GCSE-Surds.pptx for IGCSE grade 9 and 10

IGCSE surds grade 9

1. GCSE: Surds

2. Types of numbers Real Numbers Real numbers are any possible decimal or whole number. Rational Numbers Irrational Numbers are all numbers which can be expressed as some fraction involving integers (whole numbers), e.g. , , -7 .   are real numbers which are not rational. !

3. Types of numbers Activity : Copy out the Venn diagram, and put the following numbers into the correct set. 3 0.7 π . 1.3 √ 2 -1 3 4 √ 9 e Edwin’s exact height (in m) Integers Rational numbers Real numbers (Click the blue boxes above)

4. What is a surd? Vote on whether you think the following are surds or not surds. Therefore, can you think of a suitable definition for a surd? A surd is a root of a number that cannot be simplified to a rational number.    Not a surd Surd    Not a surd Surd    Not a surd Surd    Not a surd Surd ?      Not a surd Surd        

5. Laws of Surds     ? ? The only two things you need to know this topic… Basic Examples:     ? ?   ?

6. Simplifying Surds   ? Could we somehow use to break the 8 up in a way that one of the surds will simplify?   Bro Tip : Find the largest square factor of the number, and put that first . ?         ? ? ? ?

7. Test Your Understanding           ? ? ? ? ?

8. Multiplying Surds   Bro Tip : Be very careful in observing whether both of the terms are surds or just one is. ? ? ? ? ?   Bro Tip : Just multiply the non- surdey things first, then the surdey things. ?   ?   ?

9. Test Your Understanding   ? ? ? ? ? ? ? ?

10. Exercise 1 Simplify the following: Simplify the following: Simplify the following:   Simplify the following: Express the following as a single square root (hint: do the steps of simplification backwards!) Express the following as a single square root:   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 a b c d e a b c d e a b c d e f g a b c d a b c d a b

11. Adding Surds   Think of it as “if I have one lot of and I add another lot of , I have two lots of ”. It’s just how we collect like terms in algebra, e.g.   ?   ? ? ? ?

12. Test Your Understanding   ? ? ? ?

13. Brackets and Surds         ? ? ? ?

14. Test Your Understanding         ? ? ? ? ?

15. Exercise 2 Simplify the following: Expand and simplify the following, leaving your answers in the form   Expand and simplify: Determine the area of :                               Find the length of . (Using Pythagoras)   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 a b c d e f 2 a b c d e 3 a b c d e f 4 a b c 5

16. Here’s a surd. What could we multiply it by such that it’s no longer an irrational number? Rationalising The Denominator   ? ? In this fraction, the denominator is irrational. ‘ Rationalising the denominator ’ means making the denominator a rational number. What could we multiply this fraction by to both rationalise the denominator, but leave the value of the fraction unchanged?   ? ? Bro Side Note : There’s two reasons why we might want to do this: For aesthetic reasons, it makes more sense to say “half of root 2” rather than “one root two- th of 1”. It’s nice to divide by something whole! It makes it easier for us to add expressions involving surds.

17.   More Examples ? ? ? ? ?   ? ? ? Test Your Understanding:

18. Further Maths ! :: More Complex Denominators You’ve seen ‘rationalising a denominator’, the idea being that we don’t like to divide things by an irrational number. But what do we multiply the top and bottom by if we have a more complicated denominator?   ? ? We basically do the same but with the sign reversed (this is known as the ‘conjugate’). That way, we obtain the difference of two squares. Since , any surds will be squared and thus we’ll end up with no surds in the denominator.  

19. More Examples   You can explicitly expand out in the denominator, but remember that so we get Just remember: ‘difference of two squares’!   ?   ? ? ?   ? ? ?

20. Test Your Understanding AQA FM June 2013 Paper 1 Solve Give your answer in the form where and are integers.     Rationalise the denominator and simplify     ? ? Rationalise the denominator and simplify     ?

21. Exercise 3 Rationalise the denominator and simplify the following:   Expand and simplify: Rationalise the denominator, giving your answer in the form . Solve giving your answer in the form . Solve Simplify:   ? ? ? ? ? ? ? ? ? 1 2 3 4 5 a b c d e ? 6

22. A final super hard puzzle Solve   N But   ?