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.

2. 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.

3. Identify Natural Numbers
Outcome check: can you identify the natural numbers? -4
3.14 9 42
1 2
Outcome check: list any ten natural numbers and check with the person beside you.

4. 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)

5. 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.

6. 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

7. Use the Number Line to Illustrate Addition
Outcome check: use the number lines to find the sums.
1+3
4+2
5+9

8. 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.

9. 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

10. Use the Number Line to Illustrate Subtraction
Outcome check: use number lines to find the differences:
4−1
6−2
13−8

11. 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 =
e.g. 12 × 6 =

12. 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.

13. Use Area Model to Illustrate Multiplication
Outcome check: use the area model to find:
2 × 6
4 × 2
3 × 3

14. 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

15. Use Skip Counting to Illustrate Multiplication
Outcome check: use skip counting to find:
2 × 6
4 × 2
3 × 3

16. 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: 12 16 90

17. 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:
12, 18
63, 42
90, 135
3, 5, 9
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?

18. 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: 4 11 12

19. 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:
4, 9
2, 6
12, 8
6, 10, 12
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?

20. 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: 6 7 8 23

21. 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= 2 2 ×5
e.g. 24=2×2×2×3= 2 3 ×3

22. Solve Problems about Factors
Outcome check: Express each of the following numbers as a product of prime factors (i.e. prime factorise): 12 32 48 59 90

23. Use Order of Operations to Solve Problems
Evaluate: 2+5×3 21 17
Evaluate: 6÷2(1+2) 9 1

24. 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.

25. Use Order of Operations to Solve Problems
Evaluate
𝟔+𝟕×𝟖
G – groups
=6+7×8
E – exponents
MD – multiplication, division
=6+56
AS – addition, subtraction
=62

26. Use Order of Operations to Solve Problems
Evaluate
𝟒 𝟐 −(𝟒×𝟑)
G – groups
= 4 2 −12
E – exponents
=16−12
MD – multiplication, division
AS – addition, subtraction =4

27. Use Order of Operations to Solve Problems
Evaluate
𝟗− 𝟓 𝟖−𝟑 ×𝟐+𝟔
G – groups
=9− 5 5 ×2+6
E – exponents
MD – multiplication, division
=9−1×2+6
=9−2+6
AS – addition, subtraction
=7+6
=13

28. Use Order of Operations to Solve Problems
Outcome check: Evaluate:
9+6×(8−5)
14−5 ÷(9−6)
36−6 12+3
36−3×4 15− 9 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.

29. 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.

30. Solve Problems using Commutativity
Evaluate:
5 – 3
3 – 5
6 ÷ 2
2 ÷ 6
Subtraction and division are not commutative.

31. 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.

32. Solve Problems using Associativity
Evaluate:
6 – (3 – 2)
(6 – 3) – 2
24 ÷ (4 ÷ 2)
(24 ÷ 4) ÷ 2
Subtraction and division are not associative.

33. Solve Problems using Distribution
Distribution allows operations to break GEMDAS without changing the answer.
e.g. 2 × (1 + 3)
2×1+2×3
2×4
Multiplication distributes over addition, but not the other way around.

34. Solve Problems Using Distribution
Outcome check: Use distribution to evaluate:
3× 5+6
2× 5+3
7× 7+3
10× 8+2
12×(11+3)