Question
Write an expression that represents the slope of this line. You can draw a slope triangle to help you. Explain how you know your expression represents the slope of this line. as an 8th grader
Answers
GPT-4o mini
To represent the slope of a line, we can use the formula:
\[
\text{slope (m)} = \frac{\text{rise}}{\text{run}}
\]
### Step 1: Understand the Terms
- **Rise**: The vertical change between two points on the line (how much you go up or down).
- **Run**: The horizontal change between those same two points (how far you go left or right).
### Step 2: Create a Slope Triangle
Imagine you have a coordinate plane and a straight line that passes through two points, (x₁, y₁) and (x₂, y₂). You can visualize a right triangle (called a slope triangle) to help find the rise and run.
1. Start at the first point (x₁, y₁).
2. Move vertically to the second point (x₂, y₂) to find the rise (this is the vertical side of the triangle).
3. Then move horizontally from that point to find the run (this is the horizontal side of the triangle).
### Step 3: Calculate Rise and Run
- **Rise** = \( y₂ - y₁ \) (the change in y)
- **Run** = \( x₂ - x₁ \) (the change in x)
### Step 4: Write the Expression
So, you can write the expression for the slope of the line as:
\[
m = \frac{y₂ - y₁}{x₂ - x₁}
\]
### Why This Expression Represents the Slope
This expression gives you the **rate of change** of y with respect to x, which is what the slope tells you. It shows how much the y-value increases (or decreases) when the x-value increases by 1 unit. A larger absolute value of the slope means the line is steeper, while a slope of 0 means the line is flat (horizontal).
### Conclusion
By using the rise over run formula, you're calculating how steep the line is, and this helps to understand how two variables are related in a linear relationship.
\[
\text{slope (m)} = \frac{\text{rise}}{\text{run}}
\]
### Step 1: Understand the Terms
- **Rise**: The vertical change between two points on the line (how much you go up or down).
- **Run**: The horizontal change between those same two points (how far you go left or right).
### Step 2: Create a Slope Triangle
Imagine you have a coordinate plane and a straight line that passes through two points, (x₁, y₁) and (x₂, y₂). You can visualize a right triangle (called a slope triangle) to help find the rise and run.
1. Start at the first point (x₁, y₁).
2. Move vertically to the second point (x₂, y₂) to find the rise (this is the vertical side of the triangle).
3. Then move horizontally from that point to find the run (this is the horizontal side of the triangle).
### Step 3: Calculate Rise and Run
- **Rise** = \( y₂ - y₁ \) (the change in y)
- **Run** = \( x₂ - x₁ \) (the change in x)
### Step 4: Write the Expression
So, you can write the expression for the slope of the line as:
\[
m = \frac{y₂ - y₁}{x₂ - x₁}
\]
### Why This Expression Represents the Slope
This expression gives you the **rate of change** of y with respect to x, which is what the slope tells you. It shows how much the y-value increases (or decreases) when the x-value increases by 1 unit. A larger absolute value of the slope means the line is steeper, while a slope of 0 means the line is flat (horizontal).
### Conclusion
By using the rise over run formula, you're calculating how steep the line is, and this helps to understand how two variables are related in a linear relationship.