Quadratics
Introducing Quadratics – small steps
Start by looking at rectangles as arrays and rectangles made from unit squares.
Make as many rectangles as you can with 12 ones.
This draws out the concept of factors.
factor x factor = product
Linear Expressions
Distributive property
Give students a selection of x's and ones.
Show me 3x + 6
Show me 3x + 7
Which expression will factorise? Why?
factor x factor = product
3 x (x+2) = 3x + 6
Show me 5 expressions that will factorise and write them in expanded and factorised form.
Quadratic Expressions
Explore
Show me 5 other expression that will factorise.
Compare the expanded form and factorised form – what do you notice?
Make lots of rectangles and write the factorised and expanded form on the board. Only use positives at this stage. Students will begin to make generalisations.
Generalise
(x + a)(x + b) x2 + (a + b)x + ab
Compare the grid method to algebra tiles. You could also compare to FOIL and other methods for expanding brackets.
What’s the same, what’s differerent. Which method do you prefer? Why?
See base blocks to compare these methods with standard column multiplication.
Introducing negatives
At this stage students should be ready to use the grid method and do not have to think about negative area.
(x + 5)(x + 2) x² + 7x + 10
(x + 5)(x - 2) x² + 3x – 10
(x - 5)(x + 2) x² - 3x – 10
(x - 5)(x - 2) x² - 7x + 10
What’s the same and what’s different?
Keeping the numbers the same and changing the sign allows students to focus on the effect of changing the sign. Working memory is freed up allowing space for mathematical thought rather than procedural calculations.
Expand Three Linear Expressions
Expand (x + 3)(x + 4)(x + 1) using Dienes blocks (10 represents x)
Plan: (x + 4)(x + 3) = x2 + 7x + 12
Front: (x + 3)(x + 1) = x2 + 4x + 3
Side: (x + 4)(x + 1) = x2 + 5x + 4
(x + 3)(x + 4)(x + 1) ≡ x3 + 8x2 + 19x + 12
First step to factorise quadratics
Are there any more quadratic expressions with a constant term of 24?
How many quadratic expressions can you make with constant term 12?
How many quadratic expressions can you make with constant term -12?
When is the coefficient of x positive?
Generalise
x2 + ax – 12 What values of a will give you a quadratic that factorises?
x2 - bx + 12 What values of b will give you a quadratic that factorises?
How many ways can you make quadratics that factorise for each expression?
Factorise each expression.
What is special about the circled quadratic expressions?
Difference of Two Squares
Use the sliders to move a+b and a-b to the correct place on the diagram to the right.
Use the a and b sliders to change the size of square a and square b.
What two values of x sum to 0?
You may need to remind students about zero pairs.
x² – 9 x² + 0x – 9 (x – 3)(x + 3)
Generalise
(x + a)(x - b) = x2 + px - q when a > b
(x + a)(x - b) = x2 - px - q when a < b
(x + a)(x - b) = x2 - q when a = b
Compare the graph to the expanded and factorised form of the quadratic.
x^2 + x - 6 (x + 3)(x - 2)
Completing the Square
In both sets of questions, start with the perfect square and then add on or take away 1’s. This is an example of variation (change one thing) or intelligent practice. The practice leads to a generalisation.