Vertical and Horizontal Distance: A Step Toward the Pythagorean Theorem

🔑 Key Concepts

  • If two ordered pairs have the same x-value, the points form a vertical line.
  • If two ordered pairs have the same y-value, the points form a horizontal line.
  • For vertical distance, find the difference between the y-values.
  • For horizontal distance, find the difference between the x-values.
  • Absolute value helps show that distance is always positive.
  • This skill prepares students for diagonal distance, the Pythagorean theorem, and the distance formula.

✏️ Worked Examples

🧠 Math Vocabulary

  • Coordinate plane: A grid formed by the x-axis and y-axis where ordered pairs can be graphed.
  • Ordered pair: A pair of numbers written as \( (x,y) \) that gives the location of a point.
  • Vertical line: A line that goes up and down. Points on a vertical line have the same x-value.
  • Horizontal line: A line that goes left and right. Points on a horizontal line have the same y-value.
  • Distance: The number of units between two points.
  • Absolute value: A number’s distance from zero. It helps make distance positive.
  • Diagonal line: A line where both the x-values and y-values change.

💡 Main Idea

To find the distance between two points on a vertical or horizontal line, look for the coordinate that stays the same. If the x-values are the same, subtract the y-values. If the y-values are the same, subtract the x-values. Since distance is always positive, absolute value can be used to find the distance between the two different coordinates.

📚 What You Should Already Know

Students should know how to graph ordered pairs, identify x- and y-coordinates, read points on the coordinate plane, and subtract integers. It is also helpful for students to understand that distance is always positive.

🚀 What Comes Next

Finding distance on vertical and horizontal lines is an important step toward understanding diagonal distance. When a line is diagonal, both the x-values and y-values change, so students cannot find the distance by subtracting just one pair of coordinates. Instead, they can use the horizontal and vertical changes to build a right triangle. This leads naturally into the Pythagorean theorem and eventually the distance formula.

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