Division
Division on number line
1. Understanding Division:
– Division means splitting a number into equal parts.
– For example, if you have 12 candies and you want to share them equally with 3 friends, you divide 12 by 3.
2. Number Line Basics:
– A number line is a straight line with numbers placed at equal intervals.
– The numbers usually start from 0 and go up in steps like 1, 2, 3, and so on.
3. Using the Number Line for Division:
– To divide on a number line, you can think of it as repeated subtraction or “hopping back” in equal steps.
Example: 12 ÷ 3
1. Start at 0: Draw a number line from 0 to 12.
2. Mark the number 12: This is where we want to start.
3. Make jumps of 3: Since we’re dividing by 3, we hop back by 3 each time.
– First jump: From 12 to 9 (1st hop)
– Second jump: From 9 to 6 (2nd hop)
– Third jump: From 6 to 3 (3rd hop)
– Fourth jump: From 3 to 0 (4th hop)
4. Count the jumps: It took 4 jumps of 3 to get from 12 to 0.
5. Result: So, 12 ÷ 3 = 4.
Explaining division of a 4-digit number on a number line.
1. Understanding the Concept:
– Just like with smaller numbers, division for larger numbers means splitting the number into equal parts.
– The number line can still help, but instead of drawing a line up to 1,000 or 2,000, we use **larger jumps**.
2. Setting Up the Number Line:
– Since the number is large, we won’t draw every single number from 0 to the 4-digit number. Instead, we’ll use bigger steps.
– For example, if we’re dividing 2000 by 500, we can make our number line with jumps of 500.
Example: 2000 ÷ 500
1. Draw a Number Line: Start by drawing a number line with marks at 0, 500, 1000, 1500, and 2000.
2. Start at 0: Just like before, we start at 0.
3. Make Jumps of 500: Since we’re dividing by 500, we hop forward by 500 each time.
– First jump: From 0 to 500 (1st hop)
– Second jump: From 500 to 1000 (2nd hop)
– Third jump: From 1000 to 1500 (3rd hop)
– Fourth jump: From 1500 to 2000 (4th hop)
4. **Count the Jumps**: It took 4 jumps of 500 to reach 2000.
5. Result: So, 2000 ÷ 500 = 4.
3. Visualizing Larger Numbers:
– Explain that when numbers are very big, we use bigger steps. Just like jumping forward by 1’s, 2’s, or 5’s for smaller numbers, we now jump by hundreds or thousands.
– You can relate it to walking a long distance: If you want to get somewhere quickly, you take big steps instead of tiny ones.
Division of a two digit number by one digit with or without remainder
1. Understanding Division:
– Division means splitting a number into equal parts.
– When you divide a two-digit number by a one-digit number, you are figuring out how many times the one-digit number fits into the two-digit number.
2. Using a Number Line:
– A number line can help visualize the process, especially when dealing with smaller divisions.
Example 1: 25 ÷ 4 (with remainder)
1. Draw a Number Line:
– Draw a number line from 0 to 25.
2. Start at 0:
– Start at 0 on the number line.
3. Make Jumps of 4:
– Since you’re dividing by 4, make jumps of 4.
– First jump: From 0 to 4.
– Second jump: From 4 to 8.
– Third jump: From 8 to 12.
– Fourth jump: From 12 to 16.
– Fifth jump: From 16 to 20.
– Sixth jump: From 20 to 24.
4. Count the Jumps:
– You’ve made 6 jumps of 4, reaching 24.
5. Check What’s Left:
– After reaching 24, you have 1 left to get to 25.
– This leftover number (1) is the remainder.
6. Result:
– So, 25 ÷ 4 = 6 with a remainder of 1.
– Conclusion: 25 can be divided into six groups of 4, with 1 left over.
Division of a three digit number by one digit
1. Understanding the Concept:
– Division is about splitting a number into equal parts.
– When dividing a three-digit number by a one-digit number, you’re figuring out how many times the one-digit number fits into each part of the three-digit number.
2. Step-by-Step Long Division:
– For three-digit numbers, long division is a helpful method. Let’s go through an example to show how it works.
Example: 432 ÷ 3
1. Write the Numbers:
– Write 432 (the dividend) under the division bar and 3 (the divisor) outside the bar.
2. Divide the Hundreds Place:
– Look at the first digit of 432, which is 4 (in the hundreds place).
– Ask: How many times does 3 fit into 4? It fits **1 time**.
– Write 1 above the division bar, above the 4.
3. Multiply and Subtract:
– Multiply 1 by 3, which equals 3.
– Subtract this 3 from the 4, which leaves 1.
– Now, bring down the next digit (3 in the tens place).
4. Divide the Tens Place:
– Now, look at 13 (the 1 we got from subtracting, plus the 3 we brought down).
– Ask: How many times does 3 fit into 13? It fits 4 times (because 3 × 4 = 12).
– Write 4 above the division bar, above the 3.
5. Multiply and Subtract Again:
– Multiply 4 by 3, which equals 12.
– Subtract 12 from 13, which leaves 1.
– Now, bring down the last digit (2 in the ones place).
6. Divide the Ones Place:
– Now, look at 12.
– Ask: How many times does 3 fit into 12? It fits 4 times (because 3 × 4 = 12).
– Write 4 above the division bar, above the 2.
7. Multiply and Subtract One Last Time:
– Multiply 4 by 3, which equals 12.
– Subtract 12 from 12, which leaves 0.
8. Result:
– The answer is 144 with no remainder.
– Conclusion: 432 divided by 3 equals 144.
Properties of division
1. Division as the Opposite of Multiplication:
– Property: Division is the reverse process of multiplication.
– Example: If 3 × 4 = 12, then 12 ÷ 4 = 3.
– Explanation: If you know how to multiply, you can also divide. Division tells you how many times a number fits into another.
2. Division by 1:
– Property: Any number divided by 1 equals the number itself.
– Example: 45 ÷ 1 = 45.
– Explanation: If you divide something into one part, it stays the same. So, dividing by 1 doesn’t change the number.
3. Division by the Number Itself:
– Property: Any number divided by itself equals 1.
– Example: 9 ÷ 9 = 1.
– Explanation: If you divide a group into a number of parts that equals the group, you get just one part.
4. Division by Zero (Not Possible):
– Property: Division by zero is not possible.
– Example: You cannot do 8 ÷ 0.
– Explanation: If you try to split something into zero parts, it doesn’t make sense, so division by zero is undefined.
5. Division of Zero:
– Property: Zero divided by any number is always zero.
– Example: 0 ÷ 5 = 0.
– Explanation: If you have nothing to divide, no matter how many parts you try to divide it into, you still have nothing.
6. Remainder Property:
– Property: Sometimes, division doesn’t divide evenly, and there’s something left over. This leftover is called the remainder.
– Example: 17 ÷ 3 = 5 with a remainder of 2.
– Explanation: When you divide 17 by 3, 3 goes into 17 five times, but there’s 2 left over because 17 is not a multiple of 3.
7. Distributive Property of Division Over Addition:
– Property: Division can be split across addition.
– Example: (20 + 10) ÷ 5 = 20 ÷ 5 + 10 ÷ 5 = 4 + 2 = 6.
– Explanation: You can divide each part of a sum by the divisor separately, then add the results.
8. Division and Repeated Subtraction:
– Property: Division is like repeated subtraction.
– Example: 12 ÷ 4 means subtracting 4 from 12 until you reach zero: 12 – 4 = 8, 8 – 4 = 4, 4 – 4 = 0. You subtracted 3 times, so 12 ÷ 4 = 3.
– Explanation: Division tells you how many times you can subtract one number from another before reaching zero.
9. No Commutative Property:
– Property: Unlike addition and multiplication, division is not commutative. This means you cannot switch the numbers around.
– Example: 12 ÷ 3 = 4, but 3 ÷ 12 ≠ 4.
– Explanation: The order of the numbers matters in division. Dividing 12 by 3 gives a different result than dividing 3 by 12.
10. No Associative Property:
– Property: Division is not associative, meaning that the grouping of numbers matters.
– Example: (24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12.
– Explanation: Changing the grouping of numbers changes the result, so you have to be careful with how you group numbers in division.
These properties provide a solid foundation for understanding division and how it interacts with other mathematical operations.
Long division
What is Long Division?
Long division is a method used to divide large numbers by breaking the process down into a series of easier steps. It involves dividing, multiplying, subtracting, and bringing down the next digit.
Steps for Long Division
Let’s work through an example: 384 ÷ 3.
Step 1: Set Up the Problem
– Write 384 (the dividend) under the division bar.
– Write 3 (the divisor) outside the division bar.
Step 2: Divide the First Digit
– Look at the first digit of 384, which is 3.
– Ask: How many times does 3 go into 3? It goes 1 time.
– Write 1 above the division bar, over the 3.
Step 3: Multiply and Subtract
– Multiply 1 (the quotient) by 3 (the divisor), which equals 3.
– Write this 3 below the 3 of 384.
– Subtract: 3 – 3 = 0.
– Since the result is 0, there’s nothing left over at this stage.
Step 4: Bring Down the Next Digit
– Bring down the next digit in the dividend, which is 8.
– Now, you’re looking at 8.
Step 5: Divide the New Number
– Ask: How many times does 3 go into 8? It goes 2 times.
– Write 2 above the division bar, next to the 1.
Step 6: Multiply and Subtract Again
– Multiply 2 by 3, which equals 6.
– Write 6 below the 8.
– Subtract: 8 – 6 = 2.
Step 7: Bring Down the Last Digit
– Bring down the final digit in the dividend, which is 4.
– Now, you’re looking at 24.
Step 8: Divide Again
– Ask: How many times does 3 go into 24? It goes 8 times.
– Write 8 above the division bar, next to the 12.
Step 9: Multiply and Subtract One Last Time
– Multiply 8 by 3, which equals 24.
– Write 24 below the 24 you have.
– Subtract: 24 – 24 = 0.
Step 10: Write the Final Answer
– Since there are no more digits to bring down and nothing is left over, the final answer is 128.
So, 384 ÷ 3 = 128.
What If There’s a Remainder?
Let’s try an example where there’s a remainder: 37 ÷ 5.
1. Divide 37 by 5:
– 5 goes into 37 7 times (because 5 × 7 = 35).
– Write 7 above the division bar.
2. Multiply and Subtract:
– 7 × 5 = 35.
– Subtract 35 from 37, which leaves 2.
3. No More Digits to Bring Down:
– There’s nothing left to bring down, and 2 is smaller than 5, so 2 is the remainder.
4. Final Answer:
– The answer is 7 remainder 2 or written as7 R2.
So, 37 ÷ 5 = 7 R2
Division with regrouping
Steps for Division with Regrouping:
Let’s divide 96 by 4 using regrouping.
1. Set up the problem:
Write 96 inside the division box (dividend) and 4 outside (divisor).
2. Divide the first digit:
– Look at the first digit of the dividend (9 in this case).
– Ask: How many times does 4 go into 9?
– Since 4 goes into 9 two times (4 × 2 = 8), write 2 above the 9.
– Subtract 8 from 9 to get a remainder of 1.
3. Bring down the next digit:
– Now, bring down the next digit of the dividend (which is 6), making it 16.
4. Divide again:
– Now divide 16 by 4. How many times does 4 go into 16?
– Since 4 goes into 16 exactly 4 times (4 × 4 = 16), write 4 above the line.
– Subtract 16 from 16, which gives a remainder of 0.
Final Answer:
The quotient is 24, and there’s no remainder.
Key Points:
– If there’s a remainder after division, you can keep dividing by bringing down the next number.
– This method is called regrouping because sometimes you have to group digits from the dividend to make it easier to divide.
Word problem
Word Problem:
A factory produced 2,568 toys in one month. The toys are packed into boxes, and each box can hold 8 toys. How many boxes are needed to pack all the toys?
Solution:
– Total number of toys: 2,568
– Number of toys per box: 8
To find how many boxes are needed, divide the total number of toys by the number of toys each box can hold:
2,568 ÷ 8 = 321
So, the factory will need 321 boxes to pack all the toys.