TimeKeeper/questionGenerators/math/fractions.php

<?php

/**
 * Fractions ("fractions"): simplify, add, subtract, multiply, divide, mixed
 * numbers and "a fraction of a number".
 *
 * Built on the Fraction class in fraction/src (it does the arithmetic and
 * puts every answer in its simplest form). The questions available grow with
 * the student's level (accuracy this month, scaled up over their first 30
 * answers):
 *
 *   from the start  simplify, add/subtract with the same denominator,
 *                   a fraction of a number
 *   level 20+       improper fraction to mixed number, add/subtract where one
 *                   denominator is a multiple of the other
 *   level 45+       add/subtract with different denominators, multiply
 *   level 70+       divide
 *
 * The expected answer is stored server-side (TkAnswerStore); the checker
 * (answerChecker2::checkFractions) accepts any correct form but wants the
 * answer in its simplest form.
 */
require_once dirname(__DIR__, 2) . '/fraction/src/Fraction.php';

class fractions {

    const SUBJECT = 'fractions';

    private $user;
    private $level = 0; // 0-100: how hard to make the questions

    /**
     * @param string $user User to supply the question for.
     */
    public function __construct($user) {
        $this->user = $user;
    }

    /**
     * Make a question, show it, and remember it as the student's open question.
     */
    public function getFractions() {
        $problem = $this->getProblem();
        $questionHtml = $this->htmlify($problem);

        $db = e107::getDb();
        $db->delete('tkLastQuestion', "user = '" . $db->escape($this->user) . "'");
        $db->insert('tkLastQuestion', ['user' => $this->user, 'LastQuestionHtml' => base64_encode($questionHtml)]);

        e107::getRender()->tablerender('Fractions', $questionHtml);
    }

    /**
     * Make a question without showing it.
     *
     * @return array ['answer1' => string, 'question' => string, 'type' => 'fractions',
     *                'format' => 'simplest'|'mixed', 'instruction' => string]
     */
    public function getProblem() {
        $this->level = $this->difficulty();

        $kinds = ['simplify', 'sameDenominators', 'fractionOf'];
        if ($this->level >= 20) {
            $kinds[] = 'mixedNumber';
            $kinds[] = 'oneDenominatorMultiple';
        }
        if ($this->level >= 45) {
            $kinds[] = 'unlikeDenominators';
            $kinds[] = 'multiply';
        }
        if ($this->level >= 70) {
            $kinds[] = 'divide';
        }

        $kind = $kinds[array_rand($kinds)];
        return $this->$kind();
    }

    /**
     * How hard to make the questions, 0-100: last month's accuracy, scaled up
     * over the first 30 answers.
     *
     * @return float
     */
    private function difficulty() {
        $db = e107::getDb();
        $stats = $db->retrieve(
            "SELECT (SUM(correct)/COUNT(correct)*100) AS percent, COUNT(correct) AS count
             FROM `" . MPREFIX . "tkReport`
             WHERE answereddatestamp >= UNIX_TIMESTAMP(NOW() - INTERVAL 1 MONTH)
             AND username = '" . $db->escape($this->user) . "' AND subject = '" . self::SUBJECT . "'"
        );

        if (!is_array($stats) || (int) $stats['count'] === 0) {
            return 0.0;
        }

        $sample = min(1.0, ((int) $stats['count']) / 30);
        return max(0.0, min(100.0, (float) $stats['percent'])) * $sample;
    }

    /**
     * A random whole number from $min up to $maxAtFull. At level 0 the top of
     * the range is a quarter of the way there; at level 100 it is all of it.
     */
    private function num($min, $maxAtFull) {
        $span = max(0, $maxAtFull - $min);
        $max = $min + (int) ceil($span * (0.25 + 0.75 * $this->level / 100));
        return mt_rand($min, max($min, $max));
    }

    private function gcd($a, $b) {
        $a = abs($a);
        $b = abs($b);
        while ($b > 0) {
            $t = $a % $b;
            $a = $b;
            $b = $t;
        }
        return $a;
    }

    /**
     * A numerator from 1 to $d - 1 that shares no factor with $d.
     */
    private function coprimeNumerator($d) {
        for ($i = 0; $i < 50; $i++) {
            $n = mt_rand(1, $d - 1);
            if ($this->gcd($n, $d) === 1) {
                return $n;
            }
        }
        return 1;
    }

    private function problem($question, Fraction $answer, $instruction, $format = 'simplest') {
        return [
            'answer1' => (string) $answer,
            'question' => $question,
            'type' => self::SUBJECT,
            'format' => $format,
            'instruction' => $instruction,
        ];
    }

    // ---- the questions

    /** Simplify 6/8 -> 3/4 */
    private function simplify() {
        $d = $this->num(3, 12);
        $n = $this->coprimeNumerator($d);
        $k = $this->num(2, 6);

        return $this->problem(
            'Simplify ' . ($n * $k) . '/' . ($d * $k),
            new Fraction($n, $d),
            'Write this fraction in its simplest form.'
        );
    }

    /** 3/8 + 1/8, 5/9 - 2/9 */
    private function sameDenominators() {
        $d = $this->num(4, 12);

        if (mt_rand(0, 1) === 1) {
            if ($this->level >= 20) {
                $a = mt_rand(1, $d - 1);
                $b = mt_rand(1, $d - 1);          // the total can be more than 1
            } else {
                $a = mt_rand(1, $d - 2);
                $b = mt_rand(1, $d - 1 - $a);     // the total stays under 1
            }
            $answer = (new Fraction($a, $d))->add(new Fraction($b, $d));
            $question = $a . '/' . $d . ' + ' . $b . '/' . $d;
        } else {
            $a = mt_rand(2, $d - 1);
            $b = mt_rand(1, $a - 1);
            $answer = (new Fraction($a, $d))->subtract(new Fraction($b, $d));
            $question = $a . '/' . $d . ' − ' . $b . '/' . $d;
        }

        return $this->problem($question, $answer, 'Work it out and write your answer in its simplest form.');
    }

    /** 3/4 of 20 -> 15 */
    private function fractionOf() {
        $d = $this->num(2, 10);
        $n = mt_rand(1, $d - 1);
        $k = $this->num(2, 12);

        return $this->problem(
            $n . '/' . $d . ' of ' . ($d * $k),
            new Fraction($n * $k),
            'Work it out. Your answer is a whole number.'
        );
    }

    /** 17/5 as a mixed number -> 3 2/5 */
    private function mixedNumber() {
        $d = $this->num(2, 10);
        $r = $this->coprimeNumerator($d);
        $whole = $this->num(1, 6);
        $n = $whole * $d + $r;

        return $this->problem(
            $n . '/' . $d . ' as a mixed number',
            new Fraction($n, $d),
            'Write this improper fraction as a mixed number, like 2 1/3.',
            'mixed'
        );
    }

    /** 1/2 + 3/8: one denominator is a multiple of the other */
    private function oneDenominatorMultiple() {
        for ($i = 0; $i < 20; $i++) {
            $d1 = $this->num(2, 6);
            $d2 = $d1 * $this->num(2, 4);
            $first = [mt_rand(1, $d1 - 1), $d1];
            $second = [mt_rand(1, $d2 - 1), $d2];
            if (mt_rand(0, 1) === 1) {
                list($first, $second) = [$second, $first];
            }

            $problem = $this->addOrSubtract($first, $second);
            if ($problem !== null) {
                return $problem;
            }
        }
        return $this->addOrSubtract($first, $second, true);
    }

    /** 2/3 + 1/4: different denominators, neither a multiple of the other */
    private function unlikeDenominators() {
        for ($i = 0; $i < 20; $i++) {
            $d1 = $this->num(2, 8);
            $d2 = $this->num(2, 8);
            if ($d1 === $d2 || $d1 % $d2 === 0 || $d2 % $d1 === 0) {
                continue;
            }
            $first = [mt_rand(1, $d1 - 1), $d1];
            $second = [mt_rand(1, $d2 - 1), $d2];

            $problem = $this->addOrSubtract($first, $second);
            if ($problem !== null) {
                return $problem;
            }
        }
        return $this->addOrSubtract([1, 3], [1, 4], true);
    }

    /** 2/3 × 3/4 */
    private function multiply() {
        $b = $this->num(2, 9);
        $d = $this->num(2, 9);
        $a = mt_rand(1, $b - 1);
        $c = mt_rand(1, $d - 1);

        return $this->problem(
            $a . '/' . $b . ' × ' . $c . '/' . $d,
            (new Fraction($a, $b))->multiply(new Fraction($c, $d)),
            'Work it out and write your answer in its simplest form.'
        );
    }

    /** 1/2 ÷ 1/4 */
    private function divide() {
        $b = $this->num(2, 9);
        $d = $this->num(2, 9);
        $a = mt_rand(1, $b - 1);
        $c = mt_rand(1, $d - 1);

        return $this->problem(
            $a . '/' . $b . ' ÷ ' . $c . '/' . $d,
            (new Fraction($a, $b))->divide(new Fraction($c, $d)),
            'Work it out and write your answer in its simplest form.'
        );
    }

    /**
     * Add or subtract two fractions, each given as [numerator, denominator].
     * For a subtraction the larger fraction goes first so the answer is above
     * zero; two equal fractions give null (nothing to subtract).
     *
     * @return array|null
     */
    private function addOrSubtract(array $first, array $second, $forceAdd = false) {
        $add = $forceAdd || (mt_rand(0, 1) === 1);

        if (!$add) {
            $left = $first[0] * $second[1];
            $right = $second[0] * $first[1];
            if ($left === $right) {
                return null;
            }
            if ($left < $right) {
                list($first, $second) = [$second, $first];
            }
        }

        $one = new Fraction($first[0], $first[1]);
        $two = new Fraction($second[0], $second[1]);

        if ($add) {
            $answer = $one->add($two);
            $operator = ' + ';
        } else {
            $answer = $one->subtract($two);
            $operator = ' − ';
        }

        return $this->problem(
            $first[0] . '/' . $first[1] . $operator . $second[0] . '/' . $second[1],
            $answer,
            'Work it out and write your answer in its simplest form.'
        );
    }

    private function htmlify($problem) {
        // Expected answer stays on the server; the form only carries a token.
        $token = TkAnswerStore::issue($this->user, $problem);

        $frm = e107::getForm();

        $fString = $frm->open('questionForm', 'post', '', ['autocomplete' => 'off', 'class' => 'formclass']);
        $fString .= $frm->label('<h4>Fractions</h4>') . "</br>\n";
        $fString .= $frm->label(htmlspecialchars($problem['instruction'])) . "\n";
        $fString .= '</br>';
        $fString .= '<p style="font-size:140%;font-weight:bold;">' . htmlspecialchars($problem['question']) . '</p>';
        $fString .= '<p class="smalltext">Type fractions like 3/4 and mixed numbers like 2 1/3.</p>';
        $fString .= $frm->text('a1', '', 20, ['placeholder' => 'Write answer here!', 'autocomplete' => "off", 'spellcheck' => "false"]) . "\n";
        $fString .= $frm->hidden('checkerData', $token, ['id' => 'checkerData']);
        $fString .= $frm->hidden('subject', self::SUBJECT, ['id' => 'subject']);
        $fString .= '</br>' . "\n";
        $fString .= $frm->button('Answer', 'Check answer') . "\n";
        $fString .= $frm->close();
        $fString .= '</br></br>';

        return $fString;
    }
}