Graphing on the Coordinate Plane

📍 Graphing on the Coordinate Plane

Part 1 of 5 — Meet the Coordinate Plane

Topics in This Part

🔑 Key Concept: A coordinate plane is just two number lines that cross. One goes across, one goes up — and together they let us name the exact location of any point with a pair of numbers.

The Two Number Lines

You already know a single number line — it counts $0, 1, 2, 3, \dots$ from left to right. A coordinate plane uses two of them:

• The $x$-axis is the horizontal line (it runs left-to-right, like the horizon).
• The $y$-axis is the vertical line (it runs up-and-down, like a flagpole).

The point where they cross is called the origin. The origin is the "home base" — it sits at $0$ on both axes.

💡 Memory trick: The letter x has a line you can lay flat — think "x is across." That leaves y for up.

Concept Check 🎯

The First Quadrant

When the two axes cross, they make four corners, called quadrants. In Grade 5 we work in just one of them — the first quadrant — where both numbers are zero or bigger.

Picture a grid in the corner of a page:

 y
 5 ·  ·  ·  ·  ·  ·
 4 ·  ·  ·  ·  ·  ·
 3 ·  ·  ·  ·  ·  ·
 2 ·  ·  ·  ·  ·  ·
 1 ·  ·  ·  ·  ·  ·
 0 ─────────────────  x
   0  1  2  3  4  5

• The numbers along the bottom count across — that's the $x$-axis.
• The numbers up the left side count upward — that's the $y$-axis.
• The origin $(0,0)$ sits in the bottom-left corner.

🔑 Key Idea: Every spot where a grid line crossing-point lands can be named by how far across and how far up it is from the origin.

Label the Grid 🔽

Use what you just learned to finish each sentence about the grid above.

Count on the Axes 🧮

Look again at the grid above.

1) The numbers along the bottom ($x$-axis) go from $0$ up to what biggest number shown? 2) Starting at the origin, how many units would you climb up to reach the top of the grid (the highest number on the $y$-axis)?

You've Got the Map

You now know the parts of the coordinate plane:

• $x$-axis — the horizontal line (across)
• $y$-axis — the vertical line (up)
• Origin — the crossing point at $(0, 0)$

In Part 2 we'll learn how to write a point's address using two numbers — an ordered pair — and how to put a dot exactly where it belongs.

📍 Graphing on the Coordinate Plane

Part 2 of 5 — Ordered Pairs: An Address for Every Point

🔑 The Idea: A point's location is written as an ordered pair $(x, y)$. The order matters — the first number is always across, the second is always up.

What Is an Ordered Pair?

An ordered pair is two numbers inside parentheses, separated by a comma:

$(x, y) \quad\Longrightarrow\quad (\,\text{across},\ \text{up}\,)$

• The first number is the $x$-coordinate — how far to move across (right) from the origin.
• The second number is the $y$-coordinate — how far to move up from there.

💡 Say it like a story: "Start at the origin. Walk across $x$ steps, then climb up $y$ steps." That's exactly where the point lives.

Example: Plot $(3, 2)$

1. Start at the origin $(0,0)$.
2. Walk across to $3$ on the $x$-axis.
3. Climb up $2$ units.
4. Put a dot. That dot is the point $(3, 2)$.

Order Matters!

The point $(3, 2)$ is not the same as $(2, 3)$.

⚠️ The #1 mistake is swapping the numbers. Always read $x$ first, then $y$ — across before up. Some students remember it as "crawl before you climb."

Concept Check 🎯

Walk It Out 🧮

Imagine plotting the point $(6, 3)$ by starting at the origin.

1) How many units do you move across (to the right)? 2) How many units do you then move up?

Across or Up? 🔽

For each ordered pair, choose what the bold number means.

📍 Graphing on the Coordinate Plane

Part 3 of 5 — Reading Points Off a Graph

🔑 Why it works: Plotting puts a dot from a pair. Reading does the reverse — you find the pair from a dot by checking how far across and how far up it sits.

How to Read a Point

To name a point you see on the grid:

1. Drop straight down to the $x$-axis and read the number → that's your $x$-coordinate.
2. Slide straight left to the $y$-axis and read the number → that's your $y$-coordinate.
3. Write the pair as $(x, y)$.

Worked Example

Here is a grid with three points labeled $A$, $B$, and $C$:

 y
 5 ·  ·  ·  ·  ·  ·
 4 ·  ·  C  ·  ·  ·
 3 ·  ·  ·  ·  ·  ·
 2 ·  ·  ·  ·  B  ·
 1 ·  A  ·  ·  ·  ·
 0 ─────────────────  x
   0  1  2  3  4  5

• Point $A$: drop down → $x = 1$; slide left → $y = 1$. So $A = (1, 1)$.
• Point $B$: drop down → $x = 4$; slide left → $y = 2$. So $B = (4, 2)$.
• Point $C$: drop down → $x = 2$; slide left → $y = 4$. So $C = (2, 4)$.

⚠️ Watch out: Points $B = (4, 2)$ and $C = (2, 4)$ use the same two digits but land in totally different spots — proof that order matters.

Concept Check 🎯

Use the grid from the example above (points $A$, $B$, $C$).

Read the Coordinates 🧮

A new point $D$ sits 5 across and 3 up from the origin.

1) What is the $x$-coordinate of $D$? 2) What is the $y$-coordinate of $D$?

Points That Sit On an Axis

A coordinate can be $0$. When that happens, the point lands right on an axis:

💡 Shortcut: If the $y$-coordinate is $0$, you never climb up — so the point stays on the $x$-axis. If the $x$-coordinate is $0$, you never go across — so it stays on the $y$-axis.

On Which Axis? 🔽

Decide where each point lands.

📍 Graphing on the Coordinate Plane

Part 4 of 5 — Distances, Paths & Patterns

🔑 Big Payoff: Once points have addresses, we can do math with them — measure how far apart two points are and spot patterns when points line up.

Distance Along a Grid Line

When two points share the same $y$-coordinate, they sit on the same horizontal line. To find the distance between them, subtract their $x$-coordinates.

Example: Distance from $(2, 3)$ to $(7, 3)$

Both points are $3$ up, so they're on the same horizontal line. Subtract the across-values:

$7 - 2 = 5 \text{ units apart}$

When two points share the same $x$-coordinate, they sit on the same vertical line. Then you subtract their $y$-coordinates.

Example: Distance from $(4, 1)$ to $(4, 6)$

Both are $4$ across, so they're on the same vertical line. Subtract the up-values:

$6 - 1 = 5 \text{ units apart}$

💡 Rule of thumb: Same up value? Subtract the across numbers. Same across value? Subtract the up numbers. Always take bigger minus smaller so the distance is positive.

Concept Check 🎯

Measure the Distance 🧮

Find how many units apart each pair of points is.

1) $(2, 4)$ and $(2, 9)$ — they share the same $x$. Distance $= \,?$ 2) $(0, 6)$ and $(8, 6)$ — they share the same $y$. Distance $= \,?$

Spotting a Pattern

Sometimes a list of ordered pairs follows a rule. Look at this set of points:

Do you see it? In every pair, the up value is double the across value: $y = 2 \times x$. If you plotted these points, they would all line up in a straight, slanting path.

🔑 Key Idea: A rule like "$y$ is double $x$" lets you find the next point without a picture. If $x = 5$, then $y = 2 \times 5 = 10$, so the next point is $(5, 10)$.

Continue the Pattern 🔽

A pattern follows the rule "$y$ is $3$ more than $x$" — that is, $y = x + 3$. Complete the table.

📍 Graphing on the Coordinate Plane

Part 5 of 5 — Real-World Maps & Mastery Check

You can now (1) name the parts of the plane, (2) write and plot ordered pairs, (3) read points off a grid, and (4) measure distances and find patterns. Let's use it all on a real map.

A City Map on a Grid

Maps use coordinates all the time. Here is a tiny town where each block is one unit. The origin $(0,0)$ is the bus stop.

Because School $(2,5)$ and Store $(6,5)$ share the same up-value ($5$), the distance between them is $6 - 2 = 4$ blocks. And Store $(6,5)$ to Home $(6,1)$ share the same across-value ($6$), so that distance is $5 - 1 = 4$ blocks.

💡 Notice the four places form a square — each side is $4$ blocks long. Coordinates let you measure a shape without ever leaving your chair.

Map Practice 🎯

Use the town map above.

Mixed Practice 🧮

1) A point is $9$ across and $0$ up. Write its $x$-coordinate, then its $y$-coordinate. ($x = ?$, $y = ?$) 2) How far apart are $(3, 7)$ and $(3, 2)$? (distance in units)

Quick Reference

⚠️ Remember the golden rule: across before up — always read and plot the $x$-coordinate first.

Exit Quiz ✅

Answer all three to finish the lesson.