

Answer: The length of time (number of years in this case) for the sine curve showing population for specific ages to complete one wave length or period.

Q4. What is the interpretation of the parameter ‘phase shift’phase’ for this curve?

Answer: The phase shift influences the shift of the sine curve.

Q5. Do the values of amplitude and displacement concur with your answers above? Yes.

Q6. How does changing the value of period affect your sine curve?

Answer: as period increases the length of time to replicate the sinusoidal cycle increases.

Q7. How does changing the value of the phase shift slider affect your sine curve?

Answer: as phase shift changes the shape does not change, but the curve shifts horizontally.

Q8. What are your best values for displacement, amplitude, period and phase shift from your analysis using sliders?

Answer: approximately displacement = 2,258,600 amplitude= 273,428

period = 23.9 phase shift= 1.72

Q9. How is the quality of the fit of the sine curve to our data changing over time?

Answer: it does not fit as well as time goes on, because people are having children at different ages and hence the effect of the baby boom is becoming more diffuse in the second cycle. It could be modeled by a damped trigonometric curve where the amplitude of the sine curve is reducing over time.

Q10. We have succeeded to approximate the population of 20-24 year olds over time using a sine curve. But this only explains part of the variation. By how many percentage points (to the nearest 10%) does the population of 20-24 year olds actually vary over the time period selected?

Answer: the amplitude of the sine curve is only about 12% of the displacement. Hence the variation over time is roughly 10%.

Enrichment:

Q11. What is the equation of the trigonometric regression curve of best fit? Excel does not include trigonometric regression lines.

Q12. What is the computed r^2 value? ________________________

Q13. How well does the equation of your sine curve from using the sliders above compare with the computed trigonometric regression equation? Discuss briefly.

1. Q14. Does a sine curve model this population as well as it modeled the 20-24 year old population? Give a possible

explanation. your findings

Answer: See second graph below and note the large residuals from the sine curve for more recent time periods While there is a significant “baby boom echo”, the births from baby boomers are much more spread out over time and hence the curve is more shallow and less concentrated.

Q15. Is the shape of the curve replicated when the baby-boomers start to have their own children? Compare the 2 curves.

Q16. How might you interpret the period of time between the 2 peaks?

Answer: the period.

Q17. What is another trigonometric formula that would produce the same curve?

Answer: similar formula using the cosine function. Period, displacement and amplitude would be the same, but the phase would differ by 90 degrees.

Figure 1: Sine curve fit to the population of the 20-24 year old age cohort since 1971

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Extracted from Fathom collection “Trig Power – Modeling 20-24 Age Cohort & Births”

Source: E-STAT data table 051-0001

Figure 2: Sine curve fit to the population of the 15-19 year old age cohort since 1971

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Extracted from Fathom collection “Trig Power – Modeling 15-19 Age Cohort”

Source: E-STAT data table 051-0001

File: Kingston/ Trig Tools - Modeling Age Cohorts April 2007

Folder: E-STAT/Joel/Modelling Lessons

Last updated: May 1, 2007

On the web: at http

Original Folder: Supports for Grade 7-11 / Grade 11

This was developed partly from a baby boom quadratic activity and related dataset, which are available at

Revised May 27 by Terri Blackwell, Peel District School Board

Tips from Tim Erickson & textbox incorporated on August 9, 2005

Answer for Part B of Enrichment Section:

See second graph below and note the large residuals from the sine curve for more recent time periods While there is a significant “baby boom echo”, the births from baby boomers are much more spread out over time and hence the curve is more shallow and less concentrated.

• Is the shape of the curve replicated when the baby-boomers start to have their own children? Compare the 2 curves.

• How might you interpret the period of time between the 2 peaks?

• Ask your students to specify another trigonometric formula that would produce the same curve. Answer: high level students could use the cosine version of the equation or the sine version with a different phase shift shift.

Your Task

In the real world, a graph of human population rarely approximates the shape of a sine curve with any accuracy. However, occasionally the underlying conditions lead to population trends that closely follow a sinusoidal shape over specific time periods, as shown below. The so-called “baby boom” occurred right after World War 2. Canadian soldiers returning in great numbers after the war got married and/or started having children within a few years after returning to Canada. The result was a huge increase in births in Canada beginning in 1946. Births rose for several years to a peak around 1960 and then began to decline. As a result of this baby boom, distribution of specific age populations in Canada follows a curved somewhat sinusoidal shape. In this activity, we will determine if a trig function can be a useful for modeling the population of 20-24 years olds and 15-19 year olds in Canada.

Questions marked with Ν must be answered.

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Procedure

To Access Data on Canada’s Population by Age

1. Open the E-STAT website at

2. Select your language of choice. Click “Accept and enter” at the bottom of this screen. You may be prompted to enter a password (available from you teacher)

3. Click Search CANSIM on the left side bar

4. Click Browse by Subject and click Continue

5. Click on the subject Population and demography

6. Click on the subtopic - Population characteristics

7. From the list of tables, select 051-0001 Estimates of population, by age group and sex, Canada, provinces and territories, annual.

8. On the subset selection page, choose as follows:

• Under Geography select Canada

• Under Sex select Both sexes

• Under Age group scroll down and select '15-19 years' and '20-24 years' (**while holding down the Ctrl key on a Windows platform or the Command key on a Macintosh computer)

• Under From select 1971

• Under To select the most recent year

Click the Retrieve as individual Time Series button.

9. In the Output specification screen under SCREEN OUTPUT formats, select, Plain text: Table, time as rows.

10. Press the Retrieve now button.

Fathom Analysis

Copying and Pasting the Data

11. Click and drag in the table so that you have selected all the data, but not the legend at the top of the page, or the source line at the bottom of the page

12. From the Edit pull-down menu, choose Copy.

13. Switch to Fathom. (If Fathom isn’t already running, you will need to launch it)

14. Create a new collection by dragging the collection icon into the window.

15. With the collection selected, choose Paste Cases from the Edit pull-down menu.

16. Create a case table by choosing Case table from the Insert pull-down menu.

17. Rename the attributes to Year, Pop_15_19 and Pop_20_24.

18. Change the name of the collection to ‘Population by age'.

19. Save your Fathom document by choosing Save from the File pull-down menu.

Graphing and Modeling the Data

20. Create a scatter plot of the attributes Pop_ 20_24 vs. Year.

21. Create a new slider by dragging the Slider icon (second button from the right on the tool bar at the top of the page) onto the workspace.

22. Rename the slider (named V1 by default) to displacement.

23. Create 3 more sliders and rename them amplitude, period and phase shift respectively.

24. From the Graph pull-down menu, select Plot Function.

25. Enter the sinusoidal relation in parametric form as follows. You do not have to type these words. They are available when you access the Plot Function option (see below):

displacement + amplitude* sin (2*Pi/period * (Year- 1970) - phase shift)

[Notes: you can select the slider names directly from a list after clicking 'Global Values' in the pop-up box. You can select Pi directly from a list after clicking 'Special' in the pop-up box.

You do not have to enter the dependent variable.]

Ν Q1. What is the interpretation of the parameter 'displacement' for this curve? __________

Q2. What is the interpretation of the parameter 'amplitude' for this curve? ______________

26. Since we know that displacement is the centre of the values for the population data, double click on the slider value (5.00 to start with by default) and then type in the central value as a starting value for your displacement slider. Then you should see a coloured horizontal line appear on your graph.

27. Next, estimate the approximate value for your sine curve height above and below this central displacement value. Then type this in as the starting value for the amplitude slider. Now you will see the horizontal line turn into a sine curve.

28. Right click in the graph and select and turn on the ‘Show Squares’ option. This will display a square for each year for which we have a data value. Each square displays the vertical distance between a single data point and the computed function value for the corresponding year. Seeing these squares gives us visual information; we want to select slider values which appear to minimize the sum of the areas of the squares. Fathom also displays the actual Sum of the squares. This is what you want to minimize by adjusting our sliders.

29. Drag the sliders to adjust the slider values first for period and then for phase until you generate a sine curve that fits this data. Then fine tune all your slider values to adjust the sine curve to best model the data points.

30.

Tip: You can adjust the slider variables by dragging the number line, or you can double click the slider and define the range and value.

Ν Q3. What is the interpretation of the parameter ‘period’ for this curve? ______________

Q4. What is the interpretation of the parameter ‘phase shift’phase’ for this curve? ______________

31. From the Graph pull-down menu, select Make Residual Plot. Use the residuals as well as the squares to help you find the most appropriate sine curve using the sliders.

Ν Q5. Do the values of amplitude and displacement concur with your answers above?

Q6. How does changing the value of period affect your sine curve? ________________

Q7. How does changing the value of the phase shift slider affect your sine curve? ________________

Q8. What are your values for the sliders when the function best fits the data shift? ?

Displacement: ________ amplitude: _______ Period: __________ phase: _____

Q9. displacement , amplitude , period and phase shift

How is the quality of the fit of the sine curve to our data changing over time? ________

Q10. We have succeeded to approximate the population of 20-24 year olds over time using a sine curve. But this only explains part of the variation. By how many percentage points (to the nearest 10%) does the population of 20-24 year olds actually vary over the time period selected? __________

Enrichment Ν

Enter the data in a spreadsheet package or a graphing calculator and perform a trigonometric regression analysis.

Ν Q11. What is the equation of the trigonometric regression curve of best fit? ___________________

Q12. What is the computed r^2 value? ________________________

Q13. How well does the equation of your sine curve from using the sliders above compare with the computed trigonometric regression equation? ______________ Discuss briefly.

Ν Part A

1. From the Graph pull-down menu, select Make Residual Plot.

2. Examine what happens to your Residual Plot as you move the sliders.

3. How could you use the residuals to help you find the most appropriate sine curve?

Ν Part B

Repeat the procedurethis process for the Population of 15-19 year olds in Canada.

2. Q14. Does a sine curve model this population as well as it modeled the 20-24 year old population? _______ Explain the results

. your findings

Q15. Is the shape of the curve replicated when the baby-boomers start to have their own children? __________________________________ Compare the 2 curves.

Q16. How might you interpret the period of time between the 2 peaks? _______________________

Q17. What is another trigonometric formula that would produce the same curve? _______________

| |Level 1 |Level 2 |Level 3 |Level 4 |

| |- Follows the investigation |- Follows the investigation |- Follows the investigation |- Follows the investigation |

|TIPS |instructions with considerable |instructions with some assistance |instructions with very little |instructions with no assistance |

| |assistance |-Draws conclusions and inferences |assistance |-Draws insightful conclusions and |

| |-Draws conclusions and |with some reference to the data |-Draws conclusions and |inferences with reference to the |

| |inferences with little | |inferences with reference to |data |

| |reference to the data | |the data | |

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Modeling Canada’s Youth Cohorts using E-STAT Data & Fathom

Testing Terrific Trigonometry Tools

Modeling Canada’s Youth Cohorts using E-STAT Data & Fathom

Testing Terrific Trigonometry Tools

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Statistics Canada and Fathom

This lesson plan was prepared by Statistics Canada to facilitate the use of Statistics Canada data by mathematics teachers. This lesson requires the use of the Fathom software. Fathom is licensed by the Ministry of Education and used by schools in some provinces. Providing the Fathom format is in no way an endorsement or recommendation of the Fathom software by Statistics Canada.

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