Monday, October 18, 2021

Algebra 1 PARCC question: exponential websites

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The following multiple-droplist question, explained here in hopes of helping algebra students in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2015 test in algebra 1, here:

Two websites launched on the same day. At the end of the first week, the number of visitors to each website was 25. For the first eight weeks, the number of visitors to each site increased according to the corresponding rules.

• Website A: The number of visitors doubled each week.
• Website B: The number of visitors increased by 150 each week.

Part A

Complete the table to show the number of visitors to each website for the first eight weeks.

Website A Website B
Week 1: 25 25
Week 2:
Week 3:
Week 4:
Week 5:
Week 6:
Week 7:
Week 8:

Part B

Based on the data for the first eight weeks, Jose claims that the number of visitors to each website can be modeled as a linear function of the number of weeks online. For each website, decide if Jose’s claim is correct. If it is correct, explain why. If it is not correct, explain why and describe a more appropriate model.

Correct answers: Both parts are hand-scored. Part A: Website A: 25, 50, 100, 200, 400, 800, 1,600, 3,200. Website B: 25, 175, 325, 475, 625, 775, 925, 1,075. Part B: Website A is better as an exponential model; website B is linear.

PARCC evidence statement(s) tested: HS.C.10.1:

Express reasoning about linear and exponential growth.

Content scope: F-LE.1a

The evidence statement above references no Math Practice in the Common Core.

The question tests students’ understanding of the high school Common Core algebra standard HSF.LE.A.1.A, found under Functions: Linear, Quadratic, & Exponential Models (construct and compare linear, quadratic, and exponential models and solve problems), which states that they should be able to “Distinguish between situations that can be modeled with linear functions and with exponential functions” including the ability to prove that [A.1.A.1] “linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals” and to [A.1.A.3] “recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.”

Example of a solution strategy (there are others)

Show the linear equation for website B, identify factor for website A.

The increase in website B can be modeled with the function

$v_w = (150)(w-1) + 25$

where vw is the number of visits in week w and w is the week number, starting at 1. Because the first week was called week “1,” you have to subtract 1 from the value before multiplying it by the slope. Note that this equation has the same structure as the slope-intercept form of the equation of a line:

$y = mx + b$

Any equation that can be written in slope-intercept form describes a line, so the function is linear.

As for website A, the (long) paragraph tells you that the number of visitors doubles every week, which means the number is multiplied by the same factor, 2, each week. That’s the definition of an exponential function:

$v_w = (25)(2^{w-1})$

where vw is the number of visits in week w. Let’s show both functions on a graph, with a line (or exponential curve) drawn between the discrete points for each week:

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has several pages on exponential functions, including filling out tables with values for exponential functions and graphing them. Her lessons start here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a number of them about exponential functions, including this one entitled “Exponential Growth & Decay Word Problems.”

Chapter 6 of Paul A Foerster’s book Algebra and Trigonometry deals with exponential functions, in addition to logarithmic functions. He notes that an exponential function is a function with an equation of the form

$y = a \cdot b^x$

where a stands for a constant of proportionality, b stands for a positive constant base, and x and y are independent and dependent variables, respectively.

Complete reference: Foerster, Paul A. Algebra and Trigonometry: Functions and Applications, revised edition. Addison-Wesley, 1980, 1984. The book is used in several algebra classes taught in middle and high schools in both Illinois and Maryland.

Analysis of this question and online accessibility

The question measures knowledge of the Common Core standard it purports to measure and tests students’ ability recognize situations that can be modeled by linear as well as exponential functions. It is considered to have a median cognitive demand.

(The models aren’t perfect, since linear and exponential functions are both continuous while the context of the problem, website visitors each week, is discrete. But I think that’s a superfluous gap in terms of getting credit on the test: if you ask students about a linear model, they tend to answer with a linear function. The question does that. The problem comes for website A, since there are significant gaps between the real-world data and the exponential model used. The model is continuous, the data discrete. This might trip up students who think the continuous nature of an exponential function makes it unfit as a mathematical model for this situation.)

The question can be tested online and should yield results that are as valid and reliable as those obtained on paper. Students online may experience difficulties using the equation editor to show their math or reasoning. We have previously described problems with the user interface of the equation editor tool (see here), and we will not repeat that argument here. However, we will use the opportunity to remind those of you taking the PARCC math test to spend a little extra time on problems with the equation editor to:

• Make sure you have typed in all the work necessary to make your case
• Transfer all your work from scratch paper to the computer so you can receive credit for it

No special accommodation challenges can be identified with this question, so the question is considered fair.

Challenge

According to Newton’s Law of Cooling, the temperature difference between your morning cup of coffee and the room you’re in decreases exponentially with time. Model the temperature of your coffee over time if you’re in a room at 20°C and your coffee is scorching hot after three minutes, 85°C, but much more palatable five minutes after that at 72°C. Then, determine how long you can let your coffee sit before it just becomes too cold to warm you up (maybe about 55°C). Use equations, graphs, tables, whatever, to construct the model.

Purpose of this series of posts

Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.

Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.

Paul Katulahttps://news.schoolsdo.org
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

Now US News ranks elementary, middle schools

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A magazine has launched national and state rankings for elementary and middle schools based on tests. Good idea or irrelevant?