Arithmetic expressions are one of the obligatory and most important topics in the course of school mathematics. Insufficient knowledge of this topic will lead to difficulties in studying almost any other material related to algebra, geometry, physics or chemistry.
Features of working with arithmetic expressions in elementary school
In primary grades, the first arithmetic operations are introduced immediately after studying the ordinal count.
As a rule, the first two operations that are studied almost simultaneously are addition and subtraction. These actions are most in demand in the practical life of any person: when going to the store, paying bills, setting deadlines for work, and in many other everyday situations.
The main difficulty that a child may encounter is a rather high level of arithmetic abstraction. Often, children are much easier to cope with tasks when it comes to counting specific objects, such as apples or sweets.
The teacher’s task is to help move on to the concept of number, that is, to the addition and subtraction of quantities not directly tied to the physical world.
The second goal in the initial study of arithmetic expressions is the assimilation of terminology by students.
Basic terms of arithmetic in elementary school
For the operation of addition, the basic concepts are the term and the sum.
In the correct equality, 10 + 15 = 25: 10 and 15 are the terms, and 25 is the sum. In this case, the arithmetic expression on the left side of the sign "=" 10 + 15 is also correctly called the sum.
The numbers 10 and 15 are called the same word, since the sum will not be affected by their permutation.
The general rule in the form of a formula is written as follows:
a + c = c + a,
where any numbers can be in place a and c. Independence from the order is preserved not only for two, but also for any number of terms (finite).
Otherwise, the situation is with subtraction, for which you have to remember three terms at once: diminished, subtracted and difference.
In the example 25-10 = 15:
- reducible is 25;
- deductible - 10;
- and the difference is 15 or the expression 25-10.
Addition and subtraction are inverse operations to each other.
The following two inverse actions, studied in the elementary grades, multiplication and division, have a slightly greater computational complexity, so they go through them later.
In equality with a multiplication of 10 × 15 = 150: 10 and 15 are the factors, and 150 or 10 × 15 is the product.
For the permutation of the factors, the same rule applies as for the permutation of the terms: the result does not depend on the order of their sequence in the record of the arithmetic expression.
At school, the multiplication sign today is often indicated by a dot, not a cross or an asterisk.
To indicate division, a colon or a fraction sign is used (but this is in higher classes):
15: 3 = 5.
Here 15 is a dividend, 3 is a divisor, 5 is a quotient. The expression 15: 3 is also called the ratio or ratio of two numbers.
Procedure
To successfully complete tasks related to arithmetic expressions, you need to remember the order of operations:
- If an operation is enclosed in parentheses, it is executed first.
- Then multiplication or division is performed.
- Addition and subtraction are the last steps.
- If an expression contains several operations with the same priority, then they are carried out in the order in which they are written (from left to right).
Types of tasks
The most common types of arithmetic problems in elementary school are tasks to determine the order of actions, calculate and record numerical expressions according to a given verbal formulation.
Before calculating expressions of a complex construction, the child should be taught to independently arrange the order of actions, even if the assignment does not explicitly say so.
To calculate means to find the value of an arithmetic expression in the form of a number.
Examples of tasks
Task 1. Calculate: 3 + 5 × 3 + (8-1).
Before proceeding to the calculation itself, you need to understand the order of operations.
First step: Subtraction is performed because it is in parentheses.
1) 8-1 = 7.
Second action: the product is found, since the priority of this operation is higher than that of addition.
2) 5 × 3 = 15.
It remains to do the addition twice in the order in which the "+" signs are placed in the example.
3) 3 + 15 = 18.
4) 18 + 7 = 25.
The result of the calculations is written in response: 25.
Many teachers require at the beginning of training to write out each action separately. This allows the child to better navigate the course of the decision, and the teacher to identify the error in the verification.
Problem 2. Write down the arithmetic expression and find its meaning: the difference of two and the difference between the quotient of ninety and nine and the product of two triples.
In such tasks, you need to move from expressions consisting only of numbers to more complex ones.
In the above example, the condition clearly indicates the numbers for the quotient and the product.
The quotient of ninety and nine is written as 90: 9, and the product of two triples is 3 × 3.
The difference between the quotient and the work is required: 90: 9-3 × 3.
We return to the original difference between the two and the resulting expression: 2-90: 9--3 × 3. As you can see, the first of the subtractions is performed earlier than the second, which contradicts the condition. The problem is solved by placing parentheses: 2- (90: 9--3 × 3).
The resulting expression is calculated in the same way as in the first example considered.
- 90: 9 = 10.
- 3 × 3 = 9.
- 10-9 = 1.
- 2-1 = 1.
Answer: 1.