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Expression Properties

Grade 6 · Algebra · Worksheet 1

  1. Noah is helping his school's gardening club create rectangular flower beds. The club has a rectangular plot that is 18 meters long and 12 meters wide. They want to divide the plot into two smaller rectangular sections by placing a fence parallel to the width. One section will be 7 meters long, and the other will be 11 meters long. Noah needs to write an expression for the total area of the plot using the distributive property. If he writes the area as 18 × 12, which equivalent expression using the distributive property shows the sum of the areas of the two sections? Write your answer in the form a(b + c).
    Answer: ______________
  2. Liam is designing a rectangular garden for his school's community project. The garden's length is 3.5 times its width. If the total area of the garden is 126 square meters, what is the width of the garden in meters? Answer: ______________
  3. Olivia draws a diagram of a rectangular swimming pool that is divided into two sections: a shallow end and a deep end. The entire pool has a width of 15 meters. The length of the shallow end is 25 meters, and the length of the deep end is represented by the variable x meters. Using the distributive property, write an equivalent expression for the total area of the pool in square meters. Answer: ______________
  4. (-3)² + 4 × (-2) - 18 ÷ 3 = ? Answer: ______________
  5. Use the distributive property to rewrite: 6(11x + 16y) Answer: ______________
  6. Use the distributive property to rewrite: 9(8x + 11) Answer: ______________
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Answer Key & Explanations

Expression Properties · Grade 6 · Worksheet 1

  1. Noah is helping his school's gardening club create rectangular flower beds. The club has a rectangular plot that is 18 meters long and 12 meters wide. They want to divide the plot into two smaller rectangular sections by placing a fence parallel to the width. One section will be 7 meters long, and the other will be 11 meters long. Noah needs to write an expression for the total area of the plot using the distributive property. If he writes the area as 18 × 12, which equivalent expression using the distributive property shows the sum of the areas of the two sections? Write your answer in the form a(b + c). Answer: 18(7 + 5) or equivalent Solution: Identify the total length and width. The plot is 18 m long and 12 m wide. The total area is 18 × 12.
    Full step-by-step solution

    Step 1: Identify the total length and width. The plot is 18 m long and 12 m wide. The total area is 18 × 12. Step 2: The plot is divided into two sections with lengths 7 m and 11 m. So, 18 = 7 + 11. Step 3: Using the distributive property, the total area can be written as 12 × (7 + 11) or 12 × 7 + 12 × 11. Step 4: Since the problem asks for the expression in the form a(b + c), and the width is the common factor, the equivalent expression is 12(7 + 11). The answer is 12(7 + 11).

  2. Liam is designing a rectangular garden for his school's community project. The garden's length is 3.5 times its width. If the total area of the garden is 126 square meters, what is the width of the garden in meters? Answer: 6 Solution: Let the width be w meters. Since the length is 3.5 times the width, the length is 3.5w meters.
    Full step-by-step solution

    Step 1: Let the width be w meters. Step 2: Since the length is 3.5 times the width, the length is 3.5w meters. Step 3: The area of a rectangle is length × width, so: (3.5w) × w = 126 Step 4: This simplifies to: 3.5w² = 126 Step 5: Divide both sides by 3.5: w² = 126 ÷ 3.5 Step 6: Calculate 126 ÷ 3.5 = 36 Step 7: So w² = 36 Step 8: Take the square root of both sides: w = sqrt(36) = 6 Step 9: The width of the garden is 6 meters.

  3. Olivia draws a diagram of a rectangular swimming pool that is divided into two sections: a shallow end and a deep end. The entire pool has a width of 15 meters. The length of the shallow end is 25 meters, and the length of the deep end is represented by the variable x meters. Using the distributive property, write an equivalent expression for the total area of the pool in square meters. Answer: 375 + 15x Solution: The total length of the pool is the sum of the shallow end and deep end lengths: 25 + x. The area of a rectangle is length times width. 15 × 25 = 375 15 × x = 15x Write the sum of these two products: 375 + 15x.
    Full step-by-step solution

    Step 1: The total length of the pool is the sum of the shallow end and deep end lengths: 25 + x. Step 2: The area of a rectangle is length times width. So the area of the entire pool is width times total length: 15 × (25 + x). Step 3: Apply the distributive property: multiply 15 by each term inside the parentheses. 15 × 25 = 375 15 × x = 15x Step 4: Write the sum of these two products: 375 + 15x. The equivalent expression for the total area is 375 + 15x square meters.

  4. (-3)² + 4 × (-2) - 18 ÷ 3 = ? Answer: -5 Solution: Calculate the exponent: (-3)² = 9 Calculate the multiplication: 4 × (-2) = -8 Calculate the division: 18 ÷ 3 = 6 Substitute back into the expression: 9 + (-8) - 6 Add 9 and -8: 9 + (-8) = 1 Subtract 6: 1 - 6 = -5 The answer is -5.
    Full step-by-step solution

    Step 1: Calculate the exponent: (-3)² = 9 Step 2: Calculate the multiplication: 4 × (-2) = -8 Step 3: Calculate the division: 18 ÷ 3 = 6 Step 4: Substitute back into the expression: 9 + (-8) - 6 Step 5: Add 9 and -8: 9 + (-8) = 1 Step 6: Subtract 6: 1 - 6 = -5 The answer is -5.

  5. Use the distributive property to rewrite: 6(11x + 16y) Answer: 66x + 96y Solution: Identify the expression: 6(11x + 16y) Apply the distributive property: Multiply 6 by each term inside the parentheses. 6 × 11x = 66x 6 × 16y = 96y Combine the results: 66x + 96y The answer is 66x + 96y.
    Full step-by-step solution

    Step 1: Identify the expression: 6(11x + 16y) Step 2: Apply the distributive property: Multiply 6 by each term inside the parentheses. Step 3: 6 × 11x = 66x Step 4: 6 × 16y = 96y Step 5: Combine the results: 66x + 96y The answer is 66x + 96y.

  6. Use the distributive property to rewrite: 9(8x + 11) Answer: 72x + 99 Solution: Here, a = 9, b = 8x, c = 11. Multiply 9 by 8x: 9 × 8x = 72x. Multiply 9 by 11: 9 × 11 = 99.
    Full step-by-step solution

    Step 1: Apply the distributive property: a(b + c) = ab + ac. Here, a = 9, b = 8x, c = 11. Step 2: Multiply 9 by 8x: 9 × 8x = 72x. Step 3: Multiply 9 by 11: 9 × 11 = 99. Step 4: Combine the terms: 72x + 99. The answer is 72x + 99.