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

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! 

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! 

(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!

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! 

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. 5 7 is not the same as 6 8 !) 𝑎+𝑏 𝑐+𝑏 → 𝑎 𝑐 

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. ? ?

! 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.

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

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

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

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 1 2 + 1 3 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 .

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 = 1 1 + 1 𝑥−1 = 𝑥 𝑥−1 ? ?

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

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

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 3 2 2 = 𝑥 6 4 ?  =

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

( ) ( ) ( ) ( ) ( ) ( ) 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 ? ? ?  =

vs Head To Head Head Table 8 9 Rear Table 2 7 10 15 3 6 11 14 4 5 12 13

Question 1 1 𝑥 + 3 𝑥 Answer: 4 𝑥

Question 2 𝑥 2 + 𝑥 4 Answer: 3𝑥 4

Question 3 2 3 + 𝑥+1 9 Answer: 𝑥+7 9

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

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

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

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

Question 8 1 𝑥 ÷3 Answer: 1 3𝑥

Question 9 2÷ 1 𝑥 2 Answer: 2 𝑥 2

Question 10 1 𝑥 ÷ 1 𝑥 2 𝑦 Answer: 𝑥𝑦

Question 11 𝑥 𝑦 3 2 Answer: 𝑥 2 𝑦 6