ParametricEquationAnswers.pg

Context("Numeric")->variables->are(t => "Real");
Context()->variables->set(t => { limits => [ -5, 5 ] });

$x  = Formula("cos(t)");
$y  = Formula("sin(t)");
$t0 = Compute("0");
$t1 = Compute("pi/3");

($x0, $y0) = (1, 0);
($x1, $y1) = (1 / 2, sqrt(3) / 2);

$multians = MultiAnswer($x, $y, $t0, $t1)->with(
    singleResult => 0,
    checker      => sub {
        my ($correct, $student, $self) = @_;
        my ($xstu, $ystu, $t0stu, $t1stu) = @{$student};
        if ((($xstu**2 + $ystu**2) == 1)
            && (($xstu->eval(t => $t0stu)) == $x0)
            && (($ystu->eval(t => $t0stu)) == $y0)
            && (($xstu->eval(t => $t1stu)) == $x1)
            && (($ystu->eval(t => $t1stu)) == $y1))
        {
            return [ 1, 1, 1, 1 ];

        } elsif ((($xstu**2 + $ystu**2) == 1)
            && (($xstu->eval(t => $t0stu)) == $x0)
            && (($ystu->eval(t => $t0stu)) == $y0))
        {
            return [ 1, 1, 1, 0 ];

        } elsif ((($xstu**2 + $ystu**2) == 1)
            && (($xstu->eval(t => $t1stu)) == $x1)
            && (($ystu->eval(t => $t1stu)) == $y1))
        {
            return [ 1, 1, 0, 1 ];

        } elsif ((($xstu**2 + $ystu**2) == 1)) {
            return [ 1, 1, 0, 0 ];

        } else {
            return [ 0, 0, 0, 0 ];
        }
    }
);

Setup

A MultiAnswer is used to verify that the student answers satisfy the equation for the circle and have the required starting and ending points. This answer checker will allow students to enter any correct parametrization. For example, both x = \cos(t), y = sin(t), 0 ≤ t ≤ pi/3 and x = cos(2t), y = sin(2t), 0 ≤ t ≤ pi/6 will be marked correct.

When evaluating student’s answers, it is important not to use quotes. For example, if the code were $xstu->eval(t => "$t1stu") with quotes, then if a student enters pi the answer checker will interpret it as the string “pi” which will need to be converted to a MathObject Real and numerical error will be introduced in the conversion. The correct code to use is $xstu->eval(t => $t1stu) without quotes so that the answer is interpreted without a conversion that may introduce error.

The first if statement checks that the answer is fully correct, that is the parametric functions are on the unit circle and the initial and final points are correct. The next three elsif statements check the cases that the parametric functions are on the unit circle, but one of the second point, first point, or both points are incorrect.

Parametric Equation Answer Checker

Complete Code

POD for Macro Files

PG problem file

Explanation