What you should learn

To compare and order rational numbers

To find a nubmer between two rational numbers

NCTM Curriculm Standards 2, 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Rational Numbers

Density Property

Introduction: Many people all over the wrold recycle their aluminum cans in order to help our environment. The chart below represents the fraction of aluminum cans that have been recycled over the years.

The numbers shown in the chart are examples of rational numbers.

Definition of a Rational Number: A rational number is a number than can be expressed in the form a/b, where a and b are integers and b is not equal to 0.

 

Examples of rational numbers expressed in the form a/b are shown below.

Rational numbers can be graphed on a number line in the same manner as integers. The number line below is separated into fourths to show the graphs of some common fractions and decimals.

You can compare rational numbers by graphing them on a number line. Recall that a mathematical sentence that uses < and > to compare two expressions is called an inequality.

The following statements can be made about the graphs of -5, -2, 1 1/2, and 3.5 shown on the number line below.

a. The graph of -2 is to the left of the graph of 3.5

b. The graph of 1 1/2 is to the right of -5

 

These examples suggest the following rule.

Comparing Numbers on the Number Line: If a and b represent any numbers and the graph of a is to the left o fthe graph of b, then a < b. If the graph of a is to the right of the grpah of b, then a > b.

 

If <, >, and = are used to compare two numbers, then the following properties applies.

Comparison Property: For any two numbers a and b, exactly one of the following sentences is true.

The symbols , , and can also be used to compare numbers. The chart below shows several inequality symbols and their meanings.

Exercise 1: Replace each ?? with <, >, or = to make each sentence true.

a. -75 ?? 13

Since any negative number is less than any positive number, the true sentence is -75 < 13

b. -14 ?? -22 + 9

c. 3/8 ?? -7/8

 

 

You can use cross products to compare two fractions with different denominators. When two fractioins are compared, the cross products are the products of the terms on the diagonals.

Comparison Property for Rational Numbers: For any rational numbers a/b and c/d, with b > 0 and d > 0:

1. if a/b < c/d, then ad < bc, and

2. if ad < bc, then a/b < c/d.

Exercise 2: Replace each ?? with <, >, or = to make each sentence true.

a. 7/13 ?? 4/15

7(15) ?? 4(13)

105 > 52

The true sentence is 7/13 > 4/15.

 

b. 7/8 ?? 8/9

Every rational number can be expressed as a terminating or repeating decimal. You can use a calculator to write rational numbers as decimals.

Exercise 3: Use a calculator to write the fractions 3/8, 4/5, and 1/6 as decimals. Then write the fractions in order from least to greatest.

3/8 = 0.375

 

 

You can use a calculator to compare the cost per unit, or unit cost, of two similar items. Many people comparison shop to find the best buys at the grocery store. The item that has the lesser unit cost is the better buy.

unit cost = total cost number of units

Activity: Erica and Gabriella are running in a 5-kilometer race to raise money for The Multiple Sclerosis Foundation. They want to buy a sports beverage to drink after they finish the race. Boby Quencher costs $1.69 for 32 oz and New Sport costs $1.09 for 20 oz. Which is the better buy?

After this activity finish with this...

Can you always find another rational number between two rational numbers? One point that lies between any two points is their midpoint. Consider 1/3 and 1/2.

To find the coordinate of the midpoint for 1/3 and 1/2, find the average, or mean, of the two numbers.

This process can be continued indefinitely. The pattern suggests the density property.

Density Property for Rational Numbers: Between every pair of distinct rational numbers, there are infinitely many rational numbers.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 15 - 41 odd, 43 - 49

Alternative Homework: Enriched: 14 - 36 even, 38 - 49

Extra Practice: Students book page 760 Lesson 2-4

Extra Practice Worksheet: Click Here.

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