Identifying Constant Rate — I can decide whether a...

SECTION 1

Historical Context & Motivation

People have been thinking about rates for thousands of years. A rate is just a way to describe how one quantity changes compared to another. For example, how far you walk in one hour is a rate. Ancient traders needed to know these things to run their businesses.

When a rate stays the same no matter what, we call it a constant rate. Understanding constant rates helped people build roads, plan trips, and trade fairly. Let's look at how this idea developed over time.

~2000 BCE

Babylonian Trade Tables

Ancient Babylonians carved tables into clay tablets to track how much grain they could trade for a fixed amount of silver. These early tables showed constant exchange rates.

~300 BCE

Euclid and Proportions

The Greek mathematician Euclid wrote about proportions — the idea that two ratios can be equal. This is the heart of constant rates.

1600s

Galileo Studies Motion

Galileo rolled balls down ramps and measured distances at equal time intervals. He discovered that some motion has a constant rate (constant speed) while other motion does not.

1700s

Graphs Become Common

Scientists began plotting data on coordinate graphs. A straight line through the origin became the visual fingerprint of a constant rate.

So here is the big question this lesson answers: How can you tell whether a situation has a constant rate, and how do you prove it? We will use two powerful tools — tables and graphs — to find out.

SECTION 2

Core Principles & Definitions

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

Rate

A rate compares two quantities with different units. Example: 60 miles per 1 hour, or 3 dollars per 1 pound.

Constant Rate

A constant rate means the rate never changes. Every time the input goes up by the same amount, the output goes up by the same amount too.

Unit Rate

A unit rate tells you how much of one quantity goes with exactly 1 unit of the other. Example: $4 per 1 gallon.

Table Test

In a table, check if dividing every output by its input gives the same ratio. If it does, you have a constant rate.

Graph Test

On a graph, a constant rate shows up as a straight line that passes through the origin (0, 0). If the line is curved or doesn't start at (0, 0), the rate is not constant in a proportional way.

✦ KEY TAKEAWAY

Think of a constant rate like a metronome (the ticking device musicians use). A metronome clicks at the exact same speed no matter how long it runs. If you walk at a constant rate of 3 miles per hour, you cover 3 miles every single hour — the first hour, the fifth hour, the hundredth hour. The pace never changes.

SECTION 3

Visual Explanation — Constant vs. Non-Constant Rates

The easiest way to see whether a rate is constant is to graph the data. Below, two graphs are shown side by side. The left graph shows a constant rate and the right graph shows a non-constant rate. Notice the difference in their shapes.

Left: A straight line through the origin means the rate is constant (30 miles every hour). Right: A curve means the rate keeps changing — the object speeds up over time.

In the left graph, the line is perfectly straight. It passes through the origin (0, 0). This tells you the rate is constant. Every hour, the distance goes up by the same amount — 30 miles. In the right graph, the line curves upward. The object travels farther and farther each hour, so the rate is not constant.

SECTION 4

Mathematical Framework

There is a simple formula hiding behind every constant rate. If you can write the relationship as the equation below, the rate is constant.

CONSTANT RATE EQUATION

y = k × x

y = the output (like distance). k = the constant rate (like speed). x = the input (like time). The letter k stays the same for every pair of values.

To find k, you divide the output by the input. If you always get the same number, you have a constant rate.

FINDING THE UNIT RATE

k = y ÷ x

Pick any row in a table. Divide the y-value by the x-value. Do this for every row. If k is always the same, the rate is constant.

You can also check the differences between rows. If the input goes up by the same amount each time, check whether the output also goes up by the same amount each time. Equal jumps in both columns mean a constant rate.

DIFFERENCE TEST

Δy ÷ Δx = constant

Δy ("delta y") means the change in the output. Δx ("delta x") means the change in the input. If this ratio is the same every time, the rate is constant.

💡 Quick Tip

The ratio test (y ÷ x) works for any row. The difference test (Δy ÷ Δx) works between any two consecutive rows. Both should give the same answer when the rate is constant.

SECTION 5

Using a Table to Identify Constant Rate

Tables are one of the best tools for checking constant rates. Let's look at two tables side by side. One has a constant rate and one does not.

Table A — Constant Rate

Every ratio equals 5, so the rate is constant: 5 miles per hour.

Hours (x)	Miles (y)	y ÷ x
1	5	5 ÷ 1 = 5
2	10	10 ÷ 2 = 5
3	15	15 ÷ 3 = 5
4	20	20 ÷ 4 = 5

Table B — Not a Constant Rate

The ratios are 4, 5, 6, and 7 — they are NOT all the same, so this is NOT a constant rate.

Hours (x)	Miles (y)	y ÷ x
1	4	4 ÷ 1 = 4
2	10	10 ÷ 2 = 5
3	18	18 ÷ 3 = 6
4	28	28 ÷ 4 = 7

This diagram shows the ratio test applied to both tables. In Table A every ratio equals 5 (constant rate). In Table B the ratios are 4, 5, 6, and 7 (not constant).

The diagram above breaks the process into two simple steps. First, divide each output by its input. Second, compare the results. If they match, you have a constant rate. If they don't match, the rate changes.

SECTION 6

Worked Example

Let's walk through a full problem. A lemonade stand sells cups of lemonade. Here is a table showing the number of cups and the total cost.

Lemonade stand data

Cups (x)	Total Cost in $ (y)
2	6
5	15
8	24
10	30

Is the cost per cup a constant rate?

Step 1 — Find the ratio for each rowDivide the total cost (y) by the number of cups (x) for every row in the table. Row 1: 6 ÷ 2 = 3. Row 2: 15 ÷ 5 = 3. Row 3: 24 ÷ 8 = 3. Row 4: 30 ÷ 10 = 3.

All ratios equal 3

Step 2 — Compare the ratiosEvery ratio is exactly 3. Since they are all the same, the cost per cup is a constant rate.

Constant rate = $3 per cup

Step 3 — Justify with a graph (optional but powerful)If you plot the points (2, 6), (5, 15), (8, 24), and (10, 30) on a graph, they form a straight line that passes through the origin (0, 0). This is the graph test for a constant rate.

Straight line through (0, 0) ✓

Step 4 — Write the equationBecause the constant rate k = 3, you can write the equation y = 3 × x. This means the total cost equals 3 times the number of cups.

y = 3x

SECTION 7

Constant Rate vs. Variable Rate

Not every situation in life involves a constant rate. It is just as important to spot when a rate is variable (changing). Here is a comparison to help you tell them apart.

Key differences between constant and variable rates

Feature	Constant Rate	Variable Rate
Table test	y ÷ x gives the same number every time	y ÷ x gives different numbers
Graph shape	Straight line through (0, 0)	Curved line, or straight line not through (0, 0)
Equation form	y = k × x	Cannot be written as y = k × x
Real-life example	A car driving at a steady 50 mph	A car speeding up from a stop sign
Differences (Δy)	Equal when Δx is equal	Unequal even when Δx is equal

✦ KEY TAKEAWAY

Imagine filling a pool with a garden hose. If the water flows at the same speed the whole time, that's a constant rate — the pool rises the same amount every minute. But if someone keeps turning the faucet, the flow changes, so the rate is variable. The table test and graph test are your tools for figuring out which kind you have.

SECTION 8

Connection to Proportional Relationships & Slope

The idea of a constant rate connects directly to bigger math topics you will study soon. Here is a preview of how this concept grows.

From constant rate to slope — a preview of Algebra 1

What You Know Now	What Comes Next
Constant rate (k) in y = k × x	Slope (m) in y = m × x + b — the rate of change in any linear equation
Graph is a straight line through (0, 0)	Graph can be a straight line that starts anywhere (y-intercept b)
Table: y ÷ x is always the same	Table: Δy ÷ Δx is always the same (even if y ÷ x is not)
Proportional relationship	Linear relationship (may or may not be proportional)

When you learn about slope in Algebra 1, you will see that the constant rate k is actually the same thing as the slope m when the line goes through the origin. So everything you are learning now gives you a head start on linear equations!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In your own words, explain what it means for a situation to have a constant rate. Give one real-life example of a constant rate and one real-life example of a rate that is NOT constant.

PROBLEM 2 — BASIC CALCULATION

A store sells apples. Use the table to decide whether the price per apple is a constant rate. Apples: 3, 6, 9, 12. Total cost ($): 6, 12, 18, 24. Show your work by computing y ÷ x for each row.

PROBLEM 3 — INTERMEDIATE

A swimmer records the following data. Minutes: 2, 4, 6, 8. Laps: 3, 6, 10, 12. Is the swimmer swimming at a constant rate? Justify your answer using the ratio test AND the difference test.

PROBLEM 4 — APPLIED

Maya is saving money for a new bike. After 1 week she has $15, after 2 weeks she has $30, after 3 weeks she has $45, and after 4 weeks she has $60. Her friend claims Maya is saving at a constant rate. Is the friend correct? If so, write the equation and predict how much Maya will have after 7 weeks.

PROBLEM 5 — CRITICAL THINKING

Two students are debating. Student A says: 'If the points on a graph make a straight line, the rate must be constant.' Student B says: 'Not always — it depends on where the line starts.' Who is right, and why? Use an example to support your answer.

SUMMARY

Lesson Summary

A constant rate means the ratio between two quantities stays the same no matter which pair of values you pick. You can prove it with the table test — divide every y-value by its x-value and check that the results are all equal. You can also prove it with the graph test — plot the points and see if they form a straight line through the origin (0, 0). The constant rate is the unit rate k in the equation y = k × x.

If the ratios in a table are not all equal, or the graph is curved (or does not pass through the origin), then the rate is not constant. This skill prepares you for studying slope and linear equations in Algebra 1.