Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
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
Two expressions can look different and still always give the same value. Sixth graders produce such equivalent expressions on purpose, using the properties of operations they have used with numbers since elementary school. The distributive property does most of the work: 3(2 + x) expands to 6 + 3x because the 3 multiplies each term inside the parentheses. Run backwards, it factors: 24x + 18y becomes 6(4x + 3y), pulling out the greatest common factor just as students did with numbers in 6.NS.B.4.
Combining like terms is the other main move. y + y + y is three groups of y, so it equals 3y; 5a + 2a = 7a because 5 of something plus 2 of the same thing is 7 of it. The commutative and associative properties let students reorder and regroup, so 4 + 2x + 6 can be rewritten as 2x + 10. Unlike terms, such as 3x and 4, cannot be combined, and noticing that is just as important as combining the ones that can.
Writing 4(x + 3) = 4x + 3 forgets to multiply the 3. Area models with two rectangles side by side show both parts must be multiplied.
Students write 3x + 5 = 8x. A quick substitution (x = 1 gives 8 on both sides, but x = 2 gives 11 and 16) shows the expressions are not equivalent.
Writing 6x + 9 = 3(2x + 9) leaves the 9 unchanged. Students should check by distributing back.
Write an equivalent expression for 3(2x + 5) + 4x with as few terms as possible, then check with x = 2.
Answer: 3(2x + 5) + 4x = 10x + 15.
Algebra tiles make both moves concrete: three copies of (2 + x) laid out show 6 unit tiles and 3 x-tiles, and like terms are simply the same shaped tiles. Encourage a substitution check after every rewrite.
Expect test items that ask which expression is equivalent to a given one, often with distractors built from the distributing and combining errors above. Factoring with the GCF is a frequent second part.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) 5x + 15
Multiply each term by 5: 5 × x = 5x and 5 × 3 = 15.
Answer: B) 6(4x + 3y)
The GCF of 24 and 18 is 6. 24x = 6 × 4x and 18y = 6 × 3y, so 24x + 18y = 6(4x + 3y).
Answer: 10a + 2 (also accepted: 10a+2, 2 + 10a)
Combine the a-terms: 7a + 3a = 10a. The constant 2 stays: 10a + 2.
Answer: 26
2(12 - 1) + 4 = 2 × 11 + 4 = 22 + 4 = 26. The simplified form 7x - 2 also gives 28 - 2 = 26.
Answer: C) Commutative property of addition
Swapping the order of terms being added uses the commutative property of addition.
A full lesson with slides, activities and an exit ticket on generating equivalent expressions, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.EE.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn generating equivalent expressions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Mainly the distributive property, plus the commutative and associative properties of addition and multiplication, to expand, factor and combine like terms.
Yes, in a simple form: pulling out a common factor, as in 24x + 18y = 6(4x + 3y). Factoring quadratics is a high school topic.