Representing Ratios — I can represent ratio relationships...

SECTION 1

Historical Context & Motivation

People have been comparing quantities for thousands of years. Ancient traders needed to know how many coins to exchange for a bag of grain. Builders needed to mix the right amounts of sand and water to make strong bricks. All of these situations involve ratios — a way of comparing two quantities.

Over time, mathematicians found smarter ways to organize and display ratios. They created tables to keep track of values and graphs to spot patterns quickly. Let's look at how these tools developed.

~2000 BCE

Ancient Babylon

Babylonian scribes carved ratio tables into clay tablets. They listed how much grain a worker earned per day of labor.

~300 BCE

Euclid's Elements

The Greek mathematician Euclid wrote formal rules about proportions. He showed that equal ratios form a special relationship.

1637

Descartes & the Coordinate Plane

René Descartes invented the x-y coordinate plane. This let people plot number pairs as points — a perfect tool for showing ratios visually.

Today

Data Everywhere

We use ratio tables and graphs in science labs, sports stats, cooking recipes, and business reports every single day.

Here's the big question this lesson answers: How can you take a ratio and show it as a table or a graph, and what do those representations tell you?

SECTION 2

Core Principles & Definitions

Before we build tables and graphs, let's nail down a few key ideas. These are the building blocks you'll use throughout the lesson.

Ratio

A ratio compares two quantities. For example, "3 red marbles for every 5 blue marbles" is the ratio 3 : 5.

Equivalent Ratios

Equivalent ratios are different pairs of numbers that express the same comparison. 3 : 5 and 6 : 10 are equivalent because you multiply both parts by the same number.

Ratio Table

A ratio table is an organized list of equivalent ratios in rows or columns. It makes patterns easy to spot.

Coordinate Graph

A coordinate graph uses an x-axis and y-axis to plot points. Each equivalent ratio becomes one point on the graph.

Origin Point (0, 0)

For a proportional relationship, the line of points always passes through the origin (0, 0). Zero of one quantity means zero of the other.

✦ KEY TAKEAWAY

Think of a ratio like a recipe. If the recipe says 2 cups of flour for every 1 cup of sugar, you can double it (4 : 2), triple it (6 : 3), or halve it (1 : 0.5). The flavor stays the same because the ratio stays the same. A table lists all those recipe sizes. A graph shows them as dots in a straight line.

SECTION 3

Seeing Ratios: From Tables to Graphs

Let's start with a simple example. Suppose you earn $3 for every car you wash. The ratio of cars washed to dollars earned is 1 : 3. The diagram below shows both the ratio table and the matching graph side by side.

The ratio table (left) lists each pair of equivalent ratios. The coordinate graph (right) plots each pair as a point. Notice that all points form a straight line through the origin (0, 0).

Each row in the table becomes a point on the graph. The point (2, 6) means "2 cars washed, $6 earned." The point (4, 12) means "4 cars washed, $12 earned." Because the ratio stays the same, every point lines up perfectly.

💡 What Does the Origin Mean?

The point (0, 0) tells us that if you wash 0 cars, you earn $0. That makes sense! In a proportional relationship, the graph always starts at the origin.

SECTION 4

The Math Behind Ratio Tables and Graphs

Every ratio can be written as a fraction. When you build a table, you multiply both parts of the ratio by the same number. Here is the formula that connects every row.

EQUIVALENT RATIO RULE

a : b = (a × n) : (b × n)

Where a and b are the original ratio values, and n is any whole number you multiply by (called the scale factor).

On a graph, each equivalent ratio becomes an ordered pair (x, y). The relationship between x and y follows a simple equation.

GRAPHING EQUATION

y = k × x

Where k is the unit rate (how much y increases for every 1 unit of x). In our car-wash example, k = 3 because you earn $3 per car.

This equation tells you that the graph is always a straight line through (0, 0). The steeper the line, the larger the unit rate.

FINDING THE UNIT RATE

k = y ÷ x

Pick any point (x, y) on the graph (other than the origin) and divide. If (2, 6) is a point, then k = 6 ÷ 2 = 3.

SECTION 5

What Do the Points on the Graph Mean?

Every point on a ratio graph tells a mini-story. Let's look at a different scenario. A lemonade recipe uses 2 lemons for every 5 cups of water. The diagram below plots several equivalent ratios and labels what each point means in real life.

Each point represents a specific number of batches. The point (2, 5) means "2 lemons and 5 cups of water" — one batch. The point (6, 15) means "6 lemons and 15 cups of water" — three batches. Every point on this line keeps the same taste!

Notice three important things about this graph.

Straight line: All the points line up. If they didn't, the ratio would be changing.
Passes through (0, 0): Zero lemons means zero cups of water. This confirms it's a proportional relationship.
Every point is a real situation: You can read any point and know the exact amounts of lemons and water.

SECTION 6

Worked Example: Building a Table and Graph

A bakery uses 3 eggs for every 2 cakes it bakes. Let's create a ratio table, plot the points on a graph, and explain what each point means.

Eggs and Cakes Ratio

Step 1 — Identify the RatioThe ratio is 3 eggs for every 2 cakes. Write this as 3 : 2. The first quantity (eggs) will go on the x-axis, and the second quantity (cakes) will go on the y-axis.

Ratio = 3 : 2

Step 2 — Build the Ratio TableMultiply both parts of the ratio by 1, 2, 3, 4, and 5 to get equivalent ratios. For example, multiply by 2: 3 × 2 = 6 eggs and 2 × 2 = 4 cakes. Keep going for all five multipliers.

(3, 2), (6, 4), (9, 6), (12, 8), (15, 10)

Step 3 — Plot the PointsDraw an x-axis (Eggs) and a y-axis (Cakes). Plot each ordered pair. Start at the origin (0, 0). Then plot (3, 2): go right 3 and up 2. Plot (6, 4): go right 6 and up 4. Continue for all points.

All points form a straight line through (0, 0).

Step 4 — Explain the PointsEach point tells a story. The point (3, 2) means the bakery uses 3 eggs to make 2 cakes. The point (9, 6) means 9 eggs makes 6 cakes — that's three times the original batch.

Every point represents a batch size that keeps the ratio 3 : 2.

Step 5 — Find the Unit RateDivide y by x at any point. Using (6, 4): k = 4 ÷ 6 = 2/3. This means each egg produces 2/3 of a cake. You can also say it takes 3/2 (or 1.5) eggs per cake.

Unit rate: 2/3 cake per egg (or 1.5 eggs per cake)

SECTION 7

Tables vs. Graphs: Strengths and Limitations

Both tables and graphs show ratio relationships, but each has its own strengths. Choosing the right one depends on what you need to do.

Comparison of ratio tables and coordinate graphs

Feature	Ratio Table	Coordinate Graph
Shows exact values	Yes — every number is written out clearly	Harder — you have to read carefully off the axes
Shows patterns	You can see patterns in the numbers but must look closely	Yes — a straight line instantly shows a constant ratio
Predicting new values	You must calculate each new row	Extend the line to predict values you haven't calculated
Comparing two ratios	Need two separate tables side by side	Plot both on the same graph — steeper line = larger ratio
Best for	Listing specific quantities, quick calculations	Seeing the big picture, making predictions

✦ KEY TAKEAWAY

Think of a table like a detailed shopping list — it gives you exact numbers for each item. Think of a graph like a map — it shows you the big picture and lets you predict where you're headed even for values you haven't listed yet.

SECTION 8

Connection to Proportional Reasoning & Beyond

What you've learned about ratio tables and graphs is the foundation for bigger ideas. Here's how this topic connects to what comes next.

From ratios to proportional reasoning and algebra

What You Know Now	What Comes Next
Equivalent ratios form a straight line through (0, 0)	Proportional relationships: y = kx, where k is the constant of proportionality
The steepness of the line relates to the unit rate	Slope of a line in algebra (rise ÷ run = k)
Building ratio tables by multiplying	Solving proportions using cross-multiplication
Reading points on a ratio graph	Interpreting real-world graphs in science, economics, and statistics

In algebra, the straight line you graphed today becomes the equation y = mx + b. For proportional relationships, b = 0 and m equals the unit rate. So you're already doing algebra without even realizing it!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A graph of a proportional relationship passes through the points (0, 0) and (4, 12). In your own words, explain what the point (4, 12) means if the x-axis represents hours worked and the y-axis represents dollars earned.

PROBLEM 2 — BASIC CALCULATION

The ratio of apples to oranges is 4 : 6. Complete the ratio table for 1, 2, 3, 4, and 5 batches. Then write the five ordered pairs you would plot.

PROBLEM 3 — INTERMEDIATE

A ratio table shows these pairs: (2, 7), (4, 14), (6, 21), (8, 28). Find the unit rate. Then predict the y-value when x = 10 and explain how you could find this on a graph.

PROBLEM 4 — APPLIED

A pet store mixes 5 gallons of freshwater with 2 cups of salt for a fish tank. The store needs to fill a tank that requires 20 gallons of freshwater. Build a ratio table to figure out how many cups of salt are needed. Then describe what the point (20, 8) would mean on a graph.

PROBLEM 5 — CRITICAL THINKING

Two runners track their distances. Runner A's graph passes through (2, 10) and (4, 20). Runner B's graph passes through (2, 8) and (4, 16). Both start at (0, 0). Which runner is faster? How can you tell by looking at the graphs? What would happen if the two lines crossed — could that happen in a proportional relationship?

SUMMARY

Lesson Summary

A ratio compares two quantities. You can organize equivalent ratios in a ratio table by multiplying both parts by the same number. Each row of the table becomes an ordered pair that you can plot on a coordinate graph. In a proportional relationship, all points line up in a straight line through the origin (0, 0).

Every point on the graph has a real-world meaning — it tells you the specific amounts of each quantity. The unit rate (k = y ÷ x) determines the steepness of the line. Tables are best for exact values and quick calculations. Graphs are best for spotting patterns, comparing ratios, and making predictions beyond the values you've already calculated.