Use variables to represent quantities in a situation. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Where Did Variables Come From?

Have you ever played a guessing game? Imagine someone says, "I'm thinking of a number. If you add 3 to it, you get 10." You'd figure out the number is 7! People have been solving puzzles like this for thousands of years. Long ago, mathematicians needed a way to write down "the mystery number" without knowing what it was yet. That's how variables were born!

1800 BC • Ancient Babylon

People in Babylon solved word problems about land and crops. They used words like "the thing" for unknown amounts.

250 AD • Diophantus of Alexandria

A Greek mathematician started using a special symbol to stand for an unknown number. He is sometimes called the "Father of Algebra."

820 AD • Al-Khwarizmi

An Arab scholar wrote the first algebra textbook. The word "algebra" comes from the Arabic title of his book!

1637 • René Descartes

A French thinker started using letters like x, y, and z for unknowns. This is why we still use letters today!

So here's the big idea: a variable is just a letter that holds the place for a number we don't know yet. On the ISEE, you'll see letters like n, x, or y used to stand for a mystery number. Let's learn how to use them!

SECTION 2

Core Principles: What Are Variables?

A variable is a letter that represents a quantity — a number we want to find. Think of it like a box with a question mark inside. Once you figure out the number, you can replace the letter. Here are the key ideas you need to know.

A Variable Is a Placeholder

A letter like n stands for a number you don't know yet. It's like a blank in a fill-in-the-blank sentence.

Variables Can Be Any Letter

You might see x, y, n, or even a. The letter doesn't change the math. It just names the mystery number.

Words Become Math

"A number plus 4" becomes n + 4. You translate words into math using variables.

One Variable = One Unknown

On the ISEE, you usually only have one mystery number. Your job is to set up or solve for it.

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KEY TAKEAWAY

Think of a variable like a name tag at a party. Before you meet someone, the tag just says "Hello, my name is ___". The blank holds a spot for a real name. A variable is a blank that holds a spot for a real number. Once you figure out the number, you fill in the blank!

SECTION 3

See It! Variables in Action

Let's look at a picture that shows how a word problem becomes a math expression using a variable. Follow the arrows from the story to the math!

This diagram shows how "Sam has some apples" becomes n. When you add information ("gets 5 more"), you build the expression n + 5. If you know the total, you can write a full equation like n + 5 = 12.

See how each part of the story matches a part of the math? The word "some" tells us we don't know the number. That's our signal to use a variable! "Gets 5 more" tells us to add 5. And the total gives us the other side of the equation.

SECTION 4

Turning Words Into Math

On the ISEE, you'll see word problems that use everyday language. Your job is to figure out what math operation the words describe. Here are the most common translations from words to math symbols.

ADDITION WORDS

n + 5

Words like "more than," "added to," "increased by," or "total" mean you add. Example: "5 more than a number" → n + 5.

SUBTRACTION WORDS

n − 3

Words like "less than," "fewer," "decreased by," or "take away" mean you subtract. Example: "3 less than a number" → n − 3.

MULTIPLICATION WORDS

4 × n

Words like "times," "each," "per," or "groups of" mean you multiply. Example: "4 times a number" → 4 × n.

EQUALS WORDS

n + 5 = 12

Words like "is," "equals," "the result is," or "gives" tell you where the equals sign goes. Example: "A number plus 5 is 12" → n + 5 = 12.

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ISEE Test Tip

On the ISEE, always read the whole problem first. Circle or underline the key words that tell you which operation to use. Then write the math expression. This helps you avoid careless mistakes!

SECTION 5

Word-to-Variable Cheat Sheet

Let's organize the clue words you'll see on the ISEE into a handy chart. Think of these as a decoder ring for word problems!

Use this decoder to match word clues with math operations. On test day, look for these words to decide whether to add, subtract, or multiply!

Here's a tip: watch out for "less than" because it flips the order! "4 less than n" means n − 4, not 4 − n. You start with n and then take away 4. The ISEE loves to test this tricky spot!

SECTION 6

Let's Solve One Together!

Here's a problem just like the ones you'll see on the ISEE. Let's work through it step by step. You've got this!

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Sample Problem

Maria has some stickers. Her friend gives her 8 more stickers. Now Maria has 20 stickers in all. Which equation shows how to find the number of stickers Maria started with?

Step-by-Step Solution

1
Step 1 — Find the Unknown

What don't we know? We don't know how many stickers Maria started with. Let's call that mystery number n.

n = number of stickers Maria started with
2
Step 2 — Spot the Clue Words

"Gives her 8 more" means we add 8. "Now she has 20 in all" means the total equals 20.

"more" → add, "in all" → equals
3
Step 3 — Write the Equation

Start with what Maria had (n). Add 8. Set it equal to 20.

n + 8 = 20
4
Step 4 — Check by Solving

To find n, think: what number plus 8 equals 20? That's 12! Check: 12 + 8 = 20. ✓ It works!

n = 12

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PRO TIP FOR THE ISEE

After you set up the equation, you can often solve it by working backward. Ask yourself: "What number makes this equation true?" You can also try each answer choice to see which one works. Since there's no penalty for guessing on the ISEE, always pick an answer — even if you're not 100% sure!

SECTION 7

Strategies: When to Use Each Approach

On the ISEE, variable problems come in different forms. Some ask you to write an expression, some ask you to write an equation, and some ask you to find the value of the variable. Here's how to tell which is which.

What the Question Asks	What You Do	Example
"Which expression shows…"	Write the math without solving. No equals sign needed.	n + 5
"Which equation shows…"	Write the math with an equals sign and a total.	n + 5 = 12
"What is the value of n?"	Set up the equation AND solve for the number.	n = 7
"If n = 4, what is…"	Plug in the given number and calculate.	Replace n with 4, then compute: 4 + 5 = 9

Four types of variable questions on the ISEE

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STRATEGY SPOTLIGHT

Think of it like a recipe. An expression is a list of ingredients (n + 5). An equation is the full recipe with the final dish (n + 5 = 12). Solving is figuring out how much of each ingredient you need (n = 7).

SECTION 8

Variables: Just the Beginning!

Right now, you're learning to use variables in simple situations. But guess what? Variables are the building blocks for ALL of algebra! Here's a peek at how what you're learning now connects to bigger math later.

What You Learn Now	What It Leads To
Use a letter for one unknown number	Use letters for two unknowns (like x and y together)
Write expressions like n + 5	Write formulas like Area = length × width
Solve simple equations like n + 8 = 20	Solve multi-step equations like 3n + 8 = 20
Translate one word problem	Model real-world situations with equations and graphs

How today's skills build toward future math

Every time you use a variable on the ISEE, you're practicing a skill that will help you all through school. So give yourself a high five — you're building a strong math foundation right now! 🎉

SECTION 9

Practice Problems

Time to practice! Try each problem on your own first. Remember: on the ISEE, there is no penalty for a wrong answer, so always pick the best answer you can. Let's go!

Question 1: CONCEPTUAL

Jake has some baseball cards. Which letter could be used to stand for the number of baseball cards Jake has?

A. +
B. =
C. n
D. 5

Show answer (C)

The correct answer is C. A variable is a letter that stands for an unknown number. The letter n is a variable. Choice A (+) is an operation symbol, not a variable. Choice B (=) is an equals sign, not a variable. Choice D (5) is a specific number, not a variable — a variable must be a letter.

Question 2: BASIC CALCULATION

Emma has some pencils. Her teacher gives her 6 more. Which expression shows the total number of pencils Emma has now?

A. n − 6
B. n + 6
C. 6 × n
D. 6 − n

Show answer (B)

The correct answer is B. Emma starts with some pencils (n) and gets 6 more. "More" means we add, so the expression is n + 6. Choice A (n − 6) means taking away 6, but she's getting more, not losing them. Choice C (6 × n) means 6 groups of n, which doesn't match the story. Choice D (6 − n) subtracts n from 6, which reverses the situation.

Question 3: INTERMEDIATE

A baker made some muffins in the morning. She sold 9 of them. She had 15 muffins left. Which equation can be used to find the number of muffins she made?

A. n + 9 = 15
B. n − 9 = 15
C. 9 × n = 15
D. n − 15 = 9

Show answer (B)

The correct answer is B. The baker started with n muffins and sold (took away) 9. She had 15 left. So n − 9 = 15. You can check: 24 − 9 = 15. ✓ Choice A adds 9 instead of subtracting. Choice C multiplies, but there's no grouping in this problem. Choice D subtracts 15 from n and equals 9, which gives n = 24 too, but the equation doesn't match the story (she subtracted 9, not 15). On the ISEE, pick the equation that best matches the story's order.

Question 4: APPLIED

At a pet store, each fish tank holds the same number of fish. There are 4 tanks. There are 32 fish in all. If n stands for the number of fish in each tank, which equation can be used to find n?

A. n + 4 = 32
B. n − 4 = 32
C. 4 × n = 32
D. 4 + n = 32

Show answer (C)

The correct answer is C. There are 4 tanks with the same number of fish (n) in each. "Each" is a clue for multiplication. So 4 groups of n fish equals 32 total: 4 × n = 32. Check: 4 × 8 = 32, so n = 8. Choice A (n + 4 = 32) and Choice D (4 + n = 32) use addition, but equal groups means multiplication. Choice B (n − 4 = 32) uses subtraction, which doesn't match the story at all.

Question 5: CRITICAL THINKING

Liam had some marbles. He gave 7 marbles to his sister and then found 3 more marbles on the ground. Now he has 16 marbles. Which equation shows how to find the number of marbles Liam started with?

A. n − 7 + 3 = 16
B. n + 7 − 3 = 16
C. n + 7 + 3 = 16
D. n − 7 − 3 = 16

Show answer (A)

The correct answer is A. This is a two-step problem. Liam started with n marbles. First, he gave away 7 (subtract 7). Then he found 3 more (add 3). The result is 16. So the equation is n − 7 + 3 = 16. Check: n − 7 + 3 = 16 means n − 4 = 16, so n = 20. Verify: 20 − 7 + 3 = 13 + 3 = 16. ✓ Choice B reverses the operations (adds 7 and subtracts 3). Choice C adds both 7 and 3, but Liam gave away 7 (that's subtraction). Choice D subtracts both, but Liam found 3 more (that's addition).

SUMMARY

Let's Wrap It Up!

Summary

A variable is a letter (like n, x, or y) that stands for a number you don't know yet. You use clue words in word problems — like "more than" (add), "less than" (subtract), and "times" or "each" (multiply) — to translate stories into expressions and equations. An expression (like n + 5) has no equals sign. An equation (like n + 5 = 12) has an equals sign and can be solved.

On the ISEE, read each problem carefully, find the unknown, pick a variable, and match the clue words to math operations. Remember: there is no penalty for guessing, so always choose an answer! You can check your work by plugging your answer back in to see if it makes the equation true. You've got the tools — now go crush it! 🌟