Algebraic fractions Algebraic fractions are like normal fractions, but they contain algebraic expressions as the numerator and/or denominator. 3x 4x2 2a 3a + 2 and are two examples of algebraic fractions. The rules that apply to numerical fractions also apply to algebraic fractions. For example, if we multiply or divide the numerator and the denominator of a fraction by the same number or term, we produce an equivalent fraction. Teacher notes It is important to realize that, like numerical fractions, multiplying or dividing the numerator and the denominator of an algebraic fraction by the same number, term or expression does not change the value of the fraction. This fact is used both when simplifying algebraic fractions and when writing algebraic fractions over a common denominator to add or subtract them. 3x 4x2 3 4x 6 8x 3y 4xy 3(a + 2) 4x(a + 2) = = = =

Equivalent algebraic fractions Teacher notes It is important to realize that, like numerical fractions, multiplying or dividing the numerator and the denominator of an algebraic fraction by the same number, term or expression does not change the value of the fraction. This fact is used both when simplifying algebraic fractions and when writing algebraic fractions over a common denominator to add or subtract them.

Simplifying algebraic fractions Just like normal fractions, it is possible to simplify or cancel algebraic fractions. This is done using the same method of dividing the numerator and the denominator by common factors. 6ab 3ab2 How would you simplify the fraction ? 2 6ab 3ab2 = 6 × a × b 3 × a × b × b = 2 b

Simplifying algebraic fractions

Simplifying with brackets When dealing with brackets, it is important that all information has been dealt with before attempting any simplification. 5a2(x + y) 10a(x + y)2 How would you simplify the fraction ? 5a2(x + y) 10a(x + y)2 5 × a × a × (x + y) 10 × a × (x + y) × (x + y) = 2 a 2(x + y) =

Simplifying by factorizing Sometimes there is a need to factorize the numerator and the denominator before simplifying an algebraic fraction. 2a + a2 8 + 4a How would you simplify the fraction ? 2a + a2 8 + 4a a (2 + a) 4(2 + a) a 4 = = 6x + x2 12 + 6x How would you simplify ? 6x + x2 12 + 6x x (6 + x) 2(6 + x) x 2 = =

Spotting a pattern Algebraic fractions may appear that contain information that we know how to easily factorize. b2 – 36 3b – 18 How would you simplify the fraction ? b2 – 36 is the difference between two squares. b2 – 36 3b – 18 (b + 6)(b – 6) 3(b – 6) b + 6 3 = = If required, this can also be written as: Teacher notes Pupils should be encouraged to spot the difference between two squares whenever possible. b + 6 3 b 3 6 3 b 3 = + = + 2

Adding algebraic fractions Algebraic fractions can be added together using the same method that we use for numerical fractions. 1 a 2 b How would you complete this calculation: + ? The fractions need to be written over a common denominator before they can be added together. 1 a + 2 b = b ab + 2a = b + 2a ab Teacher notes If necessary, you may wish to review the method for adding numerical fractions. In general, the rule for adding together algebraic fractions is: a b c d ad + bc bd + =

Adding algebraic fractions 3 x y 2 How would you complete this calculation: + ? We need to write the fractions over a common denominator before we can add them. 3 x + y 2 = + 3 × 2 x × 2 y × x 2 × x + 6 2x xy = = 6 + xy 2x

Adding algebraic fractions

Addition pyramid Teacher notes Start by revealing the three fractions on the bottom row of the wall. Add the fractions together to find the missing values in the blocks above. Each block is the sum of the two fractions below it. Ask pupils if it is true to say that the fraction in the top row is the sum of the three fractions in the bottom row. Conclude that if the three fractions on the bottom row are a, b and c then the fraction on the top row is a + 2b + c. The activity can be varied by revealing one fraction in each row and using subtraction to find those that are missing.

Subtracting algebraic fractions Subtracting algebraic fractions uses the same method as is used when dealing with numerical fractions. p 3 q 2 How would you complete this calculation: – ? We need to write the fractions over a common denominator before we can subtract them. – = p 3 q 2 – = 2p 6 3q 2p – 3q 6 In general, the rule for subtracting algebraic fractions is: a b c d ad – bc bd – =

Subtracting algebraic fractions 2 + p 4 3 2q How would you complete this calculation: – ? = – 2 + p 4 3 2q – (2 + p) × 2q 4 × 2q 3 × 4 2q × 4 = – 2q(2 + p) 8q 12 6 Remember to cancel both parts of the numerator. = 2q(2 + p) – 12 8q Teacher notes The denominators in this example share a common factor. That means we will either have to cancel at the end of the calculation (as shown) or use a common denominator of 4q in the first step. 4 = q(2 + p) – 6 4q

Subtracting algebraic fractions

Manipulating algebraic fractions If two terms are added or subtracted in the numerator, the fraction can be split in two over a common denominator. 1 + 2 3 1 3 2 3 For example, = + a + b c b c a + can be written as If two terms are added or subtracted in the denominator of a fraction, we cannot split the fraction into two. Teacher notes Stress that if two terms are added or subtracted in the numerator of a fraction, we can split the fraction into two fractions written over a common denominator. The converse is also true. However, if two terms are added or subtracted in the denominator of a fraction, we cannot split the fraction into two. Verify these rules using the numerical example. 3 1 + 2 = 2 1 + For example, c a + b cannot be written as b a +

Fraction rules

Multiplying algebraic fractions Algebraic fractions can be multiplied using the same rules that are used for numerical fractions. 3p 4 2 (1 – p) How would you complete this calculation: × ? 3 3p 4 2 (1 – p) 3p × 2 4 × (1 – p) 6p 4(1 – p) = 3p 2(1 – p) × = = 2 In general, the rule for multiplying algebraic fractions is: Teacher notes Point out to pupils that in the example we could multiply out the brackets in the denominator. However, it is usually preferable to leave expressions in a factorized form. a b × = c d ac bd

Dividing algebraic fractions Algebraic fractions can be divided using the same rules that are used for numerical fractions. 2 3y – 6 4 y – 2 How would you complete the sum: ÷ ? 2 3y – 6 ÷ = 4 y – 2 2 3y – 6 × 4 y – 2 This is the reciprocal of: 4 y – 2 2 3(y – 2) × = 4 y – 2 1 6 = 2 In general, the rule for dividing algebraic fractions is: a b ÷ = c d × = ad bc

Multiplying and dividing

What answers can you get? When working with algebraic fractions, it is possible to combine them in a variety of different ways. How many different answers can you make from the three fractions below? p 6 p 4 p q Only use the operators: +, –, × and ÷ in your calculations. Teacher notes Some possible results that the students may come up with will include: Adding: (1) p/6 + p/4 = 5p/12; (2) p/6 + p/q = (pq+6p)/6q; (3) p/4 + p/q = (pq+4p)/4q; (4) p/6 + p/4 + p/q = (5pq+12p)/12q Subtracting: (1) p/4 – p/6 = p/12; (2) p/6 – p/q = (pq-6p)/6q; (3) p/4 – p/q = (pq-4p)/4q Multiplying: (1) p/6 × p/4 = p2/24; (2) p/6 × p/q = p2/6q; (3) p/4 × p/q = p2/4q; (4) p/6 × p/4 × p/q = p3/24q Dividing: (1) p/6 ÷ p/4 = 2/3; (2) p/6 ÷ p/q = q/6; (3) p/4 ÷ p/q = q/4 Students should be encouraged to combine operations, eg: p/6 × p/4 ÷ p/q = pq/24 Encourage students to substitute the given values of p and q into their calculations to come up with a numerical value for their answers. If p = 2 and q = 3, what is the value of each calculation? Can you create a calculation with an answer of 1?

Large and small Tom has two algebraic fractions he must combine to see what combinations he can achieve. p + q 4 p q When p = 3 and q = 5, what is: the largest result that can be achieved using only one operation? the smallest value possible using one operation? the largest possible value if one fraction can be used again and combined with a second operation? Teacher notes The different possible combinations are (using only one operation): (p+q)/4 × p/q = p(p+q)/4q (p+q)/4 + p/q = (2p+q)/4q (p+q)/4 – p/q = q(p + q) – 4p/4q (p+q)/4 ÷ p/q = q(p+q)/4p Again, encourage students to substitute the given values into the algebraic fractions. If you wish, you can change the values of p and q by editing the PowerPoint slide.