Further Algebra KUS objectives BAT construct proofs Starter
Further Algebra • KUS objectives BAT construct proofs Starter: show that
Proof by exhaustion • List all the possible cases • Show the statement is true for every case WB 1 Prove that the sum of two consecutive square numbers between 100 and 200 is an odd number Possible square numbers are 121, 144, 169, 196 Cases are: 121 + 144 = 265 which is odd 144 + 169 = 265 which is odd 169 + 196 = 265 which is odd So the sum of two consecutive squares between 100 and 200 is odd QED! ‘I’ve proved it !’
Proof by deduction • Start from known facts or definitions • Use logical steps to reach the desired conclusion Prove algebraically that the sum of two odd numbers is an even number WB 2 So two odd numbers can be written as and So the sum of these two odd number is An even number (in the 2 x table) QED! ‘I’ve proved it !’
WB 3 Prove algebraically that the product of two odd numbers is an even number So two odd numbers can be written as and So the product of these two odd number is ‘I’ve proved it !’ An even number + 1 = odd number QED!
WB 4 We can show this is always even ‘I’ve proved it !’
WB 5 Prove algebraically that the difference between the squares of any two consecutive integers is always an odd number Two consecutive integers are and The squares of consecutive integers are and The difference between the squares (think of a sensible order) ‘I’ve proved it !’
Disproof by counter example • A counter example is one example that does not work for the statement. One counter example is all you need WB 6 Prove that the following statement is not true The sum of two consecutive prime numbers is always even 2 and 3 are prime 2+ 3 = 5 which is odd So the statement is not true QED! ‘I’ve proved it !’
KUS objectives BAT construct proofs self-assess One thing learned is – One thing to improve is –
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