Let’s see how to use properties correctly to write equivalent expressions.
Select all the statements that are true. Be prepared to explain your reasoning.
Some students are trying to write an expression with fewer terms that is equivalent to $8-3(4-9x)$.
Noah says, “I worked the problem from left to right and ended up with $20 - 45x$.”
$$8 - 3(4-9x)$$
$$5(4 - 9x)$$
$$20 - 45x$$
Lin says, “I started inside the parentheses and ended up with $23x$.”
$$8 - 3(4-9x)$$
$$8 - 3(\text-5x)$$
$$8 + 15x$$
$$23x$$
Jada says, “I used the distributive property and ended up with $27x - 4$.”
$$8 - 3(4-9x)$$
$$8 - (12 - 27x)$$
$$8 - 12 - (\text-27x)$$
$$27x - 4$$
Andre says, “I also used the distributive property, but I ended up with $\text-4 - 27x$.”
$$8 - 3(4-9x)$$
$$8 - 12 - 27x$$
$$\text-4 - 27x$$
Jada’s neighbor said, “My age is the difference between twice my age in 4 years and twice my age 4 years ago.” How old is Jada’s neighbor?
Another neighbor said, “My age is the difference between twice my age in 5 years and and twice my age 5 years ago.” How old is this neighbor?
A third neighbor had the same claim for 17 years from now and 17 years ago, and a fourth for 21 years. Determine those neighbors’ ages.
Diego was taking a math quiz. There was a question on the quiz that had the expression $8x - 9 - 12x + 5$. Diego’s teacher told the class there was a typo and the expression was supposed to have one set of parentheses in it.
Combining like terms allows us to write expressions more simply with fewer terms. But it can sometimes be tricky with long expressions, parentheses, and negatives. It is helpful to think about some common errors that we can be aware of and try to avoid:
If we think about the meaning and properties of operations when we take steps to rewrite expressions, we can be sure we are getting equivalent expressions and are not changing their value in the process.
