Hmm mura'g layo ra gituyukan ang topic .. actually ok ra imo first code, all it needed are 2 small fixes. To demonstrate a point i'll explain later, i changed the function name to ADD (take note, all CAPS)
Code:
1. function ADD()
2. {
3. x=calculator.input1.value;
4. y=calculator.input2.value;
5. sum= x + y;
6. if (document.getElementById("Add").checked==true)
7. document.getElementById("output").innerHTML= sum;
8. else
9. document.getElementById("output").innerHTML= " ";
10. }
11.
12. ...
13. <input type=checkbox Id="Add" onClick="ADD()">
14. ...
</script>
The fixes are:
line 1 - Add -> ADD
line 6 - True -> true
line 13 - Add -> ADD
1) sa javascript, lahi ang ADD sa Add kay cAsE-sEnSeTiVe man ni nga language. what you had in your first code was a clash in identifier names. this happens kay sa javascript everything are first-class objects (pero lahi na na nga topic)
2) True isn't valid .. any idea why? 
Of course in real programming "ok ra" is a bad mindset. You have to be defensive and give careful attention to input verification. That's why parseFloat and isNaN came up in the discussion. Ang point lang nako we almost missed learning subtle points from your first code and the fact that it could have "worked" kay natabunan na sa ubang concerns 
so anyway, this is how i would code this exercise-
Code:
<html>
<head>
<script type="text/javascript">
window.onload = function() {
document.getElementById('Add').onclick = function () {
x = parseFloat(calculator.input1.value);
y = parseFloat(calculator.input2.value);
if (!x || !y) {
alert('Gai sad ta numbers iadd oi.');
return;
}
sum = x + y;
// ok ra to imoha gamit ang "if" kung maoy gusto ni prof
document.getElementById('output').innerHTML = this.checked ? sum : '';
}
}
</script>
</head>
<body>
<form name=calculator>
<input type=checkbox Id="Add">
<input type=text name=input1>
<input type=text name=input2>
<div id=output> </div>
</form>
</body>
</html>