ALEX Learning Activity

  

The Spread of a Virus: Does Social Distancing Matter?

A Learning Activity is a strategy a teacher chooses to actively engage students in learning a concept or skill using a digital tool/resource.

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  This learning activity provided by:  
Author: David Dai
System:Mobile County
School:Alma Bryant High School
  General Activity Information  
Activity ID: 2620
Title:
The Spread of a Virus: Does Social Distancing Matter?
Digital Tool/Resource:
Virus Outbreak Behavior Task Sheet
Web Address – URL:
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.
  Associated Standards and Objectives  
Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra I
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]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.SP.HS.45- Given a scatter plot with data with a line of best fit that can be represented by a linear function, describe what is happening to the y-values in reference to the x-values (x- and y- values limited positive numbers).
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
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.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
  • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
  • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

  • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
Teacher 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.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.24.1: Define linear function and exponential function.
ALGI.24.2: Distinguish between graphs of a line and an exponential function.
ALGI.24.3: Identify the graph of an exponential function.
ALGI.24.4: Identify the graph of a line.
ALGI.24.5: Plot points on a coordinate plane from a given table of values. a.
ALGI.24.6: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.7: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.8: Apply rules for adding, subtracting, multiplying, and dividing integers. b.
ALGI.24.9: Define constant rate of change as slope.
ALGI.24.10: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.11: Recognize the calculated difference is the constant rate of change.
ALGI.24.12: Apply rules for adding, subtracting, multiplying, and dividing integers. c.
ALGI.24.13: Define exponential growth and decay.
ALGI.24.14: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.15: Apply the rules of multiplication and division of integers.

Prior Knowledge Skills:
  • Recognize ordered pairs.
  • Identify ordered pairs.
  • Recognize linear equations.
  • Recall how to solve problems using the distributive property.
  • Define linear and nonlinear functions, slope, and y-intercept.
  • Analyze the graph to determine the rate of change.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
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
Evidence Of Student Attainment:
Students:
Given a contextual situation shown by a graph, a description of a relationship, or two input-output pairs,
  • Create a linear or exponential function that models the situation.
  • Create arithmetic and geometric sequences from the given situation.
  • Justify the equality of the sequences and the functions mathematically and in terms of the original sequence.
Teacher 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.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.25.1: Define linear function and exponential function.
ALGI.25.2: Define arithmetic sequence, geometric sequence, and input-output pairs.
ALGI.25.3: Define sequences and recursively-defined sequences.
ALGI.25.4: Recognize that sequences are functions whose domain is the set of all positive integers and zero.
ALGI.25.5: Given a chart, write an equation of a line.
ALGI.25.6: Given a graph, write an equation of a line.
ALGI.25.7: Given two ordered pairs, write an equation of a line.

Prior Knowledge Skills:
  • Given a function, create a rule.
  • Recognize numeric patterns.
  • Recall how to complete input/output tables.
  • Demonstrate how to plot points on a Cartesian plane using ordered pairs.
  • Define function, ordered pairs, input, output.
  • Graph a linear equation given the slope-intercept form of an equation.
  • Graph a function given the slope-intercept form of an equation.
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Generate the slope of a line using given ordered pairs.
  • Recall the rules for multiplying integers.
  • Define quotient, divisor, and integer.
  • Solve addition and subtraction of multi-digit whole numbers.
  • Solve addition and subtraction of multi-digit decimal numbers (emphasis on alignment).
  • Recall basic multiplication and division facts.
  • Solve multiplication problems involving multi-digit whole numbers and decimal numbers.
  • Solve division problems involving multi-digit whole numbers and decimal numbers.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
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 abx.

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.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation that may be modeled by a linear or exponential function,
  • Create a function that models the situation.
  • Define and justify the parameters (all constants used to define the function) in terms of the original context.
Teacher 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.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.27.1: Recall the formula of an exponential function.
ALGI.27.2: Recall the slope-intercept form of a linear function.
ALGI.27.3: Define b as growth or decay factor in the context of an exponential problem.
ALGI.27.4: Define k as the initial amount in the context of an exponential problem.
ALGI.27.5: Define m as the rate of change in the context of a linear problem.
ALGI.27.6: Define b as the initial amount in the context of a linear problem.

Prior Knowledge Skills:
  • Solve problems with exponents.
  • Discuss strategies for solving real-world and mathematical problems.
  • Recognize ordered pairs.
  • Identify parts of the Cartesian plane.
  • Recall how to plot points on a Cartesian plane.
  • Distinguish the difference between linear and nonlinear functions.
  • Define qualitative, increase, and decrease.
  • Recall how to name points from a graph (ordered pairs).
  • Recall how to find the rate of change (slope) in a linear equation.
  • Recall how to complete an input/output function table.
  • Analyze real-world situations to identify the rate of change and initial value from a table, graph, or description.
  • Define function, rate of change, and initial value.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
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.
  Strategies, Preparations and Variations  
Phase:
During/Explore/Explain
Activity:

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. 
Advanced 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.
Variation Tips (optional):
 
Notes or Recommendations (optional):

ALCOS 2019

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.

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).

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.
Example: If the function V(t) = 19885(0.75)^t describes the value of a car after it has been owned for t years, 19885 represents the purchase price of the car when t = 0, and 0.75 represents the annual rate at which its value decreases.
  Keywords and Search Tags  
Keywords and Search Tags: Disease, Exponential Functions, Social Distancing, Virus