Introduction to Algebra for primary school

Getting Used to Algebra
Algebra is where you use letters to
represent numbers
Aims:
• To get to grips with the algebra idea
• To solve simple algebraic problems
Sunday, 28 July 2024 Page 1

Example
Alex has some sweets, we do not know
how many sweets Alex has….
so we can say ‘Alex has x sweets’
If Alex is given 5 more sweets, how many
sweets has he got?
x + 5

Example
Bob has some toys, we do not know how
many toys he has….
so we can say ‘Bob has m toys’
If Bob buys 6 more toys, how many toys
has he now got?
m + 6

Another Example
Bill catches y fish.
Ben takes 3 away from him.
How many fish does Bill now have?
y - 3

A Third Example
Fred has x DVDs
Frank has y DVDs
How many DVDs do they have altogether?
x + y

Questions
Use algebra to write:
1) 2 less than w
2) 3 more than d
3) 5 together with c
4) f more than g
5) p less than q
6) m less than 7
w – 2
d + 3
c + 5
f + g
q - p
7 - m

Adding and Subtracting with
Letters
a + a + a =
b + b + b + b + b =
c + c – c + c =
3a
5b
2c

Questions
1) a + a =
2) c + c + c + c =
3) p - p + p =
4) v + v + v + v - v + v =
5) b + b + b – b =
6) n - n + n + n + n - n + n =
7) h - h =
8) g + g + g - g
2a
4c
p
4v
2b
3n
0
2g

Adding Expressions & Terms
2a + 4a =
6a – 5a =
7a – 4a + 10a =
6a
a
13a

Questions
1) 5c + 7c
2) 9d – 4d
3) 2s + 12s
4) 13a – 5a
5) 4e – e + 3e
6) 6e – 2e + 5e
7) 12e – 10e
8) 3h – 2h + h
9) 4r – r + 5r
10) 3w + 9w – 5w
11) 2g + 5g – 3g
12) 7f – 4f + 9f
13) 5b + 50b
14) 75j – 43j
15) 34p + 12p – 5p
16) 3m – m + m + 5m
17) d – d + d - d
18) 4f + 10f - 13f
12c
5d
14s
8a
6e
9e
2e
2h
8r
7w
4g
12f
55b
32j
41p
8m
0
f

Going Negative
3a – 5a =
5p + 4p – 12p =
-5a + 2a – 7a =
-2a
-3p
-10a

Working with Algebra
Aims:
 To be able to collect like terms in
order to simplify algebraic expressions
 To be able to multiply terms together
and expand the brackets from an
expression

Example 1
3a + 4b + 2a + 6b =
firstly collect like-terms…
3a + 2a + 4b + 6b =
5a + 10b

Example 2
-3a + 2b – 4a + 6b
collect like-terms
-3a – 4a + 2b + 6b
-7a + 8b

Example 3
-4a + 9b + 3a - 12b
collect like-terms
-4a + 3a + 9b – 12b
- a – 3b

Aims
 To be able to simplify expressions
(including expressions with indices)
 To be able to expand brackets and
simplify
 To be able to understand the 3 laws of
indices

Simplify each expression…
1) 5a – 4y – 11a + 2y
2) -4f + 6g – 7f – 3g
3) 4m + 9n – 6m - 14n
4) -4t + 7y – 10t + 5y
5) 1 + 2r – 7 – 7r
6) 3d – 5e – 4d + 2e
7) p + 8r – 7p + 2r + 3p
8) 9x + 3x – 8 - 2x + 3
9) a – 6b + 2a + 3b - 3a
10) 3g + 2h – 3h + 2g
-6a - 2y
-11f + 3g
-2m – 5n
-14t + 12y
-6 – 5r
-d – 3e
-3p + 10r
10x – 5
-3b
5g – h

Harder Simplification…
(remember, a, a2 and a3 are completely different terms)
1) 4a – 5a2 + 2a + 3a2 =
2) 5h2 + 2h3 – 10h2 – 3h3 =
3) 3x2 – 4x – 5x2 + x =
4) -4t2 + 7t – 10t2 + 5t3 =
5) r2 + 2r – 7r2 – 7r2 – 2r =
6a – 2a2
-5h2 –h3
– 3x-2x2
7t -14t2 + 5t3
-13r2

Multiplying Terms Together
When two terms in algebra are being
multiplied together, they are simply
written next to each other.
e.g. 3a means 3 x a
and efg means e x f x g

Multiplying Terms Together
Simplify:
2a x 4b =
10c x 2de =
3w x 4y x 5z =
8ab
20cde
60wyz

Questions – simple multiplication
1) 7ab x 8pq
2) 3f x 2h x 5a
3) 3abc x 4def
4) 5g x 25
5) 5mn x 2pq x 4
6) 5f x g x h x 2j
7) 3w x 4d x 2h x yz
8) 7pqr x 4abc x 2h
56abpq
12abcdef
56abchpqr
125g
40mnpq
10fghj
24dhwyz
30afh

Questions – reverse multiplication
1) 7ab x ___ = 70abc
2) 3f x ___ x 5h = 60fgh
3) 3abc x ___ = 3abcd
4) 5g x ___ = 20g
5) 5mn x ___ x 4 = 20mnp
6) 5f x ___ x 2g = 40fgh
7) 3w x ___ x 2v x yz = 24vwyz
8) 4bc x ___ x 2ad = 80abcde
10c
d
10e
4
p
4h
4
4g

Dividing Algebraic Terms
Simplify:
8a ÷ 4 =
10ab ÷ 2a =
20pq
pq =
2a
5b
20

Powers
Aims:
To remember how to work out the HCF &
LCM from any given pair of numbers
To be able to understand how indices
(powers) work in algebra
To be able to manipulate the powers in an
expression in order to simplify it

Powers Rules
a x a =
a x a x a x a =
But, what is a2 x a4?
It’s (a x a) x (a x a x a x a) =
What did you do with the powers?
You added them!
a2
a4
a6
Rule 1:
When you multiply
powers of the same
letter or number you
add the indices…

Indices Questions
a3 x a6 x a2 =
c3 x c5 x c =
2y2 x 4y5 =
3m3 x 4m5 =
1
a11
c9
8y7
12m8

Harder Indices Questions
2a3b6 x a6b2 =
3c5d4 x 5c2d4 =
2ab4 x 4a2b3 =
3mnp3 x 4mn2p5 =
2a9b8
15c7d8
8a3b7
12m2n3p8
Q1.
Q2.
Q3.
Q4.

Powers
a x a x a x a x a =
a x a x a =
But, what is a5 a3?
It’s (a x a x a x a x a) (a x a x a) =
What did you do with the powers?
You subtracted them!
a3

a5
 a2
Rule 2:
When you divide
powers of the same
letter or number you
subtract the indices…

Indices Questions
a6  a2 =
c3  c5 =
3y2 x 4y5 =
2y3
a4
c-2
6y4

Harder Indices Questions
4a6 x 8a2 =
2a9
30y2 x 4y5 =
6 x y4
16a-1
20y3
Q1.
Q2.

Powers
a x a x a =
What do you think is…
(a3)2
= a3 x a3 =
a3 Rule 3:
When you raise a
power by another
power (separated by
brackets) you multiply
the indices…
a6
what did you do
with the powers
here?
multiplied them!

Indices Questions
(a2)4 x (a3)2 =
4(a2)3 =
(4a2)3 =
(5ab3)2 =
a14
4a6
64a6
25a2b6
Q1.
Q2.
Q3.
Q4.

Have you really understood
indices??
We’ll see…
Simplify this…
3
9
19
7
2
6
5
6
3
5
4
)
2
(
)
)
((
)
(
4
)
2
(
a
a
a
a
a
a





25.4.2024
Simplify algebra with brackets
Aims:
 To be able to multiply out the brackets
from expressions…
 … and then collect like-terms and
simplify

Expand these expressions:
3(a + b) =
4(y + 6) =
2(2a + 3b – 4c) =
3a + 3b
4y + 24
4a + 6b – 8c

Adding Bracketed Expressions
4(a + 5b) + 3(7a – b) =
4a + 20b + 21a – 3b =
25a + 17b

Adding Bracketed Expressions
2(7a – 6b) - 6(2a + b) =
14a – 12b - 12a - 6b =
14a – 12a -12b -6b
2a – 18b

Expand the brackets…
1) 4(2p – 12) =
2) 5(2a + 4b) =
3) 3(4c – 4b) =
4) 7(3e + 2f – 4g) =
5) 9(6w + 2y – 7z) =
6) 10(3b – 9m) =
7) 6(-6j – 7m) =
8) 5(4a – 3b) =
8p - 48
10a + 20b
12c – 12b
21e + 14f – 28g
54w + 18y – 63z
30b – 90m
-36j – 42m
20a – 15b

Questions
1) 7(2a – 4b) + 2(2a + b)
2) 2(3y – w) + 3(2y + 5w)
3) 4(p + q) + 5(2p – 3q)
4) 6(2n + m) - 2(n – m)
5) 2(4s + r) - 7(2s – 3r)
18a – 26b
12y + 13w
14p – 11q
14n + 4m
22s – 19r

Subtracting Bracketed
Expressions
3(2a – 4b) – 2(a + 2b) =

Expressions
3(2a – 4b) – 2(a + 2b) =
6a – 12b – 2a – 4b =

Expressions
3(2a – 4b) – 2(a + 2b) =
6a – 12b – 2a – 4b =
4a – 16b

Expressions
3(2p + 3q) – 4(p – 2q) =

Expressions
3(2p + 3q) – 4(p – 2q) =
6p + 9q - 4p + 8q =

Expressions
3(2p + 3q) – 4(p – 2q) =
6p + 9q - 4p + 8q =
2p + 17q

Invisible 1
7(m – 2n) – (m – 3n) =

Invisible 1
7(m – 2n) – (m – 3n) =
7m – 14n – m + 3n =

Invisible 1
7(m – 2n) – (m – 3n) =
7m – 14n – m + 3n =
6m - 11n

Questions
1) 3(4n + 5m) – 5(2n – 4m)
2) 5(n – 2m) – (m + 2n)
3) 3(2n + 6m) – 6(n – 2m)
4) 2(3n – m) – 7(n – 5m)
5) 8(5n – m) – 2(2n + m)

Introduction to Algebra for primary school

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