Unit 3: Rational Numbers
See Skill Builder 3.1 from practice book and decimal work from Canadian Mathematics
3.1. What is a Rational Number?
A Rational Number, - Any number that CAN be written in the form where and are integers but . That is, all numbers that can be written in fraction form.
As a decimal they either terminate or repeat.
Examples:
Non-terminating and non-repeating decimals (numbers that go on forever without repeating digits) belong to the set of irrational numbers, .
Examples:
Note: .
All of these numbers are negative.
Placing Numbers on a Number Line
Remember that negative numbers are to the left of, and therefore less than, 0. Note that opposites are the same distance from zero but on opposite sides.
Examples:
a) We use and to help you place and on a number line:
b) We use and to help you place and on a number line:
c) We use and to help you place and on a number line:
d) We use and to help you place and on a number line:
A negative number is always less than a positive number.
Fractions with a positive common denominator can be compared by noting that the larger numerator implies a larger fraction.
If denominators differ, write them with a common denominator before comparing. Sometimes it may not necessarily be the lowest common denominator that will lead directly to the solution.
Ex.
Fractions with a positive common numerator can be compared by noting that the smaller denominator indicates the larger fraction.
On a number line, numbers to the right are always greater while numbers on the left are always less.
Fractions can be compared to known reference points such as 1, ½, -1, - ½. etc.
When comparing numbers given in different forms, it is often easier to compare if we first change them to the same form, usually decimal form.
Examples:
1) Give a rational number between each pair of numbers:
a) and
Note: Answer:
b) and
Note: Answer:
c) and Answer: 0
d) and
Note: and Answer:
e) and
Note: and Answer:
2) List all integers between and
3) Replace with <, >, or =. Note:
4) Find three number that lie between each of the following pairs of numbers.
(a) and
(b) and
Note: and
(c) and
(d) and Note:
(e)
3 2
and
Note:
5) Replace ? with a value to make each statement true. (a) Possible answers:
(b) Possible answers:
(c) Possible answers:
6) Place each set of numbers in order from least to greatest: (a)
Answer:
(b)
Answer:
(c)
Answer:
Answer:
Also, see examples 1-3 pp. 97-100
Complete practice sheet and p. 101 #5-11
Adding integers
Remember integer chips:
Adding the same sign gives you a total of that sign.
Adding opposite signs means you have to cancel zero pairs so you really keep the sign of the one you have more of and find the difference.
Subtracting integers
Remember integer chips:
When possible “take away”
Cancel zeros by adding opposite when necessary.
Adding & subtracting Fractions
Find a common denominator and add or subtract the numerators
3.2. Adding Rational Numbers
Note 1: Adding rational numbers in decimal form combines the rules for adding positive decimals with the rules for adding integers.
Line up place value
Find total of positives when appropriate
Find total of negatives when appropriate
“Cancel” zeros when adding opposites
Recall Adding Integers
To add two positive or two negative integers, find the total. To add opposite integers take the sign of the larger and find the difference
Examples:
Note 2: Adding rational number in fraction form combines the rules for adding positive fractions with the rules for adding integers.
We need a common denominator which becomes the denominator of our answer
Add numerators
Find total of positives when appropriate
Find total of negatives when appropriate
“Cancel” zeros when adding opposites
Recall Adding Fractions
To add fractions, use equivalent fractions with common denominators.
Examples:
Adding Rational Numbers
Examples. Calculate:
1) Move all negatives to the numerator Find common denominator
Adding same sign so we find a total of negatives
2)
Adding different signs so we keep the sign of the larger and find the difference
3) Change all mixed numbers to improper fractions when dealing with positives and negatives.
Write answers as mixed fractions where possible
Complete Practice from #3-6 from Practice book See examples 1-3 pp. 108-110 without number lines
3.3. Subtracting Rational Numbers
Note 1: Subtracting rational numbers in decimal form combines the rules for subtracting positive decimals with the rules for subtracting integers.
Line up place value
Add the opposite when necessary
Recall Subtracting Integers
Examples:
Note 2: Subtracting rational number in fraction form combines the rules for subtracting positive fractions with the rules for subtracting integers.
We need a common denominator which becomes the denominator of our answer
Add the opposite when necessary
Recall Subtracting Fractions
To subtract fractions, use equivalent fractions with common denominators.
Example:
Subtracting Rational Numbers
Examples. Calculate:
1)
Find common denominators Change to adding opposite Move negative to numerator Reduce all answers
2) Put any negatives in the numerator
Find common denominators Add opposite
Move negative to numerator
Write answers in mixed form where possible
3)
5)
Change all mixed numbers to improper fractions when dealing with positives and negatives.
6) In Alberta, the lowest temperature ever recorded was 61.1 0C at Fort Vermilion in 1911. The highest temperature was 43.3 0C at Bassano Dams in 1931. What is the difference between these temperatures?
Solution:
The difference between the temperatures is 104.4 0C
Complete #3-6 from Practice book
Also see examples 1-3 pp. 115-118 without models Discuss p. 120 #10 [ex. 1.3 - (-3.5) = 5.8]
3.4. Multiplying Rational Numbers
Note 1: Multiplying rational numbers in decimal form combines the rules for multiplying positive decimals with the rules for multiplying integers.
Note placement of decimal
The product of two numbers with the same sign is positive
The product of two numbers with different signs is negative
Recall Multiplying Integers
Examples:
Note 2: Multiplying rational numbers in fraction form combines the rules for multiplying positive fractions with the rules for multiplying integers.
The product of two numbers with the same sign is positive
The product of two numbers with different signs is negative
To multiply fractions multiply the numerators and multiply the denominators
Recall Multiplying Fractions
See skill builders from practice book
Go through Examples 1, 2 & 3 and checks from Practice book Also see examples 1-3 pp. 124-127
Complete #1-5 from the practice book
3.5. Dividing Rational Numbers
Note 1: Dividing rational numbers in decimal form combines the rules for dividing positive decimals with the rules for dividing integers.
Note placement of decimal
The quotient of two numbers with the same sign is positive
The quotient of two numbers with different signs is negative
Recall Dividing Decimals
Examples: is the same as is the same as
Note 2: We move the decimal the same number of places in the same direction so that there is no decimal in the second number.
Recall Dividing Integers
Examples:
Note 3: Dividing rational number in fraction form combines the rules for dividing positive fractions with the rules for dividing integers.
The quotient of two numbers with the same sign is positive
The quotient of two numbers with different signs is negative
To divide fractions change to multiplying by the reciprocal of the second OR write common denominators and get the first numerator divided by the second numerator.
Recall Dividing Fractions
Example: OR
Examples:
means how many ’s are in 2 12
means how much would be in each set of we divided into equal sets
Go through Examples 1 & 2 and check from Practice book Complete #1-5 from Practice book
Also see examples 1-3 pp. 131-134
Complete pp. 134-136 #3-5, 6, 7adf, 8, 9, 11a, 12, 14, 15, 17, 18
3.6. Order of Operations with Rational Numbers
Go through Examples 1, 2 & 3 and checks from Practice book Complete Practice from Practice book
Also see pp. 138-139 #1-3
Set up an appropriate expression and use the order of operations to answer each of the questions below.
Sample Problem 1: Thomas was given $125 cash for his Birthday. He wants to buy a video game that costs $74.47. If he buys the game, how much will he have left? Remember that he has to pay GST.
Sample Problem 2: Mary has $35.95 in her bank account. She uses her bankcard to buy a $23.99 DVD and a $19.49 shirt. She has to pay HST. How much does Mary have left in her account?
Sample Problem 3: A room measures 5.2m by 3.9m. (a) Calculate the area of the room.
Mathematical Properties
Many of the mathematical properties that apply to integers also apply to rational numbers.
Ex. and and
and and
and and
Commutative (order)
Adding and multiplying is commutative. That is, you can add or multiply in any order. Commutativity applies to integers and it also applies to rational
Associative (grouping)
Adding and multiplying is associative. That is, you can regroup when adding or multiplying. Associativity applies to integers and it also applies to rational numbers.
Ex.
Sometimes regrouping makes it easier to find the answer.
Example 1. Example 2.
= =
= =
= =
= =
Distributive
Rational numbers can be distributed over addition or subtraction. Example.
= = = = =
The reverse is also true and this sometimes makes it easier to find an answer. Example 1.
