Mathematical inquiry processes: Explore different cases; create examples; classify and reason. Conceptual field of inquiry: Solve quadratic equations; the quadratic formula; real, repeating, and complex roots.
The prompt acts as a bridge from the factorising quadratic expressions inquiry to solving quadratic equations. Students place numbers in the spaces and determine if it is possible to solve the resulting equation.
During the question, notice, and wonder phase, students who have carried out the factorisation inquiry can identify similarities and difference with the new prompt:
There is an equation instead of an expression because we can see an equals sign.
The equation is equal to zero.
If we put numbers into the left-hand side, we know that sometimes it will factorise and sometimes it won't.
If a = 1, the left-hand side factorises when there are two numbers whose sum is b and product is c.
The teacher might continue by using the example from the previous inquiry to show the class how to find the value of x by factorising:
x2 + 5x + 6 = 0 can be rewritten as (x + 2)(x + 3) = 0
Either x + 2 = 0 and x = -2 or x + 3 = 0 and x = -3
To convince students, the teacher substitutes the two solutions (or 'roots') back into the equation: (-2)2 + 5(-2) + 6 = 4 - 10 + 6 = 0 and (-3)2 + 5(-3) + 6 = 9 - 15 + 6 = 0.
After students have created their own examples, they will often ask what happens if it is not possible to factorise the equation. Are there still solutions?
The teacher then introduces the quadratic formula into the inquiry, explaining that it can be derived by rearranging ax2 + bx + c = 0 to make x the subject of the formula (although the formal derivation is best left until students have practised using the formula and also depends on the prior attainment of the class).
Students then create more examples and begin to speculate that quadratic equations have one or two roots (including whole numbers and decimals) except those that give rise to a negative square root in the formula. In the latter case, students maintain that the roots cannot be calculated.
There are sets of equations for students to solve in the slides (see below). In being made up of the same coefficients of x2 and x and the same constant but with different operations, they encourage students to identify and explain patterns.
In these sets of four equations, each set has two pairs of the same type of solution for x:
Either two equations with a repeating root and two equations with two real roots;
Or two equations with two real roots and two equations without real roots.
This leads onto the question posed in the structured inquiry for students to explore:
Is it possible to create a set of four equations of the same type so that either all the equations have two real roots or all the equations have no real roots?
The inquiry can lead into the use of the discriminant to determine the number of roots and also into comparing graphs of quadratic equations that have different types of roots (see 'Extending the inquiry' below).
The teacher can also link the roots of an equation to its coefficients and constant through the coefficients inquiry.
A quadratic equation can have two real roots, one repeating root or two complex roots. The contention in the prompt is true if the repeating root is counted as two. It is false if students count a 'solution' as a real root only.
During the line of inquiry, students classify equations depending on whether they have two, one repeating or no real roots.
Using the quadratic formula can lead them to realise the importance of the discriminant (b2 - 4ac) in determining the number of roots. An equation has:
Two real roots if b2 - 4ac > 0;
A repeating root if b2 - 4ac = 0: and
No real roots if b2 - 4ac < 0.
The teacher should not expect students to discover the discriminant through inductive reasoning. That would involve constructing three lists of equations based on the types of roots and then searching for a connection between the values of a, b, and c in each list.
Such a process would not only be time consuming but also offer little chance of a successful outcome.
Ideally, the teacher would be able co-construct a formal understanding of the discriminant based on students' observations from using the quadratic formula. Failing that, the teacher should, when necessary and appropriate, introduce the concept to the class and explain its origins in the formula.
One further line of inquiry involves graphing quadratic equations with different numbers of roots. Students study the properties of the graphs and link the number of roots to the x-intercepts (at which point y = 0). If a graph crosses the x-axis twice, for example, the equation has two real roots; if the graph does not cross the x-axis, there are no real roots.
Nichola Sowinska, a teacher of mathematics in Peterborough (UK), devised the prompt (see picture) for a two-day course she was running at London Metropolitan University. The participants on the course 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 promoted reasoning and communication: "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 Maths prompts, Nichola remarked: "Every time I teach in this way I am always surprised at what my students can do."
Andrew Blair compares the aims of Nichola's prompt to those of the solving quadratic equations prompt.
Nichola's prompt is designed to be open. She hoped to draw out and dig deeper into the participants' existing knowledge about equations.
In leaving the exponent undefined and using a variable (y) as the subject of the equation, the prompt could lead into different lines of inquiry beyond simply solving equations. Indeed, one question on the board (Can we plot this equation?) already points towards comparing the graphs of polynomials of different orders.
In comparison, the prompt on this page is closed. It is designed to focus secondary school students' attention on one specific aspect of the topic - that is, solving quadratic equations.
The exponent is, therefore, restricted at the outset of the inquiry. The equation is already equal to zero so no rearrangement is required. Both of these could change as the inquiry develops, but only once students have met the teacher's aim.
That the prompt suggests a single line of inquiry also reinforces its closed nature. Students fill in the missing digits and attempt to solve for x algebraically. Again, they might move onto finding solutions graphically after the main aim has been achieved.