Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for..
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Equivalent expressions name the same number no matter what value the variable takes. y + y + y and 3y are equivalent because whatever y is, adding it three times gives the same result as multiplying it by 3. Sixth graders learn to recognize equivalence and to justify it, which is a slightly different skill from producing equivalent forms in 6.EE.A.3.
There are two kinds of evidence. A single value can prove two expressions are not equivalent: if x = 3 makes 2(x + 4) equal 14 but makes 2x + 4 equal 10, the expressions differ. Agreement at one value proves nothing on its own, though, because different expressions can coincide at a single point (x + 2 and 2x both give 4 when x = 2). To show expressions really are equivalent, students use properties of operations, rewriting one into the other step by step, or reason about what the expressions mean, for example with a diagram.
Expressions such as 3x and x + 6 agree when x = 3. Students need at least a properties argument, or several values plus reasoning, before calling them equivalent.
Students may reject 4(x + 1) and 4x + 4 because they look different, or accept 4x + 1 because it looks similar. The value is what counts.
Equivalent means equal for every value of the variable, including fractions and zero, not just the few numbers tried.
Are 3(x + 2) - x and 2x + 6 equivalent? Are 3(x + 2) - x and 2x + 2 equivalent?
Answer: 3(x + 2) - x is equivalent to 2x + 6 but not to 2x + 2.
Use tables: list several x-values in the first column and evaluate each expression in the next columns. Equivalent expressions produce identical columns, which is a powerful visual, but always follow it with a properties argument.
Multi-select items ('choose all expressions equivalent to 6x + 9') are a common assessment format, so practice should include lists with several correct choices and several near misses.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: D) 4y
Four groups of y is 4 × y, written 4y. y⁴ means y × y × y × y, which is different.
Answer: A) 6(x + 12)
6(x + 12) = 6x + 72. The other three all expand to 6x + 12.
Answer: 12
2(1 + 5) = 12, while 2(1) + 5 = 7. Different values, so the expressions are not equivalent.
Answer: C) They are not equivalent, because x = 3 gives 9 and 7
Agreement at one value is not enough. At x = 3 they give 9 and 7, so they are not equivalent.
Answer: 36
4(8) + 4 = 32 + 4 = 36, and 4 × 9 = 36. In fact 4(a - 1) + 4 = 4a - 4 + 4 = 4a for every a.
A full lesson with slides, activities and an exit ticket on identifying equivalent expressions, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.EE.A.4 with an answer key, ready in about a minute.
Make a worksheet →Turn identifying equivalent expressions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Rewrite one into the other using properties of operations, such as the distributive property and combining like terms. Checking a few values is good evidence, but properties make it certain.
Find one value of the variable that gives different results. A single counterexample is enough.