Expression Properties

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Expression Properties · Grade 6 · Worksheet 1

1. Hana is helping her school's kapa haka group design a stage banner. The banner is rectangular, with a length that is 4 times its width plus an extra 6 meters for decorative borders. The width of the banner is represented by the variable w. Hana needs to write an expression for the total perimeter of the banner using the distributive property. Write an equivalent expression for the perimeter by applying the distributive property to the expression 2(w + 4w + 6). Answer: 2(5w + 6) = 10w + 12 Solution: Write the expression for the perimeter using the given information. The width is w, and the length is 4w + 6. The perimeter is P = 2(w + (4w + 6)).
   Full step-by-step solution

   Step 1: Write the expression for the perimeter using the given information. The width is w, and the length is 4w + 6. The perimeter is P = 2(w + (4w + 6)). Step 2: Combine like terms inside the parentheses: w + 4w + 6 = 5w + 6. So P = 2(5w + 6). Step 3: Apply the distributive property: 2(5w + 6) = 2 * 5w + 2 * 6 = 10w + 12. The equivalent expression is 10w + 12.

2. 3² × (8 - 2) ÷ 3 + 15 = ? Answer: 33 Solution: Solve the operation inside the parentheses: 8 - 2 = 6. The expression becomes 3² × 6 ÷ 3 + 15. Calculate the exponent: 3² = 9.
   Full step-by-step solution

   Step 1: Solve the operation inside the parentheses: 8 - 2 = 6. The expression becomes 3² × 6 ÷ 3 + 15. Step 2: Calculate the exponent: 3² = 9. The expression becomes 9 × 6 ÷ 3 + 15. Step 3: Perform multiplication and division from left to right: 9 × 6 = 54, then 54 ÷ 3 = 18. The expression becomes 18 + 15. Step 4: Perform the addition: 18 + 15 = 33. The answer is 33.

3. Use the distributive property to rewrite: 5(2x + 7) = ? Answer: 10x + 35 Solution: Multiply 5 by 2x: 5 × 2x = 10x. Multiply 5 by 7: 5 × 7 = 35. Combine the results: 10x + 35.
   Full step-by-step solution

   Step 1: Apply the distributive property: a(b + c) = ab + ac. Step 2: Multiply 5 by 2x: 5 × 2x = 10x. Step 3: Multiply 5 by 7: 5 × 7 = 35. Step 4: Combine the results: 10x + 35. The answer is 10x + 35.

4. (-3)² × (4 + 2) - 18 ÷ (-3) = ? Answer: 60 Solution: Evaluate the exponent: (-3)² = 9 Calculate inside the parentheses: (4 + 2) = 6 Multiply: 9 × 6 = 54 Divide: 18 ÷ (-3) = -6 Subtract: 54 - (-6) = 54 + 6 = 60 The answer is 60.
   Full step-by-step solution

   Step 1: Evaluate the exponent: (-3)² = 9 Step 2: Calculate inside the parentheses: (4 + 2) = 6 Step 3: Multiply: 9 × 6 = 54 Step 4: Divide: 18 ÷ (-3) = -6 Step 5: Subtract: 54 - (-6) = 54 + 6 = 60 The answer is 60.

5. Liam is organizing a school field trip to the science museum. There are 180 students going on the trip, and each chaperone can supervise 12 students. The museum charges $8.50 per student and $12.00 per chaperone for admission. If the school budget for admission fees is $1,650, how many chaperones can they afford to bring? Answer: 10 Solution: Calculate the minimum number of chaperones needed for supervision: 180 students ÷ 12 students per chaperone = 15 chaperones. Calculate the total cost for all students: 180 students × $8.50 = $1,530.
   Full step-by-step solution

   Step 1: Calculate the minimum number of chaperones needed for supervision: 180 students ÷ 12 students per chaperone = 15 chaperones. Step 2: Calculate the total cost for all students: 180 students × $8.50 = $1,530. Step 3: Subtract student cost from budget to find remaining money for chaperones: $1,650 - $1,530 = $120. Step 4: Calculate how many chaperones can be paid for with remaining money: $120 ÷ $12 per chaperone = 10 chaperones. Step 5: Compare with minimum needed: The school can afford 10 chaperones, which is less than the 15 needed for full supervision. The answer is 10 chaperones.

6. Emma is organizing a school craft fair. She has 13 identical tables that she needs to arrange in rows. She decides to set up 7 tables in the first row and the rest in the second row. To find the total number of chairs needed, she multiplies the number of tables by 9 chairs per table. Write an equivalent expression using the distributive property to represent the total number of chairs. Answer: 9(7) + 9(6) Solution: Emma has 13 tables total. She puts 7 tables in the first row, so the second row has 13 - 7 = 6 tables. Each table needs 9 chairs.
   Full step-by-step solution

   Step 1: Emma has 13 tables total. She puts 7 tables in the first row, so the second row has 13 - 7 = 6 tables. Step 2: Each table needs 9 chairs. The total number of chairs is 9 * 13. Step 3: Using the distributive property, 9 * 13 can be written as 9 * (7 + 6). Step 4: Distribute: 9 * (7 + 6) = 9(7) + 9(6). Step 5: So the equivalent expression is 9(7) + 9(6). The answer is 9(7) + 9(6).

7. Kaia is helping her school's gardening club design a rectangular flower bed. The length of the flower bed is 12 meters, and the width is 9 meters. She wants to create an equivalent expression for the perimeter using the distributive property. If the perimeter formula is 2 times (length + width), which expression shows the perimeter using the distributive property, and what is the perimeter in meters? Answer: 42 Solution: Write the expression for the perimeter using the formula: 2(12 + 9). Calculate each product: 2 × 12 = 24, and 2 × 9 = 18. Add the results: 24 + 18 = 42.
   Full step-by-step solution

   Step 1: Write the expression for the perimeter using the formula: 2(12 + 9). Step 2: Apply the distributive property: 2(12 + 9) = 2 × 12 + 2 × 9. Step 3: Calculate each product: 2 × 12 = 24, and 2 × 9 = 18. Step 4: Add the results: 24 + 18 = 42. The equivalent expression is 2 × 12 + 2 × 9, and the perimeter is 42 meters.