Cogito
Integrated Math 1 · Chapter 7 · Lesson 1
Distance and Midpoint
Measuring on a grid.
12 problems · about 21 minutes · G-GPE.B.7, G-GPE.B.6
What this lesson teaches
The student computes distances and midpoints between points on the coordinate plane.
- Distance is √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras.
- The midpoint averages each coordinate.
- Distance subtracts; midpoint adds.
Warm Up
Straightforward practice. Get the method working first.
5 problemsFrom (0, 0) to (8, 6). What is the distance?
Answer 10
Why 10.
Where does the distance formula come from?
Answer The Pythagorean Theorem applied to the horizontal and vertical gaps.
Why It is Pythagoras in coordinates.
From (0, 0) to (3, 4). Distance?
Answer 5
Why √25.
From (0, 0) to (6, 8). Distance?
Answer 10
Why √100.
From (1, 1) to (1, 9). Distance?
Answer 8
Why A vertical segment.
Build It Up
The same ideas with more to keep track of.
3 problemsFrom (2, 6) to (10, 6). Midpoint x-coordinate?
Answer 6
Why (2 + 10) ÷ 2.
From (3, 1) to (3, 11). Midpoint y-coordinate?
Answer 6
Why (1 + 11) ÷ 2.
From (−2, 0) to (4, 0). Distance?
Answer 6
Why The horizontal gap.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each operation by which formula uses it.
Answer Distance formula: Subtract the coordinates, Take a square root · Midpoint formula: Add the coordinates, Divide by 2
Why One formula subtracts and the other averages.
From (0, 0) to (5, 12). Distance?
Answer 13
Why √169.
The Distance: From (1, 2) to (4, 6). What is the distance?
Answer 5
Why 5.
The Middle: From (2, 4) to (8, 10). What is the x-coordinate of the midpoint?
Answer 5
Why 5.