Students represent letters with numbers and numbers with letters in Topic C. Students realize that nothing has changed because the properties still remain statements about numbers. They are not properties of letters, nor are they new rules introduced for the first time. Now, students can extend arithmetic properties from manipulating numbers to manipulating expressions. In particular, they develop the following identities:
a times b = b times a
a + b = b + a
g times 1 = g
g + 0 = g
g divided by 1 = g
g divided by g = 1
1 divided by g = 1/g
Students understand that a letter in an expression represents a number. When that number replaces that letter, the expression can be evaluated to one number. Similarly, they understand that a letter in an expression can represent a number. When that number is replaced by a letter, an expression is stated.
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