Graphs
Grade 2-4Coordinates & Midpoints
Coordinates are the foundation every other graphs topic is built on, and the skills in this lesson, reading points, finding a midpoint, measuring a distance, come up again and again once straight line graphs and geometry problems get involved. This lesson covers reading and plotting coordinates, finding the midpoint of two points, finding the distance between two points, and working backwards from a midpoint to find a missing endpoint.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Reading and plotting coordinates
A coordinate is written as (x, y), where x tells you how far to move along the horizontal axis, and y tells you how far to move up or down the vertical axis. Always read or plot the x-coordinate first, then the y-coordinate.
The point (3, -2) is 3 units to the right of the origin and 2 units below it. The point (-4, 1) is 4 units to the left of the origin and 1 unit above it.
Finding the midpoint of two points
The midpoint is exactly halfway between two points, and is found by averaging the x-coordinates and averaging the y-coordinates separately.
The midpoint of A(2, 3) and B(8, 7): average the x-coordinates, (2 + 8) ÷ 2 = 5, and average the y-coordinates, (3 + 7) ÷ 2 = 5, giving the midpoint (5, 5).
Finding the distance between two points
Find the horizontal and vertical distances between the two points, then use Pythagoras' theorem to find the straight-line distance, treating these two distances as the shorter sides of a right-angled triangle.
The distance between A(1, 2) and B(4, 6): the horizontal distance is 4 - 1 = 3, and the vertical distance is 6 - 2 = 4. Using Pythagoras, the distance is √(3² + 4²) = √25 = 5.
Finding a missing endpoint from a midpoint
If the midpoint and one endpoint are known, the missing endpoint can be found by reversing the midpoint calculation: double the midpoint coordinate and subtract the known endpoint coordinate, doing this separately for x and y.
M(4, 5) is the midpoint of A(2, 3) and B: for the x-coordinate, 2 × 4 - 2 = 6, and for the y-coordinate, 2 × 5 - 3 = 7, giving B(6, 7).
Worked Examples
Three exam-style questions, fully solved.
Find the coordinates of the midpoint of A(2, 3) and B(8, 7).
Easy- 1.Average the x-coordinates: (2 + 8) ÷ 2 = 5
- 2.Average the y-coordinates: (3 + 7) ÷ 2 = 5
Answer: (5, 5)
Find the distance between A(1, 2) and B(4, 6).
Medium- 1.Find the horizontal distance: 4 - 1 = 3
- 2.Find the vertical distance: 6 - 2 = 4
- 3.Use Pythagoras' theorem: √(3² + 4²) = √25
Answer: 5
M(4, 5) is the midpoint of A(2, 3) and B. Find the coordinates of B.
Hard- 1.Double the midpoint's x-coordinate and subtract the known x-coordinate: 2 × 4 - 2 = 6
- 2.Double the midpoint's y-coordinate and subtract the known y-coordinate: 2 × 5 - 3 = 7
Answer: (6, 7)
Avoid These
The most common mistakes students make.
Writing the y-coordinate before the x-coordinate, instead of always reading or plotting the x-coordinate first.
Adding the two x-coordinates or two y-coordinates when finding a midpoint but forgetting to divide the total by 2.
Losing track of negative signs when a coordinate is negative, especially when adding or subtracting during a midpoint or distance calculation.
Forgetting to take the square root at the end of a distance calculation, and leaving the answer as the sum of the squares instead of the actual distance.
When finding a missing endpoint from a midpoint, doubling the known endpoint instead of doubling the midpoint and subtracting the known endpoint.
FAQ
Questions parents and students ask.
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