Rational NumbersCHAPTER 3

Chapter 3

3.1 – WHAT IS A RATIONAL NUMBER?

WHAT ARE SOME NUMBERS BETWEEN -11 AND -12?

-10-11-12-13

EVALUATE

It doesn’t matter if the negative sign is on the numerator, or on the denominator—the fraction is still negative. They are equivalent.

The same way that positive integers all have negative counterparts (or opposites), each fraction has a negative opposite as well.

RATIONAL NUMBERS

A rational number is any number that can be written in the form of a/b where a and b are integers, and b ≠ 0.In other words, the set of rational numbers includes all integers, fractions and terminating or repeating decimals.

Rational Numbers Non-Rational Numbers

EXAMPLE

Find three rational numbers between each pair of numbers.

a) –0.25 and –0.26 b) –1/2 and –1/4

a) Remember, we can always add a zero to the end of a decimal, without changing the value. So, –0.25 and –0.26 can also be written as –0.250 and –0.260.

What numbers are between 250 and 260? 251, 252, 253, etc…

So possible answers are –0.251, –0.252, –0.253. b) Try it!

What are some different ways that we could solve this?

EXAMPLE

Order these rational numbers from least to greatest:

1.13, –10/3, –3.4, 2.777… , 3/7, –2 2/5

Putting all of the numbers into decimal form may be the easiest way to do these types of questions.

–10/3 = –3.333…3/7 = 0.429–2 2/5 = –2.4

Remember, negatives are smaller than positives: –3.4, –3.333…, –2.4, 0.429, 2.777…

Now, put the original numbers back in: –3.4, –10/3, –2 2/5, 3/7, 2.777…

BINGO

Number Line Handout

Independent practice

PG. 101-103, # 8, 9, 10, 16, 21, 22, 24

Chapter 3

3.2 – ADDING RATIONAL NUMBERS

3.3 – SUBTRACTING RATIONAL NUMBERS

EXAMPLE

Evaluate:

Method 1:

-3 -2 -1 0

+1 +1 +5/6

So, the answer is –3/6 = –1/2.

Method 2:

EXAMPLE

Evaluate:

3.1 + (–1.2)

Or:

3.1 + (–1.2) = 3.1 – 1.2 = 1.9

EXAMPLE

A diver jumps off a cliff that is 14.7 metres above sea level. After hitting the water, he plunges 3.8 metres below the surface of the water before returning to the surface. a) Use rational numbers to represent the difference in heights from the

top of the cliff to the bottom of his dive. Sketch a number line.b) The water is 5.6 metres deep. What is the distance from the ocean

floor to the bottom of the dive?a)

0

–3.8–5.6

14.7

14.7 – (–3.8) = 14.7 + 3.8

= 18.5 m

b) –3.8 – (–5.6) = –3.8 + 5.6

= 1.8 m

What would have happened if we had subtracted –3.8 from –5.6 instead? What would our answer be?

Chapter 3

PG. 111, # 3AC, 4AC, 9, 11, 12, 15, 20.

PG. 119, # 11, 12, 14, 15, 16

Chapter 33.4 – MULTIPLYING

RATIONAL NUMBERS

MULTIPLICATION

MULTIPLYING RATIONAL NUMBERS

When multiplying rational numbers:• use the procedures for determining the sign of

the product of 2 integers• for fractions, use the procedures you already

know about multiplying fractions• for decimals, use the procedures you already

know about multiplying decimals

What happens when we multiply two negative numbers? How about a positive and a negative?

EXAMPLE

Evaluate:

a) b)

a) Look for common factors to cancel.

b) Turn them into improper fractions.

DIVIDING RATIONAL NUMBERS

When dividing rational numbers:• use the procedures for determining the sign of

the quotient of 2 integers• for fractions, use the procedures you already

know about dividing fractions• for decimals, use the procedures you already

know about dividing decimals

HANDOUT

Answer the questions on the handout to the fullest of your ability, because this is a summative assessment.

EXAMPLE

Simplify, and represent as a mixed fraction.

EXAMPLE

Solve for x:

a) x ÷ (–2.6) = 9.62 b)

a) x ÷ (–2.6) = 9.62

x ÷ (–2.6) × (–2.6) = 9.62 × (–2.6)

x = –25.012

b)

Independent Practice

PG. 127, # 6, 10, 11, 12, 14, 15, 18

PG. 134, # 6, 9, 11, 12, 17, 19, 21

3.6 – ORDER OF OPERATIONS WITH

RATIONAL NUMBERS

ORDER OF OPERATIONS

What’s the acronym used to remember the order of operations?

B E D M A S Brackets

ExponentsDivisionMultiplicatio

nAdditionSubtraction

EXAMPLE

Evaluate:

Brackets

Exponents

Division

Subtraction

EXAMPLE

To convert a temperature in degrees Fahrenheit to degrees Celsius, we use the formula:

In Fort Simpson, the mean temperature in December is –9.4°F. What is this temperature in degrees Celsius?

In cases like this, it’s as if there are invisible brackets around the numerator and denominator.

The mean temperature in Fort Simpson in December is –23° Celsius.

Independent Practice

PG. 140, # 4, 6, 7, 9, 11, 12, 16, 18, 19,

21