Solve for an unknown in a simple equation. - ISEE Lower Level Mathematics Achievement

SECTION 1

Where Did Equations Come From?

People have been solving for missing numbers for thousands of years! Long ago, farmers needed to figure out things like: "I had some sheep. 5 ran away. Now I have 7. How many did I start with?" That's really just solving an equation!

An equation is a math sentence that uses an equals sign (=). It says two sides are the same amount. Let's see how this idea grew over time.

1800 BC • Ancient Babylon

People in Babylon (modern-day Iraq) wrote missing-number puzzles on clay tablets. They didn't use letters yet!

300 AD • Diophantus of Alexandria

A Greek mathematician started using symbols to stand for unknown numbers. He's called the "Father of Algebra."

820 AD • Al-Khwarizmi's Big Book

A scholar in Baghdad wrote a famous book about solving equations. The word "algebra" comes from the title of his book!

1557 • The Equals Sign Is Born

Robert Recorde invented the = sign. He said two parallel lines are the most equal things he could think of!

Today, you use equations all the time! When you figure out how much more money you need to buy a toy, you're solving an equation. Let's learn how to do it step by step.

SECTION 2

What You Need to Know First

Before we start solving, let's learn a few important ideas. These are like the rules of the game!

What Is an Equation?

An equation is a math sentence with an equals sign. It means the left side and the right side have the same value. Example: 3 + 4 = 7.

What Is the Unknown?

The unknown is the missing number you need to find. It might look like a box (□), a question mark (?), or a letter like n.

The Balance Rule

Both sides of the equals sign must balance — just like a seesaw. Whatever is on the left must equal what is on the right.

Opposite Operations

Addition and subtraction undo each other. Multiplication and division undo each other. We use this to find the unknown!

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KEY TAKEAWAY

Think of an equation like a balanced seesaw on a playground. If you take something off one side, the seesaw tips. To keep it balanced, you must do the same thing to both sides. That's exactly how equations work!

SECTION 3

See It: The Balance Model

Let's picture the equation n + 3 = 8 as a balance scale. The left side and right side must weigh the same!

The balance scale shows the equation n + 3 = 8. The left pan holds n and 3. The right pan holds 8. Since we need to "undo" adding 3, we subtract 3 from both sides. That leaves n = 5.

Notice how we always check our answer at the end. Plug 5 back in: 5 + 3 = 8. It works! Always check by putting your answer back into the original equation.

SECTION 4

The Opposite Operation Trick

Here's the big secret to solving equations: use the opposite operation to get the unknown all by itself. Addition and subtraction are opposites. Multiplication and division are opposites.

ADDITION EQUATION

n + 5 = 12 → n = 12 − 5 → n = 7

The equation has + 5, so we use the opposite: − 5. Subtract 5 from both sides to find n.

SUBTRACTION EQUATION

n − 4 = 9 → n = 9 + 4 → n = 13

The equation has − 4, so we use the opposite: + 4. Add 4 to both sides to find n.

MULTIPLICATION EQUATION

n × 3 = 15 → n = 15 ÷ 3 → n = 5

The equation has × 3, so we use the opposite: ÷ 3. Divide both sides by 3 to find n.

DIVISION EQUATION

n ÷ 2 = 6 → n = 6 × 2 → n = 12

The equation has ÷ 2, so we use the opposite: × 2. Multiply both sides by 2 to find n.

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ISEE Test Tip

On the ISEE, you can also try each answer choice! Plug each one into the equation and see which one makes both sides equal. This is called backsolving, and it's a great trick when you're stuck.

SECTION 5

Four Types of Missing-Number Equations

On the ISEE, you'll see different kinds of equations. The unknown might show up in different spots. Let's look at all four types and how to handle each one.

This chart shows the four types of simple equations. Each type uses a different operation, and each is solved by using the opposite operation. Memorize these four patterns and you'll be ready!

Sometimes the ISEE puts the unknown in a different spot. For example: 15 − □ = 9. That's okay! You can still think: "What number do I subtract from 15 to get 9?" The answer is 15 − 9 = 6. So □ = 6.

SECTION 6

Worked Example: Step by Step

Let's solve a problem together, just like you'd see on the ISEE. Follow each step carefully!

If n × 4 = 36, what is the value of n?

1
Step 1 — Read the Equation

The equation says: some number (n) times 4 equals 36. We need to find n.
2
Step 2 — Identify the Operation

The unknown n is being multiplied by 4. So we need the opposite operation.
3
Step 3 — Use the Opposite Operation

The opposite of multiplication is division. So we divide both sides by 4.

n = 36 ÷ 4
4
Step 4 — Calculate

36 ÷ 4 = 9. So n = 9.

n = 9
5
Step 5 — Check Your Answer

Plug 9 back in: 9 × 4 = 36. ✓ Both sides match! We got it right!

9 × 4 = 36 ✓

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ISEE Test Tip

If you forget the opposite operation trick, try each answer choice! For n × 4 = 36, test each option: does 7 × 4 = 36? No. Does 8 × 4 = 36? No. Does 9 × 4 = 36? Yes! That's your answer.

SECTION 7

Three Ways to Solve: Pick Your Favorite!

There are different strategies you can use to find the unknown. Each one works great! Let's compare them so you can pick your favorite.

Strategy	How It Works	Best For
Opposite Operation	Undo what's done to the unknown. If the equation has + 5, subtract 5.	All equation types. This is the fastest method!
Backsolving	Try each answer choice in the equation. The one that makes both sides equal is correct.	When you're not sure how to solve. Great ISEE strategy!
Fact Families	Think of related facts. If 7 + 5 = 12, then 12 − 5 = 7 and 12 − 7 = 5.	Addition and subtraction problems with smaller numbers.

Three strategies for solving equations

✦

KEY TAKEAWAY

Think of backsolving like trying on shoes. You don't need to know your exact shoe size — just try a few and see which one fits! On the ISEE, you always have four choices to try. One of them will make the equation true.

SECTION 8

From Simple Equations to Bigger Ideas

Right now you're solving one-step equations. That means you only need one step to find the answer. As you grow as a math student, equations will get a little bigger — but the same rules apply!

What You Know Now	What Comes Next
n + 5 = 12 (one step)	2 × n + 5 = 17 (two steps)
The unknown is shown as □ or n	The unknown uses letters like x and y
Use opposite operations	Still use opposite operations — same idea!
Check by plugging in your answer	Still check by plugging in — always works!

How simple equations lead to bigger math

The awesome thing is that the opposite operation trick you're learning right now will be used all the way through high school and college math. You're building a super important skill!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own first. Remember: find the operation, use the opposite, then check. You've got this!

Question 1: CONCEPTUAL

If n + 6 = 10, what is the value of n?

A. 3
B. 4
C. 6
D. 16

Show answer (B)

The correct answer is B (4). The equation has + 6, so use the opposite: subtract 6 from both sides. n = 10 − 6 = 4. Check: 4 + 6 = 10 ✓. Choice A (3) gives 3 + 6 = 9, which is too small. Choice C (6) gives 6 + 6 = 12, which is too big. Choice D (16) adds 10 + 6 instead of subtracting — be careful not to add when you should subtract!

Question 2: BASIC CALCULATION

What number makes this equation true? □ − 8 = 15

A. 7
B. 13
C. 23
D. 25

Show answer (C)

The correct answer is C (23). The equation has − 8, so use the opposite: add 8 to both sides. □ = 15 + 8 = 23. Check: 23 − 8 = 15 ✓. Choice A (7) comes from subtracting 15 − 8 instead of adding. Choice B (13) comes from adding 8 + 5 (a common mistake reading 15). Choice D (25) comes from adding 8 + 17 or a calculation error.

Question 3: INTERMEDIATE

If n × 6 = 42, what is n?

A. 6
B. 7
C. 8
D. 36

Show answer (B)

The correct answer is B (7). The equation has × 6, so use the opposite: divide both sides by 6. n = 42 ÷ 6 = 7. Check: 7 × 6 = 42 ✓. Choice A (6) gives 6 × 6 = 36, not 42. Choice C (8) gives 8 × 6 = 48, not 42. Choice D (36) comes from subtracting 42 − 6 instead of dividing — remember, the opposite of multiplication is division!

Question 4: APPLIED

Sara had some stickers. She gave 12 stickers to her friend. Now she has 27 stickers left. How many stickers did Sara start with?

A. 15
B. 29
C. 37
D. 39

Show answer (D)

The correct answer is D (39). Write the equation: n − 12 = 27. The equation has − 12, so add 12 to both sides: n = 27 + 12 = 39. Check: 39 − 12 = 27 ✓. Choice A (15) comes from subtracting 27 − 12 instead of adding. Choice B (29) comes from adding 27 + 2 (misreading 12). Choice C (37) comes from adding 27 + 10 instead of 27 + 12.

Question 5: CRITICAL THINKING

If 56 ÷ □ = 8, what number goes in the box?

A. 6
B. 7
C. 8
D. 48

Show answer (B)

The correct answer is B (7). This one is tricky because the unknown is the divisor (the number you divide by). Think: "56 divided by what number gives 8?" You can also think of it as 8 × □ = 56. What times 8 equals 56? That's 7, since 8 × 7 = 56. Check: 56 ÷ 7 = 8 ✓. Choice A (6) gives 56 ÷ 6 = 9 remainder 2. Choice C (8) gives 56 ÷ 8 = 7, which puts the answer on the wrong side. Choice D (48) comes from subtracting 56 − 8 instead of dividing.

SUMMARY

Lesson Summary

Summary

An equation is a math sentence with an equals sign. The unknown is the missing number you need to find. To solve, use the opposite operation: addition undoes subtraction, and multiplication undoes division. Think of the equation like a balanced seesaw — both sides must always stay equal.

Always check your answer by plugging it back into the original equation. On the ISEE, you can also try backsolving — test each answer choice to see which one makes the equation true. Remember, there's no penalty for guessing, so always pick an answer!