**Algebraic Proof**

Starter

Writing algebraically

Match the algebraic expression to the description

Prove it

Add up the 8 shaded dates. How does the sum link to the centre number?

Try moving the shaded pattern around the grid. What happens?

Starter

Add up the 8 shaded dates. How does the sum link to the centre number?

It is 8 times more

Try moving the shaded pattern around the grid. What happens?

It is always 8 times more

Proof and Explanation

What do the outer 8 expressions simplify to when you add them? How does this prove the result we spotted?

Can we explain from each grid why this is happening?

Proof and Explanation

What do the outer 8 expressions simplify to when you add them? How does this prove the result we spotted?

n-8+n-7+n-6+n-1+n+1+n+6+n+7+n+8 = 8n which is 8x the middle.

Can we explain from each grid why this is happening?

The pairs opposite each other add to 2n, so with 4 pairs this makes 8n.

Prove that the top number can be predicted from the

bottom left number.

Prove it

The top number is just the first number add 2 then multiplied by 9 (or multiplied by 9 and add 18)

Proving results

A two digit number is made by reversing the digits of a different two digit number. Prove the the difference between two such numbers will always be a multiple of 9.

**L.O. - To prove results using algebra**

(Grade 8/9)

(Grade 8/9)

Proving results

A two digit number is made by reversing the digits of a different two digit number. Prove the the difference between two such numbers will always be a multiple of 9.

The two numbers are 10x + y and 10y + x.

The difference is (10x + y) - (10y + x) = 10x - x + y - 10y

= 9x - 9y = 9(x - y).

Which is a multiple

of 9.

Proving results

Prove the sum of the squares of 2 consecutive numbers is odd.

Proving results

Prove the sum of the squares of 2 consecutive numbers is odd.

n + (n+1) =

n + n + 2n + 1 =

2n + 2n + 1 =

2(n + n) + 1

Which is an odd number (of the form

2 x something + 1)

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Main Activity 2

Prove the results on the algebraic proof sheet.

Writing algebraically

Main Activity 2

1). (2n+1) - (2n - 1) = (4n + 4n + 1) - (4n - 4n + 1) = 8n

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2). (2n+1) + (2n - 1) = (4n + 4n + 1) + (4n - 4n + 1) = 8n +2 = 4(2n ) + 1

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3). (10x + y) + (10y + x) = 11x + 11y = 11(x + y)

4). (2n) + (2n - 1) = 4n + 4n + 4n + 1 = 8n + 4n + 1 = 4(2n + n) + 1

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5). n + (n+1) + (n+2) + (n+3) + (n+4) = n + n + 2n + 1 + n + 4n + 4 +

n + 6n + 9 + n + 8n + 16 = 5n + 20n + 30 = 5(n + 4n + 6)

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6). 1000x + 100y + 10x + y = 1010x + 101y = 101(10x + y)

**Key**

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