Introduction to Addition
Addition is a method of joining things together. When you add two numbers together, you count them as one larger number. In real life, addition occurs frequently.
What if there were three more apples, for example?
As you can see, if you start with two apples and add three more, you will have a total of five apples. You could put it this way:
Addition can be shown as an equation: 2 + 3 = 5. It can be written as two plus three equals five. A mathematical equation is a math sentence. Instead of words, it employs numbers and symbols. When writing addition equations, we use two symbols: “+” and “=”. The plus sign (+) denotes the addition of two items. This is why we placed it in the middle of the apples. We started with two apples and then added three more.
The equals sign (=) is the other symbol in the equation. In an equation, the equals sign indicates that two or more things are equal or equivalent. Equivalent things do not always look or sound the same, but they mean the same thing. The equal sign in math indicates that two numbers or expressions mean the same thing, despite their appearance.
Any addition operation can be expressed in writing. Let us say you had 12 friends over for a birthday celebration. You invite four more people at the last minute. You could write something like this to get the total number of guests coming to your house:
The expression is simply another way of demonstrating the situation: a birthday party will be held for 12 friends plus four more.
Strategies for the addition operation
Certain strategies are used to perform operations in a fixed way. By using these strategies, we can perform mathematical operations easily and efficiently. There are five main strategies of addition. Let us take a look at these strategies:
Addition using an open number line
An open number line has no numbers or markings on it. Open number lines are useful for working with place values while adding numbers.
Let us see a few ways of applying this strategy.
Consider 17 + 38 for the following cases:
For example,
- Number Line 1:
The tens (10 + 30) were added first, followed by the ones (7 + 8). The number line begins at 10 (the tens from the first number) and continues till 40 by adding the three tens from the second number. Then we added 7 + 8 to get 15. The two results are then added to get 55.
- Number Line 2:
We must leave 17 as a whole and add the three tens from the second number. The 8 ones were then broken down into 3 + 5 Then the 3 ones were used to make 47 + 3 = 50. Finally, we added the last five to get to 50 + 5 = 55.
- Number Line 3:
We know that 38 can be broken down into 30 + 8. Further, 8 can be broken down into 3 + 5, so 38 is finally 30 + 3 + 5. Now, we have taken three ones from the 8 in 38. Three is then added to 17 to make 20 (17 + 3 = 20). Then we took the three tens from 30 to get to 50. Finally, we added the remaining 5 to a total of 55.
Addition using partial sum
In comparison to all the other methods of addition, the partial sum is the simplest. The partial sum method, as the name suggests, calculates partial sums of different place value columns at a time. Then we add all of the partial sums to find the total.
Partially added sums can be added in any order, but the most common method is to work from left to right. Since we read from left to right, this order seems natural, and it also prioritizes the most important place value to start from, in the addends, such as thousands before hundreds, hundreds before tens, and so on.
For mental arithmetic, the method of partial sums addition can be easily adapted.
For example, adding 7701 and 243.
Addition using regrouping
When adding two or more numbers of any size, the technique of addition with regrouping can be used. It is used with the column method of addition, which arranges sums vertically and adds numbers one at a time. Regrouping is sometimes referred to as “carrying over”. When you add all the numbers in a column together and the total is ten or more, the number in the tens place is carried over to the next place value column.
For example, if the ones column contains a 2 and a 9, the total will be 11. You would put the 1 in the ones column and then carry 1 over to the tens column.
When the sum of the values in one place value column exceeds nine, regrouping is applied.
There is no need to use the regrouping method if the sum of the values in each place value column is nine or less.
For example, let’s consider the addition equation, 28 + 14.
As with any addition with the column method, we can line these numbers up vertically in their place value columns.
The numbers in the ones column can then be added. This is the far right column, which contains the 8 and the 4. These two numbers add up to 12. We would write the 2 in the ones column beneath the line, and carry the 1 over to the tens column, writing it above the other two numbers, following the regrouping method. After this stage, you should have something similar to this:
After this, you can add the digits in the tens column together – the 2 and 1 from the original numbers, as well as the 1 you just carried over. Then you will get 4 as the answer from the sum of 1 + 2 + 1. This should be written in the tens column of your answer.
That is how the regrouping method works. The answer to this question is 42.
Addition using the compensation method
Rounding up a number (to make adding easier) and then subtracting the extra number after you have completed the calculation, is known as “compensation”.
For example,
Adding 29 and 16.
Writing this as an expression, 29 + 16
It is simpler to solve 30 + 16 = 46.
Then subtract the extra one, which turned 29 into 30, to get 45.
For example,
Adding 695 and 116.
As an expression we have, 695 + 116
It is simpler to solve 700 + 116 = 816.
Then we subtract the extra 5 that has made 695 into 700, to get 811.
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Addition using block models
Base ten blocks are visual representations that help us understand the base 10 system by representing the place value of numbers. We use these blocks to visualize the regrouping process so that we can fully comprehend the common ways used to add numbers. This method also helps us identify errors in addition.
Let us understand this by using models. Each cube is a one. This stack contains ten ones.
Let us consider that there are ten stacks in total.
Ten stacks of ten ones each is equal to one hundred.
1 hundred = 10 tens = 100.
For example, adding 44 and 37.
Solved Examples on Different Addition Methods
1) As part of an experiment, a group of children toss a coin. They get 13 heads and 32 tails. How many times did they toss the coin?
Solution: Add 13 and 32 to find out how many times the coin was tossed. Use an open number line for working on adding numbers 13 and 32. When the numbers in an addition operation are simpler, we can use the number line method for addition.
The tens (10 + 30) were added first, followed by the ones (3 + 2). The number line begins at 10 (the tens from the first number) and continues to 40 by adding the three tens from the second number. Then we added 3 + 2 to get 5, which we then added to 40 to get 45.
Hence, they tossed the coin 45 times.
2) There are 1525 frogs in a lake, and while in another lake, there are 251. How many frogs are there in total?
Solution: Adding 1525 and 251 frogs to find frogs in all. The partial sum method can be adopted for this question because it makes the calculations easy by using partial sums one place value column at a time. It then adds all of the partial sums to find a total. When we have to divide large numbers, we must use the partial sum method.
Therefore, the total number of frogs in both lakes is 1776.
3) In one field, Joseph plants 16 trees, while in another, he plants 18. What is the total number of trees he has planted?
Solution: Add 16 and 18 to find the total trees Joseph planted. When the sum of the values in one place value column exceeds nine, regrouping is used.
Here, the sum of the values in one place value = 6 + 8 = 14, which exceeds nine. Hence, the regrouping method is used.
As with any addition operation that involves the column method, we can line these numbers up vertically according to their place value columns.
The numbers in the ones column can then be added. This is the far right column, which contains the 6 and the 8. These two numbers add up to 14. We would write the 4 in the ones column beneath the line, and carry the 1 over to the tens column, writing it above the other two numbers, following the regrouping method. After this stage, you should have something similar to this:
Now, you can add the digits in the tens column together – the 1 and 1 from the original numbers, as well as the 1 you just carried over. You will get 3 as the answer from the sum of 1 + 1 + 1. This should be written in the tens column of your answer.
Therefore, Joseph planted 34 trees overall.
Math Curriculum for all Grades
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- Count up from a single digit
- The jumping technique
- Counting to ten
- How long do the tens last? (break big numbers into tens and units, add the units, then add on the tens)
- Set a goal of ten (when a number is close to ten, we can “borrow” from the other number so it reaches ten)
- The compensation strategy
- When the numbers are the same, double them.
- If the numbers are close, double them and then fix them.
Some numbers are easier to work with than others. It may be easier to add or subtract ten than nine or eleven. Compensation is a method of simplifying the addition or subtraction of numbers. If you add or subtract too much or too little, you must also add or subtract this in the result as the compensation.