Multiplication
Multiplication as repeated addition
Multiplication as repeated addition means that when you multiply a number by another, you are essentially adding that number to itself multiple times. Let’s explain this concept with a 4-digit number.
Example: 1234 x 3
This multiplication problem can be understood as adding the number 1234 three times:
1234 x 3 = 1234 + 1234 + 1234
Now, let’s break it down step by step:
1. First Addition: 1234 + 1234
– 4 + 4 = 8
– 3 + 3 = 6
– 2 + 2 = 4
– 1 + 1 = 2
So, 1234 + 1234 = 2468
2. Second Addition: Now, add 1234 to the result:
– 2468 + 1234
– 8 + 4 = 12 (write 2 and carry over 1)
– 6 + 3 + 1 = 10 (write 0 and carry over 1)
– 4 + 2 + 1 = 7
– 2 + 1 = 3
So, 2468 + 1234 = 3702
Therefore, 1234 x 3 = 3702 \).
General Concept:
When you multiply any 4-digit number (or any number, for that matter) by another number, you’re adding that number to itself as many times as the multiplier indicates. For example:
– 5678 x 4 = 5678 + 5678 + 5678 + 5678
– 8901 x 5 = 8901 + 8901 + 8901 + 8901 + 8901
This is how multiplication can be seen as repeated addition.
Multiplication facts
When working with 4-digit numbers, the multiplication properties remain the same as with smaller numbers. Here’s how they apply:
1. Commutative Property:
– The order of multiplication doesn’t matter, even with 4-digit numbers.
– Example:
1234 x 5678 = 5678 x 1234
2. Associative Property:
– The grouping of numbers can be changed without changing the product.
– Example:
(1000 x 123) x 4 = 1000 x (123 \times 4)
3. Identity Property:
– Multiplying any number by 1 will give you the same number.
– Example:
4321 x 1 = 4321
4. Zero Property:
– Any number multiplied by 0 is 0, no matter how large the number is.
– Example:
9876 x 0 = 0
These properties help simplify multiplication problems and are especially useful when dealing with larger numbers.
Multiplication of 3-digit number by two or three digits
Example 1: Multiplying a 3-digit number by a 2-digit number
Let’s multiply 234 by 56.
1. Step 1: Multiply the ones place (6 × 234):
– 6 × 4 = 24 (write down 4, carry over 2)
– 6 × 3 = 18, plus 2 = 20 (write down 0, carry over 2)
– 6 × 2 = 12, plus 2 = 14 (write down 14)
So, 6 × 234 = 1404.
2. Step 2: Multiply the tens place (50 × 234):
– Since we’re multiplying by 50 (5 in the tens place), add a zero first.
– 5 × 4 = 20 (write down 0, carry over 2)
– 5 × 3 = 15, plus 2 = 17 (write down 7, carry over 1)
– 5 × 2 = 10, plus 1 = 11 (write down 11)
So, 50 × 234 = 11700.
3. Step 3: Add the results:
– 1404
– +11700
– 13104 is the final product.
Example 2: Multiplying a 3-digit number by another 3-digit number
Let’s multiply 123 by 456.
1. Step 1: Multiply the ones place (6 × 123):
– 6 × 3 = 18 (write down 8, carry over 1)
– 6 × 2 = 12, plus 1 = 13 (write down 3, carry over 1)
– 6 × 1 = 6, plus 1 = 7
So, 6 × 123 = 738.
2. Step 2: Multiply the tens place (50 × 123):
– Add a zero first because it’s the tens place.
– 5 × 3 = 15 (write down 5, carry over 1)
– 5 × 2 = 10, plus 1 = 11 (write down 1, carry over 1)
– 5 × 1 = 5, plus 1 = 6
So, 50 × 123 = 6150.
3. Step 3: Multiply the hundreds place (400 × 123):
– Add two zeros because it’s the hundreds place.
– 4 × 3 = 12 (write down 2, carry over 1)
– 4 × 2 = 8, plus 1 = 9 (write down 9)
– 4 × 1 = 4
So, 400 × 123 = 49200.
4. Step 4: Add the results:
– 738
– + 6150
– +49200
– 56088 is the final product.
This method, called the “long multiplication” method, helps break down the process into manageable steps, making it easier for students to understand and perform the calculations.
Multiplication with or without carrying
Multiplying 4-digit numbers can be more challenging, but it’s an important skill. Let’s break down the process for both cases: without carrying and with carrying.
1. Multiplication Without Carrying:
This involves multiplying each digit without needing to carry over any digits. Here’s an example:
– Example: 1234 x 2
– Multiply each digit by 2:
– 4 x 2 = 8
– 3 x 2 = 6
– 2 x 2 = 4
– 1 x 2 = 2
– Result: 2468
2. Multiplication With Carrying:
This involves carrying over digits when the product exceeds 9. Here’s an example:
– Example: 4567 x 4
– Multiply each digit by 4:
– 7 x 4 = 28
– Write down 8 and carry over 2.
– 6 x 4 = 24
– Add the carried over 2 to get 26.
– Write down 6 and carry over 2.
– 5 x 4 = 20
– Add the carried over 2 to get 22.
– Write down 2 and carry over 2.
– 4 x 4 = 16
– Add the carried over 2 to get 18.
– Result: 18268
Word problem
Word Problem:
“A toy store ordered 1,245 boxes of action figures. Each box contains 6 action figures. How many action figures did the store order in total?”
Step-by-Step Explanation:
1. Understanding the Problem:
– The toy store has 1,245 boxes.
– Each box has 6 action figures.
– We need to find the total number of action figures.
2. Identify the Numbers:
– The number of boxes is 1,245.
– The number of action figures in each box is 6.
3. Set Up the Multiplication:
– We want to find the total number of action figures by multiplying the number of boxes by the number of action figures per box.
– This means we calculate 1,245 x 6.
4. Perform the Multiplication:
– To multiply 1,245 by 6:
– Multiply 6 by each digit in 1,245, starting from the right (units place) to the left (thousands place).
– 6 x 5 = 30 (write down 0, carry over 3)
– 6 x 4 = 24, then add the 3 you carried over: 24 + 3 = 27 (write down 7, carry over 2)
– 6 x 2 = 12, then add the 2 you carried over: 12 + 2 = 14 (write down 4, carry over 1)
– 6 x 1 = 6, then add the 1 you carried over: 6 + 1 = 7 (write down 7)
So, 1,245 x 6 = 7,470.
5. Answer:
– The store ordered a total of 7,470 action figures.