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

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

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

4cp

4v2b

3n0

2g

Adding Expressions & Terms

2a + 4a =

6a – 5a =

7a – 4a + 10a =


6a

a

13a

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

6e9e

2e2h8r

7w4g12f

55b32j

41p8m

0f

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 + 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

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

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 =


8ab

20cde

60wyz

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

30afh

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


10c

d

10e

4p

4h

4

4g

Dividing Algebraic Terms

Simplify:

8a ÷ 4 =

10ab ÷ 2a =

20pq pq =


2a

5b

20

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

a2

a4

a6

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 =


1

a11

c9

8y7

12m8

Harder Indices QuestionsHarder Indices Questions

2a3b6 x a6b2 =

3c5d4 x 5c2d4 =

2ab4 x 4a2b3 =

3mnp3 x 4mn2p5 =


2a9b8

15c7d8

8a3b7

12m2n3p8

Q1.

Q2.

Q3.

Q4.

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…


a6 a2 =

c3 c5 =

3y2 x 4y5 = 2y3


a4

c-2

6y4

Harder Indices QuestionsHarder Indices Questions

4a6 x 8a2 = 2a9

30y2 x 4y5 = 6 x y4


16a-1

20y3

Q1.

Q2.

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 =


a14

4a6

64a6

25a2b6

Q1.

Q2.

Q3.

Q4.

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

We’ll see…Simplify this…


39197

2656354

)2(

))(()(4)2(

aaa

aaa

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

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


Factorising

Reminder of how to expand brackets:

4(2m – 5) =

What is factorising then?

Q1. 8m – 20 =


8m - 20

4(2m- 5)

Factorising

Q2.

25a – 30b =

Q3.

40a + 6a2 =

Thursday 20 April 2023

Page 38

5(2m- 5)

2a

(20+3a

)

Adding Bracketed Expressions

Aims:

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) =



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) =



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)


Documents

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