Algebra 1
Quadratics, Simultaneous Equations, Cubics, Surds Indices
Why we study this
Musicians practice scales and exercises so that playing their instrument is to some extent automated, like when driving a car. Algebra plays a similar role for mathematicians: you have to become fluent at manipulating symbolic expressions, so that doing this correctly becomes natural. Algebra is the bedrock of advanced mathematics just as arithmetic is the bedrock of elementary maths.
The process of solving real-world context problems often involves translating the problem into algebraic equations. These can then be manipulated and solved using pure mathematics, devoid of any requirement to consider the context. This is where algebra comes in.
Some learners find this easier than others - recognising patterns, applying the rules consistently, avoiding errors.
Algebra 1
Quadratics, Simultaneous Equations, Cubics, Surds Indices
Why we study this
Musicians practice scales and exercises so that playing their instrument is to some extent automated, like when driving a car. Algebra plays a similar role for mathematicians: you have to become fluent at manipulating symbolic expressions, so that doing this correctly becomes natural. Algebra is the bedrock of advanced mathematics just as arithmetic is the bedrock of elementary maths.
The process of solving real-world context problems often involves translating the problem into algebraic equations. These can then be manipulated and solved using pure mathematics, devoid of any requirement to consider the context. This is where algebra comes in.
Some learners find this easier than others - recognising patterns, applying the rules consistently, avoiding errors.
Algebra 1
Quadratics, Simultaneous Equations, Cubics, Surds Indices
Why we study this
Musicians practice scales and exercises so that playing their instrument is to some extent automated, like when driving a car. Algebra plays a similar role for mathematicians: you have to become fluent at manipulating symbolic expressions, so that doing this correctly becomes natural. Algebra is the bedrock of advanced mathematics just as arithmetic is the bedrock of elementary maths.
The process of solving real-world context problems often involves translating the problem into algebraic equations. These can then be manipulated and solved using pure mathematics, devoid of any requirement to consider the context. This is where algebra comes in.
Some learners find this easier than others - recognising patterns, applying the rules consistently, avoiding errors.
History of Algebra
Algebra has a history that spans thousands of years. It started in ancient Babylon and Egypt, where people used basic maths to trade goods and build pyramids. But the 'Father of Algebra' is usually considered to be Muhammad al-Khwarizmi, a scholar in Baghdad back in the 9th century. He wrote a book that gave us the word 'al-jabr,' which basically means 'restoring' or 'rebalancing' parts of an equation. It’s like a puzzle where you’re just trying to find the missing piece.
For a long time, mathematicians didn't even use symbols like 'x' or 'y'—they wrote everything out in long sentences! Eventually, in the 16th century, they started using letters to make things quicker and clearer. This changed everything, allowing scientists like Isaac Newton to map out gravity and the motion of the planets.
So, why does this matter to you now? Because those same rules developed by ancient scholars are the secret code behind computers. From the algorithms that rank your social media feed to the physics behind video games, algebra is the language of problem-solving. Learning it now isn't just about passing a test; it’s about training your brain to break down complex challenges in the real world into simple, solvable steps.
Algebra 1 Exam Tips
When faced with a complicated piece of algebra, I find that making a conscious effort to write clearly helps. In an exam, don't correct mistakes by writing over them - start the whole process again. Knowing how to use brackets correctly is very important. Be particularly careful with negative signs - putting brackets round negative numbers and expressions. Don't try to do too many steps at once - mistakes can creep in if you oveload your brain!
Quadratics
Mastering quadratics involves fluency in factorising, using the quadratic formula, and completing the square. Pay close attention to the discriminant to determine the nature of roots before proceeding with full solutions. Click here for a brief video on my approach to factorising [text].
Completing the square is an important technique to master as it enables you to find the turning point and sketch a quadratic function easily. It is also used in equations of circles.
Here's a quick reminder on how to complete the square [text].
When solving a quadratic inequality, the failsafe way is to sketch the curve, rather than figure it out algebraically. See here [text]
Simultaneous Equations
Cubics
Understanding the factor theorem is vital for factorising cubic polynomials. Look for integer roots first to facilitate synthetic division or long division. As with factorising quadratics, I prefer a more natural 'working backwards' approach to synthetic methods of long division - see here.[text]
But as long as you are confident in your method, that's fine. To check, multiply any linear term with a quadratic to get a cubic, then use your division method to factorise.
Surds
The key to simplifying surds to to recognise the square factor and take it out of the surd. Text. With fractions, you need to rationalise the denominator. Here's a reminder.[text]
Indices
The laws of indices must be second nature. Be especially careful with fractional and negative exponents, as these often muddled . Remember that a^(-p) = 1/a^p, whereas a^(p/q) = qth root of a^p. A common mistake also is to confuse the multiplication law with the power of a power. So a^p x a^q = a^(p + q) whereas (a^p)^q = a^(pq). A technique to use when solving index equation questions is to change each index to the same base, like here. [text]
Another common method is to write a^(2x) as a^(x^2) as in this problem. [Text]