English:Integers and Rational Numbers
Integers and Rational Numbers
Introduction
Welcome to Integers and Rational Numbers. In this aiMOOC, you will extend your number sense from whole numbers to negative numbers, fractions, and decimals. You will learn how these numbers are related, how to place them on a number line, how to compare them, and how to calculate with them accurately.
These ideas matter far beyond the classroom. Temperatures can fall below zero, bank balances can increase or decrease, elevators move above and below ground level, and measurements are often expressed as fractions or decimals. By the end of the course, you should be able to model such situations with signed numbers and explain why your calculations make sense.
The diagram shows how familiar number sets fit inside larger sets. Every integer is a rational number because any integer can be written as a fraction with denominator 1.
Learning Goals
After working through this aiMOOC, you should be able to explain the difference between integers and rational numbers, locate and compare signed numbers on a number line, use absolute value, convert among fractions and decimals, perform the four arithmetic operations with rational numbers, apply sign rules, and solve multi-step problems in realistic contexts.
You should also be able to justify your reasoning with words, diagrams, estimates, and calculations rather than only giving a final answer.
Integers
An integer is zero, a positive whole number, or a negative whole number. Examples are −12, −3, 0, 5, and 81. Numbers such as 2.5 and 3/4 are not integers because they are not whole numbers.
The integers extend indefinitely in both directions. Positive integers are greater than zero, negative integers are less than zero, and zero is neither positive nor negative.
On a number line, numbers farther to the right are greater. For example, −2 is greater than −7 because −2 lies to the right of −7. This remains true even though 7 has a larger digit than 2.
Opposites and Absolute Value
Two numbers are opposites if they are the same distance from zero but lie on opposite sides of zero. The opposite of 6 is −6, and the opposite of −11 is 11. Zero is its own opposite.
The absolute value of a number is its distance from zero, so it is never negative. In symbols, |−8| = 8 and |5| = 5. Absolute value is useful when you care about the size of a change or distance but not its direction.
A common mistake is to think that a number with a larger absolute value is always greater. For negative numbers this is false: |−20| is greater than |−3|, but −20 is less than −3.
Rational Numbers
A rational number is any number that can be written in the form a/b, where a and b are integers and b is not zero. The denominator cannot be zero because division by zero is undefined.
Examples include 3/4, −7/5, 6, 0, 1.25, and 0.333.... The integer 6 is rational because 6 = 6/1. The terminating decimal 1.25 equals 5/4, and the repeating decimal 0.333... equals 1/3.
This fraction model represents three quarters. Fractions are one important representation of rational numbers.
Rational Numbers on the Number Line
Rational numbers fill the spaces between integers. Between 0 and 1 you can place 1/2, 1/4, 3/4, 0.1, 0.25, and infinitely many other rational numbers. In fact, between any two distinct rational numbers there is another rational number.
The number line also contains numbers that are not rational, such as √2. Those are called irrational numbers. For Grades 7–8, the key idea is that integers form part of the rational numbers, while rational and irrational numbers together belong to the real number system.
Fractions, Decimals, and Equivalent Forms
The same rational number can have many equivalent forms. For example, 1/2 = 2/4 = 0.5. Multiplying or dividing the numerator and denominator by the same nonzero integer creates an equivalent fraction.
To convert a fraction to a decimal, divide the numerator by the denominator. A rational number written as a decimal either terminates, such as 3/8 = 0.375, or eventually repeats, such as 2/3 = 0.666....
Fraction bars can help you see equivalence. Different pieces may represent the same total length even when their numerators and denominators differ.
Comparing and Ordering Rational Numbers
To compare rational numbers, first choose a representation that makes comparison easy. You can place them on a number line, convert them to decimals, or rewrite fractions using a common denominator.
For example, compare −3/4 and −0.6. Since −3/4 = −0.75, we compare −0.75 and −0.6. The number −0.75 lies farther left, so −3/4 < −0.6.
When both numbers are negative, the number closer to zero is greater. This is why −1/5 is greater than −2/3.
Adding and Subtracting Integers
For addition, signs and direction matter.
If two integers have the same sign, add their absolute values and keep the common sign. For example, −4 + −7 = −11.
If two integers have different signs, subtract the smaller absolute value from the larger absolute value and use the sign of the number with the larger absolute value. For example, −9 + 5 = −4.
Subtraction can be rewritten as addition of the opposite. For example, 6 − 10 = 6 + −10 = −4, while 6 − −10 = 6 + 10 = 16.
Adding and Subtracting Rational Numbers
The same sign ideas apply to fractions and decimals. For fractions, you also need a common denominator before adding or subtracting.
Example: −2/3 + 5/6. Rewrite −2/3 as −4/6. Then −4/6 + 5/6 = 1/6.
Example: 1.4 − 2.1 = −0.7. Thinking about the number line, you start at 1.4 and move 2.1 units to the left.
An estimate is a powerful check. If you calculate −5.8 + 2.2 and obtain a positive answer, the sign should make you suspicious because the negative amount has the larger absolute value.
Multiplying and Dividing Signed Numbers
For multiplication and division, the sign rule is consistent: numbers with the same sign produce a positive result, while numbers with different signs produce a negative result.
For example, −6 × −4 = 24, −6 × 4 = −24, and −24 ÷ 6 = −4.
For fractions, multiply numerators together and denominators together, then simplify. When dividing by a nonzero fraction, multiply by its reciprocal. For example, 3/5 ÷ −2/7 = 3/5 × −7/2 = −21/10.
Order of Operations
When an expression contains several operations, use the standard order of operations: work inside grouping symbols first, then evaluate exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.
For example, −3 + 2 × 5 = −3 + 10 = 7. Multiplication happens before addition.
Be especially careful with negative signs. The expressions −4² and (−4)² are not the same. In −4², the exponent applies to 4 first, giving −16. In (−4)², the entire negative number is squared, giving 16.
Real-World Applications
Signed rational numbers help describe quantities that can move above or below a reference point.
Temperature: A change from 3°C to −5°C is a decrease of 8°C.
Elevation: A diver at −12.5 m who rises 4 m reaches −8.5 m.
Money: A balance of €18.50 followed by a €25.00 payment becomes −€6.50 if no other funds are added.
Sports and games: Point gains and penalties can be represented with positive and negative values.
Science: Measurements may include positive and negative changes, while fractions and decimals express values between whole numbers.
When modeling a context, always decide what zero means and what positive and negative directions represent.
Problem-Solving Strategy
Before calculating, read the situation and identify the quantities, units, and direction of change. Choose a useful representation such as a number line, fraction, decimal, or signed expression. Estimate the size and sign of the answer, calculate carefully, and then compare the result with your estimate.
A correct answer should make sense in context. If a temperature falls by 7 degrees from 2°C, for example, the result must be below zero.
Common Misconceptions
Misconception: A negative number with larger digits is greater. Correction: On the number line, −12 is less than −4 because it lies farther left.
Misconception: Absolute value makes every number positive. Correction: Absolute value gives a nonnegative distance from zero; |0| is 0.
Misconception: A rational number must look like a fraction. Correction: Integers, terminating decimals, and repeating decimals can all be rational.
Misconception: Subtracting always makes a number smaller. Correction: Subtracting a negative number increases the value because it is equivalent to adding the opposite.
Misconception: Division by any rational number is allowed. Correction: Division by zero is never allowed.
Visual Enrichment: Fractions Are Dense
The following image shows Ford circles. Each visible circle corresponds to a reduced fraction. The picture is an enrichment activity rather than a required Grade 7–8 skill, but it offers a striking visual reminder that fractions can be arranged in rich patterns and that there are many rational numbers between whole numbers.
Ask yourself: What fractions can you recognize in the image? Which pairs have denominators that differ by only a small amount? How might the picture change if more fractions were included?
Interactive Tasks
Quiz: Test Your Knowledge
Which number is an integer but is neither positive nor negative? (Zero) (!One half) (!Three point five) (!Positive seven)
Which value is a rational number? (Three quarters) (!Square root of two) (!Pi) (!Square root of five)
Which comparison is correct? (Negative seven is less than negative three) (!Negative seven is greater than negative three) (!Negative seven equals negative three) (!Negative three is less than negative seven)
What is the absolute value of negative nine? (Nine) (!Negative nine) (!Zero) (!Eighteen)
What is negative four plus seven? (Three) (!Negative three) (!Eleven) (!Negative eleven)
What is five minus eight? (Negative three) (!Three) (!Thirteen) (!Negative thirteen)
What is negative six multiplied by negative three? (Eighteen) (!Negative eighteen) (!Nine) (!Negative nine)
What is negative twenty four divided by six? (Negative four) (!Four) (!Negative thirty) (!Thirty)
Which fraction is equal to zero point three seven five? (Three eighths) (!Three fifths) (!Five eighths) (!Seven eighths)
Which statement about rational-number decimals is correct? (They terminate or eventually repeat) (!They are always whole numbers) (!They never contain negative values) (!They must always terminate)
Memory Game
| Integer | A whole number that may be negative, zero, or positive |
| Rational number | A number expressible as a quotient of two integers with a nonzero denominator |
| Opposite | A value the same distance from zero on the other side of the number line |
| Absolute value | The distance of a number from zero |
| Numerator | The top part of a fraction |
| Denominator | The bottom part of a fraction that cannot be zero |
| Reciprocal | A fraction formed by interchanging numerator and denominator of a nonzero fraction |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Move right | Adding a positive quantity on a number line |
| Move left | Adding a negative quantity on a number line |
| Add the opposite | Rewriting a subtraction expression |
| Same signs | A positive product or quotient |
| Different signs | A negative product or quotient |
...
Crossword Puzzle
| Integer | What do you call a whole number that may be negative, zero, or positive? |
| Rational | What kind of number can be written as a quotient of two integers with a nonzero denominator? |
| Opposite | What do you call a number the same distance from zero on the other side? |
| Absolute | Which word begins the term for distance from zero? |
| Numerator | What is the top part of a fraction called? |
| Denominator | What is the bottom part of a fraction called? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Number line poster: Create a poster showing integers from −10 to 10 and add at least six rational numbers between the integers; explain how you chose their positions.
- Temperature diary: Record or invent temperatures for five times in a day, including at least one value below zero, and describe each change using signed numbers.
- Fraction and decimal cards: Make eight pairs of cards in which one card shows a fraction and the matching card shows an equivalent decimal; include both positive and negative values.
- Integer story: Write a short real-life story that can be modeled with at least three integer operations, then show and explain the calculation.
Standard
- Rational number interview: Interview a classmate or family member about where they use negative numbers, fractions, or decimals, then classify each example and explain what zero represents.
- Signed arithmetic tutorial: Produce a one-page guide or short video teaching how to add and subtract signed numbers, with diagrams and at least four original examples.
- Budget challenge: Design a one-week fictional budget with income, spending, and one negative balance; calculate the final balance and explain the effect of each transaction.
- Comparison investigation: Choose ten rational numbers in mixed forms, such as fractions and decimals, order them from least to greatest, and justify your ordering using two different methods.
Advanced
- Repeating decimal investigation: Explore at least six fractions whose decimal forms terminate or repeat, identify patterns in the results, and present a reasoned conjecture.
- Error analysis project: Create four believable incorrect solutions involving signed rational numbers, diagnose each error, and write a corrected explanation that would help another learner.
- Real-world data model: Collect a small data set involving gains and losses, temperature changes, elevations, or another signed quantity; represent it with rational numbers and analyze total and average change.
- Mathematical explainer video: Produce a three-to-five-minute video explaining why a negative multiplied by a negative is positive, using a pattern, model, or logical argument rather than memorized sign rules.
Learning Assessment
- Number-line reasoning: Place −1.4, 3/5, −7/4, 0, and 1.2 on a number line, order them, and explain how the positions support your inequalities.
- Representation transfer: Convert four rational numbers among fraction, decimal, and mixed-number forms, then explain which form is most useful for comparing each pair.
- Operation justification: Solve a multi-step expression containing addition, subtraction, multiplication, and division of signed rational numbers, and justify the sign of each intermediate result.
- Context modeling: Build and solve an equation or numerical expression for a situation involving a quantity that crosses zero, then interpret the answer with correct units.
- Misconception diagnosis: Analyze the claim that −8 is greater than −3 because 8 is greater than 3, identify the reasoning error, and correct it with a number-line argument.
- Strategy comparison: Solve the same rational-number comparison problem using decimals and common denominators, then evaluate which method is clearer and why.
Evidence of Learning
Knowledge: You can distinguish integers from other rational numbers, explain the role of zero, describe opposites and absolute value, and recognize terminating and repeating decimals as rational.
Skills: You can locate, compare, and order signed rational numbers; convert between fractions and decimals; calculate accurately with all four operations; apply the order of operations; estimate; and check whether an answer is reasonable.
Products: Your evidence may include annotated number lines, calculation records, posters, interviews, data displays, written explanations, budgets, or short instructional videos.
Reasoning: You can explain why a method works, identify and correct common errors, and use more than one representation when useful.
Transfer: You can select and apply signed rational numbers in unfamiliar contexts such as finance, measurement, temperature, elevation, science, or games.
OERs on the Topic
Use the following English Wikipedia article as an additional open reference for definitions, properties, examples, and links related to rational numbers.
Linked Learning Areas
Integers and rational numbers connect arithmetic with algebra, geometry, statistics, science, finance, and everyday quantitative reasoning. Number lines support later work with coordinates and inequalities. Fraction and decimal operations support proportional reasoning, percentages, probability, rates, equations, and measurement. Signed numbers prepare you for algebraic expressions, linear equations, graphs, and scientific quantities.