Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
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
Expressions such as 3(2x - 5) - (x - 4) and 5x - 11 look nothing alike, yet they give the same value for every x. They are equivalent, and seventh graders learn to move from one to the other using the properties of operations: the distributive property to expand or factor, the commutative and associative properties to reorder and regroup, and combining like terms to tidy the result.
The expressions here are linear, meaning the variable is never squared or multiplied by itself, and the coefficients can be any rational number: -2, 0.5, 3/4. Students expand -2(3a - 4) into -6a + 8, factor 6x + 15 as 3(2x + 5) by pulling out the greatest common factor, and subtract a whole expression by distributing the negative sign across every term. A powerful habit is checking equivalence by substituting a number: if two expressions give different values for x = 2, they cannot be equivalent.
Writing 4(x + 3) as 4x + 3 is very common. Drawing arrows from the 4 to every term inside the parentheses, or using an area model, makes the missing product visible.
In 7y - (2y - 3), the subtraction applies to both terms, giving 7y - 2y + 3. Students who write - 3 have only negated the first term.
Students sometimes simplify 2x + 7 to 9x. A quick check with x = 1 (2 + 7 = 9, but 9x = 9) can hide the error, so test a second value such as x = 2.
Writing 6x + 15 = 6(x + 15) undoes incorrectly. Expanding the factored form straight away shows whether it returns the original expression.
Simplify 3(2x - 5) - (x - 4). Then check your answer by substituting x = 2.
Answer: 3(2x - 5) - (x - 4) = 5x - 11.
Algebra tiles and area models are worth the time: a rectangle 3 wide and (2x - 5) long shows exactly why every term gets multiplied. When students factor, the same model runs backward, asking what common width fits both parts.
Test items often present four expressions and ask which are equivalent to a given one, sometimes selecting more than one. Teach students to simplify each option fully or to substitute two different values before deciding.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: B) 2x + 12
Expand: 4x + 12 - 2x. Combine the x terms: 4x - 2x = 2x. The result is 2x + 12.
Answer: 3(2x + 5) (also accepted: 3(2x+5), 3(5 + 2x))
The greatest common factor of 6 and 15 is 3. 6x = 3 × 2x and 15 = 3 × 5, so 6x + 15 = 3(2x + 5).
Answer: D) -6a + 8
Multiply each term by -2: -2 × 3a = -6a and -2 × -4 = +8. The result is -6a + 8.
Answer: 1.75x
Both are x terms, so add the coefficients: 0.5 + 1.25 = 1.75. The result is 1.75x.
Answer: B) 5(x - 2) and 5x - 10
Distributing gives 5 × x - 5 × 2 = 5x - 10. The other expressions forget to multiply a term or change a sign.
Answer: 5y + 3 (also accepted: 5y+3, 3 + 5y)
Distribute the negative sign: 7y - 2y + 3. Combine like terms: 5y + 3.
A full lesson with slides, activities and an exit ticket on equivalent linear expressions, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.EE.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn equivalent linear expressions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →They are the numbers multiplying the variable when those numbers are fractions, decimals or negatives, such as -3/4 in -3/4 x or 0.6 in 0.6n.
Simplify both into the same form, or substitute at least two different values for the variable. Equivalent expressions always give equal results.