Custom PHP App with a wordpress site

Assuming you have a page with the slug "teaching-portal", this plugin adds a rewrite rule for that page that allows one part to be added after it in the URL, what you refer to as "mygetdata". I've called it "lesson".

<?php
/**
 * Plugin Name: Custom PHP app on teaching-portal page
 * Plugin URI: http://wordpress.stackexchange.com/questions/75991
 */

// Flush rewrite rules on plugin activation
register_activation_hook(__FILE__, 'wpse75991_activate');
function wpse75991_activate()
{
    wpse75991_add_rewrite_rule();
    flush_rewrite_rules();
}

// Flush rewrite rules on plugin deactivation
register_deactivation_hook(__FILE__, 'wpse75991_deactivate');
function wpse75991_deactivate()
{
    flush_rewrite_rules();
}

// Add a custom rewrite rule for the teaching portal section
add_action('init', 'wpse75991_add_rewrite_rule');
function wpse75991_add_rewrite_rule()
{
    $pagename = 'teaching-portal'; // Slug
    add_rewrite_rule(
        $pagename.'/?([^/]*+)',
        'index.php?pagename='.$pagename.'&lesson=$matches[1]',
        'top'
    );
}

// Register the custom query var so WP recognizes it
add_filter('query_vars', 'wpse75991_add_query_var');
function wpse75991_add_query_var($vars)
{
    $vars[] = 'lesson';
    return $vars;
}

With this plugin activated you should be able to access the lesson variable in your page template. Here's a quick example:

$lesson = get_query_var('lesson');

if (empty($lesson))
{
    // Show all lessons
}
else
{
    // Show a single lesson
    var_dump($lesson);
}

Also see the Rewrite API in the codex.