6th Grade Math Worksheets - Free Printable

Let's solve the problems step by step for both sections: "WE DO TOGETHER" and "YOU DO (INDIVIDUAL)".

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WE DO TOGETHER

#### 1. Which property is shown below?
\[ 5(x + 7) = 5x + 35 \]

- Solution: This demonstrates the Distributive Property.
- The Distributive Property states that \( a(b + c) = ab + ac \).
- Here, \( 5(x + 7) = 5x + 35 \) shows that 5 is distributed over the sum \( x + 7 \).

#### 2. Classify the following as an expression, equation, or inequality and solve if possible.

##### A.) \( -4x = 32 \)
- Classification: Equation
- Solution:
\[
-4x = 32
\]
Divide both sides by \(-4\):
\[
x = \frac{32}{-4} = -8
\]
So, \( x = -8 \).

##### B.) \( 5x + 20 \)
- Classification: Expression
- Solution: This is just an expression; no solving is possible.

##### C.) \( \frac{x}{3} > 21 \)
- Classification: Inequality
- Solution:
\[
\frac{x}{3} > 21
\]
Multiply both sides by 3:
\[
x > 63
\]
So, \( x > 63 \).

#### 3. Write the equation used to solve for \( x \), then solve for \( x \).
- Problem: The area of a rectangular backyard is 210 ft squared. If one side of the fence is 35 ft, how long is the other side?

- Equation: The area of a rectangle is given by \( \text{Area} = \text{length} \times \text{width} \).
Let \( x \) be the length of the other side. Then:
\[
35x = 210
\]

- Solution:
\[
35x = 210
\]
Divide both sides by 35:
\[
x = \frac{210}{35} = 6
\]
So, the other side is \( x = 6 \) ft.

#### 4. Which of the following values would make this equation true?
\[ 34 - x \geq 40 \]

- Solution: Solve the inequality:
\[
34 - x \geq 40
\]
Subtract 34 from both sides:
\[
-x \geq 6
\]
Multiply both sides by \(-1\) (and reverse the inequality sign):
\[
x \leq -6
\]

Now, check the given options:
- A. \(-4\) (not less than or equal to \(-6\))
- B. \(8\) (not less than or equal to \(-6\))
- C. \(-10\) (less than or equal to \(-6\))
- D. \(6\) (not less than or equal to \(-6\))

The correct value is \( x = -10 \).

#### 5. Determine which variable is the Independent and Dependent variable. Then, write the equation for the table.

| Miles Traveled | Hours |
|----------------|-------|
| 45 | 1 |
| 90 | 2 |
| 135 | 3 |

- Independent Variable: Hours (since it is the input that determines the miles traveled)
- Dependent Variable: Miles Traveled (since it depends on the number of hours)

- Equation: From the table, we see that the miles traveled increase by 45 for each additional hour. This suggests a linear relationship:
\[
\text{Miles Traveled} = 45 \times \text{Hours}
\]
So, the equation is:
\[
y = 45x
\]
where \( x \) is the number of hours and \( y \) is the miles traveled.

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YOU DO (INDIVIDUAL)

#### 1. Which property is shown below?
\[ 4 + (6 + x) = (4 + 6) + x \]

- Solution: This demonstrates the Associative Property of Addition.
- The Associative Property states that \( a + (b + c) = (a + b) + c \).
- Here, the grouping of the numbers does not affect the result.

#### 2. Classify the following as an expression, equation, or inequality and solve if possible.

##### A.) \( 4 - x \)
- Classification: Expression
- Solution: This is just an expression; no solving is possible.

##### B.) \( \frac{x}{4} = 12 \)
- Classification: Equation
- Solution:
\[
\frac{x}{4} = 12
\]
Multiply both sides by 4:
\[
x = 12 \times 4 = 48
\]
So, \( x = 48 \).

##### C.) \( (-9) + x \leq 21 \)
- Classification: Inequality
- Solution:
\[
-9 + x \leq 21
\]
Add 9 to both sides:
\[
x \leq 30
\]
So, \( x \leq 30 \).

#### 3. Write the equation used to solve for \( x \), then solve for \( x \).
- Problem: The area of a hockey rink is 435 ft squared. If one side of the rink is 15 ft, how long is the other side?

- Equation: The area of a rectangle is given by \( \text{Area} = \text{length} \times \text{width} \).
Let \( x \) be the length of the other side. Then:
\[
15x = 435
\]

- Solution:
\[
15x = 435
\]
Divide both sides by 15:
\[
x = \frac{435}{15} = 29
\]
So, the other side is \( x = 29 \) ft.

#### 4. Which of the following values would make this equation true?
\[ 5(x) < -20 \]

- Solution: Solve the inequality:
\[
5x < -20
\]
Divide both sides by 5:
\[
x < -4
\]

Now, check the given options:
- A. \( 25 \) (not less than \(-4\))
- B. \(-4\) (not less than \(-4\))
- C. \(-20\) (less than \(-4\))
- D. \(4\) (not less than \(-4\))

The correct value is \( x = -20 \).

#### 5. Determine which variable is the Independent and Dependent variable. Then, write the equation for the table.

| Number of miles | Hours |
|-----------------|-------|
| 240 | 4 |
| 120 | 2 |
| 60 | 1 |

- Independent Variable: Hours (since it is the input that determines the number of miles)
- Dependent Variable: Number of miles (since it depends on the number of hours)

- Equation: From the table, we see that the number of miles decreases by half for each halving of the hours. This suggests a linear relationship:
\[
\text{Number of miles} = 60 \times \text{Hours}
\]
So, the equation is:
\[
y = 60x
\]
where \( x \) is the number of hours and \( y \) is the number of miles.

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Final Answers

#### WE DO TOGETHER
1. Distributive Property
2.
- A.) Equation, \( x = -8 \)
- B.) Expression
- C.) Inequality, \( x > 63 \)
3. Equation: \( 35x = 210 \), Answer: \( x = 6 \)
4. \( x = -10 \)
5. Independent: Hours, Dependent: Miles Traveled, Equation: \( y = 45x \)

#### YOU DO (INDIVIDUAL)
1. Associative Property of Addition
2.
- A.) Expression
- B.) Equation, \( x = 48 \)
- C.) Inequality, \( x \leq 30 \)
3. Equation: \( 15x = 435 \), Answer: \( x = 29 \)
4. \( x = -20 \)
5. Independent: Hours, Dependent: Number of miles, Equation: \( y = 60x \)

Boxed Final Answers

\[
\boxed{
\begin{array}{ll}
\text{WE DO TOGETHER:} & \text{YOU DO (INDIVIDUAL):} \\
1. \text{Distributive Property} & 1. \text{Associative Property of Addition} \\
2. \text{A.) } x = -8, \text{B.) Expression, C.) } x > 63 & 2. \text{A.) Expression, B.) } x = 48, \text{C.) } x \leq 30 \\
3. \text{Equation: } 35x = 210, \text{Answer: } x = 6 & 3. \text{Equation: } 15x = 435, \text{Answer: } x = 29 \\
4. x = -10 & 4. x = -20 \\
5. \text{Independent: Hours, Dependent: Miles Traveled, Equation: } y = 45x & 5. \text{Independent: Hours, Dependent: Number of miles, Equation: } y = 60x \\
\end{array}
}
\]