Teaching quadratic functions (re-engineered)

My book Elegance with Substance (EWS) (2009, 2015) (pdf online) has the theme that mathematics education requires fundamental re-engineering. Mathematicians are trained to think abstractly and are not trained for the empirical science of didactics. When they meet real life students in class, mathematicians suffer from cognitive dissonance, and resolve this by sticking to traditional ways of teaching, which tradition has not been designed for optimal didactics. In this manner, mathematics education is in shambles for some 5000 years.

EWS documented this with a long list of examples and hence I should not be surprised anymore to find another example. However, I was very surprised to discover the following for the quadratic function. This function is so familiar for at least some 3000 years, and you would expect that mathematicians had optimized its teaching. However, be amazed as well.

A starting point for all approaches

The quadratic function is introduced in all cases by presenting the square of x. The formula is supported by text (explaining what the function does), table (first row for effect y, second row for cause x, namely allow for Δy / Δx), and graph. Write functions with square brackets.

f[x] = x2

Properties are:

  • The vertex or turning point can be found at the origin {0, 0}.
  • There is symmetry for left and right along the line x = 0 because f[x] = f[-x].
  • There are no outcomes f[x] < 0, unless by use of the complex plane with = i2 = -1.
  • Also f[x] and –f[x] mirror, but one would say that this requires a coefficient a = -1.
Continue re-engineered: a focus on understanding

Changing and moving (transforming and translating) the function gives the basic form.

f[x] = a (x h)2 + v

  • The vertex or turning point can be found at {h, v}.
  • Coefficient a stretches or squeezes, while a negative value flips or mirrors upside down. Making coefficient a negative turns a convex form into a concave form. (Seen from the origin, concave is hollow or h-shaped, and convex is bulging or b-shaped.)
  • The value h shifts the parabola horizontally. For example, x must have a higher value if some positive h is substracted from it. If h is negative, then the graph moves leftwards.
  • The line x = h is also the mirror-axis for left and right.
  • The value v shifts the parabola vertically. If v is negative, then it shifts downwards.

A key question for a parabola is where it intersects with the horizontal axis. Even when we want to find the points of intersection with a line y = d, then this only means an adjustment to v’ = v d. Finding the solution to this quadratic equation is straightforward.

f[x] = a (x h)2 + v = 0

(x h)2 = – v / a

x1,2 = h ± √(- v / a)

It is easy to check that all solutions must have this symmetric form, since substitution back first eliminates h and then squares the remainder, and reduces to zero. This solution thus also gives the rule:

  • When v = 0 then there is a single solution, or it is touching the horizontal axis.
  • When – v / a > 0 then there are two solutions.
  • When – v / a < 0 then there is no intersection with the horizontal axis. A solution can be found in the complex plane however. There is no need to be squirmish about complex numbers because these would have been discussed before, when discussing the system of co-ordinates (see here).

These graphs show the solutions for f[x] = 1 (x – 0)2 + v, for values v = -2, 0, 2. The relevant intersection value is √2 ≈ 1.414…. The complex solution can be found by flipping the function, giving g[x] = -1 (x – 0)2 + 2, solve for the intersections, and circle these a quarter turn. Here x1,2 = ± i √2.

parabola

Given the (two) solution(s) x1,2 = h ± √(- v / a), let us write for ease or w, and observe that the function can be written in the factored form: f[x] = a (x – u)(x – w). The vertex or turning point lies in the middle, so that = (u + w) / 2. Substitution of x = h gives v =  – a (u – w)2 / 4.

The basic form can be expanded into the standard (polynomial) form.

f[x] = a (x h)2 + v

= a (x2 – 2 h x + h2) + v

= a x2 – 2 a h x + a h2 + v

f[x] = a x2 + b x + c in which b = – 2 a h and c = a h2 + v

Thus, conversely, whenever one meets the form f[x] = a x2 + b x + c then h and v can be solved as:

h = – b / (2 a)

v = c – a h2 = – (b2 – 4 a c) / (4 a)

Acknowledgements.

  • After re-engineering the above, I knew what to look for, and found this video by MIT graduate Nancy (mathbff) who calls the basic form the “vertex form” but who still has it in 2nd place instead of 1st place. There is also this lesson plan at MARS that mentions the three forms, but their order is standard, factored, basic, and the latter is called “completed square form”. I would rather put basic first and express this in its name. The lesson is what you can directly recognise in each form, which is a fair question, but not the most relevant first question on the graph of the quadratic function.
  • The complex graph comes from Norton & Lotto 1984, reproduced by Francis Su et al. here. Su et al. don’t use the basic form whence it is difficult to see how to flip the function and why complex solutions have the real part h and the imaginary part √(- v / a). The complex solution is also mentioned by wikipedia here, but this doesn’t flip the graph and doesn’t explicitly make the quarter turn, so the solution remains mysterious.
The shamble tradition: starting from theory

Mathematicians have developed a theory of polynomials. The traditional approach in teaching the quadratic function is to treat it as a special case of such polynomials.

Thus students are presented with the polynomial format, called the standard form.

f[x] = a x2 + b x + c

The standard form is intransparant. Teaching thus proceeds by presenting tricks, only to recover what is already obvious for the basic form.

(1) Students are presented with the quadratic formula and must learn it by heart. This is the fail-safe approach for students without much interest in or understanding of mathematics.

x1,2 = (-b ± √D) / (2 a) with discriminant D = b2 – 4 a c.

The possible solutions are cataloged with D < 0, D = 0 and D > 0.

  • It is not clear where D comes from and what it means. (It is only a freak result of the polynomial form.)
  • The traditional approach takes so much time in general that there often hasn’t been time to explain about the complex plane, and thus it is often said that D < 0 has no solution rather than that it has a complex solution.

(2) More interested or advanced students can be shown where the quadratic formula comes from. This generates the ritual called derivation of the quadratic formula. There are a number of approaches (wikipedia), but, “surprisingly” the following approach may not be mentioned (not in wikipedia today), and the reason must be that the tradition has lost track of the basic form.

a x2 + b x + c = a (x h)2 + v

For x = 0: c = a h2 + v hence v = c – a h2

For x ≠ 0: a x2 + b x + c = a x2 – 2 a h x + a h2 + v

b x = – 2 a h x and hence h = – b / (2 a) and substitute this in v again

Subsequently resort to above disucssion, and find x1,2 = h ± √(- v / a)

Didactic conclusions

Didactic conclusions are:

  • The basic form is transparant, provides a clear path to the standard (polynomial) form, and also provides clarity for the return path from the standard form back to the basic form.
  • Starting with the standard form is convoluted. It starts from some distant theory about polynomials in general and creates clutter and mystery about solutions. This tradition has lost track of the basic form and thus also requires more intricate solution methods or reliance on memory for the “quadratic formula”.
  • Making math opaque for such simple issues puts a heavy burden on more intricate issues.
  • Making math opaque for such simple issues also causes a flight into alternative approaches, such as reliance on applications (“realistic mathematics education” (RME)). Interesting applications are here, but my impression is that students will study those applications with more interest once they have mastered the basic form first. The general format for teaching is: (a) psychologically prime notions with a basic example (here the square of x), (b) develop the theory, (c) apply the theory.

PM. Wikipedia is not a didactic environment but a portal. When you want to know what the “quadratic formula” is, then wikipedia presents it, and wikipedia doesn’t give a didactic presentation of the underlying issue. Wikipedia has other entries like on the quadratic equation, but following these links doesn’t generate a didactic exposition. Thus beware: mathematics in wikipedia is created by math students who copy their textbooks.

Examples: two English websites and a Dutch textbook for grade 9 (age 14-15)

A traditional presentation of the quadratic function that “completes the square” without giving the general form is here. Another website is here, and it gives the basic (“vertex”) form somewhere at the end.

A Dutch textbook for grade 9 at subtop-level (“Moderne wiskunde”, HAVO 3ab, together 400 pages, edition 8, 2005) includes the following approach to the quadratic function.

  • Ch 3 (some 25 pages): Transformation and translation of functions (linear, quadratic, hyperbolic, powers and roots). It makes sense to review the different functions. However, for the quadratic function only a and c are manipulated. Given above discussion, it would make more sense to discuss each function separately with its transformations and translations.
  • Ch 5 (some 25 pages): Factoring polynomials. This is a basic algebraic skill. It is enlightened in the textbook with graphs of rectangles and indeed a parabola. When the treatment of the parabola is as convoluted as it is now, then this order can be understood, for the solution of a quadratic equation can be found by factoring. A more didactic approach however would be to (a) discuss factors using rectangles, (b) discuss the parabola in the re-engineered fashion, and (c) only later practice also on the skill of factoring. The advantage of the re-engineered approach is that we can have text, formula, table and graph in a single review at a much earlier stage, which enhances understanding.
  • Ch 8 (some 30 pages): Quadratic formulas. The traditional f[x] = a x2 + b x + c with vertex and symmetry, quadratic formula, the use of D, graphical form, and solution of quadratic equations. In the re-engineered approach, this would be integrated in the earlier chapters 3 and 5.
  • Ch 9 (some 25 pages): Manipulating more functions: Addition and multiplication of graphs. Periodic formulas. This deepens earlier notions and provides more practice. This might be less needed in the re-engineered approach when issues are transparant.
  • Ch 11ab (some 50 pages): Graphs and equations. Intersection of graphs and finding solutions. However, finding the intersection of parabola a x2 + b x + c and line b’ x + c’ reduces to a x2 + b x + c = b’ x + c’ or solving a x2 + (b – b’) x + (c – c’) = 0, which is just solving for another parabola, which has already been discussed. This split chapter only practices earlier notions, which might be less needed when the presentation was transparent enough.

The conclusion is that this textbook provides key ingredients, but in less didactic format and order, such that the textbook resorts to a lot of practice to allow students to replace insight by routine. (Routine should not be called skill when insight is lacking.)

Conclusion

There is scope for re-engineering the didactics of quadratic functions into a much more transparant manner. The focus must be on what is essential to understand (at this level) and on starting from some abstract general theory (relevant for understanding issues at another level).

This re-engineered approach can likely already be used in elementary school. See the other book A child wants nice and no mean numbers (CWNN) (2015) (pdf online).

It remains to be tested whether the re-engineered approach indeed is as transparant as suggested here, and whether pupils would have a fast road to insight and skill and improved attitude. It always are the students themselves who show what works for them.

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