There continues to be a sharp distinction in this curriculum between justification and proof, such as justifying the identity (𝑎 + 𝑏) 2 = 𝑎 2 + 2𝑎𝑏 + 𝑏 using area properties and proving the identity using the distributive property.
The key point is that the area of a figure is always a nonnegative quantity and so cannot be used to prove an algebraic identity where the letters can stand for negative numbers (there is no such thing as a geometric figure with negative area). This is one of many reasons that manipulatives such as Algebra Tiles need to be handled with extreme care: depictions of negative area actually teach incorrect mathematics. (A correct way to model expressions involving the subtraction of two positive quantities using an area model is depicted in the last problem of the Problem Set.)
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