8.10: Quantway Core 4.2 Workforce (IT)- Student Lesson

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INTRODUCTION

Quick Check-in:

Please discuss the following questions with your group:

What specific concept from Unit 4.1 do you feel fairly confident about?
Are there any concepts from Unit 4.1 that you find challenging?

Completing the CaS Chart

You may want to use the Comprehension and Synthesis (CaS) Chart in this collaboration. You may have used a CaS Chart in a previous collaboration. Remember, using the CaS Chart helps you have a deeper understanding of the problem situation. CaS Charts will help you to identify the main issue(s) that need(s) to be resolved and will help you to recognize the quantitative information that is available in the problem situation that can help you to solve the problem. You will use CaS Charts in some of the Quantway collaborations to “unpack” problem situations and support problem-solving.

As you read the problem situation, complete the CaS Chart. You may wish to return to these steps as you complete the CaS Chart.

Read through the problem situation, Problem Situation: Analyzing Instagram and Twitter Trends. As you are reading:

Complete Column A. Hint: What issues in the problem situation do you need to understand in order to solve the problem? Is there contextual information that you need to know in order to understand the problem situation?
Complete Column B. Hint: What quantitative information is provided in the problem situation that will help you solve the problem? Hint: Quantitative information is often a number, but also could be a number word, like “two”.
Complete Column C. Hint: It is not necessary to solve the problem or use calculations right now. In this column, brainstorm ways you might address the issues presented in the problem situation (Column A) using the quantitative information in Column B. There are no wrong answers.

Column A	Column B	Column C
What is/are the main issue(s) in this problem situation?	What is the key quantitative information you need to solve the issue(s) in the problem situation?	Describe in writing how the information in Column B will help you address the issue(s) in Column A later in the lesson.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand

that linear models are appropriate when the situation has a constant rate of increase/decrease or can be approximated by a constant rate.
that the rate of change (slope) has units in context.
the difference between a positive slope and a negative slope.
that the linear models for authentic situations have limitations in using them to make predictions.

By the end of this collaboration, you should be able to

make a linear model when given data or information in context.
calculate a slope given data or information in context.
estimate the value that makes two linear models equivalent.

SPECIFIC LANGUAGE AND LITERACY OBJECTIVES

By the end of this collaboration, you should be able to

read and comprehend the problem situation.
complete the CaS Chart with quantitative and IT information from the problem situation.
demonstrate an understanding of mathematics by writing complete and correct responses to questions.
demonstrate the ability to describe, analyze, and synthesize information using linear models and slope.
use appropriate quantitative and information technology vocabulary to discuss mathematics in this collaboration.

PROBLEM SITUATION: ANALYZING INSTAGRAM AND TWITTER TRENDS

Ben recently graduated from college with a degree in Computer Science. He is very interested in working in social media. Social media are websites and apps where users create, share, and participate in an online community. Facebook, Twitter, and Instagram are some examples of social media sites.

Ben wants to work at a popular social media company. But, he is concerned about getting a job at a company that will continue to be successful. For example, Ben knows that Myspace was once the most visited website in the United States, and hired over 1,600 employees. Today, Myspace only employs 150 people.

Ben is particularly interested in working at Twitter or Instagram. He analyzes user statistics to help make his decision about where to apply. Ben wants to use these data to predict which company will be more successful in the future so he can decide which company to join.

Ben decides to analyze the number of Twitter and Instagram followers. Companies report this information or data by Quarter. Businesses often refer to “quarters”, which are three-month periods of time. For example, Quarter 1 (Q1) is the first three months of the year (January, February, March). There are four quarters in the year, which we will refer to as Q1, Q2, Q3, Q4.

The data show the number of Twitter and Instagram users by Quarter. The data indicate that in the first quarter (Q1) of 2012, Twitter had 138 million users. In the first quarter (Q1) of 2014, the number of Twitter users had increased to 255 million. In the first quarter (Q1) of 2012, Instagram had only 27 million users. However, in the first quarter (Q1) of 2014, Instagram had 200 million users.

(1) Without doing any calculations, to which company do you think Ben should apply? Explain your answer. Write your answer in 1–2 complete sentences. Be sure to include both numbers and facts from the problem situation to support your answer.

Will Instagram ever catch up to Twitter in terms of its number of users? If so, when would we expect that to happen? Let’s find out!

For this problem, we will assume that the change in membership for both sites is linear. We will build linear models for the membership of both sites. To develop a linear model, you will need to calculate the rate of change. Linear models have a constant rate of change. The rate of change is also referred to as the slope, because it describes how steeply the graph is increasing or decreasing.

The rate of change is the amount of change in one quantity (in this case the number of users) per change in the amount of another quantity (in this case, time). It can be calculated as

\(\large{rate\;of\;change} = \dfrac{absolute\;change\;in\;number\;of\;users}{absolute\;change\;in\;time}\)

Here, we can define two variables:

U = number of users (in millions)

t = number of years

(2) Find the rates of change for both Twitter and Instagram users. Round to the nearest tenth of a million users. Be sure to specify the units of the rates.

Twitter =

Instagram =

(3) Based on the rates of change, predict the number of Twitter and Instagram users in Quarter 1 (Q1) of 2019.

(4) To predict when Instagram will overtake Twitter, create linear models for the membership of these websites. Use the subscripts (shown below) to distinguish between the number of users of Twitter and Instagram:

UT = Number of Twitter users

UI = Number of Instagram users

These variables will represent the output of the models. The input variable is time. However, in mathematical models, time is not represented as it is on a clock or calendar. Instead, time is represented in elapsed time. Elapsed time is the amount of time that has passed since some starting point. In this case, the starting point will be 2012. Therefore,

t = the number of years since 2012

Using these variables, develop linear models for the number of users of Instagram and Twitter.

(5) Use the models developed above to fill out the following table:

Year	Years Since Q1 2012 (t)	Twitter Users (millions)	Instagram Users (millions)
2012	0
2014	2
2015	3
2016	4
2017	5
2018	6
2019	7

(6) Plot both sets of data on one graph. Note: If completing this problem online, follow the instructions given online to create your graph.

(7) Based on the graph, when would we predict that Instagram will overtake Twitter in membership?

(8) Now, using the models you created for Twitter and Instagram users, find an algebraic solution for the approximate year and quarter (Q1, Q2, Q3, or Q4) when Instagram will surpass (that is, overtake) Twitter in membership.

FURTHER APPLICATIONS

(9) In 2000, China had approximately 18 million internet users. However, by 2012 China had about 580 million internet users. In 2000, the United States had 116 million internet users, and by 2012 that figure had risen to 260 million internet users.

Create linear models to estimate the year in which the number of internet users in China surpassed the number of internet users in the United States.

(a) China

(b) US

(10) The table of values below shows the relationship between the velocity of a car in miles per hour (mph) and the braking distance in feet.

(a) Calculate the slope between each of the two points in the table. The first answer is shown as an example.

Velocity (mph)	Braking Distance (ft)	Slope Between Two Points
5	0.98
15	8.84	The slope between (5, 0.98) and (15, 8.84) is 0.786.
20	15.72	Find the slope between (15, 8.84) and (20, 15.72).
30	35.37
50	98.24

(b) Is this a linear relationship? Explain your answer.

(i) At speeds between 5 and 15 miles per hour, the braking distance increases exactly 0.786 feet for every increase of 1 mile per hour in speed.

(ii) On average, at speeds between 5 and 15 miles per hour, the braking distance increases 0.786 feet for every increase of 1 mile per hour in speed.

(iii) At speeds between 5 and 15 miles per hour, the braking distance decreases exactly 0.786 feet for every increase of 1 mile per hour in speed.

(iv) On average, at speeds between 5 and 15 miles per hour, the braking distance decreases 0.786 feet for every increase of 1 mile per hour in speed.

(d) Use the trend of the data in the table to make predictions about the braking distance for speeds between 50 and 70 mph. Which of the following is a correct statement?

(i) The braking distance between 50 and 70 mph will increase by about 3.144 feet per mile.

(ii) The braking distance between 50 and 70 mph will increase by exactly 3.144 feet per mile.

(iii) The braking distance between 50 and 70 mph will increase by more than 3.144 feet per mile.

(iv) The braking distance between 50 and 70 mph will increase by less than 3.144 feet per mile.

(e) Explain your answer to Question 2(d) in 1–2 complete sentences.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

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