Middle School Math Quiz: Generate Equivalent Expressions
20 questions · exam conditions
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Generate Equivalent ExpressionsQuestion 1 of 20

Which expression is NOT equivalent to 12x+18y12x + 18y?

6(2x+3y)6(2x + 3y) when factoring out the greatest common factor of the coefficients
3(4x+6y)3(4x + 6y) when factoring out a common factor from both terms
2(6x+8y)2(6x + 8y) when factoring out 2 from both terms in the expression
18y+12x18y + 12x when using the commutative property of addition to rearrange terms
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Middle School Math Quiz

Middle School Math Quiz: Generate Equivalent Expressions

Practice Generate Equivalent Expressions in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Generate Equivalent Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which expression is NOT equivalent to 12x+18y12x + 18y?

  1. 6(2x+3y)6(2x + 3y) when factoring out the greatest common factor of the coefficients
  2. 3(4x+6y)3(4x + 6y) when factoring out a common factor from both terms
  3. 2(6x+8y)2(6x + 8y) when factoring out 2 from both terms in the expression (correct answer)
  4. 18y+12x18y + 12x when using the commutative property of addition to rearrange terms
Explanation: Let's check each option: A) 6(2x+3y)=12x+18y6(2x + 3y) = 12x + 18y ✓. B) 3(4x+6y)=12x+18y3(4x + 6y) = 12x + 18y ✓. C) 2(6x+8y)=12x+16y≠12x+18y2(6x + 8y) = 12x + 16y ≠ 12x + 18y ✗. D) 18y+12x=12x+18y18y + 12x = 12x + 18y ✓. Choice C is incorrect because 2×9=182 × 9 = 18, not 2×8=162 × 8 = 16.

Question 2

Consider the expressions 8a+12b8a + 12b and 4(2a+3b)4(2a + 3b). A student claims these are equivalent and wants to verify by substituting a=2a = 2 and b=1b = 1. What values should the student get for both expressions if they are truly equivalent?

  1. Both expressions should equal 20 when the given values are substituted correctly
  2. Both expressions should equal 24 when the given values are substituted correctly
  3. Both expressions should equal 32 when the given values are substituted correctly
  4. Both expressions should equal 28 when the given values are substituted correctly (correct answer)
Explanation: When you encounter algebraic expressions that claim to be equivalent, you're testing whether they simplify to the same form and produce identical results when you substitute values. This is a fundamental skill for working with algebraic expressions. Let's verify these expressions by substituting a=2a = 2 and b=1b = 1. For the first expression 8a+12b8a + 12b: 8(2)+12(1)=16+12=288(2) + 12(1) = 16 + 12 = 28. For the second expression 4(2a+3b)4(2a + 3b): 4(2(2)+3(1))=4(4+3)=4(7)=284(2(2) + 3(1)) = 4(4 + 3) = 4(7) = 28. Both expressions equal 28, confirming they are equivalent. You can also verify this algebraically by distributing the 4 in the second expression: 4(2a+3b)=4⋅2a+4⋅3b=8a+12b4(2a + 3b) = 4 \cdot 2a + 4 \cdot 3b = 8a + 12b, which matches the first expression exactly. Looking at the wrong answers: Choice A gives 20, which might result from calculation errors like 8(2)+12(1)=16+4=208(2) + 12(1) = 16 + 4 = 20 if you mistakenly calculated 12×1=412 \times 1 = 4. Choice B gives 24, which could come from errors like 8(2)+8(1)=16+8=248(2) + 8(1) = 16 + 8 = 24 if you confused the coefficient 12 with 8. Choice C gives 32, which might result from 8(2)+16(1)=16+16=328(2) + 16(1) = 16 + 16 = 32 if you doubled the coefficient 12 incorrectly. When verifying equivalent expressions through substitution, always double-check your arithmetic carefully. Small calculation errors can lead you to conclude that equivalent expressions are different, or vice versa.

Question 3

A student claims that 3(2x+5)3(2x+5) is equivalent to 6x+156x+15. Which expression is the correct result of expanding 3(2x+5)3(2x+5) using the distributive property?

  1. 6x+156x+15 (correct answer)
  2. 2x+152x+15
  3. 6x+86x+8
  4. 6x+56x+5
Explanation: This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression using the distributive property to expand 3(2x + 5) is 6x + 15, by multiplying 3 by 2x and 3 by 5. A common error is incomplete distribution, like 3(2x + 5)=6x + 5 missing the 15, or arithmetic wrong like 3×5=8. Expanding: distribute multiplier to every term inside parentheses (a(b+c)=ab+ac, don't miss any terms). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

Question 4

A student rewrites 6x+2y+4x6x+2y+4x using multiple properties. First, use the commutative property to reorder terms, then combine like terms. Which expression is equivalent?

  1. 10x+2y10x+2y (correct answer)
  2. 6x+6y6x+6y
  3. 12xy12xy
  4. 10xy+210xy+2
Explanation: This question tests using properties of operations to generate equivalent expressions, combining commutative property and like terms. First, use commutative property to reorder 6x + 2y + 4x as 6x + 4x + 2y (order doesn't matter), then combine like terms: (6 + 4)x + 2y = 10x + 2y. For example, reordering groups the x terms: 6x + 4x = 10x, with 2y unchanged, resulting in 10x + 2y. The correct equivalent expression using these properties is 10x + 2y. A common error is combining unlike terms, like adding x and y to 12xy, or misadding coefficients. Apply commutative to reorder, then combine only same variables by adding coefficients. To verify equivalence, test with x = 1, y = 1: 6(1) + 2(1) + 4(1) = 12 and 10(1) + 2(1) = 12, which matches.

Question 5

A student simplifies 7y−2y+47y-2y+4 by combining like terms.

Which expression is equivalent to 7y−2y+47y-2y+4?

  1. 9y+49y+4
  2. 5y+45y+4 (correct answer)
  3. 5y5y
  4. 4y+54y+5
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically combining like terms in 7y-2y+4. Combining like terms: 7y-2y=(7-2)y=5y, constant +4 remains, so 5y+4. For example, combine 2x+3x+5=5x+5. The correct equivalent expression by combining like terms is 5y+4. A common error is wrong subtraction, like 7-2=9 leading to 9y+4, or combining unlike terms like 5y+4=9y. Combining like terms: only same variables, (7-2)y+4=5y+4, not y with constants. Arithmetic errors in coefficients are common at grade 6.

Question 6

In a science lab, the total length of wire used is written as 24x+18y24x+18y (in centimeters). Factor the expression by finding the greatest common factor (GCF). Which expression is equivalent?

  1. 6(4x+3y)6(4x+3y) (correct answer)
  2. 4(6x+18y)4(6x+18y)
  3. 2(12x+9y)2(12x+9y)
  4. 6(3x+4y)6(3x+4y)
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically factoring by finding the greatest common factor (GCF). The distributive property in reverse allows factoring: for 24x + 18y, find the GCF of 24 and 18, which is 6, then divide each term by 6 (24x ÷ 6 = 4x, 18y ÷ 6 = 3y) and write 6(4x + 3y). For example, to factor 24x + 18y, list factors of 24 (1, 2, 3, 4, 6, 8, 12, 24) and 18 (1, 2, 3, 6, 9, 18), identify the greatest common as 6, then factor out: 6(4x + 3y). The correct equivalent expression using factoring with GCF is 6(4x + 3y). A common error is choosing a smaller common factor like 2 or 3 instead of the greatest 6, or misdividing terms like writing 6(3x + 4y) which equals 18x + 24y. When factoring, first find the GCF of all coefficients by listing factors and picking the largest common one, then divide each term accurately. To verify equivalence, test with x = 1, y = 1: 24(1) + 18(1) = 42 and 6(4(1) + 3(1)) = 6(7) = 42, which matches.

Question 7

Verify that the two expressions are equivalent by substituting x=2x=2 and y=1y=1: Expression A: 6(x+y)6(x+y) and Expression B: 6x+6y6x+6y. What value do both expressions equal?

  1. 66
  2. 2424
  3. 1212
  4. 1818 (correct answer)
Explanation: This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. To verify equivalence, substitute x=2 and y=1 into 6(x + y) = 6(2 + 1) = 6×3 = 18, and 6x + 6y = 6×2 + 6×1 = 12 + 6 = 18, so both equal 18. A common error is arithmetic mistakes in substitution, like 6(2+1)=12 instead of 18, or not distributing in the second expression. Equivalence: expressions equal for ANY variable value (test: if x=1: 3(2+1)=9 and 6+3(1)=9✓, if x=0: 3(2)=6 and 6+0=6✓, equivalent). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

Question 8

A science club orders xx packs of stickers and xx packs of labels. Their total is written as x+x+x+xx+x+x+x. Combine like terms to write an equivalent expression.

  1. x+4x+4
  2. 2x2x
  3. x4x^4
  4. 4x4x (correct answer)
Explanation: This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression by combining like terms in x + x + x + x is 4x, adding the four x terms as (1 + 1 + 1 + 1)x. A common error is like terms wrong calculation, such as exponentiating to x^4 instead of adding coefficients to get 4x, or treating it as addition without combining. Combining like terms: 2x+3x+5=(2+3)x+5=5x+5 (only combine same variables, not x with constants). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

Question 9

A student simplifies the expression 2x+3x+52x+3x+5 by combining like terms.

Which expression is equivalent to 2x+3x+52x+3x+5?

  1. 5x5x
  2. 6x+56x+5
  3. 5x+55x+5 (correct answer)
  4. 5x2+55x^2+5
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically combining like terms in 2x+3x+5. Combining like terms means adding coefficients of the same variable: 2x+3x=(2+3)x=5x, and the constant 5 remains, so 5x+5. For example, combine y+y+y=(1+1+1)y=3y. The correct equivalent expression by combining like terms is 5x+5. A common error is combining unlike terms, like treating 5x+5 as 10x, or wrong calculation like 2+3=6 leading to 6x+5. When combining like terms: 2x+3x+5=(2+3)x+5=5x+5, only combine same variables, not x with constants. Mistakes like arithmetic errors in adding coefficients are common at grade 6.

Question 10

A student writes the expression 9x+69x+6 for a total score. Factor the expression by taking out the greatest common factor (GCF). Which expression is equivalent?

  1. 9(x+6)9(x+6)
  2. 2(9x+3)2(9x+3)
  3. 3(3x+2)3(3x+2) (correct answer)
  4. 6(32x+1)6(\tfrac{3}{2}x+1)
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically factoring by the greatest common factor (GCF). To factor 9x + 6, find the GCF of 9 and 6, which is 3, then divide each term: 9x ÷ 3 = 3x, 6 ÷ 3 = 2, so 3(3x + 2). For example, factors of 9 are 1, 3, 9 and of 6 are 1, 2, 3, 6, so GCF is 3; factor out to 3(3x + 2). The correct equivalent expression using GCF factoring is 3(3x + 2). A common error is using a larger number like 9, giving 9(x + 6) = 9x + 54 which is not equivalent, or using fractions unnecessarily. When factoring, identify the GCF by listing factors, divide each term, and write the factored form accurately. To verify equivalence, test with x = 1: 9(1) + 6 = 15 and 3(3(1) + 2) = 3(5) = 15, which matches.

Question 11

A student claims that 2x+3x+x2x + 3x + x is equivalent to 6x26x^2. What is the correct equivalent expression?

  1. 6x6x because you add the coefficients when terms have the same variable (correct answer)
  2. 6x36x^3 because you multiply the coefficients and add the exponents together
  3. 5x+x5x + x because you can only combine two like terms at one time
  4. 2x+4x2x + 4x because 3x+x=4x3x + x = 4x and you combine from left to right
Explanation: When combining like terms, you add the coefficients: 2x+3x+x=(2+3+1)x=6x2x + 3x + x = (2 + 3 + 1)x = 6x. The student incorrectly squared the result. Choice B incorrectly applies exponent rules. Choice C shows a misunderstanding of combining like terms. Choice D incorrectly combines only some terms.

Question 12

During a class fundraiser, the money collected is modeled by 18x+12y18x+12y. Factor the expression completely by finding the greatest common factor (GCF).

  1. 2(9x+6y)2(9x+6y)
  2. 3(6x+4y)3(6x+4y)
  3. 12(1.5x+y)12(1.5x+y)
  4. 6(3x+2y)6(3x+2y) (correct answer)
Explanation: This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression by factoring 18x + 12y completely using the GCF of 6 is 6(3x + 2y). A common error is using a wrong GCF like 3, resulting in 3(6x + 4y), which is not completely factored since 6 and 4 share a further factor of 2. Factoring: (1) find GCF of all coefficients (24 and 18: list factors, identify greatest common: 6), (2) divide each term by GCF (24x÷6=4x, 18y÷6=3y), (3) write GCF(quotients): 6(4x+3y). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

Question 13

During a class fundraiser, the money collected is modeled by 24x+18y24x+18y, where xx and yy are different types of donations. Factor the expression by finding the greatest common factor.

Which expression is equivalent to 24x+18y24x+18y?

  1. 6(4x+3y)6(4x+3y) (correct answer)
  2. 4(6x+18y)4(6x+18y)
  3. 2(12x+18y)2(12x+18y)
  4. 6(3x+4y)6(3x+4y)
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically factoring 24x+18y by finding the greatest common factor (GCF). The distributive property allows factoring: for 24x+18y=6(4x+3y), pull out the GCF of 6 from the coefficients 24 and 18, then divide each term: 24x/6=4x, 18y/6=3y. For example, factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6 + 18y/6)=6(4x+3y). The correct equivalent expression using factoring is 6(4x+3y). A common error is using a wrong GCF, like 2 or 3 instead of 6, or arithmetic mistakes in division, such as 24/6=3 instead of 4. For factoring: (1) find GCF of all coefficients (24 and 18: 6), (2) divide each term by GCF (24x÷6=4x, 18y÷6=3y), (3) write GCF(quotients): 6(4x+3y). Equivalence means expressions are equal for any variable value, like testing x=1, y=1: 24(1)+18(1)=42 and 6(4(1)+3(1))=42.

Question 14

Lisa needs to simplify 5(x−2)+3(2x+1)−4x5(x - 2) + 3(2x + 1) - 4x. She decides to work step by step. After distributing but before combining like terms, what should her expression look like?

  1. x−2+2x+1−4xx - 2 + 2x + 1 - 4x with the distributive property applied incorrectly to variables
  2. 5x−2+6x+1−4x5x - 2 + 6x + 1 - 4x with the distributive property applied to coefficients only
  3. 5x−7+6x−4x5x - 7 + 6x - 4x with some terms already combined during distribution
  4. 5x−10+6x+3−4x5x - 10 + 6x + 3 - 4x with all multiplication completed before combining (correct answer)
Explanation: When you see an expression with parentheses like this, you need to apply the distributive property step by step before combining like terms. The distributive property means you multiply the number outside the parentheses by each term inside. Let's work through 5(x−2)+3(2x+1)−4x5(x - 2) + 3(2x + 1) - 4x systematically. First, distribute the 5: 5⋅x=5x5 \cdot x = 5x and 5⋅(−2)=−105 \cdot (-2) = -10, giving us 5x−105x - 10. Next, distribute the 3: 3⋅2x=6x3 \cdot 2x = 6x and 3⋅1=33 \cdot 1 = 3, giving us 6x+36x + 3. The last term, −4x-4x, stays as is since there are no parentheses. After distributing everything, you get 5x−10+6x+3−4x5x - 10 + 6x + 3 - 4x. Choice A incorrectly ignores the coefficients outside the parentheses entirely, treating 5(x−2)5(x - 2) as just x−2x - 2. Choice B partially applies the distributive property but forgets to multiply the constants inside the parentheses—it should be 5⋅(−2)=−105 \cdot (-2) = -10, not just −2-2. Choice C jumps ahead by combining some terms during the distribution step, but the question specifically asks what the expression looks like "after distributing but before combining like terms." Choice D correctly shows all multiplication completed: 5x−10+6x+3−4x5x - 10 + 6x + 3 - 4x. This is exactly what you should have after applying the distributive property completely but before the final step of combining like terms. Remember: distribute first, then combine like terms. Don't skip steps or you'll make careless errors.

Question 15

Sarah wants to factor the expression 15a+25b−10c15a + 25b - 10c by taking out the greatest common factor. Which of the following shows the correct factorization?

  1. 5(3a+5b−2c)5(3a + 5b - 2c) because 5 divides evenly into all three coefficients (correct answer)
  2. 5(3a+5b−c)5(3a + 5b - c) because the greatest common factor of the coefficients is 5
  3. 15(a+b−c)15(a + b - c) because 15 is the largest coefficient in the original expression
  4. 10(a+2b−c)10(a + 2b - c) because 10 is close to the average of the three coefficients
Explanation: The greatest common factor of 15, 25, and 10 is 5. Factoring out 5: 15a+25b−10c=5(3a+5b−2c)15a + 25b - 10c = 5(3a + 5b - 2c). Choice B incorrectly shows −c-c instead of −2c-2c. Choice C uses 15 as a factor, but 15 doesn't divide evenly into 25 or 10. Choice D uses 10 as a factor, but 10 doesn't divide evenly into 15 or 25.

Question 16

Simplify the expression 2x+3x+52x+3x+5 by combining like terms. Which expression is equivalent?

  1. 5x+55x+5 (correct answer)
  2. 5x2+55x^2+5
  3. 5x5x
  4. 6x+56x+5
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically combining like terms. Combining like terms involves adding coefficients of similar terms: 2x + 3x + 5 = (2 + 3)x + 5 = 5x + 5, where x terms combine but the constant 5 stays separate. For example, in 2x + 3x + 5, group the like terms 2x + 3x = 5x, then add the constant: 5x + 5. The correct equivalent expression by combining like terms is 5x + 5. A common error is combining unlike terms, like adding 5 to 5x to make 6x + 5, or squaring to 5x^2 + 5. When combining like terms, only add coefficients for terms with the same variable, leaving constants alone. To verify equivalence, test with x = 1: 2(1) + 3(1) + 5 = 10 and 5(1) + 5 = 10, which matches; errors like 5x give 5, which doesn't.

Question 17

Simplify by using multiple properties (combine like terms): 4y+3+2y4y+3+2y.

  1. 4y+5y4y+5y
  2. 9y+39y+3
  3. 6y6y
  4. 6y+36y+3 (correct answer)
Explanation: This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression by combining like terms in 4y + 3 + 2y is 6y + 3, adding the y terms (4 + 2 = 6) and leaving the constant 3. A common error is combining unlike terms, like adding 6y + 3 to get 9y, or leaving it as 4y + 5y. Combining like terms: 2x+3x+5=(2+3)x+5=5x+5 (only combine same variables, not x with constants). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

Question 18

Use the commutative property of addition to rewrite the expression x+5x+5 as an equivalent expression.

  1. x+0x+0
  2. x−5x-5
  3. 5+x5+x (correct answer)
  4. 5x5x
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically the commutative property of addition. The commutative property states that a + b = b + a, so the order of terms doesn't matter: x + 5 is equivalent to 5 + x. For example, rewriting x + 5 using commutative property simply switches the order to 5 + x, as addition is commutative. The correct equivalent expression using the commutative property is 5 + x. A common error is changing to multiplication like 5x or subtraction like x - 5, or simplifying unnecessarily to x + 0. When applying the commutative property, only reorder the terms without changing operations or values. To verify equivalence, test with x = 3: 3 + 5 = 8 and 5 + 3 = 8, which matches; other options like 5x = 15 do not.

Question 19

A student groups numbers to make mental math easier: (x+3)+7(x+3)+7. Use the associative property of addition to rewrite the expression and simplify.

  1. (x+7)+3=x+21(x+7)+3=x+21
  2. x+(3+7)=x+10x+(3+7)=x+10 (correct answer)
  3. x+(3⋅7)=x+21x+(3\cdot 7)=x+21
  4. (x+3)⋅7=7x+21(x+3)\cdot 7=7x+21
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically the associative property of addition. The associative property states that (a + b) + c = a + (b + c), so regrouping (x + 3) + 7 as x + (3 + 7) = x + 10 doesn't change the value. For example, to simplify (x + 3) + 7, use associative property to regroup: x + (3 + 7) = x + 10. The correct rewritten and simplified expression using the associative property is x + (3 + 7) = x + 10. A common error is mixing with multiplication, like turning addition into x + 21 or distributing as (x + 3) × 7 = 7x + 21. When applying the associative property, only change the grouping of addition without altering operations. To verify equivalence, test with x = 1: (1 + 3) + 7 = 11 and 1 + 10 = 11, which matches; errors like x + 21 give 22, which doesn't.

Question 20

A student records repeated terms as y+y+y+yy+y+y+y. Combine like terms.

Which expression is equivalent to y+y+y+yy+y+y+y?

  1. y4y^4
  2. 4y4y (correct answer)
  3. y+4y+4
  4. 3y3y
Explanation: This question tests using properties of operations to generate equivalent expressions, specifically combining like terms in y+y+y+y. Combining like terms: y+y+y+y=(1+1+1+1)y=4y, adding coefficients of the same variable. For example, combine y+y+y=(1+1+1)y=3y. The correct equivalent expression by combining like terms is 4y. A common error is exponentiating, like y+y+y+y=y^4, or wrong count like 3y. When combining like terms, add the coefficients and keep the variable, only for same variables. Mistakes like treating addition as multiplication or exponentiation are common at grade 6.