# Mathematics

**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.

From the last page, for your reference:

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

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

#1 Points possible: 10. Total attempts: 5

In 2000, the population of Nevada was 1,998,257. In 2010, the population had grown to 2,700,551. Compute the absolute and relative change in the population from 2000 to 2010.

The absolute change was: people

The relative change was: % (rounded to 2 decimal places)

#2 Points possible: 24. Total attempts: 5

Compute the absolute and relative change for the states below.

State | 2000 Population | 2010 Population | Absolute Change | Relative Change (to 2 decimal places) | |

New York | 18,976,457 | 19,378,102 | % | ||

Texas | 20,851,820 | 25,145,561 | % | ||

Florida | 15,982,378 | 18,801,310 | % | ||

Michigan | 9,938,444 | 9,883,640 | % |

#3 Points possible: 10. Total attempts: 5

Of the five states you’ve now calculated the absolute and relative change for,

a) which has had the largest absolute change in population?

b) which has had the largest relative change in population?

#4 Points possible: 8. Total attempts: 5

Why are the answers to the two parts of the last question different? Select all that are true.

· A large absolute change may not be a large relative change if the starting population was large.

· A large absolute change may not be a large relative change if the starting population was small.

· A large relative change may not be a large absolute change if the starting population was large.

· A large relative change may not be a large absolute change if the starting population was small.

#5 Points possible: 6. Total attempts: 5

Michigan’s population changed from 9,938,444 to 9,883,640. Which are correct ways to describe the change? (select all that are correct)

· Michigan’s population increased by 54,804 people

· Michigan’s population increased by -54,804 people

· Michigan’s population decreased by 54,804 people

· Michigan’s population decreased by -54,804 people

· Michigan’s population changed by 54,804 people

· Michigan’s population changed by -54,804 people

The number of Representatives each state has in the House of Representatives is based on the size of the population in the state. Since the number of representatives is fixed at 435, when the census was done in 2010 some states gained representatives and others lost representatives. You can see which states gained and lost representatives in this map from the Census Bureau.

It’s a common misconception that a state that lost representatives must have lost population. As you can see, New York lost two representatives, even though your calculations earlier showed the population increased.

#6 Points possible: 15. Total attempts: 5

Look back at your calculations for Nevada and Florida

a) Which of the two had a larger absolute change?

b) Which of the two had a larger relative change?

Based on the 2010 census, Florida gained two representatives, and Nevada gained one.

c) Does it appear that absolute or relative change matters more when determining the number of representatives gained or lost?

**HW 2.6**

#1 Points possible: 5. Total attempts: 5

Which of the following was one of the main *mathematical* ideas of the lesson?

· Absolute change is measured as a quantity (for example, an increase of $3). Relative change is measured as a percentage compared to the reference value (for example, an increase of 3%).

· The population of a state determines how many representatives that state has in the House of Representatives.

· To calculate a percent, divide the comparison value by the reference value.

· Consider this situation: Quantity 1 increases by 15%. Quantity 2 increases by 20%. Quantity 2 must have increased by a larger amount than Quantity 1.

#2 Points possible: 8. Total attempts: 5

The following headlines all refer to change. Identify the change as *absolute *or *relative* **.**

a. “Enrollments at Northeastern University are expected to increase by 1,500!”

· Absolute change

· Relative change

b. “Another 14% tuition increase is expected.”

· Absolute change

· Relative change

c. “A new proposal has sales tax rates dropping from 3% to 1%, a drop of only two percentage points.”

· Absolute change

· Relative change

d. “A new proposal has sales tax rates dropping from 3% to 1%, a 67 percent drop!”

· Absolute change

· Relative change

Questions 3 and 4 refer to data taken from the U.S. Census1. The dollar values take into account the changes in the economy over the years (i.e., *inflation*). Inflation is a complicated issue, but for Questions 3 and 4, you do not need to worry about it.

#3 Points possible: 5. Total attempts: 5

A typical high-income household in 1980 earned $125,556. A similar household in 2009 earned $180,001. What was the relative increase in income for these households from 1980 to 2009? Round to the nearest one percent. %

#4 Points possible: 5. Total attempts: 5

A typical middle-income household in 1980 earned $34,757. A similar household in 2009 earned $38,550. What was the relative increase in income for these households from 1980 to 2009? Round to the nearest one percent. %

#5 Points possible: 8. Total attempts: 5

Due to temporary tax cuts in 2010, a person with typical deductions earning $50,000 per year would have saved 2% of their income plus $850 in federal taxes.

a. How much money would this person save? $

b. What percent did this person save on her income? Round to the nearest tenth of a percent. %

#6 Points possible: 8. Total attempts: 5

Due to the same law, a person earning $500,000 per year with typical deductions would save 2% of the first $106,800 they earned plus $14,250 in federal taxes. Fill in the blanks to complete the statement below. Round to the nearest dollar and to the nearest tenth of a percent. A person earning $500,000 a year saved $ or % of their income.