Choose an Equation That Models a Situation - ISEE Middle Level Mathematics Achievement

SECTION 1

Why Do We Write Equations?

People have been solving word problems for thousands of years. Ancient merchants needed to figure out prices, farmers needed to split land, and builders needed exact measurements. The trick was always the same: turn a real-life situation into math you can work with.

For most of history, people described problems in long sentences. It was not until mathematicians invented variables (letters that stand for unknown numbers) that writing equations became fast and powerful. Today, writing an equation is like creating a shortcut for a word problem.

1800 BCE • Babylonian Word Problems

Ancient Babylonians wrote math problems on clay tablets. They solved them with words and steps, but had no symbols for unknowns.

250 CE • Diophantus Uses Symbols

The Greek mathematician Diophantus started using shorthand symbols to represent unknown numbers, an early step toward algebra.

820 CE • Al-Khwarizmi and Algebra

The Persian scholar al-Khwarizmi wrote the first textbook on algebra. The word 'algebra' comes from the Arabic title of his book.

1637 • Descartes Introduces x, y, z

René Descartes popularized using letters like x, y, and z for unknowns. This is the notation we still use on the ISEE today!

On the ISEE, you will not need to solve the equation every time. Sometimes the question just asks you to pick the equation that correctly matches the story. That means the most important skill is translating words into math. Let's learn how!

SECTION 2

Core Principles of Modeling a Situation

An equation is a math sentence that uses an equal sign (=) to show that two things have the same value. When a problem describes a situation with an unknown number, you can write an equation to represent it. Here are the key ideas you need.

Identify the Unknown

Find what the problem is asking you to figure out. This becomes your variable (like x or n). Ask yourself: "What number don't I know yet?"

Spot the Operations

Certain words signal math operations. "More than" means add. "Less than" means subtract. "Times" or "each" often means multiply. "Split" or "shared" means divide.

Find the Total or Result

Look for the number that everything equals. Words like "is," "was," "equals," or "gives" usually point to where the equal sign goes.

Build and Check

Put the pieces together into an equation. Then re-read the problem to make sure your equation tells the same story as the words.

✦

KEY TAKEAWAY

Think of writing an equation like writing a recipe. A word problem gives you the ingredients (numbers and operations) and the finished dish (the result). Your job is to put them together in the right order. If the recipe says "add 5 cups of flour," you write "+ 5." If it says "the total is 20," you write "= 20."

SECTION 3

From Words to Equations — A Visual Guide

The diagram below shows how a word problem breaks apart into pieces. Each piece becomes part of an equation. Follow the arrows to see how the English words turn into math symbols.

This diagram breaks down how the word problem about Maria's earnings becomes the equation 8h − 15 = 49. Notice how each phrase in the story maps to a specific part of the equation.

The purple box identifies the unknown (what we want to find). The cyan box shows the operations (multiply and subtract). The pink box gives us the result (what everything equals). Once you find these three pieces, putting the equation together is straightforward.

SECTION 4

Key Word-to-Symbol Translations

Certain words almost always mean the same math operation. Learning these keyword translations will help you move quickly on the ISEE. Below are the most common ones you will see.

ADDITION KEYWORDS

"more than," "increased by," "total," "sum," "plus," "added to" → +

Example: "5 more than a number" becomes n + 5

SUBTRACTION KEYWORDS

"less than," "fewer," "decreased by," "minus," "remaining" → −

Example: "7 less than a number" becomes n − 7. Watch the order!

MULTIPLICATION KEYWORDS

"times," "each," "per," "of," "product," "double," "triple" → ×

Example: "$6 per ticket" with t tickets becomes 6t (or 6 × t)

DIVISION KEYWORDS

"divided by," "split among," "shared equally," "per" → ÷

Example: "24 stickers split among n friends" becomes 24 ÷ n

⚠

ISEE Tip: Watch Out for "Less Than"!

"7 less than a number" is written as n − 7, not 7 − n. The phrase "less than" flips the order. Always put the variable first, then subtract. This is one of the trickiest translations on the test!

SECTION 5

Common Equation Patterns on the ISEE

Most ISEE word problems follow a handful of patterns. If you recognize the pattern, you can write the equation quickly. The diagram below shows the five patterns you are most likely to see.

Five common equation patterns you will see on the ISEE. Each card shows an example situation and the matching equation. The strategy box at the bottom reminds you to use process of elimination.

The most common ISEE pattern is number 3: the two-step equation. It combines multiplication with addition or subtraction. When you see a rate (like dollars per item) and a flat fee or starting amount, you are dealing with a two-step equation.

SECTION 6

Worked Example: Choosing the Right Equation

Let's walk through a full ISEE-style problem step by step.

ℹ

Problem

A gym charges a $20 membership fee plus $5 for each class. Tanisha paid a total of $55. Which equation could be used to find the number of classes, c, Tanisha took?

Step-by-Step Solution

1
Step 1 — Identify the Unknown

The problem asks for the number of classes Tanisha took. The variable is c (the number of classes).

Unknown: c = number of classes
2
Step 2 — Spot the Operations

"$5 for each class" tells us to multiply: 5 × c, or 5c. "Plus a $20 membership fee" tells us to add 20. So the total cost is 5c + 20.

Expression: 5c + 20
3
Step 3 — Find the Result

"Paid a total of $55" tells us what everything equals. The word "total" and "paid" both signal the equal sign.

Result: = 55
4
Step 4 — Build the Equation

Put the expression and the result together.

5c + 20 = 55
5
Step 5 — Verify by Re-reading

Read the equation like a sentence: "5 dollars times the number of classes, plus a 20-dollar fee, equals 55 dollars total." That matches the story perfectly!

✓ Equation matches the situation

💡

ISEE Test Strategy

On the real test, you can also check your equation by solving it. If 5c + 20 = 55, then 5c = 35, so c = 7. Does it make sense that 7 classes at $5 each ($35) plus $20 equals $55? Yes! This quick check confirms you picked the right equation.

SECTION 7

Common Mistakes and How to Avoid Them

The ISEE is designed so that wrong answer choices match common mistakes. If you know what mistakes to watch for, you can avoid traps and eliminate wrong answers faster.

Common Mistake	What It Looks Like	How to Fix It
Reversing subtraction	Writing 7 − n instead of n − 7 for "7 less than n"	"Less than" means you subtract FROM the variable. Think: "7 less than 20" is 20 − 7 = 13.
Wrong operation	Using addition when the problem says "times" or "each"	Circle the keyword in the problem. Match it to your operation chart before writing the equation.
Putting the result on the wrong side	Writing 55 = 5c − 20 instead of 5c + 20 = 55	Both sides of an equation are equal, so 55 = 5c + 20 is actually the same as 5c + 20 = 55. Check the operation, not just the order.
Forgetting a step in a two-step problem	Writing 5c = 55 and leaving out the $20 fee	After writing your equation, check: does it include ALL the numbers from the problem? If a number is missing, you probably skipped a step.

Common mistakes students make when choosing equations on the ISEE

✦

KEY TAKEAWAY

Think of each wrong answer choice as a different "trap door." The test-makers know which mistakes students commonly make, and they include those mistakes as answer choices. If you know the traps ahead of time, you can walk right past them. Always re-read the problem after choosing your answer to make sure it matches the story.

SECTION 8

From Equations to Inequalities and Beyond

Right now, you are learning to write equations using an equal sign. As you move into higher math, you will also see inequalities (using symbols like < and > instead of =). The skills you build now — identifying unknowns, spotting operations, and translating carefully — are the exact same skills you will use later.

What You Learn Now	What Comes Next
One-step equations (x + 5 = 12)	Multi-step equations (3x + 5 − 2 = 18)
Two-step equations (2n + 7 = 21)	Equations with variables on both sides (2n + 7 = n + 14)
Modeling with = (equals)	Modeling with <, >, ≤, ≥ (inequalities)
Translating words into one equation	Systems of equations (two equations at once)

How today's equation skills connect to future math topics

The great news is that the ISEE Middle Level focuses on the skills in the left column. If you master these, you are not only ready for the test — you are also building a strong foundation for algebra in high school.

SECTION 9

Practice Problems

Try these five problems on your own. They go from easier to harder. Remember: there is no penalty for guessing on the ISEE, so always pick an answer! Use process of elimination if you are not sure.

Question 1: CONCEPTUAL

Sam has some stickers. He gives away 9 stickers and has 15 left. Which equation can be used to find the number of stickers, s, Sam started with?

A. s + 9 = 15
B. s − 9 = 15
C. 9 − s = 15
D. 9s = 15

Show answer (B)

The correct answer is B. Sam started with s stickers and gave away 9, which means subtract: s − 9. He has 15 left, so s − 9 = 15. Choice A is wrong because adding 9 would mean he gained stickers, not gave them away. Choice C reverses the subtraction (9 − s), which does not match the story because Sam starts with s, not 9. Choice D uses multiplication (9s), but the problem says nothing about groups or rates.

Question 2: BASIC CALCULATION

A book costs $7. Which equation can be used to find the number of books, b, you can buy with $42?

A. b + 7 = 42
B. 7b = 42
C. 42b = 7
D. b − 7 = 42

Show answer (B)

The correct answer is B. Each book costs $7, so the total cost is 7 times the number of books: 7b. The total you have is $42, so 7b = 42. Choice A uses addition, but buying multiple books at $7 each requires multiplication, not adding 7 once. Choice C has the numbers in the wrong positions — 42b would mean each book costs $42. Choice D subtracts 7, which does not represent buying multiple books at $7 each.

Question 3: INTERMEDIATE

A taxi ride costs $3 plus $2 for each mile driven. If the total fare is $17, which equation can be used to find the number of miles, m?

A. 2m − 3 = 17
B. 2m + 3 = 17
C. 3m + 2 = 17
D. 5m = 17

Show answer (B)

The correct answer is B. The flat fee is $3 (added once) and the per-mile rate is $2 (multiplied by the number of miles). So the total is 2m + 3, and this equals 17. Choice A subtracts the $3 instead of adding it, but the problem says the $3 is added to the cost. Choice C swaps the numbers: it multiplies by 3 and adds 2, which reverses the rate and the flat fee. Choice D adds 3 and 2 to get 5m, but the flat fee is not multiplied by the number of miles.

Question 4: APPLIED

Elena is saving money for a $120 bicycle. She already has $45 and earns $15 each week from babysitting. Which equation can be used to find the number of weeks, w, until she has enough money?

A. 15w = 120
B. 15w − 45 = 120
C. 15w + 45 = 120
D. 45w + 15 = 120

Show answer (C)

The correct answer is C. Elena earns $15 per week, so after w weeks she earns 15w dollars. She also already has $45, so her total savings is 15w + 45. She needs this to equal $120, giving 15w + 45 = 120. Choice A forgets the $45 she already has. Choice B subtracts 45 instead of adding it, which would mean she loses $45 instead of starting with it. Choice D swaps the 15 and 45: it says she earns $45 per week, which is incorrect.

Question 5: CRITICAL THINKING

A school is ordering pizzas for a party. Each pizza costs $12 and there is a $8 delivery charge. The school has a $100 budget. Which equation could be used to find the greatest number of pizzas, p, the school can order?

A. 8p + 12 = 100
B. 12p + 8 = 100
C. 12p − 8 = 100
D. 20p = 100

Show answer (B)

The correct answer is B. Each pizza costs $12, so p pizzas cost 12p dollars. The $8 delivery charge is a one-time flat fee that gets added: 12p + 8. The total equals the $100 budget, so 12p + 8 = 100. Choice A swaps the cost per pizza ($12) and the delivery charge ($8), making it seem like pizzas cost $8 each. Choice C subtracts the delivery charge instead of adding it, but a delivery charge adds to the total cost. Choice D combines 12 + 8 = 20 and multiplies by p, but the $8 delivery is charged once, not per pizza.

SUMMARY

Lesson Summary

Summary

To choose an equation that models a situation, follow four steps. First, identify the unknown and assign it a variable. Second, spot the operations by looking for keywords like "more than" (add), "less than" (subtract), "each" or "per" (multiply), and "split among" (divide). Third, find the result — the number that everything equals, usually signaled by words like "is," "total," or "left." Fourth, build and verify your equation by re-reading the problem to confirm every part matches.

Watch out for common ISEE traps: reversed subtraction ("7 less than n" is n − 7, not 7 − n), swapped numbers in the rate and the constant, and missing steps in two-step problems. Use process of elimination to cross out answers with wrong operations, and always guess if you are unsure — there is no penalty on the ISEE!