Multiplication is a fundamental arithmetic operation that combines equal groups of objects to find a total, representing repeated addition. It is essential for scaling numbers and appears in various forms, such as whole numbers, fractions, and decimals, across numerous real-world applications.
- Represents repeated addition and scaling.
- Connects to division as an inverse operation.
- Used in equal groups, arrays, and area models.
- Applies to whole numbers, fractions, and decimals.
Combining equal groups for a total
28
History and Origins
Babylonians, Egyptians, Chinese
The concept of multiplication as repeated addition was known to ancient civilizations such as the Babylonians, Egyptians, and Chinese, who developed early methods and tables for efficient calculation. The modern multiplication symbol (×) was introduced by William Oughtred in the 17th century.
- Ancient roots in Babylon, Egypt, and China.
- Originally done with tables and patterns.
- The “×” symbol was introduced in the 1600s.
William Oughtred
*
Multiplication Facts / Times Tables
Times tables
Multiplication facts, often called times tables, are the basic products for pairs of numbers usually from 1 to 12. Memorizing them helps students perform calculations quickly and accurately.
- Also known as "times tables."
- Usually cover 1 through 12.
- Foundation for higher math skills.
3 × 4 = 12, 7 × 8 = 56
Exercise:
- List the product for each pair of factors from 1 to 5 in a mini multiplication table.
- Identify the pattern in the products of the 5 times table.
- Recall the product of 7 × 8.
Properties of Multiplication
Commutative, Associative, Distributive
Multiplication has several important properties that help simplify calculations:
- Commutative: a × b = b × a
- Associative: (a × b) × c = a × (b × c)
- Distributive: a × (b + c) = (a × b) + (a × c)
Commutative
3 × 4 + 3 × 5
Exercise:
- Demonstrate the commutative property with two numbers.
- Show an example of the associative property using three factors.
- Use the distributive property to simplify 6 × (2 + 7).
Multiplication Models
Equal groups, arrays, and area models
Equal groups, Arrays, Area models
Multiplication can be represented and understood through various models:
- Equal Groups: Multiplying the number of groups by the number in each group.
- Arrays: Using rows and columns to show factors and product visually.
- Area Models: Representing multiplication as the area of a rectangle with side lengths as factors.
3 groups of 4 objects, A 3-row by 4-column grid, A rectangle with sides 3 and 4
Exercise:
- Draw an array for 5 × 3 and count the total.
- Use an area model to multiply 6 by 4.
- Describe equal groups for 7 × 2.
Multiplication of Fractions and Decimals
Multiply numerators and denominators
For fractions, you multiply the numerators together and the denominators together:
(a/b) × (c/d) = (ac)/(bd)
- Multiply numerators and denominators.
- Result is often simplified.
Ignore decimal points and multiply as whole numbers, Then place the decimal in the product
For decimals, multiply as if they were whole numbers, then place the decimal point in the product so it has the same total number of decimal places as both original numbers combined.
Exercise:
- Multiply (2/3) × (4/5) and simplify the result.
- How would you multiply 0.3 by 0.4 using the decimal rule?
- Calculate the product of 1.2 and 0.05.
Multiplication in Algebra
In algebra, multiplication extends to variables and expressions, following the same principles as numerical multiplication.
- Multiply coefficients and variables (e.g., 3x × 4 = 12x).
- Use distributive property to multiply across parentheses (e.g., a(b + c) = ab + ac).
- Multiply variables by adding exponents (e.g., x² × x³ = x^(2+3)).
3x × 4, a(b + c)
x^5
Exercise:
- Multiply: 5y × 3.
- Expand: 2(x + 4).
- Simplify: x³ × x².
Conclusion
Multiplication is a vital mathematical operation with roots in ancient civilizations, and it goes beyond simple repeated addition to include properties and models that help us understand and apply it in diverse contexts—from basic arithmetic to algebra and beyond.
- Developed by ancient cultures and symbolized by “×” in the 17th century.
- Core concepts include times tables, properties (commutative, associative, distributive), and visual models.
- Extends to fractions, decimals, and algebraic expressions for broad application.
Exercises:
- Explain the historical significance of multiplication and name one ancient civilization that contributed to its development.
- Demonstrate the distributive property with a numerical example.
- Show how multiplication applies to both whole numbers and fractions with suitable examples.