Use coordinate points to identify a rectangle or other...

SECTION 1

Where Did Coordinate Grids Come From?

Have you ever played Battleship or found a spot on a map using letters and numbers? You were using a coordinate grid! People have been using grids like this for hundreds of years. Let's see how it all started.

~300 BC

Ancient Greek Maps

Greek mapmakers drew grids on their maps to help sailors find places. This was one of the first uses of a grid system!

1637

René Descartes Invents the Coordinate Plane

A French thinker named René Descartes created a way to describe any point using two numbers. We still use his idea today!

1800s

Grids in Schools

Teachers started using coordinate grids to help students learn about shapes and geometry. Kids have been plotting points ever since!

Today

Grids Are Everywhere

Video games, GPS apps, and even treasure maps use coordinate grids. When you plot points, you're using the same tool!

On the ISEE, you might see a question that gives you coordinate points. Your job is to figure out what shape those points make. Is it a rectangle? A triangle? A square? Let's learn how to do this step by step!

SECTION 2

Core Ideas You Need to Know

Before we start plotting points, let's learn a few important ideas. These are the building blocks you'll need!

The Coordinate Plane

A flat grid made of two number lines that cross. The horizontal line is the x-axis (goes left-right). The vertical line is the y-axis (goes up-down).

Ordered Pairs

A point on the grid is written as two numbers inside parentheses, like (3, 5). The first number tells you how far to go right. The second tells you how far to go up.

The Origin

The spot where the x-axis and y-axis cross is called the origin. Its coordinates are (0, 0). It's your starting point every time!

Connecting the Dots

When you plot points and draw lines between them in order, you create a shape. Three points can make a triangle. Four points can make a rectangle or a square!

✦ KEY TAKEAWAY

KEY TAKEAWAY

SECTION 3

See It on a Grid!

Let's look at a coordinate plane with four points plotted on it. When we connect the dots, what shape do we see?

Four points are plotted: A (2, 3), B (8, 3), C (8, 6), and D (2, 6). When we connect them, we get a rectangle! Notice how opposite sides are the same length.

Look at the diagram above. Point A is at (2, 3). That means we go 2 spaces right and 3 spaces up from the origin. Point B is at (8, 3) — same height, but farther to the right. When we connect all four points, we see straight sides and four right angles. That's a rectangle!

ISEE Test Tip

SECTION 4

How to Figure Out the Shape

You don't need fancy formulas for this! Here are easy ways to figure out what shape the points make. We use simple subtraction to find the length of each side.

HORIZONTAL DISTANCE

Length = bigger x − smaller x

When two points are at the same height (same y-value), subtract the x-values. Example: from (2, 3) to (8, 3) the distance is 8 − 2 = 6.

VERTICAL DISTANCE

Length = bigger y − smaller y

When two points are directly above each other (same x-value), subtract the y-values. Example: from (2, 3) to (2, 6) the distance is 6 − 3 = 3.

Shape Clues from Side Lengths

Rectangle: 4 sides, opposite sides equal, all corners are square (right angles).
Square: A special rectangle where ALL four sides are the same length.
Triangle: 3 points connected by 3 sides.
Line segment: Only 2 points — they just make a straight line, not a shape.

✦ KEY TAKEAWAY

REMEMBER THIS

SECTION 5

A Visual Guide to Common Shapes on the Grid

Let's see what different shapes look like when we plot their corners on a coordinate grid. The ISEE might ask you about any of these!

Four common shapes you might see on the ISEE: a square (all sides equal), a rectangle (opposite sides equal), a triangle (3 sides), and a right triangle (has a square corner).

Quick reference: number of points tells you the type of shape!

Shape	Number of Points	How to Spot It
Triangle	3	Three corners, three straight sides
Square	4	Four equal sides, four square corners
Rectangle	4	Opposite sides equal, four square corners
Pentagon	5	Five corners, five straight sides

SECTION 6

Worked Example: What Shape Is It?

Let's solve a problem just like one you might see on the ISEE. Follow along step by step!

Sample Problem

Step 1 — Count the PointsWe have 4 points. That means our shape has 4 corners. It could be a rectangle, square, or another 4-sided shape.

4 points → 4-sided shape

Step 2 — Find the Bottom SideLook at (1, 1) and (5, 1). They both have y = 1, so they're at the same height. The distance across is 5 − 1 = 4 units.

Bottom side = 4 units

Step 3 — Find the Top SideLook at (1, 4) and (5, 4). They both have y = 4. The distance across is 5 − 1 = 4 units. The top matches the bottom!

Top side = 4 units ✓ Matches bottom!

Step 4 — Find the Left SideLook at (1, 1) and (1, 4). They both have x = 1, so they're stacked vertically. The distance is 4 − 1 = 3 units.

Left side = 3 units

Step 5 — Find the Right SideLook at (5, 1) and (5, 4). They both have x = 5. The distance is 4 − 1 = 3 units. The right matches the left!

Right side = 3 units ✓ Matches left!

Step 6 — Name the ShapeOpposite sides are equal (4 and 4, 3 and 3). All sides are horizontal or vertical, so all corners are square. But the sides are NOT all the same length (4 ≠ 3), so it's not a square.

The shape is a RECTANGLE!

SECTION 7

Test-Taking Tips and Tricks

Knowing the math is great, but smart test-taking strategies can help you even more. Here are some tricks for the ISEE!

Smart strategies for ISEE coordinate geometry questions

Strategy	How It Helps
Sketch on scratch paper	Draw a quick grid and plot the points. You can SEE the shape instead of guessing.
Count points first	3 points = triangle. 4 points = rectangle, square, or other quadrilateral. This narrows your choices right away.
Check for equal sides	If all 4 sides are the same length, it's a square. If only opposite sides match, it's a rectangle.
Eliminate wrong answers	Cross out choices that can't be right. If you see 4 points, you know it's not a triangle!
Always answer every question	There's no penalty for wrong answers on the ISEE! If you're stuck, make your best guess.

✦ KEY TAKEAWAY

PROCESS OF ELIMINATION

SECTION 8

Looking Ahead: Trickier Shapes

Most ISEE questions at this level keep things simple. But some questions are a little trickier! Let's look at what makes shapes harder to identify.

How questions get harder on the ISEE

Basic Version	Trickier Version
Points given with a picture of the grid	Points given as a list — no picture provided
Shape has sides that go perfectly horizontal and vertical	Shape might be tilted or rotated on the grid
They ask: "What shape is this?"	They ask: "What is the missing point?" to complete a shape
Simple whole numbers like (2, 3)	Larger numbers or numbers close together

Finding a Missing Point

SECTION 9

Practice Problems

Time to practice! Try each problem on your own first. Then check the answer. Remember, on the real ISEE, always answer every question — even if you have to guess!

Three points are plotted on a coordinate grid at (2, 1), (6, 1), and (4, 4). If you connect these three points with straight lines, what shape do you form?

Four points are plotted on a coordinate grid at (1, 2), (5, 2), (5, 6), and (1, 6). If you connect these points in order and then connect the last point back to the first, what shape do you form?

Four points are plotted on a coordinate grid at (1, 1), (7, 1), (7, 4), and (1, 4). If you connect these points in order and then connect the last point back to the first, what shape do you form?

Five points are plotted on a coordinate grid at (2, 1), (5, 1), (6, 3), (4, 5), and (1, 3). If you connect these points in order and then connect the last point back to the first, how many sides does the shape have?

A rectangle has its four corners at (2, 1), (8, 1), (8, 4), and (2, 4) on a coordinate grid. A straight line is drawn from corner (2, 1) to corner (8, 4). Into how many triangles does this line divide the rectangle?