Unit 4 — Proportional Relationships, Rational Numbers, and Algebraic Thinking

Welcome to Grade 7 Mathematics!

Mathematics allows us to recognize patterns, describe relationships, and solve real-world problems.

In Grade 7, you will move beyond arithmetic toward deeper algebraic reasoning by studying proportional relationships, rational numbers, variables, and mathematical models.

These ideas are used every day by scientists, engineers, economists, programmers, architects, and many other professionals.

Proportional Relationships

Two quantities are proportional if they maintain a constant ratio.

Example:

If one notebook costs $3:

• 2 notebooks cost $6
• 5 notebooks cost $15
• 10 notebooks cost $30

The ratio remains constant.

Recognizing proportional relationships helps solve many practical problems.

Unit Rates

A unit rate compares a quantity to one unit.

Examples:

• 60 miles per hour
• $2.50 per pound
• 120 words per minute
• 30 frames per second

Unit rates make comparisons easier and are widely used in everyday decision-making.

Rational Numbers

A rational number is any number that can be written as a fraction of two integers.

Examples include:

• positive and negative whole numbers
• fractions
• terminating decimals
• repeating decimals

Examples:

• 7
• -12
• 3/4
• -5/8
• 0.25
• 0.333...

Rational numbers allow us to describe many kinds of measurements and quantities.

Operations with Rational Numbers

You should become comfortable:

• adding rational numbers
• subtracting rational numbers
• multiplying rational numbers
• dividing rational numbers

Always pay careful attention to positive and negative signs.

Estimate your answer before calculating to check whether your result is reasonable.

Variables and Expressions

Variables represent unknown or changing quantities.

Examples:

• x
• y
• n
• a

Expressions combine:

• variables
• numbers
• mathematical operations

Examples:

• 3x + 5
• 2(y − 4)
• 7a − 9

Expressions describe mathematical relationships.

Solving Multi-Step Equations

Many equations require several steps.

Example:

3x + 8 = 26

Subtract 8 from both sides:

3x = 18

Divide both sides by 3:

x = 6

Always check your solution by substituting it back into the original equation.

Inequalities

An inequality compares quantities that are not necessarily equal.

Examples:

• x > 12
• y ≤ 5
• n ≥ -3

Many real-world situations involve ranges of possible values rather than one exact answer.

Mathematical Models

Mathematics helps represent real situations.

Examples include:

• travel time
• shopping costs
• population growth
• scientific measurements
• business planning
• engineering design

Building mathematical models allows us to solve practical problems efficiently.

Activity

Complete the following:

• solve ten proportional reasoning problems
• calculate ten unit rates
• solve fifteen multi-step equations
• graph five inequalities
• write five equations representing real-world situations

Explain the reasoning behind each solution.

Practice

Answer these questions.

What is a proportional relationship?

Why are unit rates useful?

What is a rational number?

What does a variable represent?

How do you solve a multi-step equation?

Why are mathematical models important?

Review

Today you learned:

• proportional relationships maintain constant ratios
• unit rates simplify comparisons
• rational numbers include fractions and many decimals
• variables represent unknown quantities
• multi-step equations require logical reasoning
• inequalities describe ranges of values
• mathematical models solve practical problems

These concepts strengthen algebraic thinking and prepare you for increasingly advanced mathematics.

Looking Ahead

Next we will investigate geometry, transformations, circles, probability, and statistics while continuing to strengthen mathematical reasoning and problem-solving skills.