Combining Like Terms and Distributive Property Practice for Grade 9
A multi-part practice set for 9th graders that builds fluency with combining like terms, the Distributive Property, and combined distribution-and-simplification problems, with worked examples, error analysis prompts, and a tape diagram and hanger model for visualizing equivalence. Students work through about four problems per section plus critical-thinking items, making it a solid fit for a 45 to 60 minute practice block or independent homework.
“can you make a copy like this with different numbers Combining Like Terms Key Terms Term — a single number, a variable, or numbers and variables multiplied together. Like Terms — terms that have the same variable raised to the same power. Constant numbers (like 5 and -2) are also like terms. Coefficient — the number multiplied by a variable (in −3x, the coefficient is -3). Worked Example Try It! Problem: Simplify 3x+5−x+2 Step 1: Identify like terms. We have x terms (3x and −x) and constant terms (5 and 2). Step 2: Group them. (3x−x)+(5+2) Note: −x is the same as −1x. Step 3: Combine. 2x+7 1. Simplify the expression below: 4y−3+2y+8 Problem: Simplify −5a+2b−3a+b Step 1: Group like terms. Keep the sign attached to the coefficient. (−5a−3a)+(2b+b) Step 2: Combine coefficients. −5−3=−8 2+1=3 Final Answer: −8a+3b 2. Simplify the expression below: −6m+4n−2m+n Problem: Simplify 4x2−7x+2x2+10x Annotation: Terms with x2 are NOT like terms with x. They must be grouped separately. Step 1: Group. (4x2+2x2)+(−7x+10x) Step 2: Combine. 6x2+3x 3. Simplify the expression below: 3p2−5p+4p2+12p Problem: Simplify 10−(4x+3)+7x Step 1: Distribute the negative. Subtracting a group is like multiplying by -1. 10−4x−3+7x Step 2: Group and Combine. (−4x+7x)+(10−3) 3x+7 4. Simplify the expression below: 15−(2y+6)+5y Critical Thinking: Error Analysis 5. A student simplified the expression 8x−3−10x+5 and got 18x+2. Identify the student's error. How did they arrive at 18x, and what is the correct simplified expression? Visualizing Expressions 6. Use the tape diagram below to represent the expression x+x+5+x+2. Then, write the simplified version of the expression. Name: ____________________ Date: ____________________ The Distributive Property Key Concept The Distributive Property allows you to multiply a single term by two or more terms inside a set of parentheses. You "distribute" the outside multiplier to every term inside. a(b+c)=ab+ac Watch the Signs: When distributing a negative number, remember that a negative times a negative is a positive. Worked Example Try It Yourself Problem 1: Simplify 3(2x−5) Step 1: Multiply 3 by the first term, 2x. 3×2x=6x Step 2: Multiply 3 by the second term, −5. 3×(−5)=−15 Final Expression: 6x−15 1. Simplify 4(3x+2) Problem 2: Simplify −2(x−4) Step 1: Multiply −2 by x. −2×x=−2x Step 2: Multiply −2 by −4. A negative times a negative is a positive. −2×(−4)=8 Final Expression: −2x+8 2. Simplify −5(2y−3) Problem 3: Simplify x(x+7) Step 1: Multiply x by x. x×x=x2 Step 2: Multiply x by 7. x×7=7x Final Expression: x2+7x 3. Simplify 2a(3a−4) Problem 4: Simplify −(3x−9) Note: A negative sign outside parentheses is the same as multiplying by −1. Step 1: −1×3x=−3x Step 2: −1×(−9)=9 Final Expression: −3x+9 4. Simplify −(5x+12) Critical Thinking A student simplified the expression −3(x−5) and got −3x−15. Explain the error the student made and provide the correct simplified expression. Name: ____________________ Date: ____________________ Distributing and Combining Like Terms When an expression contains both parentheses and additional terms, we follow a specific order to simplify it. First, we use the Distributive Property to remove the parentheses by multiplying the term outside by everything inside. Then, we Combine Like Terms by grouping the variable terms together and the constant terms together. Worked Example Try It Yourself Problem: Simplify 2(3x+4)−5x+1 Step 1: Distribute Multiply 2 by both 3x and 4. (2×3x)+(2×4)−5x+1 6x+8−5x+1 Step 2: Group Like Terms Move the x terms together and the constants together. (6x−5x)+(8+1) Step 3: Combine 1x+9 or x+9 1. Simplify the expression: 3(2x−1)+4x−7 Use the space below to show your distribution and combining steps. Worked Example: Negative Multipliers Problem: Simplify −4(x−2)+3x Step 1: Distribute Multiply −4 by x and −4 by −2. Remember that a negative times a negative is a positive. −4x+8+3x Step 2: Combine Group −4x and 3x. (−4x+3x)+8 −1x+8 or −x+8 2. Simplify the expression: −2(3x+5)+10x Pay close attention to the sign of the multiplier. Worked Example: Subtraction as Distribution Problem: Simplify 10−(2x+3)+5x Step 1: Distribute the Negative Treat the minus sign before the parenthesis as a −1. 10−1(2x)−1(3)+5x 10−2x−3+5x Step 2: Combine (−2x+5x)+(10−3) 3x+7 3. Simplify the expression: 15−(4x−2)−6 Remember to distribute the negative to both terms inside the parentheses. Worked Example: Complex Expressions Problem: Simplify 5(x+2)−2(x−4) Step 1: Distribute both sets (5x+10)+(−2x+8) Step 2: Combine (5x−2x)+(10+8) 3x+18 4. Simplify the expression: 4(2x+3)−3(x+1) Show each step clearly. Visualizing Equivalence We can use a balanced hanger to prove that an expression before and after simplifying is the same. If the hanger stays level, the expressions are equivalent. This hanger represents 2(x + 3) on the left and 2x + 6 on the right. 5. Error Analysis A student simplified the expression 4(x+3)−2x and got 4x+3−2x=2x+3. Identify the error the student made, explain why it is incorrect, and provide the correct simplified expression. Name: ____________________ Date: ____________________ Answer Key 1. Combining Like Terms 6y+5 Solution: (4y+2y)+(−3+8)=6y+5 Common error: Adding −3 and 8 to get −11 or −5 by ignoring the signs. −8m+5n Solution: (−6m−2m)+(4n+n)=−8m+5n Common error: Thinking n has no coefficient; it is 1n. 7p2+7p Solution: (3p2+4p2)+(−5p+12p)=7p2+7p Common error: Combining p2 and p terms together (e.g., 14p3). 3y+9 Solution: 15−2y−6+5y→(5y−2y)+(15−6)=3y+9 Common error: Forgetting to distribute the negative to the 6, resulting in 15−2y+6. Error Analysis: The student added the coefficients 8 and 10 instead of subtracting (8−10). They also likely added −3 and 5 correctly to get 2, but missed the negative sign on the 10x. Correct expression: −2x+2. Tape Diagram: The diagram shows three x segments and two constant segments (5 and 2). Simplified expression: 3x+7. 2. The Distributive Property 12x+8 Solution: 4(3x)+4(2)=12x+8 −10y+15 Solution: −5(2y)−5(−3)=−10y+15 Common error: Writing −15 at the end; remember negativenegative=positive. 6a2−8a Solution: 2a(3a)+2a(−4)=6a2−8a −5x−12 Solution: −1(5x)+−1(12)=−5x−12 Critique: The student failed to distribute the −3 to the second term (−5). They treated it as −3x−15 instead of −3x+15. Correct expression: −3x+15. 3. Distributing and Combining 10x−10 Solution: 6x−3+4x−7→(6x+4x)+(−3−7)=10x−10 4x−10 Solution: −6x−10+10x→(−6x+10x)−10=4x−10 −4x+11 Solution: 15−4x+2−6→−4x+(15+2−6)=−4x+11 Common error: Distributing the negative to 4x but not to −2. 5x+9 Solution: (8x+12)−3x−3→(8x−3x)+(12−3)=5x+9 Error Analysis: The student did not distribute the 4 to the 3 inside the parentheses. They only multiplied the first term. Correct solution: 4x+12−2x=2x+12. ”
Teacher Guide & Tips
- Use this as a guided practice set after a brief mini-lesson on like terms: have students complete the first worked example together, then work the "Try It" problems independently or in pairs before moving on to the Distributive Property section.
- Discussion question: Why can't we combine 4x² and 7x, even though both contain x? Follow up with: how does the sign of the multiplier change the result when distributing a negative?
- Scaffold for struggling readers: have them write each step on a separate line, underline like terms with matching colors, and check each distribution by substituting x = 1 into both the original and simplified expressions.
- Extension for advanced students: ask them to write their own error-analysis item (like the student work with 8x − 3 − 10x + 5) for a classmate to diagnose, or to explain why the hanger model proves two expressions are equivalent.
- Estimated time: about 15 minutes per section for the core problems, and roughly 45 minutes total including the error-analysis and visualization items.
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Try Diffit for free“can you make a copy like this with different numbers Combining Like Terms Key Terms Term — a single number, a variable, or numbers and variables multiplied together. Like Terms — terms that have the same variable raised to the same power. Constant numbers (like 5 and -2) are also like terms. Coefficient — the number multiplied by a variable (in −3x, the coefficient is -3). Worked Example Try It! Problem: Simplify 3x+5−x+2 Step 1: Identify like terms. We have x terms (3x and −x) and constant terms (5 and 2). Step 2: Group them. (3x−x)+(5+2) Note: −x is the same as −1x. Step 3: Combine. 2x+7 1. Simplify the expression below: 4y−3+2y+8 Problem: Simplify −5a+2b−3a+b Step 1: Group like terms. Keep the sign attached to the coefficient. (−5a−3a)+(2b+b) Step 2: Combine coefficients. −5−3=−8 2+1=3 Final Answer: −8a+3b 2. Simplify the expression below: −6m+4n−2m+n Problem: Simplify 4x2−7x+2x2+10x Annotation: Terms with x2 are NOT like terms with x. They must be grouped separately. Step 1: Group. (4x2+2x2)+(−7x+10x) Step 2: Combine. 6x2+3x 3. Simplify the expression below: 3p2−5p+4p2+12p Problem: Simplify 10−(4x+3)+7x Step 1: Distribute the negative. Subtracting a group is like multiplying by -1. 10−4x−3+7x Step 2: Group and Combine. (−4x+7x)+(10−3) 3x+7 4. Simplify the expression below: 15−(2y+6)+5y Critical Thinking: Error Analysis 5. A student simplified the expression 8x−3−10x+5 and got 18x+2. Identify the student's error. How did they arrive at 18x, and what is the correct simplified expression? Visualizing Expressions 6. Use the tape diagram below to represent the expression x+x+5+x+2. Then, write the simplified version of the expression. Name: ____________________ Date: ____________________ The Distributive Property Key Concept The Distributive Property allows you to multiply a single term by two or more terms inside a set of parentheses. You "distribute" the outside multiplier to every term inside. a(b+c)=ab+ac Watch the Signs: When distributing a negative number, remember that a negative times a negative is a positive. Worked Example Try It Yourself Problem 1: Simplify 3(2x−5) Step 1: Multiply 3 by the first term, 2x. 3×2x=6x Step 2: Multiply 3 by the second term, −5. 3×(−5)=−15 Final Expression: 6x−15 1. Simplify 4(3x+2) Problem 2: Simplify −2(x−4) Step 1: Multiply −2 by x. −2×x=−2x Step 2: Multiply −2 by −4. A negative times a negative is a positive. −2×(−4)=8 Final Expression: −2x+8 2. Simplify −5(2y−3) Problem 3: Simplify x(x+7) Step 1: Multiply x by x. x×x=x2 Step 2: Multiply x by 7. x×7=7x Final Expression: x2+7x 3. Simplify 2a(3a−4) Problem 4: Simplify −(3x−9) Note: A negative sign outside parentheses is the same as multiplying by −1. Step 1: −1×3x=−3x Step 2: −1×(−9)=9 Final Expression: −3x+9 4. Simplify −(5x+12) Critical Thinking A student simplified the expression −3(x−5) and got −3x−15. Explain the error the student made and provide the correct simplified expression. Name: ____________________ Date: ____________________ Distributing and Combining Like Terms When an expression contains both parentheses and additional terms, we follow a specific order to simplify it. First, we use the Distributive Property to remove the parentheses by multiplying the term outside by everything inside. Then, we Combine Like Terms by grouping the variable terms together and the constant terms together. Worked Example Try It Yourself Problem: Simplify 2(3x+4)−5x+1 Step 1: Distribute Multiply 2 by both 3x and 4. (2×3x)+(2×4)−5x+1 6x+8−5x+1 Step 2: Group Like Terms Move the x terms together and the constants together. (6x−5x)+(8+1) Step 3: Combine 1x+9 or x+9 1. Simplify the expression: 3(2x−1)+4x−7 Use the space below to show your distribution and combining steps. Worked Example: Negative Multipliers Problem: Simplify −4(x−2)+3x Step 1: Distribute Multiply −4 by x and −4 by −2. Remember that a negative times a negative is a positive. −4x+8+3x Step 2: Combine Group −4x and 3x. (−4x+3x)+8 −1x+8 or −x+8 2. Simplify the expression: −2(3x+5)+10x Pay close attention to the sign of the multiplier. Worked Example: Subtraction as Distribution Problem: Simplify 10−(2x+3)+5x Step 1: Distribute the Negative Treat the minus sign before the parenthesis as a −1. 10−1(2x)−1(3)+5x 10−2x−3+5x Step 2: Combine (−2x+5x)+(10−3) 3x+7 3. Simplify the expression: 15−(4x−2)−6 Remember to distribute the negative to both terms inside the parentheses. Worked Example: Complex Expressions Problem: Simplify 5(x+2)−2(x−4) Step 1: Distribute both sets (5x+10)+(−2x+8) Step 2: Combine (5x−2x)+(10+8) 3x+18 4. Simplify the expression: 4(2x+3)−3(x+1) Show each step clearly. Visualizing Equivalence We can use a balanced hanger to prove that an expression before and after simplifying is the same. If the hanger stays level, the expressions are equivalent. This hanger represents 2(x + 3) on the left and 2x + 6 on the right. 5. Error Analysis A student simplified the expression 4(x+3)−2x and got 4x+3−2x=2x+3. Identify the error the student made, explain why it is incorrect, and provide the correct simplified expression. Name: ____________________ Date: ____________________ Answer Key 1. Combining Like Terms 6y+5 Solution: (4y+2y)+(−3+8)=6y+5 Common error: Adding −3 and 8 to get −11 or −5 by ignoring the signs. −8m+5n Solution: (−6m−2m)+(4n+n)=−8m+5n Common error: Thinking n has no coefficient; it is 1n. 7p2+7p Solution: (3p2+4p2)+(−5p+12p)=7p2+7p Common error: Combining p2 and p terms together (e.g., 14p3). 3y+9 Solution: 15−2y−6+5y→(5y−2y)+(15−6)=3y+9 Common error: Forgetting to distribute the negative to the 6, resulting in 15−2y+6. Error Analysis: The student added the coefficients 8 and 10 instead of subtracting (8−10). They also likely added −3 and 5 correctly to get 2, but missed the negative sign on the 10x. Correct expression: −2x+2. Tape Diagram: The diagram shows three x segments and two constant segments (5 and 2). Simplified expression: 3x+7. 2. The Distributive Property 12x+8 Solution: 4(3x)+4(2)=12x+8 −10y+15 Solution: −5(2y)−5(−3)=−10y+15 Common error: Writing −15 at the end; remember negativenegative=positive. 6a2−8a Solution: 2a(3a)+2a(−4)=6a2−8a −5x−12 Solution: −1(5x)+−1(12)=−5x−12 Critique: The student failed to distribute the −3 to the second term (−5). They treated it as −3x−15 instead of −3x+15. Correct expression: −3x+15. 3. Distributing and Combining 10x−10 Solution: 6x−3+4x−7→(6x+4x)+(−3−7)=10x−10 4x−10 Solution: −6x−10+10x→(−6x+10x)−10=4x−10 −4x+11 Solution: 15−4x+2−6→−4x+(15+2−6)=−4x+11 Common error: Distributing the negative to 4x but not to −2. 5x+9 Solution: (8x+12)−3x−3→(8x−3x)+(12−3)=5x+9 Error Analysis: The student did not distribute the 4 to the 3 inside the parentheses. They only multiplied the first term. Correct solution: 4x+12−2x=2x+12. ”
Teacher Guide & Tips
- Use this as a guided practice set after a brief mini-lesson on like terms: have students complete the first worked example together, then work the "Try It" problems independently or in pairs before moving on to the Distributive Property section.
- Discussion question: Why can't we combine 4x² and 7x, even though both contain x? Follow up with: how does the sign of the multiplier change the result when distributing a negative?
- Scaffold for struggling readers: have them write each step on a separate line, underline like terms with matching colors, and check each distribution by substituting x = 1 into both the original and simplified expressions.
- Extension for advanced students: ask them to write their own error-analysis item (like the student work with 8x − 3 − 10x + 5) for a classmate to diagnose, or to explain why the hanger model proves two expressions are equivalent.
- Estimated time: about 15 minutes per section for the core problems, and roughly 45 minutes total including the error-analysis and visualization items.
Combining Like Terms and Distributive Property Practice for Grade 9
A multi-part practice set for 9th graders that builds fluency with combining like terms, the Distributive Property, and combined distribution-and-simplification problems, with worked examples, error analysis prompts, and a tape diagram and hanger model for visualizing equivalence. Students work through about four problems per section plus critical-thinking items, making it a solid fit for a 45 to 60 minute practice block or independent homework.
What is Diffit?
Teachers use Diffit to get quality materials for their unique classrooms. Adjust difficulty, add scaffolds, or change the topic to get started.
Try Diffit for free