Volume 2, Number 3, February 2016
Group 5: mathematics
Lesson plan
Jennifer Wathall
Here is another example of a learning experience that utilises the inductive approach when
learning about the discriminant. This would take around 40 minutes to work through.
The nature of roots in quadratic equations
Look at the four examples of quadratic equations and solve them using the quadratic formula.
2
1
y = 3x – x – 1
2
y = 2x – 3x – 5
3
y = 4x – 12x + 9
4
y=x –x+1
2
2
2
What does solving the above quadratic equation tell you about the associated quadratic
functions and their graphs? What is the special name for these values?
What do you notice about what is underneath the square root sign for the four examples
above?
Find out what the name for the ‘bit’ underneath the square root is called and include an
explanation.
Sketch these four parabolas here
2
1
y = 3x – x – 1
2
y = 2x – 3x – 5
3
y = 4x – 12x + 9
4
y=x –x+1
2
2
2
Complete this table:
Value of the
_______________
b2 − 4ac > 0 and a perfect
square
b2 − 4ac > 0 and not a
perfect square
b2 − 4ac = 0
b2 − 4ac < 0
Nature of the roots
Graphical sketch
As you work through these four questions you will find that each quadratic equation given provides an
example of different types of roots:
•
rational
•
irrational
•
one repeated
•
no roots
Starting with specific, numerical examples hopefully helped you to generalise about the value of the
discriminant and the nature of the roots. One possible generalisation from the above activity could be:
‘I will understand that the expression underneath the square root sign in the quadratic formula; the
discriminant, conveys the nature of the roots of quadratic which highlight geometrical features.’
This topic links to the IB mathematics standard level guide topic 2.7:
•
Solving equations, both graphically and analytically.
•
Use of technology to solve a variety of equations, including those where there is no
appropriate analytic approach.
•
Solving ax + bx + c = 0 , a ≠ 0.
•
The quadratic formula.
•
The discriminant Δ = b − 4ac and the nature of the roots, that is, two distinct real roots, two
equal real roots, no real roots.
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2
This resource is part of IB REVIEW, a magazine written for A-level students by subject experts.
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Philip Allan Publishers © 2016
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