Math 9 Unit 3 Rational Numbers and Surface Area

Lesson 3-1 – Comparing and Ordering Rational Numbers

ExploreThe percent of Canadians who live in rural areas has been decreasing since 1867. At that time, about 80% of Canadians lived in rural areas. Today, about 80% live in urban areas, mostly in cities. The table shows changes in the percent of Canadians living in urban and rural areas over four decades.

Decade Change in the Percent of Canadians in Urban Areas

(%)

Change in the Percent of Canadians in Rural Areas

(%)1966-1976 +1.9 -1.91976-1986 +1.0 -1.01986-1996 +1.4 -1.41996-2006 +2.3 -2.3

How can you tell that some changes in the table are increases and others are decreases?

1. How are the rational numbers in the table related? Explain your reasoning.

2. a) Choose a rational number in decimal form. Identify it’s opposite.

b) Choose a rational number in fraction form. Identify it’s opposite.

c) Identify another pair of opposite rational numbers.

3. Identify equivalent rational numbers from the following list:124

−84

−9−3

−424

−2123

−( 4−1 )−(−4−2 )

Did you know? An urban area has a population of 1000 or more. In urban areas, 400 or more people live in each square km. Areas that are not urban are called rural.

4. Choose a rational number in fraction form (can’t be any of the ones we’ve already used today). Switch with a partner and find 4 rational numbers that are equivalent to your chosen number.

Integers, decimals, mixed numbers, and fractions are all rational numbers. Fractions can be positive or negative. Decimals can be terminating or repeating.

A number line is a great way to organize and compare rational numbers. A number line allows you to see how the numbers are related. Recall that on a number line, the smaller number is always to the left of the larger number. For example,

−2 is larger than −5 +1 is larger than −3 but smaller than +3

 The numbers to the left of the skateboarder are smaller than the numbers to the right of the skateboarder. 

When determining the location of a number on a number line, you may find it helpful to do the following:

1. Verify the spacing and numbering on the line. For instance, does the line

go from −5 to 5, −2 to 2, or 5 to 10? Does each space represent , , 1, 2, or 5 units?

Rational Number-a number that can be written as a/b where a and b are integers and b≠0. -any number that can be written as a terminating or repeating decimal

Example: 4, -3.5, ½, -3/4 etc.

2. Estimate the position of the number. For example, the number 2.2 is a little more than 2, and the number -2.2 is a little less than −2. 

3. Express all the numbers in the same form. Depending on the given rational numbers, you may convert all fractions to decimals or you may convert all decimals to fractions.

Fractions can be expressed as mixed numbers or improper fractions. In order to compare rational numbers in fraction form, it is important that you can convert between the two!http://www.learnalberta.ca/content/mesg/html/math6web/index.html?page=lessons&lesson=m6lessonshell02.swf

Example 1: Compare and Order Rational NumbersCompare and order the following rational numbers

−1.2 4578−0.5−7

8

Show you knowCompare the following rational numbers. Write them in ascending order and descending order

0.3−0.6−341 15−1

Check your understanding

When comparing rational numbers, it is helpful to use fractions that have the same common denominator. This allows you to compare the numerators and identify which fraction is larger or smaller. 

For example, Ms. Crossey asks you to determine whether or is the greater fraction. What do you do? One method involves identifying a common denominator between the two fractions. In this case the lowest common denominator is _________.  Both of the fractions can be expressed with a denominator of _______.  

 Multiplying both the numerator and denominator by the same number does not change the value of the fraction.

Fractions having the same value are equivalent fractions. Equivalent fractions represent the same rational number. In this example each original fraction is expressed as an equivalent fraction with a denominator of 24.  

 

1. 2.

Fractions and each represent the same amount. They are equivalent fractions.

Likewise, and each represent the same amount and are equivalent fractions. Now that the fractions are written with a common denominator, you can compare the numerators. Because 8 is less than 9,

 

So, is the greater fraction.  

Example 2: Compare Rational Numbers

Which fraction is greater, −34

∨−23 ?

Method One: Use Equivalent Fractions Method Two: Use Decimals

Show you know

Which fraction is smaller −710

∨−35 ?

Identify a rational number between two given rational numbersWhen you are building things, you sometimes need to identify a specific number between two measurements. A number line can also be used in this situation. The number line helps to identify a rational number between two given rational numbers. 

For example, the instructions for a bookcase tell you to drill a hole for the handle in

a cabinet door. The hole needs to be at least from the edge of the cabinet. It

cannot be farther than from the edge of the cabinet door. It is important that all the handles be in the same spot. You will need to identify the specific measurement for drilling the holes. A number line (like a tape measure) can be used to determine where to drill the hole on each cabinet door. Both fractions have a denominator of 8. Draw a number line with eight equal ticks starting at 0 and ending at 1. This will allow you to easily

identify and on the number line.  

Looking at the number line, you will see that is between and . The number

is equivalent to .  

 

The hole for each cabinet door handle should be drilled away from the edge of the door. Example 3: Identify a Rational number between two given rational numbersIdentify a fraction between -0.6 and -0.7.

Show you knowIdentify a fraction between -2.4 and -2.5

Assignment Worksheet 3-1Lesson 3-2 - Problem Solving with Rational Numbers in Decimal Form

Get Focused

Chinooks are warm, dry winds that come down on the eastern side of the Rocky Mountains. The conditions that produce a chinook occur year-round. The effects are most noticeable during the cold winter months. The term chinook is derived from the Salishan word Tsinuk. This is the name of a First Nation group living along the Columbia River. The term was originally used to describe a wind that blew from the direction of their camp. The telltale sign of the strong, warming winds is the distinctive chinook arch. This cloud arch can run parallel to the Rocky Mountains for a few hundred kilometres. For a chinook to take place, the following two conditions must occur:

a high pressure weather system in western British Columbia a low pressure weather system in the eastern slopes of the Rocky Mountains

Chinooks can increase the temperature by 25°C or more within a few hours. The rate of temperature change caused by a chinook can be expressed by rational numbers in decimal form. Explore 

The process of creating a chinook begins with air from the Pacific Ocean. This air is warm and moist. Before the air reaches southwestern Alberta, it has to move over three mountain ranges. The air cools as it climbs up the western side of each mountain range in British Columbia. For every 100-m rise in elevation, the warm, moist air cools about

0.5°C to 0.8°C. The air then loses some of its moisture in the form of rain or snow. This cooler, drier air begins its descent down the eastern side of the mountain range. The dry air warms at a rate of about 1°C for each 100-m drop in elevation. The dry air will continue to heat up faster than the wet air as it moves over the three mountain ranges.

Try this

1. Suppose that warm, moist air cools 0.7°C for every 100 m it climbs up the western slope of the Rocky Mountains.

a. What integer would you use to represent a temperature decrease of 0.7°C?

b. What operation would you use to determine the change in elevation?

c. What is the overall temperature change of the air from an elevation of 800 m to 2500 m?

2. Assume that cold, dry air warms 0.9°C for every 100-m drop in elevation.a. What integer would you use to represent a temperature increase of 0.9°C?

b. What is the overall temperature change of the air as it descends from an elevation of 2200 m to 1100 m?

3. The temperature in Calgary during a chinook rose from −15°C to +3°C in 5 hours.a. What operation would you use to determine the average temperature change

per hour?

b. What was the average temperature change per hour?

Example 1: Add and subtract rational numbers in decimal formEstimate and calculatea) 2.65 + (-3.81)

b) -5.69 – (-6.83)

Show you knowa) -4.38 + 1.52 b) -1.25 – 3.55

Example 2: Multiply and Divide Rational Numbers in decimal form

Estimate and calculatea) 0.45 x (-1.2)

b) -2.3 x (-0.25)

Show you knowa) -1.4(-2.6) b) -2.67÷4.6

ExploreChinooks vary in strength. The rate of temperature change is one indicator of a chinook’s strength. Some chinooks will cause a more rapid temperature change than others. 

 

The following number lines show the temperature changes for each location.  

  

 

Considering the amount the temperature changes over a specific period of time is one way to compare the weather. The unit for average rate of temperature change is °C/h. Problems involving rates may require more than one operation to be performed. In this case subtraction is required to determine the change in temperature. The change in temperature then needs to be divided by the number of hours. When solving problems involving more than one operation, each step can be performed separately. The individual operations can also be combined into one expression. Order of operations is then applied to solve the expression.

The rates of temperature change for Cochrane and Grande Prairie can be represented by the following expressions: 

 The expressions can now be evaluated using order of operations. 

Cochrane Grande Prairie

The average rate of temperature change in Cochrane was 3°C/h.

The average rate of temperature change in Grande Prairie was −0.9°C/h.

Change in temperature per hour is just one example of a rate. Rates are also used to express speed, fuel consumption, and changes in altitude. Many different types of problems can be solved by comparing rates

Try This

4. The fuel consumption of a car is 12.7 L/100 km in the city and 7.7 L/100 km on the highway. In one week the car is driven 175 km around the city and 250 km on the highway.

a. Write an expression to represent the total fuel consumption for the week.

b. Evaluate your expression using order of operations.

Example 3: Apply Operations with rational numbers in decimal formOn Saturday, the temperature at the Blood Reserve near Stand Off, Alberta decreased by 1.2°C/h for 1.5 hours. It then decreased by 0.9°C/h for 1.5 hours. a) What was the total decrease in temperature?

b) What was the average rate of decrease in temperature.

Show you knowA hot air balloon climbed at 0.8 m/s for 10s. a) what was the overall change in altitude?

b) What was the average rate of change in altitude?

Assignment Worksheet 3-2