1. Year 8: Algebraic Fractions Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 11 th June 2013

2. Are these algebraic steps correct? 40 - x 3  FailWin! (Click your answer) = x + 4 40 3 = 2x + 4 2(4 – 2x) = 3x - 2 2(4) = 5x - 2  FailWin!  FailWin! Starter

3. Are these algebraic steps correct?  FailWin! (Click your answer) Starter

4. Are these algebraic steps correct? (x+3) 2  FailWin! (Click your answer) x 2 + 3 2 (3x) 2  FailWin! 32x232x2 9x 2 Starter

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

6. What did we learn? Bro Tip #1: You can’t add or subtract a term which is ‘trapped’ inside a bracket, fraction or root.

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 x 2 together: x 2 is a multiple of both denominators.

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

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

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

12. Remember that we can multiply the denominators to find something that is a multiple of both. But the ‘simplest’ thing we can use is the Lowest Common Multiple. Lowest Common Multiple of 3 and 4? 12 Lowest Common Multiple of x and y? xy Lowest Common Multiple of 2x and 3x 2 ? 6x 2 Lowest Common Multiple of x and x+1? x(x+1) ? ? ? ? More Difficult Algebraic Fractions

13. ? ? Step 1: Identify the Lowest Common Multiple. Step 2: Whatever we multiplied the denominator by, we have to do the same to the numerator. ?? Example

14. ? ? Step 1: Identify the Lowest Common Multiple. Step 2: Whatever we multiplied the denominator by, we have to do the same to the numerator. ?? Example

15. Harder Questions ? ? ? We can do a cross-multiplication type thing just as before.

16. ? ?? ? Test Your Understanding ? ? ??

17. 1 2 3 4 5 7 8 9 10 11 ? ? ? ? ? ? ? ? ? ? ? 6 Exercise 2 22 ?

18. y22y22 x3x3 ×= xy 2 6 z24z24 x3x3  = 3z 2 4x x+1 3 x+2 4  = 4(x+1) 3(x+2) ?? ? Multiplying and Dividing The same rules apply as with normal fractions. ?

19. Test Your Understanding ( ) = x22x22 4 3x ×= 2x 3 x2y3z5x2y3z5 3 x 6 y 9 z 15 2x+1 3 y+4 5  = 5(2x+1) 3(y+4) ? ? ?

20. 1 2 3 4 5 7 8 9 10 11 6 y32y32 xyxy × = xy 2 2 x 2y xyxy × = x 2 2y 2 x+1 x 2 xyxy × = x+1 xy 2x y zqzq  = 2qx yz x+1 y z+1 q  = q(x+1) y(z+1) q 2 y+1 xqxq  = q 3 x(y+1) ( ) xy2xy2 = x2y4x2y4 2 2q 5 z 3 = 4q 10 z 6 2 ( ) 3x y = 9x 2 y 2 2 ( ) 3x 2 y 3 2z 4 = 27x 6 y 9 8z 12 3 ( ) x+1 3y = (x+1) 2 9y 2 2 12 ( ) x+1 3y = (x+1) 2 9y 2 2 ? ? ? ? ? ? ? ? ? ? ? ? 13 14 15 16 18 ? ? ? ? ? 17 ? Exercise 3

21. Head Table 2 3 45 6 7 89 10 11 1213 14 15 Rear Table Head To Head

22. Question 1

23. Question 2

24. Question 3

25. Question 4

26. Question 5

27. Question 6

28. Question 7

29. Question 8

30. Question 9

31. Question 10

32. Question 11