Advanced problem solving with algebraic manipulation | Maths

Starter quiz

What expression represents the following: I start with a number ( $n$ ), subtract 3, and then double the result.

'2(n-3)' ✓
What expression represents the following: I take a number ( $n$ ), multiply it by 5, and then add 10 to the product.

'5n + 10' ✓
What expression represents the following: I triple a number ( $n$ ) and then subtract 4 from it.

'3n - 4' ✓
Which expression represents the following: I square a number ( $n$ ), then subtract twice the number from the square.

$n^2 - n$

$n^2 - 2n$ ✓

$n^2 - 2n^2$
Which expression represents the following: I take a number ( $n$ ), halve it, and then add the square of the number to this result.

$(\frac{n}{2})^2 + n$

$\frac{n}{2} + n^2$ ✓

$\frac{n^2}{2} + n$
A rectangle has side lengths of $3x + 5$ and $4x - 3$ . Write an expression for the perimeter of the shape.

'14x + 4' ✓

Exit quiz

The sum of two consecutive even integers is 506. What is the value of the smallest integer?

'252' ✓
I take a number, add 6, and multiply the result by 9. My new number is 900. What was the original number?

'94' ✓
The sum of three consecutive integers is 72. What is the largest integer?

'25' ✓
If the square of a positive number is decreased by 5, the result is 139. What is the number?

'12' ✓
The area of a square is $49 cm^2$ . What is the perimeter?

'28 cm ' ✓
The sum of a negative number and half its square is equal to 40. What is the number?

'-10' ✓

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Lesson Details

Key learning points

Quadratic equations can model real-life situations.
Solutions would need to be interpreted in context.
The difference of two squares is not only used in algebra, but can be used to quickly calculate numerical expressions.
Where there is a choice of method, you will decide the most efficient method to use.

Common misconception

Including solutions which are not valid in the context of the question.

Pupils should finish every problem by substituting their solutions back into the original problem. This will help them check their answers but also make them think about whether their solutions are valid in context.

Keywords

Partial product - A partial product refers to any of the multiplication results that lead up to an overall multiplication result.
Binomial - A binomial is an algebraic expression representing the sum or difference of exactly two unlike terms.
Factorise - To factorise is to express a term as the product of its factors.
Solution (equality) - A solution to an equality with one variable is a value for the variable which, when substituted, maintains the equality between the expressions.