Learning Activities (1) | Classroom Resources (2) |

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 45 :

[MA2019] AL1-19 (9-12) 27 :

*Example: If the function *V(t) = 19885(0.75)^{t} *describes the value of a car after it has been owned for *t* years, 1985 represents the purchase price of the car when* t *= 0, and 0.75 represents the annual rate at which its value decreases.* [MA2019] AL1-19 (9-12) 25 :

45 ) Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [S-ID6]

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. *Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.* [S-ID6a]

b. Informally assess the fit of a function by plotting and analyzing residuals. [S-ID6b]

c. Fit a linear function for a scatter plot that suggests a linear association. [S-ID6c]

[MA2019] AL1-19 (9-12) 27 :

27. Interpret the parameters of functions in terms of a context. **Extend from linear functions, written in the form ***mx + b*, to exponential functions, written in the form *ab*^{x}.

25. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

[MA2019] AL1-19 (9-12) 24 : 24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

This exploration provides students the opportunity to actively engage in creating their own knowledge about exponential models. They are given the support to conduct their own simulation and record the information, make predictions using the data that they have collected, and compare their predictions to the technology generated models. Additionally, the technology element is necessary to make sense of the data in a more efficient manner as compared to hand calculations of procedures. The emphasis is to produce a deep conceptual understanding of rates of change of exponential functions in multiple representations and use that information to build up procedural fluency.

*This activity results from the ALEX Resource Development Summit.*

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 41 :

[MA2015] AL1 (9-12) 45 :

41 ) Represent data with plots on the real number line (dot plots, histograms, and box plots). [S-ID1]

[MA2015] AL1 (9-12) 45 :

45 ) Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [S-ID6]

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. *Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.* [S-ID6a]

b. Informally assess the fit of a function by plotting and analyzing residuals. [S-ID6b]

c. Fit a linear function for a scatter plot that suggests a linear association. [S-ID6c]

These interactive tools give students an opportunity to explore more about stemplots, control charts, and histograms as covered in the Against All Odds statistics series. Simulations allow students to explore statistics methods in-depth using their own data.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 45 :

[MA2015] AL1 (9-12) 46 :

45 ) Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [S-ID6]

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. *Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.* [S-ID6a]

b. Informally assess the fit of a function by plotting and analyzing residuals. [S-ID6b]

c. Fit a linear function for a scatter plot that suggests a linear association. [S-ID6c]

[MA2015] AL1 (9-12) 46 :

46 ) Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [S-ID7]

In Module 2, Topic D, students analyze relationships between two quantitative variables using scatterplots and by summarizing linear relationships using the least-squares regression line. Models are proposed based on an understanding of the equations representing the models and the observed pattern in the scatter plot. Students calculate and analyze residuals based on an interpretation of residuals as prediction errors.