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

We use a MultiAnswer() answer checker that will verify that the students 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 = (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 is fully correct, that is the parametric functions are on the unit circle and the initial and final points are correct. The other three ifelse in the answer checker has either the second point, first point or both points wrong.

BEGIN_PGML
Find a parametrization of the unit circle from the point
[` \big(1,0\big) `] to [` \big(\frac{1}{2},\frac{\sqrt{3}}{2}\big) `].
Use [` t `] as the parameter for your answers.

[` x(t) = `] [__]{$multians}

[` y(t) = `] [__]{$multians}

for [__]{$multians} to [__]{$multians}.
END_PGML

Statement

Since the correct answer depends on all answer blanks, the MathObject $multians is input into all answer blanks.

Parametric Equation Answer Checker

Complete Code

POD for Macro Files

PG problem file

Explanation