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What is \(3 \times 3\)?

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Explanation: \(3 \times 3\) is the square of 3: \(3^2 = 9\).

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Multiplication

Multiplication Practice Quiz with a Step-by-Step Interactive Lesson

Use the quiz at the top of the page to practice multiplication. If you want a refresher, click Start lesson to open a step-by-step guide.

How this multiplication practice works

1. Take the quiz: answer the questions at the top of the page.
2. Open the lesson (optional): review the method with examples and quick checks.
3. Retry: return to the quiz and apply what you reviewed.

What you’ll learn in the multiplication lesson

Meaning & vocabulary

Multiplication as equal groups
Multiplication as repeated addition
Factors and product

Arrays & properties

Array model (rows and columns)
Commutative property: \(a\times b=b\times a\)

Fast multiplication facts

Patterns for ×0, ×1, ×2, ×5, ×10
Doubling for ×4 and ×8
×9 trick: \(9n=10n-n\)

Bigger numbers

Distributive property: \(a\times(b+c)=a\times b+a\times c\)
Split to multiply (tens + ones)
Two-digit × two-digit (area / box method)

Back to the quiz

When you’re ready, return to the quiz at the top of the page and continue practicing.

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Multiplication
Lesson

Step-by-step guide

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Multiplication Lesson

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Lesson Overview

Lesson overview

Purpose: Build a clear understanding of multiplication and learn reliable methods you can use in any problem.

Success criteria

Explain \(a\times b\) as equal groups (and as repeated addition).
Use arrays and the commutative property \(a\times b=b\times a\).
Use efficient strategies for facts (×0, ×1, ×2, ×5, ×10, doubling, ×9 trick).
Multiply larger numbers using the distributive property (split and add).
Evaluate mixed expressions by doing multiplication first.
Recognize where multiplication is used in real life and across math topics.

Key vocabulary

Factor: a number you multiply (in \(a\times b\), both \(a\) and \(b\) are factors).
Product: the result of multiplication (the product of \(a\times b\)).
Array: rows and columns that model multiplication.

Quick pre-check

Pre-check 1: Which expression means “3 groups of 4”?

\(3\times 4\) \(34\) \(3+4\) \(4-3\)

Hint: In this lesson, \(a\times b\) means “\(a\) groups of \(b\)”.

Pre-check 2: Compute \(3\times 4\).

Hint: \(3\times 4\) is \(4+4+4\).

Equal Groups

Equal groups and repeated addition

Learning goal: Translate between multiplication and repeated addition, and compute simple products.

Key idea

Multiplication represents equal groups. In this lesson we read \(a\times b\) as \(a\) groups of \(b\). That means: \(a\times b = b + b + \dots + b\) (repeated \(a\) times).

Worked example

Example: \(5\times 3\)

\(5\times 3\) means 5 groups of 3.
Repeated addition: \(3+3+3+3+3 = 15\).
So, the product is \(15\).

Try it

Try it 1: Compute \(6\times 4\).

Hint: \(6\times 4\) is 6 groups of 4: \(4+4+4+4+4+4\).

Try it 2: Which repeated addition matches \(3\times 5\)?

\(3+3+3+3+3\) \(5+5+5\) \(5+5+5+5\) \(3+5\)

Hint: The first factor tells how many groups. \(3\times 5\) is 3 groups of 5.

Summary

\(a\times b\) can be read as \(a\) groups of \(b\).
Multiplication can be written as repeated addition.

Arrays

Arrays and the commutative property

Learning goal: Use an array model and explain why \(a\times b = b\times a\).

Key idea

An array arranges objects in rows and columns. If you rotate an array, the total number of objects stays the same. This helps explain the commutative property: \(\,a\times b = b\times a\).

Worked example

Example: \(3\times 4\) and \(4\times 3\)

\(3\times 4\) can be seen as 3 rows of 4.
Rotate the array: you get 4 rows of 3, which is \(4\times 3\).
Both totals are \(12\).

Try it

Try it 1: If \(7\times 8 = 56\), what is \(8\times 7\)?

Hint: Use the commutative property \(a\times b=b\times a\).

Try it 2: Which statement is the commutative property of multiplication?

\(a\times b = b\times a\) \(a\times b = a + b\) \(a\times b = a - b\) \(a\times b = a \div b\)

Hint: “Commutative” means you can switch the order.

Summary

Arrays model multiplication using rows and columns.
\(a\times b\) and \(b\times a\) have the same product.

Facts & Strategies

Efficient strategies for multiplication facts

Learning goal: Use patterns and mental strategies to find products quickly and accurately.

Key patterns

×0: the product is 0
×1: the product is the same number
×10: append a zero (for whole numbers)
×5: half of ×10 (multiply by 10, then divide by 2)
×2: double
×4: double twice
×8: double three times
×9 trick: \(9n = 10n - n\)

Worked example

Example: \(9\times 7\)

Use \(9n = 10n - n\):
\(9\times 7 = 10\times 7 - 7 = 70 - 7 = 63\).

Try it

Try it 1: Compute \(9\times 6\).

Hint: \(9\times 6 = 10\times 6 - 6\).

Try it 2: Compute \(8\times 7\).

Hint: Doubling helps: \(7\to 14\to 28\to 56\) (double 3 times).

Summary

Use patterns first (×0, ×1, ×10, ×2, ×5).
Use doubling for ×4 and ×8; use the \(9n=10n-n\) trick for ×9.

Split to Multiply

Split to multiply (distributive property)

Learning goal: Multiply larger numbers by splitting into friendly parts and adding partial products.

Key idea

The distributive property allows you to split a factor: \(\,a\times(b+c)=a\times b + a\times c\). This creates smaller products you can do accurately.

Worked example

Example: \(23\times 4\)

Split 23 into \(20+3\).
\(23\times 4 = (20\times 4) + (3\times 4) = 80 + 12 = 92\).

Try it

Try it 1: Compute \(14\times 7\) using splitting.

Hint: Split \(14\) into \(10+4\): \(10\times 7\) and \(4\times 7\).

Worked solution

\(14\times 7 = (10+4)\times 7 = 10\times 7 + 4\times 7\).
\(10\times 7 = 70\), \(4\times 7 = 28\).
Add: \(70+28 = 98\).

Try it 2: Compute \(26\times 3\).

Hint: Split \(26\) into \(20+6\): \(20\times 3\) and \(6\times 3\).

Summary

Split one factor into tens and ones to reduce errors.
Multiply each part, then add the partial products.

Two-digit × Two-digit

Two-digit multiplication with the area model

Learning goal: Multiply two two-digit numbers by splitting both numbers and adding partial products.

Key idea

To multiply \((10+a)\times(10+b)\), multiply each part and add: \((10+a)\times(10+b)=10\times 10 + 10\times b + a\times 10 + a\times b\). This is the distributive property used twice (often taught as the “area model” or “box method”).

Worked example

Example: \(12\times 13\)

Split: \(12=10+2\), \(13=10+3\).
Partial products: \(10\times 10=100\), \(10\times 3=30\), \(2\times 10=20\), \(2\times 3=6\).
Add: \(100+30+20+6=156\). So \(12\times 13=156\).

Try it

Try it: Compute \(14\times 12\).

Hint: Split \(14=10+4\), \(12=10+2\). Multiply parts, then add.

Worked solution

\(14\times 12=(10+4)\times(10+2)\).
\(10\times10=100\), \(10\times2=20\), \(4\times10=40\), \(4\times2=8\).
Add: \(100+20+40+8=168\).

Summary

Split both numbers into tens and ones.
Compute partial products, then add them carefully.

Multiply First

Order of operations: multiply first

Learning goal: Evaluate expressions that contain multiplication and addition/subtraction.

Key idea

When an expression contains \(+\) or \(−\) and \( \times \), do multiplication first (then add or subtract).

Worked example

Example: \(18 - 3\times 4\)

Step 1: Multiply: \(3\times 4=12\).
Step 2: Subtract: \(18-12=6\).
So, \(18 - 3\times 4 = 6\).

Try it

Try it 1: Compute \(7 + 4\times 5\).

Hint: Multiply \(4\times 5\) first, then add 7.

Try it 2: Compute \(15 - 6\times 2\).

Hint: Multiply \(6\times 2\) first.

Summary

In mixed expressions, do multiplication before addition/subtraction.
Work step-by-step to avoid common mistakes.

Applications & History

Why multiplication matters

Learning goal: Connect multiplication to geometry, scaling, and everyday situations — and learn a few fun facts.

Where you use multiplication

Area (geometry): rectangle area = length × width.
Scaling: doubling/tripling a recipe, resizing a drawing.
Money: price × quantity.
Science and computing: repeated patterns, arrays, and growth models.

Worked example: area of a rectangle

Example: A rectangle is 8 cm long and 3 cm wide.

Area = length × width = \(8\times 3 = 24\).
Answer: The area is \(24\text{ cm}^2\).

Try it

Try it 1: A rectangle is 8 units by 3 units. What is the area?

Hint: Area = length × width.

Fun facts (a little history)

Tables: Multiplication tables are sometimes called a “Pythagorean table” because they form a grid of products.
Different methods: Before calculators, people developed clever ways to multiply. One famous method uses repeated doubling and addition (often called “Egyptian multiplication”).
Symbols: You might see multiplication written as \( \times \), as a dot \( \cdot \), or just by putting numbers next to parentheses, like \(3(4)\).

Try it 2: In algebra, which symbol is often used for multiplication to avoid confusing \(x\) with “×”?

\( \times \) \( \cdot \) \( + \) \( = \)

Hint: In algebra, the dot is very common (for example, \(a\cdot b\)).

Final recap

Multiplication models equal groups and can be written as repeated addition.
Arrays support \(a\times b=b\times a\).
Use strategies for facts, and use splitting (distributive property) for larger numbers.
In mixed expressions, do multiplication first.
Multiplication is used everywhere: area, scaling, money, science, and more.

Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the skill.