1. Year 8: Algebraic Fractions
Dr J Frost
Last modified: 11th June 2013

2. 40 - x 3 40 3 = x + 4 = 2x + 4 2(4) = 5x - 2 2(4 – 2x) = 3x - 2
Starter
(Click your answer)
Are these algebraic steps correct?
40 - x 3 40 3
= x + 4
= 2x + 4
Fail 
Win! 
2(4) = 5x - 2
2(4 – 2x) = 3x - 2
Ask students why they think the step might have been made (in addition to pointing out why it’s incorrect!). Emphasise that when a term is contained within either a bracket, fraction or root, applying the reverse of the term to the whole expression doesn’t generally have the desired effect.
Fail  
Win!
2 =3𝑥+2
2−𝑥 =2𝑥+3 
Fail
Win! 

3. 𝑎 2 𝑏 𝑎+𝑏 𝑎𝑏 𝑏 Starter   Are these algebraic steps correct? Fail
(Click your answer)
Are these algebraic steps correct?
𝑎 2 𝑏 𝑎+𝑏
𝑎𝑏 𝑏
Fail 
Win! 

4. (x+3)2 x2 + 32 (3x)2 32x2 9x2 Starter    
(Click your answer)
Are these algebraic steps correct?
(x+3)2
x2 + 32 
Fail 
Win!
(3x)2
32x2
9x2 
Fail 
Win!

5. y2 + x s(4 + z) 2 + x s 𝑥 2 +2 =𝑦+2 1 + r (2x+1)(x – 2) pq(r+2) + 1
Starter
To cancel or not to cancel, that is the question?
(Click your answer)
y2 + x
2 + x
s(4 + z) s
𝑥 2 +2 =𝑦+2 
Fail
Win! 
Fail 
Win! 
Fail 
Win! 
(2x+1)(x – 2)
x – 2
pq(r+2) + 1 pq
1 + r 2
- 1
Fail 
Win! 
Fail 
Win! 
Fail 
Win! 

6. What did we learn? 𝑎+𝑏 𝑐+𝑏 → 𝑎 𝑐 
Bro Tip #1: You can’t add or subtract a term which is ‘trapped’ inside a bracket, fraction or root.
2 𝑎−𝑥 =2𝑥+1 → 2 𝑎 =3𝑥+1 
Bro Tip #2: In a fraction, we can only divide top and bottom by something, not add/subtract.
(e.g is not the same as 6 8 !)
𝑎+𝑏 𝑐+𝑏 → 𝑎 𝑐 

7. Adding/Subtracting Fractions
What’s our usual approach for adding fractions? ?
Sometimes we don’t need to multiply the denominators. We can find the Lowest Common Multiple of the denominators. ? ?

8. ! Adding/Subtracting Algebraic Fractions ? ?
The same principle can be applied to algebraic fractions. ! ? ?
Bro Tip: Notice that with this one, we didn’t need to times x and x2 together: x2 is a multiple of both denominators.

9. Further Examples ? ? ?
Bro Tip: Be careful with your negatives!

10. Test Your Understanding? ? ? ? ?
“To learn the secret ways of the ninja, add fractions you must.”

11. Exercise 1 ? 1 9 ? 15 ? 2 ? ? 16 ? 10 ? 3 17 ? 11 ? 4 ? ? 18 ? 12 ? 5 ? 19 ? 13 ? 6 ? 14 7 ? 20 ? ? 21 ? 8 ?

12. Harder Questions 3 𝑥 + 2 𝑥−1 = 3 𝑥−1 +2𝑥 𝑥 𝑥−1 = 5𝑥−3 𝑥 𝑥−1 ? ? ?
We can do a cross-multiplication type thing just as before.
3 𝑥 + 2 𝑥−1 = 3 𝑥−1 +2𝑥 𝑥 𝑥−1 = 5𝑥−3 𝑥 𝑥−1 ? ? ?
If were to add say, then we could use 6 as the denominator because 2×3=6 is divisible by both 2 and 3.
This gives us a clue what we could use as a denominator .

13. 𝑥+1 𝑥 − 𝑥 𝑥+1 = 𝑥+1 2 − 𝑥 2 𝑥 𝑥+1 = 2𝑥+1 𝑥 𝑥+1
Test Your Understanding
𝑥+1 𝑥 − 𝑥 𝑥+1 = 𝑥+1 2 − 𝑥 2 𝑥 𝑥+1 = 2𝑥+1 𝑥 𝑥+1 ? ?
2 𝑥 + 3 𝑥+1 = 2 𝑥+1 +3𝑥 𝑥(𝑥+1) = 5𝑥+2 𝑥(𝑥+1) ? ?
1 𝑥−1 − 3 𝑥+3 = 1 𝑥+3 −3(𝑥−1) (𝑥−1)(𝑥+3) = −2𝑥+6 (𝑥−1)(𝑥+3) ? ?
1+ 1 𝑥−1 = 𝑥−1 = 𝑥 𝑥−1 ? ?

14. Exercise 2
1+ 1 𝑧 = 𝑧+1 𝑧
1 𝑥 + 1 𝑦 = 𝑦+𝑥 𝑥𝑦
3 𝑥 + 𝑦 = 6+𝑥𝑦 2𝑥
1 𝑥+1 − 1 𝑥−1 = −2 (𝑥+1)(𝑥−1)
1 𝑥 𝑥+2 = 2𝑥+3 𝑥+1 𝑥+2
2 𝑥−4 − 1 𝑥+4 = 𝑥+12 𝑥+4 𝑥−4 ? 1
𝑦 𝑦+1 + 𝑦 𝑦− = 2 𝑦 2 𝑦 2 −1
4− 3 𝑦− = 4𝑦−7 𝑦−1
𝑥 𝑦 − 𝑦 𝑥 = 𝑥 2 +𝑥− 𝑦 2 𝑦(𝑥+1)
𝑥 𝑥−1 − 𝑥+2 𝑥 = 2 𝑥+1 𝑥−1
1 𝑥 + 1 𝑥 𝑥+2 = 3 𝑥 2 +6𝑥+2 𝑥(𝑥+1)(𝑥+2)
1 𝑥 𝑥 = 𝑥+2 𝑥+1 10 ? 7 ? 2 ? 8 3 ? 9 ? ? 4 10 ? ? 5 N1 ? 6 ? ? N2

15. Extra Practice ? ? ? ? ? ? 𝑥 2 + 𝑥+1 3 = 5𝑥+2 6 2− 𝑥 3 = 6−𝑥 3
𝑥 2 + 𝑥 = 5𝑥+2 6
2− 𝑥 = 6−𝑥 3
5 𝑥+1 − 2 𝑥 = 3𝑥−2 𝑥 𝑥+1
3 𝑥−2 − 2 𝑥 = 𝑥+4 𝑥 𝑥−2
𝑥 𝑥 = 𝑥 2 +𝑥+4 2 𝑥+1
5 𝑥𝑦 − 3 𝑥 = 5𝑥+5−3𝑥𝑦 𝑥𝑦 𝑥+1 ? 1 ? 2 3 ? ? 4 ? 5 6 ?

16. y2 2 x 3 xy2 6 z2 4 x 3 3z2 4x × =  = x+1 3 x+2 4 4(x+1) 3(x+2)  =
Multiplying and Dividing
The same rules apply as with normal fractions. y2 2 x 3
xy2 6 ? z2 4 x 3
3z2 4x ? × =  =
x+1 3
x+2 4
4(x+1)
3(x+2) ?
𝑥 = 𝑥 6 4 ?  =

17. x2 2 4 3x 2x 3 × = Test Your Understanding? ?
2𝑥+1 3 ÷ 𝑦 = 5 2𝑥+1 3 𝑦+4 × =
𝑥 2 𝑦 3 𝑧 = 𝒙 𝟔 𝒚 𝟗 𝒛 𝟏𝟓 ?

18. ( ) ( ) ( ) ( ) ( ) ( ) Exercise 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?y3 2 x y
xy2 2 ?
( ) x y2 = x2 y4 2 ? ? 1 = 7 13 × x 2y x y x2
2y2
( ) 3x y =
9x2 y2 2 ? 8 ? 14 ? 2 = × 15 9
( )
2q5 z3 =
4q10 z6 2 ? ?
x+1 x2 x y
x+1 xy ? 3 = × 16 2x y z q
2qx yz 10
( )
3x2y3
2z4 =
27x6y9
8z12 3 ? ? ? 4  = 17
x+1 y
z+1 q
q(x+1)
y(z+1) ? 11
( )
x+1 3y =
(x+1)2
9y2 2 ? ? 5  = 18 q2
y+1 x q q3
x(y+1) 12
( )
x+1 3y =
(x+1)2
9y2 2 6 ? ? ?  =

19. vs Head To Head Head Table 8 9 Rear Table 2 7 10 15 3 6 11 14 4 5 1213

20. Question 1
1 𝑥 + 3 𝑥
Answer: 4 𝑥

21. Question 2
𝑥 2 + 𝑥 4
Answer: 3𝑥 4

22. Question 3
2 3 + 𝑥+1 9
Answer: 𝑥+7 9

23. Question 4
𝑥 𝑦 + 𝑥+1 𝑦 2
Answer: 𝑥𝑦+𝑥+1 𝑦 2

24. Question 5
1 𝑥 + 𝑥 𝑦
Answer: 𝑦+ 𝑥 2 𝑥𝑦

25. Answer: 2𝑧+2 𝑧(𝑧+2) 𝑜𝑟 2𝑧+2 𝑧 2 +2𝑧
Question 6
1 𝑧 + 1 𝑧+2
Answer: 2𝑧+2 𝑧(𝑧+2) 𝑜𝑟 2𝑧+2 𝑧 2 +2𝑧

26. Question 7
1 𝑥 + 1 𝑦 + 1 𝑧
Answer: 𝑥𝑦+𝑥𝑧+𝑦𝑧 𝑥𝑦𝑧

27. Question 8
1 𝑥 ÷3
Answer: 1 3𝑥

28. Question 9
2÷ 1 𝑥 2
Answer: 2 𝑥 2

29. Question 10
1 𝑥 ÷ 1 𝑥 2 𝑦
Answer: 𝑥𝑦

30. Question 11
𝑥 𝑦
Answer: 𝑥 2 𝑦 6