Absolute and relative change

Lesson 2.6

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 9. Compare proportional relationships represented in different ways, considering units when doing so.

Specific Objectives

Students will understand that

· a relative change is different from an absolute change.

· a relative measure is always a comparison of two numbers.

Students will be able to

· calculate a relative change.

· explain the difference between relative change and absolute change.

Measuring Change

When a quantity, such as population, changes, we can calculate the absolute change and also the relative change.

Absolute change is the new value of the quantity minus the original value.

Relative change is the absolute change divided by the original value.

Note that the “quantity” values are always positive (at least in almost all contexts). But the absolute change can turn out to be a negative number or a positive number.

· If a quantity increases (has gotten larger), then the absolute change is positive. Why? When the new value is larger, then the new value minus the original value is positive, and thus the absolute change is positive.

· If a quantity decreases (has gotten smaller), then the absolute change is negative. Why? When the new value is smaller, then the new value minus the original value is negative, and thus the absolute change is negative.

The relative change’s sign (negative or positive) is the same as the sign of the absolute change. This is true since the relative change is found by dividing the absolute change by the original value, and the original value is positive.

Another way to talk about negative change (either absolute or relative):

                A negative change can be said to be a decrease by the positive number.

Example A: In 2013 Jerry received 12 speeding tickets. Since then, his driving has improved and in 2014 he only had one ticket.

What is the absolute change in tickets?

New value – original value = 1 – 12 = −11. The absolute change in his number of tickets from 2013 to 2014 is −11.

Another way to say this is: The absolute change in his number of tickets from 2013 to 2014 is a decrease of 11.

Jerry got 11 fewer tickets in 2014 compared to his 12 tickets in 2013.

What is the relative change?

Relative change = absolute changeoriginal value=−1112≈−0.92absolute changeoriginal value=-1112≈-0.92 = −92%. In other words:

The number of tickets he received in 2014 decreased by about 92% from 2013.

Example B: Suppose that when Sasha started college her school had 8,210 students. By the time she graduated there were 9.440 students.

What was the absolute change in students?

New value – original value = 9,440 – 8,210 = 1,230

The college’s enrollment increased by 1,230 students during the years Sasha attended the school.

What was the relative change in students?

Relative change = absolute changeoriginal value=12308210absolute changeoriginal value=12308210 = .1498 = about 15%

The number of students increased by about 15% over the years Sasha was enrolled.

Representatives

Problem Situation: How the Census Affects the House of Representatives

Every 10 years, the United States conducts a census. The census tells how many people live in each state. You can also find how much population has changed over time from the census data. The original purpose of the census was to decide on the number of representatives each state would have in the House of Representatives. Census data continue to be used for this purpose, but now have many other uses. For example, governments may use the data to plan for public services such as fire stations and schools. You will be given a list of states and their populations in 2000 and 2010. You will be asked to calculate the population growth for each state. You will examine how this affects the number of representatives each state has in the House of Representatives.