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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
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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 + 5Thursday 20 April 2023 Page 2
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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 + 6Thursday 20 April 2023 Page 3
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Another Example
Bill catches y fish.
Ben takes 3 away from him.
How many fish does Bill now have?y - 3
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A Third Example
Fred has x DVDs
Frank has y DVDs
How many DVDs do they have altogether?
x + yThursday 20 April 2023 Page 5
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Questions
Use algebra to write:1) 2 less than w2) 3 more than d3) 5 together with c4) f more than g5) p less than q6) m less than 7
Thursday 20 April 2023 Page 6
w – 2
d + 3
c + 5
f + g
q - p
7 - m
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Adding and Subtracting with Letters
a + a + a =
b + b + b + b + b =
c + c – c + c =
Thursday 20 April 2023 Page 7
3a
5b
2c
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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
Thursday 20 April 2023 Page 8
2a
4cp
4v2b
3n0
2g
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Adding Expressions & Terms
2a + 4a =
6a – 5a =
7a – 4a + 10a =
Thursday 20 April 2023 Page 9
6a
a
13a
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Questions
1) 5c + 7c2) 9d – 4d3) 2s + 12s4) 13a – 5a5) 4e – e + 3e6) 6e – 2e + 5e7) 12e – 10e8) 3h – 2h + h9) 4r – r + 5r
10) 3w + 9w – 5w11) 2g + 5g – 3g12) 7f – 4f + 9f13) 5b + 50b14) 75j – 43j15) 34p + 12p – 5p16) 3m – m + m +
5m17) d – d + d - d18) 4f + 10f - 13f
Thursday 20 April 2023 Page 10
12c5d14s8a
6e9e
2e2h8r
7w4g12f
55b32j
41p8m
0f
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Going Negative
3a – 5a =
5p + 4p – 12p =
-5a + 2a – 7a =
Thursday 20 April 2023 Page 11
-2a
-3p
-10a
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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
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Example 1
3a + 4b + 2a + 6b =
firstly collect like-terms…
3a + 2a + 4b + 6b =
5a + 10b
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Example 2
-3a + 2b – 4a + 6b
collect like-terms
-3a – 4a + 2b + 6b
-7a + 8b
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Example 3
-4a + 9b + 3a - 12b
collect like-terms
-4a + 3a + 9b – 12b
- a – 3b
Thursday 20 April 2023 Page 15
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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
Thursday 20 April 2023 Page 16
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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 + 2gThursday 20 April 2023 Page 17
-6a - 2y
-11f + 3g
-2m – 5n
-14t + 12y
-6 – 5r
-d – 3e
-3p + 10r
10x – 5
-3b
5g – h
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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 =Thursday 20 April 2023 Page 18
6a – 2a2
-5h2 –h3
– 3x-2x2
7t -14t2 + 5t3
-13r2
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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 aand efg means e x f x g
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Multiplying Terms Together
Simplify:
2a x 4b =
10c x 2de =
3w x 4y x 5z =
Thursday 20 April 2023 Page 20
8ab
20cde
60wyz
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Questions – simple multiplication
1) 7ab x 8pq2) 3f x 2h x 5a3) 3abc x 4def4) 5g x 255) 5mn x 2pq x 46) 5f x g x h x 2j7) 3w x 4d x 2h x yz8) 7pqr x 4abc x 2h
Thursday 20 April 2023 Page 21
56abpq
12abcdef
56abchpqr
125g40mnpq
10fghj24dhwyz
30afh
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Questions – reverse multiplication
1) 7ab x ___ = 70abc2) 3f x ___ x 5h = 60fgh3) 3abc x ___ = 3abcd4) 5g x ___ = 20g5) 5mn x ___ x 4 = 20mnp6) 5f x ___ x 2g = 40fgh7) 3w x ___ x 2v x yz = 24vwyz8) 4bc x ___ x 2ad = 80abcde
Thursday 20 April 2023 Page 22
10c
d
10e
4p
4h
4
4g
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Dividing Algebraic Terms
Simplify:
8a ÷ 4 =
10ab ÷ 2a =
20pq pq =
Thursday 20 April 2023 Page 23
2a
5b
20
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PowersPowersAims: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
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Powers RulesPowers Rulesa 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!You added them!Thursday 20 April 2023 Page 25
a2
a4
a6
Rule 1:Rule 1:When you multiplymultiply powers of the same letter or number you addadd the indices…
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Indices QuestionsIndices Questions
a3 x a6 x a2 =
c3 x c5 x c =
2y2 x 4y5 =
3m3 x 4m5 =
Thursday 20 April 2023 Page 26
1
a11
c9
8y7
12m8
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Harder Indices QuestionsHarder Indices Questions
2a3b6 x a6b2 =
3c5d4 x 5c2d4 =
2ab4 x 4a2b3 =
3mnp3 x 4mn2p5 =
Thursday 20 April 2023 Page 27
2a9b8
15c7d8
8a3b7
12m2n3p8
Q1.
Q2.
Q3.
Q4.
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PowersPowersa 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!You subtracted them!Thursday 20 April 2023 Page 28
a3
a5
a2
Rule 2:Rule 2:When you divide divide powers of the same letter or number you subtract subtract the indices…
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Indices QuestionsIndices Questions
a6 a2 =
c3 c5 =
3y2 x 4y5 = 2y3
Thursday 20 April 2023 Page 29
a4
c-2
6y4
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Harder Indices QuestionsHarder Indices Questions
4a6 x 8a2 = 2a9
30y2 x 4y5 = 6 x y4
Thursday 20 April 2023 Page 30
16a-1
20y3
Q1.
Q2.
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PowersPowersa x a x a =
What do you think is… (a3)2
= a3 x a3 =
Thursday 20 April 2023 Page 31
a3 Rule 3:Rule 3:When you raise raise a power by another power (separated by brackets) you multiply multiply the indices…a6
what did you do with the powers here?
multiplied them!
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Indices QuestionsIndices Questions
(a2)4 x (a3)2 =
4(a2)3 =
(4a2)3 =
(5ab3)2 =
Thursday 20 April 2023 Page 32
a14
4a6
64a6
25a2b6
Q1.
Q2.
Q3.
Q4.
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Have you Have you reallyreally understood understood indices?? indices??
We’ll see…Simplify this…
Thursday 20 April 2023 Page 33
39197
2656354
)2(
))(()(4)2(
aaa
aaa
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Expanding the Brackets
Expand these expressions:
3(a + b) =
4(y + 6) =
2(2a + 3b – 4c) =
Thursday 20 April 2023 Page 34
3a + 3b
4y + 24
4a + 6b – 8c
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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) =
Thursday 20 April 2023 Page 35
8p - 4810a + 20b
12c – 12b
21e + 14f – 28g54w + 18y – 63z
30b – 90m
-36j – 42m
20a – 15b
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Factorising
Aims:
• To remember how to expand brackets, including expressions with indices
• To learn what the process factorising is and be able to apply it to any expression
Thursday 20 April 2023 Page 36
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Factorising
Reminder of how to expand brackets:
4(2m – 5) =
What is factorising then?
Q1. 8m – 20 =
Thursday 20 April 2023 Page 37
8m - 20
4(2m- 5)
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Factorising
Q2.
25a – 30b =
Q3.
40a + 6a2 =
Thursday 20 April 2023
Page 38
5(2m- 5)
2a
(20+3a
)
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Adding Bracketed Expressions
Aims:
To be able to multiply out the brackets from expressions…
… and then collect like-terms and simplify
Thursday 20 April 2023 Page 39
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Adding Bracketed Expressions
4(a + 5b) + 3(7a – b) =
4a + 20b + 21a – 3b =
25a + 17b
Thursday 20 April 2023 Page 40
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Adding Bracketed Expressions
2(7a – 6b) + 6(3a + b) =
14a – 12b + 18a + 6b =
32a – 6b
Thursday 20 April 2023 Page 41
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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)
Thursday 20 April 2023 Page 42
18a – 26b
12y + 13w
14p – 11q
14n + 4m
22s – 19r
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Subtracting Bracketed Expressions
3(2a – 4b) – 2(a + 2b) =
Thursday 20 April 2023 Page 43
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Subtracting Bracketed Expressions
3(2a – 4b) – 2(a + 2b) =
6a – 12b – 2a – 4b =
Thursday 20 April 2023 Page 44
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Subtracting Bracketed Expressions
3(2a – 4b) – 2(a + 2b) =
6a – 12b – 2a – 4b =
4a – 16b
Thursday 20 April 2023 Page 45
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Subtracting Bracketed Expressions
3(2p + 3q) – 4(p – 2q) =
Thursday 20 April 2023 Page 46
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Subtracting Bracketed Expressions
3(2p + 3q) – 4(p – 2q) =
6p + 9q - 4p + 8q =
Thursday 20 April 2023 Page 47
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Subtracting Bracketed Expressions
3(2p + 3q) – 4(p – 2q) =
6p + 9q - 4p + 8q =
2p + 17q
Thursday 20 April 2023 Page 48
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Invisible 1
7(m – 2n) – (m – 3n) =
Thursday 20 April 2023 Page 49
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Invisible 1
7(m – 2n) – (m – 3n) =
7m – 14n – m + 3n =
Thursday 20 April 2023 Page 50
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Invisible 1
7(m – 2n) – (m – 3n) =
7m – 14n – m + 3n =
6m - 11n
Thursday 20 April 2023 Page 51
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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)
Thursday 20 April 2023 Page 52
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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)
Thursday 20 April 2023 Page 53