Download Natural Numbers * Learning Outcomes

1
Natural Numbers – Learning Outcomes
 Identify and list natural numbers.
 Add, subtract, multiply, and divide natural numbers.
 Use the number line to illustrate arithmetic operations.
 Use skip counting and area models to illustrate
multiplication problems.
 Solve problems about factors and multiples.
 Identify prime factors, and prime factorise numbers.
 Use the order of operations to solve problems.
 Use the commutative, associative, and distributive
properties to solve problems.
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Identify and List Natural Numbers
 The natural numbers are the set of all positive, whole
numbers.
 ℕ = 1, 2, 3, 4, 5, 6, …
 Negative numbers and decimals are not natural
numbers.
 We also call them the counting numbers because we
use them to describe how many of something there are.
 e.g. three chairs,
 e.g. seven dwarves,
 e.g. forty thieves.
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Identify Natural Numbers
 Outcome check: can you identify the natural numbers?
a) -4
b) 3.14
c) 9
d) 42
e) -1 000 000
f)
1
2
g) 42
 Outcome check: list any ten natural numbers and
check with the person beside you.
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Use the Number Line to Illustrate Addition
 Number lines are used to represent numbers and
compare their sizes.
 Since 1 is the smallest natural number, it goes on the left.
 Numbers increase as you go to the right.
 Don’t forget the arrows at both ends! (they mean the
line keeps going past what we can see)
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Use the Number Line to Illustrate Addition
 Addition is a mathematical operation that takes two
numbers and finds their sum or total.
 We use the symbol ‘+’ (read as “plus”)to mean add.
 e.g. 3 + 5 means add 3 and 5.
 e.g. 2 + 9 means add 2 and 9.
 You can find the answer by counting, remembering your
addition tables from primary school, or by using a
number line.
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Use the Number Line to Illustrate Addition
 Use a dot for your starting point and draw an arrow to
the right.
 Count how many notches you need to travel.
 e.g. 3 + 5
 e.g. 2 + 9
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Use the Number Line to Illustrate Addition
 Outcome check: use the number lines to find the sums.
1+3
4+2
5+9
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Subtract Natural Numbers
 Subtraction is a mathematical operation that takes two
numbers and finds their difference.
 We use the symbol ‘-’ (read as “minus”) to mean
subtract.
 e.g. 5 − 3 means subtract 3 from 5.
 e.g. 10 − 6 means subtract 6 from 10.
 You can find the answer by counting, remembering your
subtraction tables from primary school, or by using a
number line.
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Use the Number Line to Illustrate Subtraction
 Use a dot for your starting point and draw an arrow to
the left.
 Count how many notches you need to travel.
 e.g. 5 − 2
 e.g. 10 − 6
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Use the Number Line to Illustrate Subtraction
 Outcome check: use number lines to find the
differences:
4−1
6−2
 13 − 8
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Multiply Natural Numbers
 Multiplication is a mathematical operation that takes
two numbers and finds their product.
 We use the symbol ‘×’ (read as “times”) to mean
multiply.
 e.g. 5 × 3 means multiply 5 and 3.
 e.g. 12 × 6 means multiply 12 and 6.
 You can find the answer by repeatedly adding one of
the numbers to itself.
 e.g. 5 × 3 = 3 + 3 + 3 + 3 + 3
 e.g. 12 × 6 = 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6
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Use Area Model to Illustrate Multiplication
 Area models can also be used to find products.
 The product of two numbers is the area of a rectangle
with those side lengths.
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Use Area Model to Illustrate Multiplication
 Outcome check: use the area model to find:
1. 2 × 6
2. 4 × 2
3. 3 × 3
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Use Skip Counting to Illustrate Multiplication
 Starting from zero, draw an arrow representing one of
the numbers.
 Draw this arrow as many times as the other number.
 e.g. 6 × 2
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Use Skip Counting to Illustrate Multiplication
 Outcome check: use skip counting to find:
1. 2 × 6
2. 4 × 2
3. 3 × 3
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Solve Problems about Factors
 Factors of a number divide evenly into that number.
 e.g. 12 ÷ 6 = 2, so 6 is a factor of 12.
 e.g. 12 ÷ 5 = 2.4, so 5 is not a factor of 12.
 Outcome check: List all the factors of:
1. 12
2. 16
3. 90
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Solve Problems about Factors
 The highest common factor (HCF) of a set of numbers is
the greatest factor they all share.
 Outcome check: Find the HCF of these sets of numbers:
1. 12, 18
2. 63, 42
3. 90, 135
4. 3, 5, 9
5. Gheed wants to plant 63 tomato plants and 81 rhubarb
plants. She wants to plant them in rows where each row
has the same number of tomato plants and the same
number of rhubarb plants. What is the greatest number
of rows she can plant?
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Solve Problems about Multiples
 Multiples of a number are the set of that number
multiplied by natural numbers.
 e.g. Multiples of 3 = {3, 6, 9, 12, 15, 18, 21, …}
 e.g. Multiples of 7 = {7, 14, 21, 28, 35, 42, 49, …}
 Outcome check: List the first seven multiples of:
1. 4
2. 11
3. 12
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Solve Problems about Multiples
 The lowest common multiple (LCM) of a set of numbers is
the least multiple they all share.
 Outcome check: Find the LCM of these sets of numbers:
1. 4, 9
2. 2, 6
3. 12, 8
4. 6, 10, 12
5. Alanah is buying burgers and buns for a barbecue.
Burgers come in packs of 4 and buns come in packs of
6. What is the smallest number of burgers that Alanah
can buy?
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Solve Problems about Factors
 Prime numbers have exactly two factors (itself and 1).
 e.g. 5 is only divisible by 5 and 1, so 5 is prime.
 e.g. 12 is divisible by 1, 2, 3, 4, 6, and 12; so 12 is not
prime.
 Outcome check: List all the factors of the following
numbers and state whether they are prime:
i.
6
ii. 7
iii. 8
iv. 23
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Solve Problems about Factors
 Prime factorisation is when you keep dividing a number
until all your factors are prime.
 e.g. 10 = 2 × 2 × 5 = 22 × 5
 e.g. 24 = 2 × 2 × 2 × 3 = 23 × 3
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Solve Problems about Factors
 Outcome check: Express each of the following numbers
as a product of prime factors (i.e. prime factorise):
1. 12
2. 32
3. 48
4. 59
5. 90
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Use Order of Operations to Solve Problems
 Evaluate: 2 + 5 × 3
A. 21
B. 17
 Evaluate: 6 ÷ 2(1 + 2)
A. 9
B. 1
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Use Order of Operations to Solve Problems
 To resolve problems, operations are tackled in order of
strength – GEMDAS
 G – groups (operations inside brackets, roots, modulus,
numerators, denominators, …) are always strongest.
 E – exponents (powers, roots) are next.
 MD – multiplication, division are the same strength.
 AS – addition, subtraction are the same strength.
 If a tie would cause different answers, work from left to
right.
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Use Order of Operations to Solve Problems
Evaluate
G – groups
E – exponents
MD – multiplication,
division
AS – addition,
subtraction
𝟔+𝟕×𝟖
=6+7×8
=6+7×8
= 6 + 56
= 62
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Use Order of Operations to Solve Problems
Evaluate
G – groups
E – exponents
MD – multiplication,
division
AS – addition,
subtraction
𝟒𝟐 − (𝟒 × 𝟑)
= 42 − 12
= 16 − 12
= 16 − 12
=4
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Use Order of Operations to Solve Problems
Evaluate
G – groups
E – exponents
MD – multiplication,
division
AS – addition,
subtraction
𝟓
𝟗−
×𝟐+𝟔
𝟖−𝟑
5
=9− ×2+6
5
5
=9− ×2+6
5
=9−1×2+6
=9−2+6
=7+6
= 13
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Use Order of Operations to Solve Problems
 Outcome check: Evaluate:
1. 9 + 6 × (8 − 5)
2.
3.
4.
14 − 5 ÷ (9 − 6)
36−6
12+3
36−3×4
9
15−3
 A plumber charges a €50 callout fee plus €20 per hour
worked. If you have a discount voucher for €30, write an
expression for the price of the plumber for 4 hours work.
 Evaluate your expression.
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Solve Problems using Commutativity
 Evaluate:
3+5
5+3
6×2
2×6
 When you can change the order of numbers without
changing the answer, the operation is commutative.
 Addition and multiplication are commutative.
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Solve Problems using Commutativity
 Evaluate:
5–3
3–5
6÷2
2÷6
 Subtraction and division are not commutative.
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Solve Problems Using Associativity
 Evaluate:
 2 + (3 + 4)
 (2 + 3) + 4
 2 × (3 × 4)
 (2 × 3) × 4
 When you can group operations together without
changing the answer, the operation is associative.
 Addition and multiplication are associative.
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Solve Problems using Associativity
 Evaluate:
 6 – (3 – 2)
 (6 – 3) – 2
 24 ÷ (4 ÷ 2)
 (24 ÷ 4) ÷ 2
 Subtraction and division are not associative.
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Solve Problems using Distribution
 Distribution allows operations to break GEMDAS without
changing the answer.
 e.g. 2 × (1 + 3)
2×4
2×1+2×3
 Multiplication distributes over addition, but not the other
way around.
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Solve Problems Using Distribution
 Outcome check: Use distribution to evaluate:
1. 3 × 5 + 6
2. 2 × 5 + 3
3. 7 × 7 + 3
4. 10 × 8 + 2
5. 12 × (11 + 3)