Input-Output Rules — Identify an input-output rule from a...

SECTION 1

Where Did Input-Output Rules Come From?

People have been looking for patterns in numbers for thousands of years. Ancient mathematicians noticed that certain actions always produce the same result. For example, doubling any number always gives you twice that number. This idea — that a consistent rule connects one number to another — is the foundation of input-output rules.

Over time, mathematicians created tables and machines to organize these patterns. Today, input-output tables show up everywhere — from computer programs to science experiments. On the SHSAT, you will see a table of numbers and need to figure out the rule that connects them.

~300 BC

Ancient Patterns

Greek mathematicians like Euclid studied number patterns and relationships, laying the groundwork for rules that connect numbers.

1600s

The Birth of Functions

Mathematicians like Leibniz introduced the idea of a function — a rule that assigns exactly one output to each input.

1800s

Tables Become Standard

Scientists and engineers began using organized tables to record input-output data in experiments and manufacturing.

Today

SHSAT & Everyday Math

Input-output tables appear on standardized tests like the SHSAT. They test your ability to spot patterns and write rules using algebra.

So here is the big question: if someone gives you a table of input and output numbers, how do you figure out the hidden rule that connects them? That is exactly what this lesson will teach you.

SECTION 2

Core Principles & Definitions

Before we dive in, let's lock down a few key ideas. An input is the number you start with. An output is the number you get after applying a rule. The rule (also called a function) is the operation or set of operations that changes the input into the output.

Input (x)

The starting number you plug into the rule. In a table, inputs are usually in the left column.

Output (y)

The result you get after the rule is applied. Outputs are usually in the right column.

Rule (Function)

The math operation(s) that turn every input into its output. Example: multiply by 3, then add 1.

Consistency

A correct rule must work for EVERY row in the table, not just one or two rows.

✦ KEY TAKEAWAY

Think of an input-output rule like a vending machine. You put in a coin (the input), the machine does something inside (the rule), and a snack pops out (the output). The same coin always gives you the same snack. Your job is to figure out what the machine is doing inside!

SECTION 3

Visualizing the Input-Output Machine

The diagram below shows how an input-output machine works. Each input number enters from the left, passes through the rule, and exits as an output number on the right. Notice that the same rule is applied to every single input.

Each input (left, in cyan) enters the rule box. The rule multiplies by 2 and then adds 3. The result appears as the output (right, in green). Notice that every row follows the same rule: y = 2x + 3.

Look at the diagram carefully. When x = 1, the machine calculates 2 × 1 + 3 = 5. When x = 2, it calculates 2 × 2 + 3 = 7. The rule never changes. This is the most important thing to remember: a true rule works for every single pair in the table.

SECTION 4

The Mathematical Framework

Most input-output rules on the SHSAT follow a pattern called a linear rule. That means the rule looks like y = mx + b, where m and b are numbers you need to find. Let's break that formula down.

GENERAL LINEAR RULE

y = m × x + b

y = output, x = input, m = the number you multiply the input by (called the rate of change), b = a number you add or subtract at the end (called the constant).

FINDING m (RATE OF CHANGE)

m = (y₂ − y₁) ÷ (x₂ − x₁)

Pick any two rows from the table. Subtract their outputs, then subtract their inputs. Divide the first result by the second. This gives you m.

FINDING b (CONSTANT)

b = y − m × x

Once you know m, pick any row. Plug in the input and output to solve for b. Then double-check with another row!

💡 SHSAT Tip

Not every rule is linear. Sometimes the rule involves squaring the input (x²) or dividing it. If your m value changes between different pairs of rows, the rule is probably not y = mx + b. Try other operations!

SECTION 5

Step-by-Step Strategy for Finding the Rule

Here is a clear, repeatable strategy you can use on every input-output table you see. The diagram below maps out each step.

Follow these six steps in order. Step 6 is the most important — always verify your rule against every row in the table before choosing your answer.

Step 1 — Look at the table and note all the input-output pairs.
Step 2 — Subtract consecutive outputs to see how much the output changes.
Step 3 — Subtract consecutive inputs to see how much the input changes.
Step 4 — Divide the output change by the input change to find m.
Step 5 — Use any row to solve for b: b = output − m × input.
Step 6 — Check your rule against every row. If it works for all, you've found it!

SECTION 6

Worked Example

Let's work through a full example together. Here is the table:

Find the rule that connects x to y.

Input (x)	Output (y)
2	11
4	19
6	27
8	35

Finding the Rule: y = ?

Step 1 — Find the change in outputsSubtract consecutive outputs: 19 − 11 = 8, 27 − 19 = 8, 35 − 27 = 8. The output goes up by 8 each time.

Change in output = 8

Step 2 — Find the change in inputsSubtract consecutive inputs: 4 − 2 = 2, 6 − 4 = 2, 8 − 6 = 2. The input goes up by 2 each time.

Change in input = 2

Step 3 — Calculate mDivide: m = 8 ÷ 2 = 4. This tells us the output is multiplied by 4 relative to the input.

m = 4

Step 4 — Solve for bUse the first row (x = 2, y = 11). Plug into b = y − m × x: b = 11 − 4 × 2 = 11 − 8 = 3.

b = 3

Step 5 — Write the rule and verifyThe rule is y = 4x + 3. Check: when x = 4, y = 4 × 4 + 3 = 19 ✓. When x = 6, y = 4 × 6 + 3 = 27 ✓. When x = 8, y = 4 × 8 + 3 = 35 ✓. It works for every row!

Rule: y = 4x + 3

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students can trip up on input-output problems. Here are the most common mistakes and how to dodge them.

Common Mistake	Why It Happens	How to Fix It
Only checking one row	Students find a rule that works for one pair and stop.	Always test your rule against every row in the table.
Confusing m and b	Students mix up the multiplier and the constant.	Find m first (using differences), then solve for b second.
Forgetting about b	Students find the multiplier but forget there may be a number added or subtracted.	After finding m, always check: does m × x exactly equal y? If not, find the difference — that's b.
Assuming the rule is always linear	Some tables use x², x³, or division.	If the output changes are NOT constant, try squaring or other operations.

✦ KEY TAKEAWAY

Think of verifying your rule like proofreading an essay. You wouldn't check just the first sentence and call it done. Go through every row to make sure your rule is correct from start to finish.

SECTION 8

Beyond Linear Rules — A Preview

Most SHSAT input-output problems use linear rules (y = mx + b). But as you move into high school algebra and beyond, you will encounter more complex rules. Here is a quick comparison.

Rule Type	Example	How to Spot It
Linear	y = 3x + 2	The output changes by the same amount each time.
Quadratic	y = x² + 1	The output changes increase. Try squaring the input.
Multiplicative	y = 2ˣ	The output doubles (or triples, etc.) each time.
Two-step	y = (x + 1) × 5	Need two operations. Try adding/subtracting first, then multiplying.

For the SHSAT, focus on mastering linear rules first. Once you can spot y = mx + b quickly, you will be ready to handle trickier patterns when they come up in high school courses like Algebra 1 and Algebra 2.

SECTION 9

Practice Problems

Try these five problems on your own. Each one gets a little harder. Work through the steps from the flowchart before checking the answer!

PROBLEM 1 — CONCEPTUAL

A table shows these pairs: (1, 5), (2, 10), (3, 15), (4, 20). What is the input-output rule?

PROBLEM 2 — BASIC CALCULATION

Given the table: Input → 3, 5, 7, 9 and Output → 10, 16, 22, 28. Find the rule.

PROBLEM 3 — INTERMEDIATE

A table shows: Input → 1, 3, 5, 7 and Output → 2, 12, 22, 32. What is the rule? If the input were 10, what would the output be?

PROBLEM 4 — APPLIED

A movie theater charges an entrance fee plus a price per ticket. The table shows: Tickets bought → 2, 4, 6, 8 and Total cost ($) → 17, 29, 41, 53. Write a rule for the total cost. How much would 10 tickets cost?

PROBLEM 5 — CRITICAL THINKING

A table shows: Input → 1, 2, 3, 4 and Output → 3, 6, 11, 18. Is this a linear rule? If not, can you figure out the actual rule?

SUMMARY

Lesson Summary

An input-output rule is a math operation that turns every input (x) into a specific output (y). Most SHSAT problems use a linear rule in the form y = mx + b, where m is the rate of change and b is the constant.

To find the rule, use the six-step strategy: examine the table, find the change in outputs, find the change in inputs, divide to get m, solve for b, and verify with every row. If the output changes are not constant, the rule may involve squaring or another operation. Always check your answer against all the data!