Translate a Word Relationship into an Equation - ISEE Lower Level Quantitative Reasoning

SECTION 1

Why Do We Turn Words into Math?

People have been solving word problems for thousands of years! Long before we had the math symbols we use today, people wrote out their math puzzles using only words. Let's look at how math language grew over time.

1800 BC • Ancient Egypt

Egyptian students solved word problems written on papyrus scrolls. They described everything in sentences — no symbols yet!

800 AD • Al-Khwarizmi's Algebra

A mathematician named Al-Khwarizmi wrote a book about solving problems using unknowns. This gave us the word "algebra"!

1600s • Math Symbols Are Born

Mathematicians started using symbols like +, −, ×, and = instead of writing everything in words. This made math much faster!

Today • Word Problems on the ISEE

On the ISEE test, you read word problems and turn them into equations. This skill helps you find the right answer quickly.

So here's the big question: how do you figure out which math operation a word problem is asking you to do? That's exactly what this lesson will teach you. By the end, you'll be a word-problem detective!

SECTION 2

Core Principles: Words Are Clues

Every word problem hides a math equation inside it. Your job is to find the clue words that tell you which operation to use. Think of it like a secret code! Certain words always point to addition, subtraction, multiplication, or division.

Find the Clue Words

Words like "more than," "times," or "left over" tell you which math operation to use. Circle or underline them!

Pick the Numbers

Find the numbers in the problem. Write them down. They are the building blocks of your equation.

Choose the Operation

Match the clue words to +, −, ×, or ÷. Then connect your numbers with the right symbol.

Write the Equation

Put it all together: numbers, operation, and an equals sign. Now you can solve it!

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KEY TAKEAWAY

Think of clue words like a recipe. Just like a recipe tells you to "add" eggs or "divide" the dough, clue words in a word problem tell you exactly which math step to take. Follow the recipe, and you'll get the right answer every time!

SECTION 3

The Clue-Word Decoder Chart

Here is your secret decoder chart! It shows the most common clue words and which math operation each one means. Study this chart — it will help you on the ISEE!

This chart shows the four main math operations and the clue words that match each one. Try to memorize at least two clue words for each operation!

Notice that some words can be tricky. The word "per" can mean multiplication or division, depending on the problem. Always read the whole sentence before deciding!

SECTION 4

How to Build an Equation from Words

Let's learn the step-by-step way to turn a sentence into an equation. An equation is a math sentence that uses numbers, symbols, and an equals sign. Here are some common patterns you'll see on the ISEE.

ADDITION PATTERN

Part + Part = Total

When a problem says "in all" or "altogether," you add the parts to find the total. Example: 7 apples plus 5 apples is 12 apples → 7 + 5 = 12

SUBTRACTION PATTERN

Total − Part = Difference

When a problem says "how many more" or "left over," you subtract. Example: 12 stickers take away 5 stickers is 7 stickers → 12 − 5 = 7

MULTIPLICATION PATTERN

Number of Groups × Size of Each Group = Total

When a problem says "each" or "every," you multiply. Example: 4 bags with 6 marbles each → 4 × 6 = 24

DIVISION PATTERN

Total ÷ Number of Groups = Size of Each Group

When a problem says "shared equally" or "split," you divide. Example: 24 cookies shared among 6 friends → 24 ÷ 6 = 4

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ISEE Test Tip

On the ISEE, there is no penalty for guessing. If you're stuck, use process of elimination — cross out answers that don't make sense, then pick from what's left. Always answer every question!

SECTION 5

From Words to Equation: A Step-by-Step Map

Let's see the whole process in a picture! This flowchart shows how to go from reading a word problem to writing a complete equation. Follow each arrow from start to finish.

Follow these five steps every time you see a word problem. With practice, you'll zoom through them!

Let's try an example with these steps. Imagine the problem says: "Sam has 8 toy cars. He gets 5 more for his birthday. How many toy cars does Sam have in all?"

Step 1: Read it. Sam starts with 8 cars and gets 5 more.
Step 2: Clue words: "more" and "in all" → addition!
Step 3: Operation: +
Step 4: Equation: 8 + 5 = ?
Step 5: Solve: 8 + 5 = 13. Sam has 13 toy cars!

SECTION 6

Worked Example: A Real ISEE-Style Problem

Let's walk through a problem just like you'd see on the ISEE. Follow along with each step!

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The Problem

Maria baked 36 cookies. She wants to share them equally among 9 friends. How many cookies will each friend get?

Step-by-Step Solution

1
Step 1 — Read the Problem Carefully

Maria has 36 cookies. She will share them equally among 9 friends. We need to find how many each friend gets.
2
Step 2 — Find the Clue Words

The clue words are "share equally" and "each." These tell us we need to divide!

Operation: Division (÷)
3
Step 3 — Find the Numbers

The total is 36 cookies. The number of groups is 9 friends.

Numbers: 36 and 9
4
Step 4 — Write the Equation

We put the total first, then the division sign, then the number of groups: 36 ÷ 9 = ?

Equation: 36 ÷ 9 = ?
5
Step 5 — Solve!

36 ÷ 9 = 4. Each friend gets 4 cookies. We can check: 9 × 4 = 36. ✓ It works!

Answer: 4 cookies each

SECTION 7

Watch Out for Tricky Words!

Some words on the ISEE can fool you if you're not careful. Let's look at a few tricky words and learn how to handle them.

Tricky Word	What It Seems Like	What It Really Means
"less than"	Subtract the second number from the first	Flip the order! "5 less than 12" means 12 − 5, not 5 − 12
"more than"	First number + second number	Also flip! "3 more than 10" means 10 + 3
"twice"	Something happens two times	Multiply by 2! "Twice 7" means 2 × 7 = 14
"how many more"	Maybe addition?	It's subtraction! Find the difference between two amounts

Common tricky words on ISEE word problems

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KEY TAKEAWAY

Think of "less than" and "more than" like giving directions. If someone says "the store is 3 blocks past the park," you start at the park. In math, "5 less than 12" means start at 12 and go down 5. Always ask yourself: where do I start?

SECTION 8

Building Toward Harder Problems

Now that you can translate one-step word problems, let's peek at what comes next. Some ISEE problems need two steps to solve. You use the same skills — you just do them twice!

One-Step Problem	Two-Step Problem
"Tom has 8 balls. He gets 4 more. How many in all?"	"Tom has 8 balls. He gets 4 more, then gives away 3. How many now?"
Equation: 8 + 4 = 12	Equations: 8 + 4 = 12, then 12 − 3 = 9
One clue word: "more"	Two clue words: "more" and "gives away"

One-step vs. two-step word problems

The great news is that two-step problems are just two one-step problems glued together. Solve the first step, use that answer in the second step, and you're done. You already have all the skills you need!

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ISEE Strategy

When you see a longer problem, break it into smaller pieces. Solve one piece at a time. Don't try to do everything in your head at once!

SECTION 9

Practice Problems

Time to practice! Remember: read the problem, find the clue words, pick the operation, and write the equation. You've got this!

Question 1: CONCEPTUAL

Emma has 7 pencils. She finds 5 more in her desk. Which equation shows how many pencils Emma has in all?

A. 7 − 5 = 2
B. 7 + 5 = 12
C. 7 × 5 = 35
D. 7 ÷ 5 = 1 R2

Show answer (B)

The correct answer is B. The clue words "more" and "in all" both signal addition. Emma starts with 7 pencils and finds 5 more, so we add: 7 + 5 = 12. Choice A uses subtraction, which would mean she lost pencils. Choice C uses multiplication, which would mean 7 groups of 5. Choice D uses division, which would mean splitting pencils into groups.

Question 2: BASIC CALCULATION

A baker made 48 muffins. He puts them into boxes of 8 muffins each. Which equation shows how to find the number of boxes?

A. 48 − 8 = 40
B. 48 + 8 = 56
C. 48 ÷ 8 = 6
D. 48 × 8 = 384

Show answer (C)

The correct answer is C. The clue is that the baker puts muffins into equal groups of 8 — this is division! 48 ÷ 8 = 6 boxes. Choice A (subtraction) would just remove 8 muffins. Choice B (addition) would add more muffins. Choice D (multiplication) would give a number much larger than the total, which doesn't make sense for splitting into groups.

Question 3: INTERMEDIATE

Lisa had some stickers. She gave 9 stickers to her friend and now has 15 stickers left. Which equation can be used to find how many stickers Lisa started with?

A. 9 + 15 = 24
B. 15 − 9 = 6
C. 15 × 9 = 135
D. 15 ÷ 9 = 1 R6

Show answer (A)

The correct answer is A. This one is tricky! Lisa "gave away" stickers, which usually means subtraction. But we need to find the original total, so we work backward. If she gave away 9 and has 15 left, she started with 9 + 15 = 24. Choice B subtracts again, which would give a number smaller than what she has now — that can't be right. Choices C and D use multiplication and division, which don't fit this story.

Question 4: APPLIED

There are 6 soccer teams in a league. Each team has 11 players. There are also 4 referees. Which expression shows the total number of people at the league?

A. (6 + 11) × 4
B. (6 × 11) + 4
C. (6 × 11) − 4
D. (6 + 11) + 4

Show answer (B)

The correct answer is B. This is a two-step problem! First, find the total number of players: 6 teams with 11 each means 6 × 11 = 66 players. Then add the 4 referees: 66 + 4 = 70 people. Choice A wrongly adds teams and players before multiplying. Choice C subtracts referees instead of adding them. Choice D adds 6 + 11 instead of multiplying, which doesn't account for all teams.

Question 5: CRITICAL THINKING

Jake's age is 3 more than twice his sister Mia's age. If Mia is 4 years old, which equation correctly shows Jake's age?

A. 3 + 4 = 7
B. 3 × 4 + 2 = 14
C. (2 × 4) + 3 = 11
D. (4 + 3) × 2 = 14

Show answer (C)

The correct answer is C. Let's decode this carefully. "Twice Mia's age" means 2 × 4 = 8. Then "3 more than" means we add 3 to that: 8 + 3 = 11. So Jake is 11 years old. Choice A only adds 3 + 4 and ignores "twice." Choice B multiplies by 3 instead of 2 and adds 2 instead of 3 — the numbers are switched. Choice D adds first and then multiplies, which changes the answer because order matters.

SUMMARY

Let's Review What You Learned!

Summary

You now know how to translate word relationships into equations! Start by reading the problem carefully and finding the clue words. Words like "in all" and "altogether" mean addition. Words like "left over" and "fewer" mean subtraction. Words like "each" and "groups of" mean multiplication. Words like "shared equally" and "split" mean division.

Watch out for tricky words like "less than" and "more than" — they flip the order of the numbers. Use the five-step process every time: Read, Find clue words, Pick the operation, Write the equation, and Solve. On the ISEE, always answer every question — there's no penalty for guessing. You've got this!