Learning Activity

The Spread of a Virus: Does Social Distancing Matter?

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

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.

Content Standards

Mathematics (2019) Grade(s): 09-12 - Geometry with Data Analysis

MA19.GDA.12
Represent data of two quantitative variables on a scatter plot, and describe how the variables are related.

Unpacked Content

Vocabulary

Quantitative variables
Scatter plot
Residuals

Knowledge

Students know:

Techniques for creating a scatter plot,
Techniques for fitting linear functions to data.
Methods for using residuals to judge the closeness of the fit of the linear function to the original data.

Skills

Students are able to:

Accurately create a scatter plot of data.
Make reasonable assessments on the fit of the function to the data by examining residuals.
Accurately fit a function to data when there is evidence of a linear association.
Use technology to find the least-squares line of best fit for two quantitative variable.

Understanding

Students understand that:

Functions are used to create equations representative of ordered pairs of data.
Residuals may be examined to analyze how well a function fits the data.
When a linear association is suggested, a linear function can be fit to the scatter plot to aid in modeling the relationship.

Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.24
Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

Unpacked Content

Vocabulary

Linear functions
Exponential functions
Constant rate of change
Constant percent rate of change
Intervals
Percentage of growth
Percentage of decay

Knowledge

Students know:

Key components of linear and exponential functions.
Properties of operations and equality

Skills

Students are able to:

Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).

Understanding

Students understand that:

Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.

Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.25
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).

Unpacked Content

Vocabulary

Arithmetic and geometric sequences
Arithmetic sequence
Geometric sequence
Exponential function

Knowledge

Students know:

That linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Properties of arithmetic and geometric sequences.

Skills

Students are able to:

Accurately recognize relationships within data and use that relationship to create a linear or exponential function to model the data of a contextual situation.

Understanding

Students understand that:

Linear and exponential functions may be used to model data that is presented as a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.

Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.27
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 $ab^x$.

Unpacked Content

Vocabulary

Parameters

Knowledge

Students know:

Key components of linear and exponential functions.

Skills

Students are able to:

Communicate the meaning of defining values (parameters and variables) in functions used to model contextual situations in terms of the original context.

Understanding

Students understand that:

Sense making in mathematics requires that meaning is attached to every value in a mathematical expression.

Interdisciplinary Connections

Phase

During/Explore/Explain

Learning Objectives

Students will simulate the spread of disease and represent the spread using multiple representations (table, graph, algebraic model).

Students will interpret the rate of change for their models and compare them to their predictions.

Students will develop an understanding of the virulent spread of disease in terms of their model's rate of change.

Activity Details

Introduce the task by reading the situation with the students. Highlighting the start time and the amount of time it takes for the disease to spread. State to students that goal of the activity is to see which types of viruses are more deadly and whether social distancing measures make a difference. To accomplish this, students will be placed in groups to model the spread of a virus using beans. All groups will receive a cup of pinto beans that represent a healthy student population. Each group will also receive a different kind of bean (Kidney, Black, Lima) that will represent an infected individual. [Note: Teachers are not limited to the suggested beans. If teachers implementing the lesson wish to use different color and size beans, they are encouraged to do so.]

To simulate the spread of the virus, all of the pinto beans will be evenly spread out in a box lid or serving tray so that no beans are on top of each other. [Note: Any flat surface that can hold the beans without the beans spilling over will suffice. Uniform sizes are encouraged so that comparisons across groups are more effective.] Next, students will then drop one infected individual into the healthy population, flattening all of the beans so that they are not on top of each other. Every pinto bean that is touching the virus bean has now become infected. Students will then remove the number of infected pinto beans from the healthy population and replace them with the same number of virus beans. Repeating the process and completing the table in the task sheet.

As students complete their data tables and begin plotting their values on the graphs provided, it is important that students understand the type of scale they are using along the vertical axis. Without an appropriate scale, students will have difficulty plotting all of their points on the graph.

Once students complete their graphs, they are asked to make sense of the key features of their graph and data table. The first part requires students to make sense of the rate of change as they are conducting their simulation and use that information to make a prediction about how many shakes (how much time) is needed before all of the healthy individuals are infected. Students are then asked to explicitly state what they believe the rate of change is based on their data table and graphs. This will be compared to their technology generated models.

Students will repeat the simulation two times. The first time is for students to see how the virus is spread without social distancing, and the second time with social distancing, modeled by using a larger serving tray or box lid.

Once students have answered the questions asking them to make sense of their table and graph features, they will rely on available technology to graph their points and create a model that best fits their data.

Teachers should model the procedure for creating a model with the available technology if students are unfamiliar.

If using the TI-84, TI-Nspire, Geogebra (https://www.geogebra.org/classic#spreadsheet), or Excel, students will recreate their table of values in a spreadsheet. It is important that the headings for each column are included so that the vertical and horizontal axis can be well-defined. Once the values are in the spreadsheet, students will need to generate a graph of their data values where the time is represented on the horizontal axis and the number of infected people is on the vertical axis. Students will then use their technology to create a regression model.

[Note: Students should be familiar with the idea of lines of best fit from their linear function knowledge. The regression model illustrates the same concept of finding the curve of best fit.] If using Desmos(https://www.desmos.com/calculator), students will need to insert a table and enter their data values. This will automatically plot their points on the graph. To generate an algebraic regression model, students will need to type into the row beneath their table. This will generate values below this entry for a and b that represent the initial value and rate of change for the model.

Once the algebraic model is generated for the curve of best fit (an exponential model) students will record their results and should be comparing their previous predictions to the technology generated rate of change. Discussion of why linear and quadratic models are not appropriate should be addressed. Discussion of whether the graph will extend infinitely should be addressed also.[Note: If appropriate, the correlation coefficient and the coefficient of determination should also be included during exploration for students to understand why exponential models are the best fit model.]

Assessment Strategies

Teachers need to closely observe students' simulation procedures to ensure that they are conducting the simulation correctly. Teachers should determine whether students are counting infected beans correctly and then correctly removing healthy beans with infected beans.

Teachers need to explicitly attend to students' interpretations of the rate of change to assess their conceptual understanding of the rate of change as a multiplicative rate and not an additive rate and how that applies to the context of the virus spread.

Teachers should plan questions that they will ask students during small group exploration to formatively assess their understanding of the spread of disease. Teachers should listen carefully to student responses and ask follow up questions to elicit further thinking or pose questions that orient students toward the path to understanding the concepts.

To assess students' progress towards mastery of the standards, the teacher should review the students' completed work on the task sheet and on the technology-based graphing tools.

Acceleration and Intervention

Approximate Duration

Total Duration

31 to 45 Minutes

Related Learning Activities

Learning Activity (Before)

The Spread of a Virus: Are All Viruses Virulent?

Learning Activity (During)

Learning Activity (After)

The Spread of a Virus: Which Virus Is More Virulent?

Background and Preparation

Background / Preparation

Teachers need to spend time gathering materials for this simulation. The materials include different kinds of beans (pinto, black, kidney, red, etc.), a flat tray-like surface of two different sizes (small and large) for the beans to be poured into, cups for bean distribution, and available technology to create an algebraic model of students' data. The task sheet for each student is linked here, note that this is the same as the previously included link.

Digital Tools / Resources

Virus Outbreak Behavior Task Sheet

https://drive.google.com/file/d/1C6RKqLukzvOyl3PT_WgBOSCLmevUTh-H/view?usp=shar…

Approved Date

2020-08-04

Author

Owner (Author)

dcdai

Learning Resource Type

The Spread of a Virus: Does Social Distancing Matter?

Subject Area

Grade(s)

Overview

MA19.GDA.12 Represent data of two quantitative variables on a scatter plot, and describe how the variables are related.

Vocabulary

Knowledge

Skills

Understanding

MA19.A1.24 Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

Vocabulary

Knowledge

Skills

Understanding

MA19.A1.25 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).

Vocabulary

Knowledge

Skills

Understanding

MA19.A1.27 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 $ab^x$.

Vocabulary

Knowledge

Skills

Understanding

Phase

Learning Objectives

Activity Details

Assessment Strategies

Total Duration

Learning Activity (Before)

Learning Activity (During)

Learning Activity (After)

Background / Preparation

URL (web address)

Approved Date

Owner (Author)

MA19.GDA.12
Represent data of two quantitative variables on a scatter plot, and describe how the variables are related.

MA19.A1.24
Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

MA19.A1.25
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).

MA19.A1.27
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 $ab^x$.