Why does a negative times a negative equal a positive?

The other way I've found useful for some students (though not as rigorous) is: +3 x -1 = -3 +2 x -1 = -2 +1 x -1 = -1 0 x -1 = 0 -1 x -1 = ?

I use an analogy from language. If I ask the students what it means when we say the phrase, “not bad.” They recognize that it generally means “good” and we try to think of other double negatives in language to build the connection.

In my exp, those struggling with "negative times negative" will struggle with that explanation too. I find the best way in is by associating negative numbers with debts. Three $5 debts, you have -$15 (3*-5=-15). Take away 3 $5 debts and you're $15 better off (ie, - 3*-5=15).

Multiplying by a positive increases something. Multiplying by a negative decreases something. Multiplying a negative by a negative thus decreases the negativeness of a negative, into a positive.

Suppose you pay 3 bills for 40 dollars each, 3⋅(−40)=−120 is added to your bank. The opposite of being billed would be billing someone else. So, if you bill 3 people for 40 dollars each, (−3)⋅(−40) is added to your account. This value should be positive since you get money.

Cool ! Now I can show my students if they ask !

Love the explanation

I linked to taking photos in the Olsen days. Take a photo and you get a negative of the image. Take a photo of the negative and you get a positive image.

I love and understand this, but can you please make another one that's more 7th grade friendly?