Every one does algebra every day. How do I fairly share 12 sweets between 4 people? If a recipe calls for 2oo grams of flour to make a dish for 4 people how many grams will I need to make the the dish for 2 people? I’ve got £30 million left and 6 players to pick to complete my fantasy football team, how much, on average, can I spend on each player?
We solve such problems instinctively but there is something about the notation of algebra (x, y, brackets etc) that is intimidating. “x” and “y” or similar terms are just shorthand for some unknown variable. “x” could refer to “number of sweets” or “grams of flour” or “average spend per player in £”. It’s just quicker to write formulas (and set questions) using terms such as “x” and “y”.
Look at this question to see how an everyday, common sense problem can be solved using words and by summarising those words into notations used in algebra.
The Old Trafford groundsman is trying to work out how long it will take to re-turf the pitch. The pitch measures 100 meters x 60 meters (6,000m²). The groundsmen and his staff can lay 60m² of turf per hour.
a) Write this problem as a formula using the terms; “Time taken to re-turf the whole pitch”, “Total pitch size” and “Turf laid per hour”
b) Re-write this problem as a formula substituting “x” for “Time taken to re-turf the whole pitch” and the appropriate numbers for “Total pitch size” and “Turf laid per hour”.
c) Calculate the number of hours to re-turf the pitch.
a) Time taken to re-turf the whole pitch = Total pitch size/Turf laid per hour.
b) x = 6,000m²/60m²
c) 100 hours
As usual here are some more questions followed by my approach (which hopefully explains methods and techniques used) and answers.
1. Write an expression for the area of a rectangle with a length of 6x + 3 and a width of x.
Expand this expression.
2. Show that 6(x+2) + 8(x+1) ≡ 2(7x +10)
3. A regular octagon has sides of length 9x + 3 metres.
Write an expression for the perimeter of the octagon.
If the perimeter of the octagon is 240 metres, what is the value of x?
How long is each side of the octagon?
4. Factorise 12pr³ + 18p²r² + 6p²r
5. Show that (4x + 4) (2x — 5) ≡ 4(2x² — 3x — 5)
6. Show that (x + y)² — (x — y)² ≡ 4xy
Use this knowledge to calculate 69² — 51²
1. & 3. Where possible you should use brackets. Remember that W(Y +Z) is equal to W x (Y + Z) and that each element in the bracket needs to multiplied by the term outside the bracket so W(Y+Z) = W x (Y + Z) = (W x Y) + (W x Z). Expanding an impression refers to this process of multiplying out the brackets.
2. The sign “≡” means identical to. There are often questions about showing that one expression is identical to another expression. Just expand each expression step by step to demonstrate they are identical.
4. Factorising is the opposite of expanding an impression. You find common factors to place outside brackets and place the remaining terms in brackets. For example 2x² + 4x becomes 2x(x + 2). You can do this in stages:
2x² + 4x. Removing the first common factor, x gives:- x(2x + 4)
x(2x + 4) now remove the final common factor, 2 gives:- 2x(x +2)
It is best to check your answer by expanding it, in this case:
2x(x + 2) =2x² + 4x.
It can help to write an expression with the powers multiplied out to help you understand the common factors so:
12pr³ + 18p²r² + 6p²r = (12 x p x r x r x r) + (18 x p x p x r) + (6 x p x p x r)
Then write down the common factors in this case :- 6 (relates to 12, 18 & 6), p & r. These factors then go outside the bracket and the remaining terms are placed in the bracket.
5. Expand each expression. This type of question is good as you know that if you can show that the expressions are identical you have got it right!
6. When you are squaring a bracket use this method (a + b) (a + b) =
a² + ab + ba + b² = a² +2ab + b²
Remember that multiplying a negative by a negative gives a positive so (a — b) (a — b) =
a² — ab + –ab + b² = a² –2ab + b²
This is another type of question that gives instant encouragement and is satisfying to solve.
1. 3x(x +1). Expanded = 6x² + 3x
2. 6(x+2) + 8(x +1) = 6x + 12 + 8x + 8 = 14x + 20
2(7x + 10) = 14x + 20
3. Octagon has 8 sides so if each side is 9x + 3 then the perimeter is:
8(9x + 3)
If the perimeter is 240 metres then:-
8(9x + 3) = 240
9x + 3 = 30
9x = 27
x = 3
Each side = (9 x 3) + 3 = 30 metres
4. Highest common factors = 6, p and r.
Answer = 6pr(2r² + 3pr + p)
5. (4x + 4) (2x — 5) = 8x² — 20x + 8x –20 = 8x² — 12x — 20
4(2x² — 3x — 5) = 8x² — 12x — 20
6. (x + y)² = (x + y) (x + y) = x² +xy + yx + y² = x² + 2xy + y²
(x — y)² = (x — y) (x — y) = x² –xy — yx + y² = x² — 2xy + y²
Therefore (x + y)² — (x — y)² = (x² + 2xy + y²) — (x² — 2xy + y²) = 4xy
and (x + y)² — (x — y)² ≡ 4xy
69² — 51² = (60 + 9)² — (60 — 9)²
Since we know (x + y)² — (x — y)² ≡ 4xy it follows that
(60 + 9)² — (60 — 9)² = 4 x 60 x 9 = 4 x 540 = 2,160
and 69² — 51² = 2,160