Math 1 · Unit 11 · Lesson 1

Distance, the Pythagorean Theorem on the Grid

Three tools, three questions: how steep (slope), how long (distance), where is the middle (midpoint). This lesson is how long. Two points on the same line are a count. A slanted segment is not, so draw the right triangle under it: the legs are countable, the segment is the hypotenuse, and the Pythagorean theorem gives its length. Subtract the x's, subtract the y's, square both, add, and take the square root to undo the squaring. That pattern is the distance formula.

About 45 minutes Undo the square with the square root

The big idea

Three tools answer three questions about a segment: slope for how steep, distance for how long, and midpoint for where the middle is. Distance comes from a picture. A horizontal or vertical segment is a count. A slanted one is the hypotenuse of a right triangle whose legs run across and up along the grid, so the Pythagorean theorem gives its length: square the legs, add, and take the square root. Do that by drawing a few times and the same steps appear every time: subtract the x's, subtract the y's, square both, add, square root. Written once for any two points, that is the distance formula, a shortcut for a picture you can always draw again. Order does not matter because squaring removes the sign, and the square root is there because the theorem gives the hypotenuse squared and the square root undoes the squaring, the inverse idea from Unit 1.

The parts of this lesson

Each part is its own short page. Go in order, or pick the one you need.

1The Words We UseThe vocabulary for this lesson, in one place.Start here 2Three tools, and the ones you can countUnit 4 ended by checking shapes with slopes, and said two tools were still missing: how long a segment is,... 3The right triangle under the segmentThe trick is to draw a right triangle under the slanted segment: one leg going across and one going up,... 4Three questions kids askDoes the order matter? 5Practice ProblemsTry them, get help one step at a time, then print how it went.

Printables

One page with counting segments, the right triangle under a slant, the Pythagorean theorem, the distance formula built from the pattern, and the answers to order and square root. Plus a worksheet with an answer key and a short lesson plan.

Study guide (PDF) Worksheet and answer key (PDF) Lesson plan (PDF)

Where this goes next

Previous lesson: Association Is Not CausationThe end of Unit 10. Next lesson: Midpoint, Meeting in the MiddleWhere is the middle? The average, across and up. Back to the Math 1 mapEvery Math 1 lesson, in course order. Tell us what you thoughtA few quick questions about this lesson. It takes a minute, and it is how the lesson gets better.

NC Math 1 standards in this lesson: NC.M1.G-GPE.4 (use coordinates to solve geometric problems), building from 8.G.8 (distance between points with the Pythagorean theorem).