GENIUS / Grades 6–8 / Mathematics / Grade 7
GENIUS PRACTICE · GRADE 7

Spot the pattern.
Explain the math.

Discover proportional relationships, operations with rational numbers, and equations through real-life problems. Explore the examples, then test your understanding with instant explanations.

Start learning ↗Try interactive practice ↗Browse key skills ↗

Grade 7 skill pathways

Three ideas that build mathematical thinking.

These connected skills make it easier to understand patterns, solve practical problems, and prepare for algebra.

1 · Relationships

Proportions & percents

Recognize constant rates, use equivalent ratios, and solve percentage problems involving discounts and increases.

Explore proportional relationships ↗
2 · Number sense

Rational numbers

Add, subtract, multiply, and divide positive and negative numbers, including fractions and decimals.

Explore rational numbers ↗
3 · Algebra

Expressions & equations

Use variables, simplify expressions, and solve two-step equations while keeping both sides balanced.

Explore equations ↗

Lesson 1 · Proportional relationships

When one quantity changes, what happens to the other?

A proportional relationship has a constant ratio between matching quantities. You can often describe it with y = kx, where k is the constant of proportionality.

Understand the relationship

What makes a relationship proportional?

Suppose notebooks cost $3 each. The total cost rises by $3 for every additional notebook.

1 notebook → $3
2 notebooks → $6
5 notebooks → $15

In every row, cost ÷ notebooks = 3. That's the constant unit rate.

Worked example

Use a proportional rate.

Four tickets cost $28. If each ticket has the same price, how much do seven tickets cost?

  1. Find one ticket: $28 ÷ 4 = $7.
  2. Multiply by seven tickets: 7 × $7 = $49.
  3. Check: 4 × $7 = $28, so the same rate works.
Answer: 7 tickets cost $49.

Lesson 2 · Rational numbers

Positive and negative numbers follow consistent rules.

Rational numbers include integers, fractions, and terminating or repeating decimals. Their signs help us reason about change, direction, and quantity.

Build the meaning

Adding a negative number

Imagine a balance of $8 and a $13 charge. The change is 8 + (−13).

8 + (−13) = −5

You finish at −$5. On the number line, you moved thirteen steps left from 8.

Worked example

Multiply signed numbers.

Find (−6) × (−4).

  1. Multiply the magnitudes: 6 × 4 = 24.
  2. Two negative factors produce a positive product.
  3. The result is positive 24.
(−6) × (−4) = 24

Lesson 3 · Expressions and equations

Keep both sides balanced.

Solving an equation means finding a value that makes its two sides equal. Undo operations one step at a time, and check your result.

Worked example

Solve 3x + 5 = 20.

  1. Subtract 5 from both sides: 3x = 15.
  2. Divide both sides by 3: x = 5.
  3. Check: 3(5) + 5 = 20.
Solution: x = 5
Think it through

What does the variable represent?

A gym charges $4 per visit plus a one-time $6 fee. The total cost is $26.

4v + 6 = 26

Subtract 6 and divide by 4: v = 5 visits. This connects the equation to a real situation.

GENIUS interactive practice

Try it. Check it. Learn why.

Choose a skill and answer three original problems. After each choice, you'll see an explanation. Your results stay in this page.

Question 1 of 3

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Choose an answer to see why it works.

Ask for the reasoning, not just the answer.

Try asking: “What stays constant?” “How can you check it?” or “Can you show the same idea another way?”

Review the lessons ↗Return to proportions, rational numbers, or algebra examples.Practice another skill ↗Switch categories to work on a different Grade 7 concept.