getting used to algebra algebra is where you use letters to represent numbers aims: to get to grips...

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

Thursday 20 April 2023 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 + 5Thursday 20 April 2023 Page 2

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

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 + yThursday 20 April 2023 Page 5

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

w – 2

Adding and Subtracting with Letters

a + a + a =

b + b + b + b + b =

c + c – c + c =

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

Adding Expressions & Terms

2a + 4a =

6a – 5a =

7a – 4a + 10a =

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

12c5d14s8a

2e2h8r

7w4g12f

55b32j

Going Negative

3a – 5a =

5p + 4p – 12p =

-5a + 2a – 7a =

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

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 + 2gThursday 20 April 2023 Page 17

-6a - 2y

-11f + 3g

-2m – 5n

-14t + 12y

-6 – 5r

-d – 3e

-3p + 10r

10x – 5

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 =Thursday 20 April 2023 Page 18

6a – 2a2

-5h2 –h3

– 3x-2x2

7t -14t2 + 5t3

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

Multiplying Terms Together

Simplify:

2a x 4b =

10c x 2de =

3w x 4y x 5z =

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

56abpq

12abcdef

56abchpqr

125g40mnpq

10fghj24dhwyz

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

Dividing Algebraic Terms

Simplify:

8a ÷ 4 =

10ab ÷ 2a =

20pq pq =

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

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

Rule 1:Rule 1:When you multiplymultiply powers of the same letter or number you addadd the indices…

Indices QuestionsIndices Questions

a3 x a6 x a2 =

c3 x c5 x c =

2y2 x 4y5 =

3m3 x 4m5 =

Harder Indices QuestionsHarder Indices Questions

2a3b6 x a6b2 =

3c5d4 x 5c2d4 =

2ab4 x 4a2b3 =

3mnp3 x 4mn2p5 =

15c7d8

12m2n3p8

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

Rule 2:Rule 2:When you divide divide powers of the same letter or number you subtract subtract the indices…

a6 a2 =

c3 c5 =

3y2 x 4y5 = 2y3

Harder Indices QuestionsHarder Indices Questions

4a6 x 8a2 = 2a9

30y2 x 4y5 = 6 x y4

PowersPowersa x a x a =

What do you think is… (a3)2

= a3 x a3 =

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!

(a2)4 x (a3)2 =

4(a2)3 =

(4a2)3 =

(5ab3)2 =

25a2b6

Have you Have you reallyreally understood understood indices?? indices??

We’ll see…Simplify this…

2656354

))(()(4)2(

Expanding the Brackets

Expand these expressions:

3(a + b) =

4(y + 6) =

2(2a + 3b – 4c) =

3a + 3b

4y + 24

4a + 6b – 8c

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 - 4810a + 20b

12c – 12b

21e + 14f – 28g54w + 18y – 63z

30b – 90m

-36j – 42m

20a – 15b

Factorising

• 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

Factorising

Reminder of how to expand brackets:

4(2m – 5) =

What is factorising then?

Q1. 8m – 20 =

8m - 20

4(2m- 5)

Factorising

25a – 30b =

40a + 6a2 =

Thursday 20 April 2023

5(2m- 5)

(20+3a

Adding Bracketed Expressions

To be able to multiply out the brackets from expressions…

… and then collect like-terms and simplify

4(a + 5b) + 3(7a – b) =

4a + 20b + 21a – 3b =

25a + 17b

2(7a – 6b) + 6(3a + b) =

14a – 12b + 18a + 6b =

32a – 6b

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) =

6a – 12b – 2a – 4b =

3(2a – 4b) – 2(a + 2b) =

6a – 12b – 2a – 4b =

4a – 16b

3(2p + 3q) – 4(p – 2q) =

6p + 9q - 4p + 8q =

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)

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)

getting used to algebra algebra is where you use letters to represent numbers aims: to get to grips...

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