Solving quadratic equations inquiry
Mathematical inquiry processes: Generate examples; analyse structure; determine necessary conditions. Conceptual field of inquiry: Expansion of brackets; factorisation of quadratic expressions.
The prompt, in presenting two equations, requires students to consider whether it is always possible to "go both ways." They will confirm that expanding the brackets is always possible, but speculate that the reverse process might not be. Questions and comments that regularly arise about the two-part prompt include:
Do numbers go in the boxes?
Can all the numbers be the same?
Can they be decimals or fractions?
What if we use a 'minus'?
Are the two sides really equal?
Are they equal whatever numbers you use?
The picture shows the initial thoughts of a year 10 class. The teacher reports that the students followed the lead of two groups by placing numbers in the boxes. The class had met the expansion of two linear expressions before and the teacher reinforces the use of a grid to show the expansion clearly. (If the class has not previously met expanding brackets, students can request teacher instruction through the regulatory cards.) The grid method has the advantage of allowing younger students to link arithmetic and algebraic multiplication. It also facilitates an analysis of the lower half of the prompt, which can be more problematic for students than the upper part. Through using the grid 'in reverse,' the class began to develop an understanding of factorisation.
Training teachers through inquiry
Nichola Sowinska, a teacher of mathematics in Peterborough (UK) devised the prompt above for a two-day course she was running at the London Metropolitan University (UK). The students were on the Subject Knowledge Enhancement programme prior to starting a teacher training course. The initial questions they posed about the prompt gave Nichola an insight into their levels of mathematical knowledge and creativity.
Nichola reports that the students assumed the equation was quadratic before realising in discussion that the power of x did not have to be two. The inquiry developed, with the use of @Desmos, into exploring quadratic equations fully and analysing the differences between graphs of equations with even and odd powers.
Jamal King, one of the students, thought the course was very well delivered: "The style of learning, being different to the traditional, really encouraged the class to engage in discussions to find solutions."
Nichola adds that the success of the inquiry in the context of teacher training was twofold:
"It opened the students' eyes to a different pedagogical approach. It also allowed them to put the formulas they remembered from school into a deeper context and understand links which they had not had the opportunity to make before."
Reflecting on the times she has used inquiry, Nichola remarked: "Every time I teach in this way I am always surprised at what my students can do."
Students find it easier to explore the prompt 'backwards' at first by expanding the bracket. Again, an understanding of solving quadratic equations by completing the square can begin to develop in the exploration of this prompt.
At the beginning of the inquiry, the teacher might restrict the parameters of the prompt by specifying that a, b and c must be (positive) integers. The prompt is false. The equation has two real roots, one repeating root or two imaginary roots.