Module 1 Order of Operations, Fractions and Decimals

Enabling Courses: Enabling Mathematics
Module 1
Order of Operations, Fractions and
Decimals
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Module Plan
o General introduction
• Welcome
• Module aims
• How to use these Modules
o Order of Operations
o Fractions
•
•
•
•
•
Proper fractions, improper fractions and mixed numbers
Adding Fractions
Subtracting Fractions
Multiplying Fractions
Dividing Fractions
o Decimals
•
•
•
•
Adding Decimals
Subtracting Decimals
Multiplying Decimals
Dividing Decimals
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General introduction
o Enabling subjects:
• Introductory level of knowledge
• Self-paced and flexible
o Mathematics is used everyday in medicine, cookery,
brewing and wine making, growing crops and animals,
architecture and building, and forensic science
o Aim: Provide a smooth transition into your introductory
courses such as chemistry.
o Structure: 4 modules
• Best to complete in order
o Use activities, and quizzes to help you learn
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Order of Operations
o Order of Operations: Follow steps in a particular order to
get correct answer.
o Certain sums in mathematics must be done before
others.
o Six operations:
•
•
•
•
•
•
Addition (+)
Subtraction (–)
Multiplication (x)
Division (÷ or /)
Brackets ( ) and
Powers (also referred to as “of” or “to the order of”) (e.g. 32, 63).
Image:
http://europeansectionarcipreste.blogspot.com.au/2012/04/multiplying-anddividing-fractions.html
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Order of Operations: BODMAS
o In mathematics we follow BODMAS rule for the order of
operations:
•
•
•
•
•
•
Brackets
Of or Order
Division
Multiplication
Addition
Subtraction
(
)
22 , 34 , 5 6
÷, or /
x
+
http://www.mentalstarters.co.uk/Key%20Stage%204
%20Foundation.htm
o Enable to compute in the correct order any mathematical
question put forward.
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Order of Operations: BODMAS
o Example:
(3 – 5) + 6 ÷ 3 – 2
B first:
(3 – 5) = -2
D next.
6÷3=2
o Therefore: -2 + 2 = 0
Here, A first (work out the
And: 0 – 2 = -2
Answer = -2
answer from left to right).
Then complete the sum
using S.
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Fractions
o An essential tool:
• To make exact measurements and we often need to work with
parts or percentages of the whole
• When preparing a mixture for a natural medicine formulation; in
many mixtures, fractions of amounts need to be added,
subtracted, multiplied or even divided.
o Includes:
•
•
•
•
•
Proper fractions, improper fractions and mixed numbers
Adding Fractions
Subtracting Fractions
Multiplying Fractions
Dividing Fractions
http://www.sweetcounter.co.uk/sc2-flip-flap-fractions-p-333.html
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Proper Fractions
o Proper fraction: has a numerator less than its
denominator.
• Numerator as the top number and Denominator as the bottom
number. So a proper fraction is not a whole number, but a
fraction of it.
o Examples:
¾
Or
-1/8
Or
𝟏𝟏/𝟏𝟐 Or
𝟑
𝟒
-
𝟏
𝟖
𝟏𝟏
𝟏𝟐
Proper Fraction
Improper Fraction
http://memoryjoggers.com/2012/08/remembering-properand-improper-fractions/
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Improper Fractions
o Improper fraction: has a numerator greater than its
denominator.
o Examples:
𝟑
Or
𝟓
𝟑
−𝟗 𝟖
Or
−
𝟏𝟓
Or
𝟏𝟓
𝟏𝟐
𝟓
𝟏𝟐
𝟗
𝟖
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Mixed Numbers
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Mixed Numbers
o Converting mixed numbers into improper fractions: The
whole number is first converted to a fraction. The
fractions are then added
o Example:
http://www.visualfractions.com/MixedtoFracC/mixedtofrCircles.html
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Mixed Numbers
http://www.visualfractions.com/MixedCircles/imixedcircles.html
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Adding Fractions
http://www2.potsdam.edu/abramovs/compenv310.htm
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Adding Fractions
o Adding fractions with different denominators:
o Example:
Multiples of 8 are 8, 16, 24, 32, 40, 48, and so on.
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and so on.
The lowest common multiple of 8 and 3 is 24. The fraction needs to
be converted to a denominator of 24. Note that the numerator and
denominator of each fraction must be multiplied by the same
number.
Multiply the numerator and denominator of the first fraction by 3 to
make the denominator 24.
Multiply the numerator and denominator of the second fraction by 8
to make the denominator 24.
Now we can add the numerators.
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Subtracting Fractions
http://www.buzzle.com/articles/subtracting-fractions-with-unlikedenominators.html
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Subtracting Fractions
o Subtracting fractions with different denominators:
o Example:
The lowest common multiple of 6 and 5 is 30.
This fraction can be simplified by dividing the numerator and
denominator by a common number. All fractions can be simplified
if there is a common number you can divide the numerator and
denominator by.
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Multiplying Fractions
o Simply need to multiply the numerators and
denominators
o Best for the fractions to be in their simplest form when
multiplying them, so that the answer is not a large
fraction.
o Two methods:
• Multiplying fractions that are in their simplest form
• Multiplying fractions that need to be simplified first
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Multiplying Fractions
http://testpreparations.com/help/learn-to-multiply-and-divide-with-fractions
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Multiplying fractions
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Dividing Fractions
o To divide fractions, flip or invert the fraction that to be
dividing by (the second fraction).
o Then change the division sign to a multiplication sign
and treat it in the same way as multiplication of fractions
o Example:
3 2

4 7
Flip the second fraction.
=
3 7

4 2
Change the  sign to x.
=
3 7
x
4 2
Multiply the numerators and denominators.
21
= 8
2
=
5
8
http://edtech2.boisestate.edu/robertsona/506/finalProject/DividingFractions.html
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Decimals
http://www.mentalstarters.co.uk/Year%205.htm
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Adding Decimals
o Place all decimals vertically keeping the decimal points
aligned.
o Example:
http://www.coolmath.com/prealgebra/02-decimals/06-decimals-adding-01.htm
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Subtracting Decimals
o Same as adding decimals.
o Example:
http://www.coolmath.com/prealgebra/02-decimals/07-decimals-subtracting-01.htm
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Multiplying Decimals
o Step 1: Multiply decimals by temporarily ignoring the
decimal point and multiply the digits.
o Step 2: The number of decimal places in the final answer
must be the same as the total number in the question.
So, insert the decimal point in the answer by counting to
the left for total number of decimals in question.
http://www.coolmath.com/prealgebra/02-decimals/08-decimals-multiplying-01.htm
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Multiplying Decimals
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Dividing Decimals
o Same method as multiplication of decimals.
o Complete the division of the two numbers first and then
include the position of the decimal point
o Example:
8.1 ÷ 9
= 81 ÷ 9
=9
o Now, in the original question, 8.1 has one decimal place
and 9 has zero decimal places. So in total move one
decimal place to the left to arrive at the final answer of
0.9.
http://www.mahalo.com/how-todivide-decimals/
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What Is Next
o In the Next module we will cover….
Percentages, Ratios and Indices (powers) and Logarithms
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