Numbers are used for counting and are the basic building blocks of math. In real-life scenarios, we often see a need to do more with numbers like, putting things together or taking a few things away and counting the rest.
Let us see a few common situations that Justin experienced on a regular day at school –
These situations need us to work with numbers, or in other words, do number operations.
These operations are also called Arithmetic Operations. There are four basic operations in mathematics which are the fundamental key aspects for most of the fields of science and technology.
The basic mathematical operations are:
Addition is a prominent arithmetic operation. Performing this operation means to combine and finding the total. We need at least two numbers to combine, which are called as ‘addends’ and the result is called as the ‘sum.’
When we try to add big values, there arises a need to use many tricks and strategies to make our task easier. There are many addition strategies that can be applied to add numbers easily. Some of them are listed below. We can perform addition using:
Pictures and models are easy to understand, interpret. They are fun to learn with. Hence, using visual and concrete models for teaching any concept is an effective way. We can count the number of objects or blocks in a picture frame to find the total.
If pictures are replaced by numerals or each picture is mapped on to a number, then counting the pictures is nothing but counting the numbers forward. To add two numbers using the method of counting forward, we choose a bigger number out of the two given numbers and count forward as many times as the smaller number. The number at which you stop is the sum.
Numbers can be plotted on a line. Such a line is called as a number line. Therefore, we can also use a number line to add two or more numbers. The steps followed to add is similar to that briefed in the count forward method, except that we count forward on a number line to perform the mathematical operation of addition.
How do you think people managed to calculate when calculators were not available back in those days? They used a device by name ‘Abacus.’ Abacus is a device with wooden pegs and beads representing digits of numbers. Simple strategies were formulated to solve problems that include complex numbers, and those strategies were in turn used in abacus. Abacus model inspired many useful inventions like computers, calculators, etc.
Components of a number bond include two parts which make a whole. Number bond is a pictorial representation of a addition sentence which includes part + part = whole. This can be used to understand that a whole is made of parts and these parts can be of different proportions. Also, this strategy can be used to formulate a number in different ways. Learn this and play around making numbers!
The concept of bar model is a replica of the number bond but the view of the pictorial representation is different. A rectangle divided horizontally in two rectangles, out of those the top one will show the part and the bottom rectangle will be further divided into two more rectangles by drawing a vertical line from one side to the other. These show parts that make the whole. Unlike number bonds bar models have a particular advantage. Bar models depict the proportion of the parts as well by different areas of the parts.
Counting tens and also performing operation with tens numbers is an easy venture. Tens frame makes is even easier as this involves a rectangular frame with 10 cells in them. This particular method is taught to young learners as a visual model to add or subtract numbers. As we know any number is can be displayed a objects or pictures. These objects when put into the tens frames to make a ten or more it is easier for a child to visualize the process of addition or subtraction and also perform them as well.
A double is the sum obtained when a number is added to itself. In other words we can say that we are doubling the number. We can easily perform addition splitting an addend into two doubles. And this strategy has a sub technique that is called as doubles plus one, which can be applied while adding two consecutive numbers.
Using base-10 blocks to represent equations is an excellent approach to teach how to solve them and provide a mental understanding of the equations. It develops a much more in-depth understanding of addition than simply memorizing facts.
A place value chart, often known as a place value table, is a tool for illustrating the numerical value of each digit. In other words, it displays the digit’s value dependent on its position in the numeral, thus the term place value.
When adding huge numbers it is necessary for us to split that big addend into parts that are easier to add. This method as involves splitting a big addend so, we call it as decomposition.
Compensation is a multi-digit addition mental math approach in which one of the addends is adjusted to make the equation easier to complete. This approach may be preferred by certain students over left-to-right addition or breaking up the second number. When it comes to making equations easier to solve, compensation is a good method.
Subtraction is the reverse arithmetic operation of addition. If addition means bringing together, subtraction means to take away. We use the “-” sign for subtracting one number from the other. This sign is called as “minus” sign. Just like addition we use many strategies to perform subtraction too. The number to be subtracted is called as the subtrahend and the number from which we subtract is called as the minuend. The answer that we obtain after we subtract is called as the difference.
They can be listed as follows:
We can use pictures and models to subtract. If we need to take away a number from another number we will cross out the first number of objects from the given set of objects and then count the remaining to find the difference. We can also use models with counters to subtract. This method is effective to perform the operation using small numbers.
The abacus is a counting device that has been used by humans for thousands of years to perform a wide range of arithmetic operations. With the appropriate understanding, almost any math problem may be solved by manipulating beads on an abacus. Where calculators or pen and paper are not readily available, merchants, dealers, and the general public nevertheless use them. You’ll notice four rows of beads behind a separating bar when you first look at the abacus. Above the bar, you’ll notice that there are rows of just one bead. We move these beads following a technique to perform subtraction.
Students may readily picture the value of each number in an equation using tens-frames, which is also a great material for teaching subtraction. They can use their imagination to transfer the number of dots from the second tens-frame to the first tens-frame and observe how many are left.
A collection of 10 small cubes put together will make a base 10 block. Given two numbers to subtract you can represent them visually using the base ten blocks and place the model of the biggest number on the top of the smallest number. Subtract the digits in the respective places by crossing out or taking away the blocks.
When studying subtraction, children are usually introduced to counting back as one of the first tactics. One can subtract by counting backward by understanding that -1 denotes the previous number, and -2 denotes the number two numbers prior in the counting sequence.
Subtraction on a number line is a way of determining the difference between two numbers that are evenly spaced on a horizontal line.
You’ll need to execute the following to find the difference on a number line:
Positive integers should be subtracted to the left.
Negative integers are subtracted by going towards the right side of the zero.
Number Bonds make it easier to teach Subtraction. We do not have to remember the subtraction facts if we use this method. Number bonds are as straightforward as demonstrating that two parts will combine to form a whole. In addition, you will be given two parts and asked to find the whole. You are given the total and one of the parts in subtraction, and you must determine the other part.
The kid can visualize the answer to addition and subtraction by illustrating the bond.
Bar models are extensively used to solve real-world problems on subtraction. As explained earlier bar models are a alternate visual representation like number bonds.
It is not difficult to subtract, however, it is tough to do it orally using the traditional way. Using place value, it is simple to mentally add the numbers. Suppose you are asked to find 277 – 146.
200 – 100 =100 (Sub place value of first digit)
70 – 40 = 30 (Sub place value of second digit)
7 – 6 = 1 (Sub place value of third digit)
Now add the results, 100 +30+1 = 131.
Answer, 277 – 146 = 131
This is sometimes called the decomposition or partitioning. This method is exactly the same as using place value expansion to subtract.
Multiplication is also one of the four fundamental arithmetic operations, produces the outcome of merging groups of equal size. Therefore, multiplication can be termed as repeated addition of the same number. The signs cross ‘×’ , asterisk ‘*’ (in coding), and dot ‘.’ are used to express multiplication. The result of multiplying two numbers is known as the ‘product.’ ‘Multiplicand’ refers to the number of items in each group, while multiplier refers to the number of such equal groupings.
The strategies/tricks adopted to ease the process of multiplying are as follows:
Equal groups practice is a visual technique of learning basic multiplication by sorting a bunch of items into equal heaps, for example. 12 mangoes can be divided into three groups of four mangoes each.
There are 2 times 3 or 3 + 3 or 6 candies overall if there are two sets of three candies. Hence, multiplication is, in other words, repeated addition.
Learning to skip count is a crucial multiplication strategy. Skip counting sequences can help students figure out their solutions to unknown information if they know how to skip count by rote.
For instance, if we need to find 6 x 6, we can skip count from a known fact (usually the 2, 5, or 10 multiplication facts) to find the answer – 16, 25, 36.
On a number line, a multiplication sentence can be represented by jumping the number of times equal to the multiplier with a size equal to the multiplicand from zero.
After we’ve covered multiplication as repeated addition or equal groups, the next natural step is to cover arrays. An array is a collection of items or images that are arranged in rows and columns.
Suppose we have a multiplication sentence like 3 × 4 = 12, this means the objects are arranged in 3 rows and 4 columns.
Number bonds are mental representations of the relationship between a number and the components that make it up. Suppose a number bond is displaying the sentence 7 × 3 = 21, then 7 and 3 are the parts and 21 is the whole.
Place value is significant because it lays the groundwork for regrouping and multiplication with multiple digits. We can find a brief description about the procedure in our page “Multiplication.”
Multiplying two big integers with Lattice multiplication is a grid-based approach. Fibonacci employed lattice multiplication in the fourteenth century after it was found in India in the tenth century. The box method is another name for this strategy.
The area of a shape is the amount of space it takes up. If a rectangle’s length is 15 units and its breadth is 18 units, the area may be calculated by multiplying 15 by 18. Therefore, we can understand that the product 15 x 18 geometrically as the area of a rectangle with a length of 15 units and a width of 18 units.
A way of dividing a bunch of things into equal portions is known as division. It’s one of the four basic arithmetic operations that produce a fair sharing result. Multiplication is the inverse process of division. In multiplication, four groups of three equal 12; in the division, the number of ways groups of 3’s from 12 is 4. The division’s main purpose is to figure out how many equal groups there are, or how many people are in each group when everyone shares equally.
The signs ÷, / (coding) are used to show division process. A division number sentence in words can be written as “Dividend ÷ Divisor = Quotient.” Where,
Dividend: The dividend is the number that is being divided.
Divisor: The number by which dividend is being divided by is called divisor.
Quotient: A quotient is a result obtained.
Here is a sample division number statement 21 ÷ 3 = 7.
The Strategies used to perform division are as follows:
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