Functions, Rates, and Quantitative Reasoning: From ...commoncoretools.me/wp-content/uploads/2016/01/JMM... · Functions, Rates, and Quantitative Reasoning: From Proportionality to

Functions, Rates, and Quantitative Reasoning:From Proportionality to Exponential Growth

Cody L. PattersonThe University of Texas at San Antonio

January 7, 2016

Cody L. Patterson The University of Texas at San Antonio

In This Talk

In this talk:

An interesting function-graphing task

The role of RoC (Rate of Change) in linear functions

Thinking about covariational reasoning

A progression: Proportional → Linear → Exponential


A Function-Graphing Task

A Ferris wheel stands several meters above the ground. You ridethe Ferris wheel, which rotates at a constant speed.. . . while apower line hovers dangerously overhead.









Sketch a graph of the relationship between your distance from theground and your distance from the power line while you are on theFerris wheel.


Linear Functions in High School Algebra

Much of the focus in algebra in Grades 8 and 9 is on developingtechniques for working with linear functions.

Many of these techniques are aimed at doing“cross-representational work” – transferring information from onerepresentation of a linear function to another.












Covariational Reasoning

Carlson et al. developed a framework for covariational reasoning, a collection ofmental actions associated with coordinating two varying quantities:

MentalAction

Description

MA1 Coordinating the value of one variable with changes in the other

MA2Coordinating the direction of change of one variable withchanges in the other

MA3Coordinating the amount of change of one variable withchanges in the other

MA4Coordinating the average rate of change of the function withuniform increments of change in the input variable(“chunky” covariation)

MA5Coordinating the instantaneous rate of change of the functionwith continuous changes in the independent variable(“smooth” covariation)

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying

covariational reasoning while modeling dynamic events: A framework and a

study. Journal for Research in Mathematics Education, 33(5), 352-378.


Covariational Reasoning

Carlson et al. developed a framework for covariational reasoning, a collection ofmental actions associated with coordinating two varying quantities:

MentalAction

Description

MA1 Coordinating the value of one variable with changes in the other

MA2Coordinating the direction of change of one variable withchanges in the other

MA3Coordinating the amount of change of one variable withchanges in the other

MA4Coordinating the average rate of change of the function withuniform increments of change in the input variable(“chunky” covariation)

MA5Coordinating the instantaneous rate of change of the functionwith continuous changes in the independent variable(“smooth” covariation)

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying

covariational reasoning while modeling dynamic events: A framework and a

study. Journal for Research in Mathematics Education, 33(5), 352-378.


Responses to the Ferris wheel task

Some teacher responses to the Ferris wheel task



Periodic function (“thematic response”)



Increasing, then decreasing



Decreasing, then increasing (coordinating the correct pair ofquantities)



Decreasing function (coordinating direction of change)



Linear function (coordinating rate of change)



Linear function with attention to domain



“Probably not linear”, but graph appears to be a line



Rate of change is −1, but some telltale signs of non-linearity?


Quantitative Covariational Reasoning

In working with certain classes of functions that appear frequentlyin the real world, it is useful to be able to make precise statementsabout how changes in the dependent variable (call it y) correspondto changes in the independent variable (call it x).

(Call this “quantitative covariational reasoning” (QCR)?)

Example: When your distance from the ground increases by someamount, your distance from the power line decreases by the sameamount.

(Alternatively, when your distance from the ground changes by∆x , your distance from the power line changes by −1 ·∆x)



This type of thinking is included among the CCSSM for HSFunctions:

F-LE.1: Distinguish between situations that can be modeled withlinear functions and with exponential functions. . . . Prove thatlinear functions grow by equal differences over equal intervals, andthat exponential functions grow by equal factors over equalintervals.



Function type Quantitative description of covariation

ProportionalWhen x changes by a fixed amount ∆x , y changes by afixed amount k∆x .

(Unit rate = k)When x grows/shrinks by a fixed scale factor, ygrows/shrinks by the same scale factor.

Linear When x changes by a fixed amount ∆x , y changes by a(RoC = m) fixed amount m∆x .

ExponentialWhen x increases by a fixed amount, y grows/shrinks by afixed factor.

Power When x grows/shrinks by a fixed factor α, y grows/shrinks(y = Axk ) by the factor αk .

LogarithmicWhen x grows/shrinks by a fixed factor, y changes by afixed amount.


A Map of Covariational Reasoning?

A possible map of covariational reasoning, including QCR:

From MA3, “Coordinating the amount of change of one variablewith changes in the other,” the existing framework of CR dealswith coordinating rates of change, first discretely, thencontinuously. QCR deals with increasing quantitative precisionwhen coordinating changes in variables.


Building Functions With QCR

For some problems, we can build formulas for functions by using“discrete” operation sense gained in Grades K–7:

At the Aerial World Amusement Park, patrons pay $45 for general admission.In order to go on a ride, each patron must present a ride ticket. Each visitorgets two ride tickets for free as part of the general admission package, but eachadditional ride ticket after the second costs $8. How much does it cost (indollars) for a visitor to attend the park and ride n rides, where n ≥ 2?

Cost = Admission + Cost of add’l rides

= 45 + 8 · (# of add’l rides)

= 45 + 8(n − 2)



For some problems, we can build formulas for functions by using“discrete” operation sense gained in Grades K–7:

At the Aerial World Amusement Park, patrons pay $45 for general admission.In order to go on a ride, each patron must present a ride ticket. Each visitorgets two ride tickets for free as part of the general admission package, but eachadditional ride ticket after the second costs $8. How much does it cost (indollars) for a visitor to attend the park and ride n rides, where n ≥ 2?

Cost = Admission + Cost of add’l rides

= 45 + 8 · (# of add’l rides)

= 45 + 8(n − 2)



But for more “continuous” problems (and for certain contexts), wegain an advantage by using quantitative covariational reasoning:

On a mountainside, the air temperature decreases by about 5 degrees

Fahrenheit for every 1000 feet of increase in elevation. If the temperature at

2000 feet above sea level is 64◦F, then what is the temperature in degrees

Fahrenheit at elevation x feet?

Temp = (Temp at 2000 ft) + (Temp difference)

= 64 + RoC · (Elevation difference)

= 64− 51000 (x − 2000)



But for more “continuous” problems (and for certain contexts), wegain an advantage by using quantitative covariational reasoning:

On a mountainside, the air temperature decreases by about 5 degrees

Fahrenheit for every 1000 feet of increase in elevation. If the temperature at

2000 feet above sea level is 64◦F, then what is the temperature in degrees

Fahrenheit at elevation x feet?

Temp = (Temp at 2000 ft) + (Temp difference)

= 64 + RoC · (Elevation difference)

= 64− 51000 (x − 2000)



What about building an exponential function?

The population of the United States is approximately 322 million, and is

currently growing at a rate of approximately 0.76% per year. Assuming this

growth rate continues, how many millions of people will live in the U.S. in the

year t?

Pop. in year t = (Pop. in year 2016) · (Copies of growth factor)

= 322 · 1.0076 · 1.0076 · · · · · 1.0076︸︷︷︸A copy of the growth factor for each intervening year

= 322 · 1.0076t−2016



What about building an exponential function?

The population of the United States is approximately 322 million, and is

currently growing at a rate of approximately 0.76% per year. Assuming this

growth rate continues, how many millions of people will live in the U.S. in the

year t?

Pop. in year t = (Pop. in year 2016) · (Copies of growth factor)

= 322 · 1.0076 · 1.0076 · · · · · 1.0076︸︷︷︸A copy of the growth factor for each intervening year

= 322 · 1.0076(t−2016)


Using QCR to Read a Formula

We can interpret a formula for an exponential function by usingalgebraic structure to make a statement about how a change in theindependent variable corresponds to a change in the dependentvariable:

The population, in thousands of people, of a small town t yearsafter 2000 can be modeled by the function

P = 3.4 · 2t/11.2

We can use algebraic structure to see that when t increases by11.2, P grows by a factor of 2. So the 11.2 in the formularepresents the doubling time of the population.


Using QCR to Read a Formula

We can interpret a formula for an exponential function by usingalgebraic structure to make a statement about how a change in theindependent variable corresponds to a change in the dependentvariable:

The population, in thousands of people, of a small town t yearsafter 2000 can be modeled by the function

P = 3.4 · 2t/11.2

We can use algebraic structure to see that when t increases by11.2, P grows by a factor of 2. So the 11.2 in the formularepresents the doubling time of the population.


A Connection Between Linear and Exponential Functions

Analyzing the U.S. Population



The data seem to conform to not one, but two different linearfunctions (one roughly between 1790 and 1900, the other roughlybetween 1900 and 2010).



The data seem to conform to not one, but two different linearfunctions (one roughly between 1790 and 1900, the other roughlybetween 1900 and 2010).


Progressions in Reasoning About Functions


Thank you!

Thank you!

[email protected]


Documents

Functions, Rates, and Quantitative Reasoning: From ...commoncoretools.me/wp-content/uploads/2016/01/JMM... · Functions, Rates, and Quantitative Reasoning: From Proportionality to