Integer Operations — Solve problems involving integer operations. - SHSAT Math

SECTION 1

Where Did Negative Numbers Come From?

Have you ever checked the weather and seen a temperature like −5°F? That little minus sign tells you it's below zero. But for thousands of years, people didn't believe numbers below zero could even exist! The story of integers (whole numbers that can be positive, negative, or zero) is a long journey through history.

Ancient Greek mathematicians actually rejected the idea of negative numbers. They thought math was about measuring real things — and how could you have less than nothing? It took centuries before people realized negative numbers were useful for tracking debts, temperatures, and much more.

200 BCE • China — First Negative Numbers

Chinese mathematicians used red rods for positive numbers and black rods for negative numbers to solve systems of equations.

628 CE • India — Brahmagupta's Rules

The Indian mathematician Brahmagupta wrote the first rules for adding, subtracting, and multiplying with negative numbers and zero.

1200s • Fibonacci Brings Ideas to Europe

Fibonacci introduced Eastern number ideas to Europe. Many European mathematicians still called negative numbers "absurd" or "fictitious."

1600s • Number Line Invented

John Wallis drew the first number line, placing negative numbers to the left of zero. This visual tool helped people accept negative numbers.

1800s • Full Acceptance

Mathematicians formally accepted integers as a complete number system. Today, integers are essential in science, finance, and everyday life.

So here's the big question this lesson answers: How do we add, subtract, multiply, and divide integers correctly every single time? On the SHSAT, you'll see integer problems everywhere. Let's build a rock-solid foundation.

SECTION 2

Core Principles of Integer Operations

Before we jump into calculations, let's nail down the key ideas. An integer is any whole number — positive, negative, or zero. Examples include −7, 0, 3, and 42. Fractions like ½ and decimals like 2.5 are not integers.

Absolute Value

The absolute value of a number is its distance from zero on the number line. It is always positive or zero. For example, |−5| = 5 and |5| = 5.

Sign Rules for Multiplication & Division

Same signs → positive result. Different signs → negative result. For example, (−3) × (−4) = +12, but (−3) × 4 = −12.

Additive Inverse

Every integer has an additive inverse — the number you add to it to get zero. The additive inverse of 7 is −7 because 7 + (−7) = 0.

Subtraction = Adding the Opposite

You can always rewrite subtraction as addition. For example, 5 − 8 is the same as 5 + (−8). This trick makes tricky problems much easier.

Order of Operations Still Applies

When a problem mixes operations, follow PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

✦

KEY TAKEAWAY

Think of integers like a tug-of-war game. Positive numbers pull to the right, and negative numbers pull to the left. When you add a positive and a negative number, the side with the stronger pull (larger absolute value) wins, and the answer takes that side's sign.

SECTION 3

The Number Line — Your Best Friend

The number line is the most powerful visual tool for understanding integers. It places zero in the center, with positive numbers stretching to the right and negative numbers stretching to the left. Let's look at how addition and subtraction work on it.

In Example A, we start at 2 and jump 3 spaces to the right, landing on 5. In Example B, we start at 3 and jump 5 spaces to the left (because we're adding a negative), landing on −2.

Here's the big idea: adding a positive number moves you right, and adding a negative number moves you left. Since subtraction is the same as adding the opposite, subtracting a positive moves you left, and subtracting a negative moves you right. If you can picture the number line in your head, you can check your answers quickly on the SHSAT.

SECTION 4

The Rules — Written as Formulas

Let's organize the rules for all four operations into clear formulas. These are the rules you should memorize for the test.

Addition Rules

SAME SIGN ADDITION

a + b = +(|a| + |b|) when both a and b are positive a + b = −(|a| + |b|) when both a and b are negative

When two numbers share the same sign, add their absolute values and keep the shared sign. Example: (−4) + (−6) = −(4 + 6) = −10.

DIFFERENT SIGN ADDITION

a + b = sign of larger | | × (|larger| − |smaller|)

When the signs are different, subtract the smaller absolute value from the larger one. The answer takes the sign of the number with the bigger absolute value. Example: (−9) + 4 = −(9 − 4) = −5.

Subtraction Rule

SUBTRACTION AS ADDITION

a − b = a + (−b)

Change the subtraction sign to addition and flip the sign of the second number. Then follow the addition rules. Example: 5 − (−3) = 5 + 3 = 8.

Multiplication & Division Sign Rules

SIGN RULES FOR × AND ÷

(+) × (+) = (+) (+) ÷ (+) = (+) (−) × (−) = (+) (−) ÷ (−) = (+) (+) × (−) = (−) (+) ÷ (−) = (−) (−) × (+) = (−) (−) ÷ (+) = (−)

Quick memory trick: Same signs → positive result. Different signs → negative result. This rule is identical for multiplication and division.

💡

SHSAT Tip

Count the number of negative signs in a multiplication or division problem. If the count is even, the answer is positive. If the count is odd, the answer is negative. Example: (−2) × (−3) × (−1) has three negatives (odd), so the answer is negative: −6.

SECTION 5

Sign Rules — A Complete Visual Guide

The sign rules can feel confusing at first, but they follow a clean pattern. The diagram below shows every combination of signs for all four operations. Study it like a cheat sheet!

This diagram summarizes every sign rule you need. Notice that multiplication and division follow the exact same sign pattern. Subtraction always converts to addition first.

Operation	Example	Step-by-Step	Result
Addition (same signs)	(−3) + (−8)	Add: 3 + 8 = 11; both negative → −11	−11
Addition (diff. signs)	7 + (−10)	Subtract: 10 − 7 = 3; 10 is bigger and negative → −3	−3
Subtraction	4 − (−6)	Rewrite: 4 + 6 = 10	10
Multiplication	(−5) × (−7)	Multiply: 5 × 7 = 35; same signs → positive	35
Division	(−24) ÷ 6	Divide: 24 ÷ 6 = 4; different signs → negative	−4

Examples of all four integer operations

SECTION 6

Worked Example — Multi-Step Integer Problem

On the SHSAT, you'll often see problems that combine multiple integer operations. Let's work through one step by step.

ℹ

PROBLEM

Simplify: (−3) × (4 − 7) + (−12) ÷ (−4)

Solving (−3) × (4 − 7) + (−12) ÷ (−4)

1
Step 1 — Parentheses First

Look inside the parentheses: 4 − 7. Rewrite subtraction as addition: 4 + (−7). Since the signs are different, subtract: 7 − 4 = 3. The larger absolute value (7) is negative, so the result is −3.

4 − 7 = −3
2
Step 2 — Rewrite with Simplified Parentheses

Replace (4 − 7) with −3. The expression is now: (−3) × (−3) + (−12) ÷ (−4).

(−3) × (−3) + (−12) ÷ (−4)
3
Step 3 — Multiplication (left to right before addition)

Multiply (−3) × (−3). The absolute values: 3 × 3 = 9. Both signs are negative (same sign), so the result is positive.

(−3) × (−3) = +9
4
Step 4 — Division (same priority as multiplication)

Divide (−12) ÷ (−4). The absolute values: 12 ÷ 4 = 3. Both signs are negative (same sign), so the result is positive.

(−12) ÷ (−4) = +3
5
Step 5 — Addition (final step)

Now add: 9 + 3. Both numbers are positive, so we simply add.

9 + 3 = 12

✦

KEY TAKEAWAY

Think of PEMDAS like a recipe. You have to follow the steps in order — you can't frost a cake before you bake it! Always handle parentheses first, then multiplication and division from left to right, and finally addition and subtraction from left to right.

SECTION 7

Common Mistakes and How to Avoid Them

Integer operations are straightforward once you know the rules, but there are traps that catch students on the SHSAT every year. Let's look at the most common mistakes and how to fix them.

Common Mistake	What Students Do Wrong	How to Fix It
Subtracting a negative	Write 5 − (−3) = 2, forgetting to flip the sign.	Rewrite as 5 + 3 = 8. Subtracting a negative always becomes addition.
Wrong sign on the answer	Write (−4) + 9 = −5 instead of +5.	The larger absolute value is 9 (positive). The answer must be positive: +5.
Multiplying before parentheses	In 2 × (3 + 5), they do 2 × 3 first to get 6 + 5 = 11.	Parentheses first! 3 + 5 = 8, then 2 × 8 = 16.
Confusing −x² with (−x)²	Think −3² = 9, but it actually equals −9.	−3² means −(3²) = −9. Only (−3)² = 9 because the parentheses include the negative.
Not going left to right	In 12 ÷ 3 × 2, they do 3 × 2 first to get 12 ÷ 6 = 2.	Multiplication and division are equal priority — go left to right. 12 ÷ 3 = 4, then 4 × 2 = 8.

Top 5 integer operation mistakes on the SHSAT

🎯

SHSAT STRATEGY

Before picking an answer, do a quick sign check. Ask yourself: "Should this answer be positive or negative?" You can often eliminate two or three answer choices just by knowing the sign. It's like narrowing down a multiple-choice question before you even finish calculating.

SECTION 8

Connecting Integers to Bigger Ideas

Integer operations are the building blocks for nearly everything else in math. Once you master them, you're ready to tackle more advanced topics. Let's see how integers connect to what's coming next.

Concept Now (Integers)	Where It Leads
Adding and subtracting integers	Solving one-step and two-step equations (e.g., x + 5 = −3)
Multiplying and dividing integers	Working with algebraic expressions (e.g., −2(x − 4))
Order of operations with integers	Evaluating complex expressions and functions
Number line understanding	Coordinate plane graphing (x-axis and y-axis use integers)
Sign rules for multiplication	Understanding slopes (positive vs. negative) in linear equations

How integer skills connect to algebra and beyond

On the SHSAT specifically, integer operations show up in arithmetic problems, algebra problems, and even geometry problems (when coordinates or measurements are negative). The sign rules you learn here will be used over and over. Think of integers as the foundation of every math topic on the test.

ℹ

Looking Ahead

When you start working with rational numbers (fractions and decimals that can be negative), all the same sign rules apply. The only new challenge is the fraction or decimal arithmetic itself. So mastering integer signs now makes rational number operations much easier later!

SECTION 9

Practice Problems

Time to test yourself! These five problems go from easy to challenging. Try each one before reading the answer. Write out your work on scrap paper, just like you would on test day.

Question 1: CONCEPTUAL

Is the product of three negative integers positive or negative? Explain why.

Show answer

The product of three negative integers is negative. When you multiply the first two negatives, you get a positive (same signs → positive). Then you multiply that positive result by the third negative, and different signs give a negative result. In general, an odd number of negative factors produces a negative product, and an even number produces a positive product.

Question 2: BASIC CALCULATION

Evaluate: (−8) + 15 + (−4)

Show answer

Work left to right. First, (−8) + 15: the signs are different, so subtract the absolute values: 15 − 8 = 7. Since 15 has the larger absolute value and is positive, the result is +7. Next, 7 + (−4): different signs again, so 7 − 4 = 3. The larger absolute value (7) is positive, so the result is +3.

Question 3: INTERMEDIATE

Simplify: (−6) × 3 − (−2) × (−5)

Show answer

Follow order of operations — multiplication comes before subtraction. First, (−6) × 3 = −18 (different signs → negative). Second, (−2) × (−5) = +10 (same signs → positive). Now the expression is −18 − 10. Rewrite subtraction as addition: −18 + (−10). Both are negative, so add absolute values: 18 + 10 = 28, and keep the negative sign. The answer is −28.

Question 4: APPLIED

At 6 AM, the temperature was −7°F. By noon, it had risen 18 degrees. By midnight, it had dropped 23 degrees from the noon temperature. What was the temperature at midnight?

Show answer

Start at −7°F. A rise of 18 degrees means add 18: −7 + 18 = 11°F at noon (different signs, 18 − 7 = 11, positive because 18 is larger). Then a drop of 23 degrees means subtract 23: 11 − 23 = 11 + (−23) = −12°F. Subtract absolute values: 23 − 11 = 12, and 23 is larger with a negative sign. The temperature at midnight was −12°F.

Question 5: CRITICAL THINKING

If a × b > 0 and a + b < 0, what must be true about the signs of a and b? Explain your reasoning.

Show answer

The condition a × b > 0 means the product is positive. This happens when a and b are both positive or both negative (same signs). The condition a + b < 0 means the sum is negative. If both were positive, their sum would be positive — that contradicts a + b < 0. So both a and b must be negative. Two negative numbers multiply to give a positive product, and they add to give a negative sum. This type of reasoning (using sign rules to narrow down possibilities) is a powerful SHSAT strategy.

SUMMARY

Integer Operations — Key Concepts Review

Summary

Integers are whole numbers that can be positive, negative, or zero. For addition, same signs mean you add the absolute values and keep the sign; different signs mean you subtract the absolute values and take the sign of the larger one. Subtraction is simply rewritten as adding the opposite: a − b = a + (−b). For multiplication and division, same signs give a positive result and different signs give a negative result.

Always follow PEMDAS (order of operations) when a problem has more than one step. Watch out for common traps like subtracting a negative (which becomes addition) and the difference between −x² and (−x)². Use the number line to visualize problems and double-check your signs. These skills are the foundation for algebra, coordinate geometry, and nearly every other SHSAT math topic.