Lesson 10 — More Location and Distance with Coordinates

Learners use coordinates to solve more location and distance problems: they plot rectangles, find side lengths from coordinates, and compute perimeters and the lengths of paths along grid lines. This deepens the coordinate grid as a tool for describing space.

D05 P3: Intellectual & Cognitive Awareness D05.S3 50 minutes Draft

Objectives

D05.S3.05.01 Graph points in the first quadrant of a coordinate grid and use them to solve problems about location and distance.

Essential question

How do I use coordinates to find distances, paths, and perimeters on a grid?

perimeterside lengthpathrectanglecoordinatesdistance

Materials

Standard materials

First-quadrant grid · 1 per pair A numbered grid for plotting shapes and measuring side lengths and paths
Shape and path cards · 1 set per group Cards with rectangle corners or paths to plot and measure (e.g., a field with corners (1,1),(5,1),(5,4),(1,4))
Distance and perimeter recording page · 1 per learner A page for recording plotted shapes, side lengths, perimeters, and path lengths

Low-tech / no-cost

A chalk or dirt grid Draw a rectangle on a ground grid and pace its sides; add the four side lengths to find the perimeter
String around a shape Mark the corners of a rectangle with pebbles on a string grid, then measure each side with a piece of string

Enriched / lab & device

A digital map with a grid · 1 per group A local map to find the perimeter of a real field or park and the length of a walking path, where devices allow
A coordinate perimeter game · 1 per group A game where partners plot rectangle corners and challenge each other to find the perimeter

Works in different contexts

large-group Build one rectangle and find its perimeter together, then have pairs plot shapes and paths and share one result
multi-age Younger learners count side lengths of a plotted rectangle while older learners compute perimeters and multi-leg paths
self-directed A learner works the recording page alone, plotting shapes and computing perimeters and path lengths
level-grouped Learners ready to extend find the perimeter of a rectangle from its four corner coordinates and a shortest path along grid lines between two far points
outdoor-only Lay out a real rectangle on a ground grid, pace each side, and add the lengths to find the perimeter

A first-quadrant coordinate grid showing a rectangle with corners at (1,1), (5,1), (5,4), and (1,4). The side lengths are labeled: width 4 units across and height 3 units up. Below the grid, the perimeter is computed as 4 plus 3 plus 4 plus 3 equals 14 units. A dashed path from (1,1) to (5,4) along grid lines is also shown. Labels and counts, not color alone, carry the meaning so it prints clearly in grayscale.

Lesson 10 — More Location and Distance with Coordinates

Summary

Learners deepen their coordinate skills by plotting rectangles, finding side lengths from coordinates, and computing perimeters and the lengths of paths along grid lines. They connect coordinates back to the measurement work of the unit.

Objectives

Graph points in the first quadrant and use them to solve problems about location and distance, including the perimeter of a rectangle. (D05.S3.05.01)

Connection

A community garden plots a rectangle of land for planting. Its four corners are at (1,1), (5,1), (5,4), and (1,4). To fence it, the gardeners need the perimeter — the distance all the way around. From the coordinates they read the sides: 4 units across and 3 units up, so the fence is 4 + 3 + 4 + 3 = 14 units. Coordinates turn a drawing into numbers a builder, a farmer, or a town planner can use to plan a fence, a path, or a plot of land.

Materials

First-quadrant grid
Shape and path cards
Distance and perimeter recording page

Preparation

Copy grids for pairs and shape/path cards for groups.
Copy the recording page.
Have worked examples ready: rectangle (1,1),(5,1),(5,4),(1,4) → perimeter 14; path length.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a rectangle’s side lengths come from the differences of its coordinates — width = difference in x, height = difference in y; (2) perimeter = add all four side lengths (or 2 × width + 2 × height); (3) a path along grid lines is found by adding each leg’s length. Model a worked example first, then guide a second (Kirschner, Sweller & Clark, 2006). Retrieval: ask learners to recall Lesson 9’s “same row / same column” distance rule and reuse it here. Common slip: adding only two sides instead of all four for perimeter, or miscounting units. Coordinates are evidence; the fairness value appears in planning shared space so everyone’s plot is measured honestly and equally (docs/philosophy.md §4).

Procedure

Gather (5 min). Recall: how far apart are (1,2) and (4,2)? Today you use coordinates to find perimeters and paths — the math of fencing and planning land.
Plot a rectangle (10 min). Plot the garden’s corners: (1,1), (5,1), (5,4), (1,4), and connect them to make a rectangle. Read the side lengths from the coordinates: the bottom side goes from x = 1 to x = 5, so it is 4 units across; the left side goes from y = 1 to y = 4, so it is 3 units up.
Find the perimeter (10 min). The perimeter is the distance all the way around: **4 + 3 + 4
- 3 = 14 units**. Check with the short rule: 2 × 4 + 2 × 3 = 8 + 6 = 14.
Measure a path (10 min). To walk from (1,1) to (5,4) along the grid lines, you go across then up — or up then across. Count the legs: 4 + 3 = 7 units. Draw the path.
Practice with a partner (10 min). Take your shape and path cards. Plot each shape, read its side lengths, find the perimeter, and measure the path. Check each other: did you add all four sides? Did you count each leg of the path?
Close (5 min). Coordinates let you measure space without a ruler — the numbers tell you the lengths. That is how maps and plans become real fences, roads, and gardens.

Differentiation

Support: Use small rectangles (2 by 3) and read side lengths aloud; keep the grid large and tactile so learners with low vision can follow by touch.
Extension: Find the perimeter of a rectangle from coordinates alone (no drawing), and find the shortest path along grid lines between two far-apart points.

Assessment

Formative (observation/self): Can the learner read side lengths from coordinates and compute a rectangle’s perimeter and a path’s length correctly?
Self-check: The learner asks, “Did I add all four sides for the perimeter, and did I count every leg of the path?”

Home connection

Find a rectangular space (a rug, a room, a garden bed) and use coordinates or steps to measure its sides, then add them to find the distance around it.

Resources

Perimeter via coordinates is standard in elementary mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The fairness value of measuring shared space equally is a commitment (docs/philosophy.md §4).