Fun Property: Math Just Works
Using clocks as an analogy, we can figure out whether the rules of modular arithmetic just work .
Lets say two times look the same on our clock . If we add the same x hours to both, what happens?
Well, they change to the same amount on the clock! 2:00 + 5 hours 14:00 + 5 hours both will show 7:00.
Why? Well, we never cared about the excess 12:00 that the 14 was carrying around. We can just add 5 to the 2 remainder that both have, and they advance the same. For all congruent numbers , adding and subtracting has the same result.
Its harder to see whether multiplication stays the same. If 14 2 , can we multiply both sides and get the same result?
Lets see what happens when we multiply by 3?
Well, 2:00 * 3 6:00. But whats 14:00 * 3?
Remember, 14 = 12 + 2. So, we can say
- 14 * 3 = * 3 = + mod 12
The first part can be ignored! The 12 hour overflow that 14 is carrying around just gets repeated a few times. But who cares? We ignore the overflow anyway.
When multiplying, its only the remainder that matters, which is the same 2 hours for 14:00 and 2:00. Intuitively, this is how I see that multiplication doesnt change relationships with modular math . See the above link for more rigorous proofs these are my intuitive pencil lines.
Working Out Division With Remainders
If a child already understands basic division, including how to divide numbers evenly, you can now proceed to the next step. This step entails working with remainders where numbers cant be divided evenly.
To elaborate the concept of remainders in division, use items such as candies and blocks. Start by counting a certain number of candies that cannot be divided into equal groups or cannot be divided evenly. For instance, ask the child to divide 9 candies into groups of 4 or divide 15 candies into groups of 6.
In the first scenario, there would be a remainder of 1 because each of the 4 groups would have two candies. There would be a remainder of 3 candies in the second example because each of the 6 groups would have 2 candies. This concept helps a child understand that some numbers are leftover in division and are called remainders.
You should then write down the division problem on a worksheet. For example, 9÷4= 2 remainder 1 and 15÷6=2 remainder 3.
To make learning division with remainders more effective, provide a child with more division problems like 7÷2, 10÷3, 15÷4, 20÷7, 25÷10, etc. The ultimate goal is for a kid/student to exercise until they can explain why they have remainders in each group without your help. You may allow them to use candies or other items if they need to do grouping.
1:1 Math Lessons
Fun With Modular Arithmetic
A reader recently suggested I write about modular arithmetic . I hadnt given it much thought, but realized the modulo is extremely powerful: it should be in our mental toolbox next to addition and multiplication.
Instead of hitting you in the face with formulas, lets explore an idea weve been subtly exposed to for years. Theres a nice article on modular arithmetic that inspired this post.
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How To Divide Numbers With Decimals
While dividing numbers with decimals, we can observe the decimal point in the divisor. As we know, there should not be decimals in the divisor. Hence, we need to remove the divisors decimal point by adding as many zeros to the dividend as there are digits after the decimal point.
Example: Divide 5 by 2.5.
In this way, we can divide whole numbers by decimals.
Use The Herding Game For Teaching Division With Remainders
Just when kids think they have division all figured out, along come remainders! Teaching division with remainders can be one of the trickiest tasks, so start by playing this fun and active game. For each round, the teacher calls out different groups of animals for students to form . Any leftover students who dont fit into the group go into the holding pen, introducing the idea of remainders.
Learn more: The Teacher Studio/The Herding Game
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How To Divide Polynomials
Polynomials are expressions which consist of unknown variables. Thus, we need to follow the steps to perform division on polynomials.
- First, we have to write the polynomials into division form, i.e.numerator and denominator
- Then cancel the like or common terms.
- Now, perform the division like fractions.
Example: Divide x2y3 by xy3.
In both numerator and denominator, xy3 is the common term. So cancelling the like terms we get
x2y3/xy3 = x
How To Divide Numbers Step By Step
Division method is nothing but the inverse of multiplication. In multiplication, we multiply the numbers into multiples but in the division, we divide the numbers into parts. Let us learn first how to divide a number with the help of examples.
Example: Divide 2 by number 4.
Solution: 2÷4 = 2/4
Since, both the number, 2 and 4 are the multiples of 2. Therefore, we can cancel out the common factor 2 from both numerator and denominator, to get
2/4 = /
= ½ = 0.5
This was the case when the digits were single. Now let us solve a case of double digits.
Example: Divide 22 by 11.
Solution: 22÷11 = 22/11
Since, 22 and 11 both are multiples of 11, therefore
22/11 = /
Cancelling the common terms from both numerator and denominator, we get
22/11 = 2
Now, let us consider a case of a three-digit number.
Example: Divide 204 by 144.
Solution: 204÷144 = 204/144
Both the numbers 204 and 144 are the multiples of 12. Therefore,
204/144 = /
To understand how to divide decimal numbers, we have to follow the below-given steps
- Step 1: Convert the decimals into equivalent fractions.
- Step 2: Cancel the like terms from numerator and denominator.
- Step 3: Write the final answer in fraction or decimal.
Let us understand the above steps with the help of an example.
Example: Divide the decimals 0.14 and 0.07.
Solution: First we have to write the decimals into fractions.
0.14 = 14/100
Now divide both the fractions.
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Try Multiplying By Zero
So let us try using our new “numbers”.
For example, we know that zero times any number is zero:
Example: 0×1 = 0, 0×2 = 0, etc
So that should also be true for 10:
0 × 10 = 0
But we could also rearrange it a little like this:
0 × 10 = 00 × 1 = 1
Arrggh! If we multiply 10 by zero we could get 0 or 1.
In fact we can’t have both possibilites, so we cannot define 10 to be a number.
So it is undefined.
Odd Even And Threeven
Shortly after discovering whole numbers we realized they fall into two groups:
- Even: divisible by 2
- Odd: not divisible by 2
Whys this distinction important? Its the beginning of abstraction were noticing the properties of a number and not just the number itself .
This is huge it lets us explore math at a deeper level and find relationships between types of numbers, not specific ones. For example, we can make rules like this:
- Even x Even = Even
- Odd x Odd = Odd
- Even x Odd = Even
These rules are general they work at the property level.
But even/odd is a very specific property: division by 2. What about the number 3? How about this:
- Threeven means a number is divisbile by 3
- Throdd means you are not divisible by 3
Weird, but workable. Youll notice a few things: theres two types of throdd. A number like 4 is 1 away from being threeven , while the number 5 is two away .
Being threeven is just another property of a number. Perhaps not as immediately useful as even/odd, but its there: we can make rules like threeven x threeven = threeven and so on.
But its getting crazy. We cant make new words all the time.
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How To Teach Division To A Child
Learning how to divide or teaching division is not complicated as it seems. A parent or teacher can master a straightforward approach to explaining division effectively. Consequently, a child can quickly grasp the concept pretty easily.
Below is a step-by-step process on how to learn division from the basic concept of division to long division. This approach involves a gradual process, making it easier to assess how a child handles division problems.
Note: Its vital to ensure your teaching is engaging and fun to enhance its effectiveness.
Uses Of Modular Arithmetic
Now the fun part why is modular arithmetic useful?
Simple time calculations
We do this intuitively, but its nice to give it a name. You have a flight arriving at 3pm. Its getting delayed 14 hours. What time will it land?
Well, 14 2 mod 12. So I think of it as 2 hours and an am/pm switch, so I know it will be 3 + 2 = 5am.
This is a bit more involved than a plain modulo operator, but the principle is the same.
Putting Items In Random Groups
Suppose you have people who bought movie tickets, with a confirmation number. You want to divide them into 2 groups.
What do you do? Odds over here, evens over there. You dont need to know how many tickets were issued , everyone can figure out their group instantly , and the scheme works as more people buy tickets.
Need 3 groups? Divide by 3 and take the remainder . Youll have groups 0, 1 and 2.
In programming, taking the modulo is how you can fit items into a hash table: if your table has N entries, convert the item key to a number, do mod N, and put the item in that bucket . As your hash table grows in size, you can recompute the modulo for the keys.
Picking A Random Item
I use the modulo in real life. Really. We have 4 people playing a game and need to pick someone to go first. Play the mod N mini-game! Give people numbers 0, 1, 2, and 3.
Now everyone goes one, two, three, shoot! and puts out a random number of fingers. Add them up and divide by 4 whoever gets the remainder exactly goes first. .
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Special Names For Numbers In Division Equation
Each part involved in a division equation has a special name.
Dividend: The dividend is the number that is being divided in the division process.
Divisor: The number by which the dividend is being divided by is called the divisor.
Quotient: The quotient is a result obtained in the division process.
Remainder: Sometimes, we cannot divide things exactly. There may be a leftover number. That leftover number is called the remainder.
The relationship between these four parts can be expressed as follows:
Dividend = Divisor x Quotient + Remainder
This is also called the division formula to check whether the answer is correct or not.
For example, lets divide 16 by 3. The leftover will be 1.
Here, dividend = 16, divisor = 3, quotient = 5 and remainder = 1
So, 16 = 3 × 5 + 1
Example: 15 Divided By 02
When we multiply the 0.2 by 10 we get a whole number:
0.2 × 10 = 2
But we must also do it to the 15:
15 × 10 = 150
So 15 ÷ 0.2 has become 150 ÷ 2 :
150 ÷ 2 = 75
And so the answer is:
15 ÷ 0.2 = 75
The number we divide by is called the divisor.
To divide decimal numbers:
Multiply the divisor by as many 10’s as we need, until it is a whole number.Remember to multiply the dividend by the same number of 10’s.
Multiplying by 10 is easy, we just shift one space over like this:
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Features Of The Math Trainer
|Designed for high speed so that you get lots of practice.|
|Timed Workout style just like athletes use.|
|Cutoff Time pushes you to quickly remember, not count to get an answer.|
|Shows you the correct answer when you get it wrong.|
|Remembers your performance so that it gives you more practice on your weaknesses.|
Create A Division House
This is such a creative way of teaching division! Kids answer a series of questions to determine the specifications of their division house. For instance, to determine the number of windows in the house, students must divide the day of the month they were born by the number of kids in their family. When theyre done with the math, its time to draw their house!
Learn more: Teaching With a Mountain View
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Division Tips And Tricks
Draw a PictureCheck Your Answer by MultiplyingDivision by SubtractionDivide by Three TrickMore Divide by Number Tricks
- Divide by 1 – Anytime you divide by 1, the answer is the same as the dividend.
- Divide by 2 – If the last digit in the number is even, then the entire number is divisible by 2. Remember that divide by 2 is the same as cutting something in half.
- Divide by 4 – If the last two digits divide by 4, then the entire number is divisible by 4. For example, we know that 14237732 can be divided evenly by 4 because 32 ÷ 4 = 8.
- Divide by 5 – If the number ends in a 5 or a 0, it is divisible by 5.
- Divide by 6 – If the rules for divide by 2 and divide by 3 above are true, then the number is divisible by 6.
- Divide by 9 – Similar to the divide by 3 rule, if the sum of all the digits is divisible by 9, then the entire number is divisible by 9. For example, we know that 18332145 is divisible by 9 because 1+8+3+3+2+1+4+5 = 27 and 27 ÷ 9 = 3.
- Divide by 10 – If the number ends in a 0, then it is divisible by 10.
Advanced Kids Math Subjects
What Is 1 Divided Half
We know that, while dividing numbers by fractions, the division operation should be converted to multiplication. Thus, 1 divided by half can be performed as given below:
1/ = 1 × = 2
To learn more about division and other mathematical operations, download BYJUS The Learning App today!
Put your understanding of this concept to test by answering a few MCQs. Click Start Quiz to begin!
Select the correct answer and click on the Finish buttonCheck your score and answers at the end of the quiz
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What Is Division In Math
Division is the opposite of multiplication. If 3 groups of 4 make 12 in multiplication, 12 divided into 3 equal groups give 4 in each group in division.
The main goal of dividing is to see how many equal groups are formed or how many are in each group when sharing fairly.
In the above example, to divide 12 donuts into 3 similar groups, you will have to put 4 donuts in each group. So, 12 divided by 3 will give the result 4.
How To Divide
Learn how to divide a number by another number. The division is a basic arithmetic operation apart from addition, subtraction and multiplication. The division is the reverse process of multiplication. This method is represented by the symbol ÷ or /. In this method, when a number x is divided by another y, we get an output result that equals to third number z, such as
x ÷ y = z
Here, we will learn to divide the numbers, fractions, decimals, polynomials, with examples.
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