The four operations and order of operations

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レッスンノート

The Four Operations

Addition (+) combines numbers to find a total or sum.
Subtraction (−) finds the difference between numbers.
Multiplication (×) is repeated addition; the result is the product.
Division (÷) splits a number into equal parts; the result is the quotient.
Each operation has an inverse: addition and subtraction are inverses, as are multiplication and division.

Order of Operations: BIDMAS/BODMAS

The conventional order is: Brackets, Indices (powers and roots), Division and Multiplication (equal priority), Addition and Subtraction (equal priority).
Perform operations from left to right when they have the same priority.
Brackets (parentheses) are always evaluated first, working from the innermost to the outermost.
Indices include powers (e.g., 3²) and roots (e.g., √9).
Multiplication and division are done before addition and subtraction.
Example: 1 + 2 × 3 = 1 + 6 = 7, not (1 + 2) × 3 = 9.

Using Brackets to Override Order

Brackets force a different order: (2 + 3) × 4 = 5 × 4 = 20.
Nested brackets: evaluate the innermost first, e.g., [2 × (3 + 4)] − 5 = [2 × 7] − 5 = 14 − 5 = 9.
Different types of brackets ( ), [ ], { } can be used to make grouping clearer.

Indices and Roots

An index (power) tells you how many times to multiply the base by itself: 3⁴ = 3 × 3 × 3 × 3 = 81.
Indices have higher priority than multiplication and division.
Example: 1 − 2 × 3⁴ = 1 − 2 × 81 = 1 − 162 = −161.
A root is the inverse of a power: √(1+3) + 5 = √4 + 5 = 2 + 5 = 7.
The radical symbol √ extends over its radicand (the expression under the root) without needing brackets.

Mental Strategies

Use known facts and place value to calculate mentally, e.g., 25 × 4 = 100.
Break numbers into parts: 47 + 38 = 47 + 30 + 8 = 85.
Use inverse operations to check: if 7 × 8 = 56, then 56 ÷ 8 = 7.
Estimate first to check the reasonableness of an answer.

Written Methods for Addition and Subtraction

Column addition: align numbers by place value, add each column from right to left, carrying when a column sum is 10 or more.
Column subtraction: align numbers, subtract each column from right to left, borrowing when needed.
For decimals, align the decimal points before adding or subtracting.

Written Methods for Multiplication

Long multiplication: multiply by each digit of the second number, then add the partial products.
Example: 23 × 14 = (23 × 10) + (23 × 4) = 230 + 92 = 322.
For decimals, multiply as if whole numbers, then place the decimal point so the total number of decimal places matches the question.

Written Methods for Division

Short division: divide each digit from left to right, carrying remainders.
Long division: use for larger divisors; the steps are divide, multiply, subtract, bring down.
Example: 322 ÷ 14 = 23 because 14 × 23 = 322.
For decimals, adjust the division to work with whole numbers where possible.

Inverse Operations and Checking

Use inverse operations to check answers: addition checks subtraction, multiplication checks division.
For example, if 56 ÷ 8 = 7, then 7 × 8 = 56 confirms the answer.
Estimating before calculating helps spot errors.

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練習問題

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1.In the conventional order of operations, which operation has the highest precedence?
Easy
- ABrackets (parentheses)
- BExponentiation (powers and roots)
- CMultiplication and division
- DAddition and subtraction
2.Work out the value of 1 + 2 × 3.
Easy
- A7
- B9
- C6
- D5
3.Work out the value of (1 + 2) × 3.
Easy
- A9
- B7
- C6
- D12
4.Work out the value of 1 − 2 × 3⁴.
Medium
- A−161
- B161
- C−81
- D−162
5.Work out the value of 1 + 2^(3 + 4).
Medium
- A129
- B2187
- C15
- D128
6.Work out the value of √(1 + 3) + 5.
Medium
- A7
- B9
- C6
- D3
7.Work out the value of [2 × (3 + 4)] − 5.
Medium
- A9
- B11
- C13
- D21
8.Work out the value of (3 + 5)².
Medium
- A64
- B16
- C28
- D36

もっと問題

9.Work out the value of 3 + 5².
Easy
- A28
- B64
- C16
- D13
10.Work out the value of 3 × 5².
Easy
- A75
- B225
- C30
- D45
11.In the expression 1 + 2 × 3, the addition is performed before the multiplication.
Easy
True or false?
12.Operations of the same precedence, such as multiplication and division, are conventionally evaluated from left to right.
Easy
True or false?
13.Which of the following statements about the order of operations are correct? (select all that apply)
Medium
- ABrackets are evaluated before exponentiation.
- BExponentiation is evaluated before multiplication.
- CAddition is evaluated before exponentiation.
- DMultiplication is evaluated before subtraction.
- ESubtraction is evaluated before division.
14.Which of the following expressions have the value 9? (select all that apply)
Medium
- A(1 + 2) × 3
- B[2 × (3 + 4)] − 5
- C1 + 2 × 3
- D(3 + 5)²
- E1 + 2^(3 + 4)
15.Match each expression to its value.
Medium
- 1 + 2 × 3
- (1 + 2) × 3
- 3 + 5²
- 3 × 5²
- 7
- 9
- 28
- 75
16.Put the four levels of the conventional order of operations in the order they are applied, starting with the first.
Easy
- Brackets (parentheses)
- Exponentiation (powers and roots)
- Multiplication and division
- Addition and subtraction
17.Which operation should be carried out first in 1 + 2 × 3²?
Medium
- AThe power 3²
- BThe multiplication 2 × 3
- CThe addition 1 + 2
- DThe addition 1 + 6
18.The student's answer of 36 is correct.
Easy
True or false?
19.Work out the value of 48 ÷ 6 × 2.
Easy
- A4
- B16
- C1.5
- D576
20.Work out the value of 20 − 3 × 4 + 2.
Medium
- A10
- B70
- C12
- D−10
21.Which of these expressions have the value 24? (select all that apply)
Medium
- A2 + 4 × 6 − 2
- B(2 + 4) × (6 − 2)
- C2 × 4 + 6 × 2
- D2 × (4 + 6) + 2
- E2 + 4 × (6 − 2)
22.In the expression 10 − 2 × 3, the subtraction is performed before the multiplication.
Easy
True or false?
23.Put these steps in the correct order to evaluate 3 + 4 × (2 + 1)².
Easy
- Work out 2 + 1 = 3
- Work out 3² = 9
- Work out 4 × 9 = 36
- Work out 3 + 36 = 39
24.Select all the statements that are true about the order of operations.
Medium
- AExponentiation is performed before multiplication.
- BAddition and subtraction have the same precedence.
- CBrackets are always evaluated last.
- DWhen operations have the same precedence, work from left to right.
- EMultiplication is performed before addition.
25.Work out the value of 100 − (2 + 3)² × 2.
Hard
- A50
- B−50
- C150
- D0
26.Work out the value of 6 + 4 × 5.
Easy
- A26
- B50
- C34
- D46
27.Work out the value of 15 − 3 × 4.
Easy
- A3
- B48
- C12
- D27
28.Work out the value of 7 + 2³.
Medium
- A15
- B17
- C23
- D13
29.Work out the value of 20 ÷ 4 + 3 × 2.
Medium
- A11
- B16
- C13
- D10
30.Work out the value of (8 + 4) ÷ 2².
Medium
- A3
- B6
- C9
- D12
31.Work out the value of √(9) × 4.
Easy
- A12
- B36
- C6
- D18
32.Which of the following expressions have the value 14? (select all that apply)
Medium
- A2 + 3 × 4
- B20 − 2 × 3
- C2 × (3 + 4)
- D12 + 6 ÷ 3
- E4² − 2
33.In the expression 8 ÷ 2 × 4, the multiplication is performed before the division.
Easy
True or false?
34.What is the correct value of the expression?
Medium
- A8
- B28
- C16
- D4
35.Which expression gives the number of pencils each class receives?
Medium
- A12 × 24 ÷ 8
- B12 ÷ 8 × 24
- C12 × (24 ÷ 8)
- D12 ÷ (24 × 8)
36.A rectangle has length 8 cm and width 5 cm. A square has side length 6 cm. Which expression gives the total area of the rectangle and the square?
Medium
- A8 × 5 + 6²
- B(8 + 5) × 2 + 6²
- C8 × 5 + 6 × 4
- D8 + 5 + 6²
37.Which of the following expressions have the value 16? (select all that apply)
Medium
- A2⁴
- B4 + 2²
- C(2 + 2)²
- D8 + 2 × 4
- E(20 − 2) × 2
38.In the expression 5 + 3 × 2², the addition is performed before the exponentiation.
Easy
True or false?
39.Match each expression to its correct value.
Medium
- 3 + 4 × 2
- (3 + 4) × 2
- 3 + 4²
- 3² + 4²
- 11
- 14
- 19
- 25
40.Put the steps in the correct order to evaluate 5 + 2 × (3² − 4).
Medium
- Evaluate 3²
- Subtract 4 from 9
- Multiply 2 by 5
- Add 5 to 10
41.Complete the statement about precedence.
Easy
An operation with a higher ____ is performed before an operation with a lower one.
42.Complete the rule for operations of equal precedence.
Easy
Operations of the same precedence are conventionally evaluated from ____ to ____.
43.Complete the calculation by filling in each missing value.
Medium
1 − 2 × 3⁴ = 1 − 2 × ____ = 1 − ____ = ____
44.What name is given to the rank of an operation, which decides how early it is performed?
Easy
45.What symbol is placed over the radicand of a square root so that no brackets are needed?
Medium
46.What is the correct value of 1 + 2 × 3²?
Medium
47.Work out the value of 5 + 3 × 2³.
Medium
48.Complete the statement about the order of operations.
Medium
In the conventional order of operations, ____ are evaluated first, then ____, then multiplication and division, and finally addition and subtraction.
49.Work out the value of 12 + 3 × 5.
Medium
50.Which operation in the expression is performed first?
Easy
51.Work out the value of 7 + 5 × 2.
Easy
52.Work out the value of (7 + 5) × 2.
Easy
53.Explain why the expression 1 + 2 × 3 has the value 7 and not 9.
Medium
54.Explain why brackets are used in expressions such as (2 + 3) × 4, and give the value of this expression.
Medium
55.Explain why the expression 2 + 3 × 4 has the value 14 and not 20.
Medium
56.Explain why the expression 4 + 2 × 3 has the value 10 and not 18.
Hard
57.Explain why the expression 7 + 4 × 3 has the value 19 and not 33.
Medium
58.What is the correct value of 1 + 2 × 3²?
Medium
- A19
- B18
- C21
- D20
59.Work out the value of 5 + 3 × 2³.
Medium
- A32
- B26
- C29
- D35
60.Work out the value of 12 + 3 × 5.
Medium
- A33
- B27
- C30
- D24
61.Work out the value of 7 + 5 × 2.
Easy
- A17
- B19
- C18
- D16
62.Work out the value of (7 + 5) × 2.
Easy
- A24
- B26
- C28
- D22

復習ノート

## The Four Operations and Order of Operations **The four operations** are addition (+), subtraction (−), multiplication (×) and division (÷). ### Written methods - **Addition and subtraction**: line up place values, then work column by column (carry or borrow as needed). - **Long multiplication**: multiply by each digit in turn, then add the partial products. - **Short division**: divide digit by digit, carrying remainders. - **Long division**: divide, multiply, subtract, bring down — repeat. - These methods work for **integers and decimals** (keep decimal points aligned). ### Mental strategies Break numbers into parts, use near-doubles, round and adjust, and use known facts (e.g. 25 × 4 = 100). ### Order of operations (BIDMAS/BODMAS) Evaluate in this order: 1. **B**rackets (parentheses) — innermost first 2. **I**ndices / **O**rders — powers and roots 3. **D**ivision and **M**ultiplication (left to right) 4. **A**ddition and **S**ubtraction (left to right) Operations of the same precedence are done **left to right**. **Examples** - 1 + 2 × 3 = 1 + 6 = 7 (multiplication before addition) - (1 + 2) × 3 = 3 × 3 = 9 (brackets first) - 1 − 2 × 3^4 = 1 − 2 × 81 = 1 − 162 = −161 (indices, then multiplication, then subtraction) - 1 + 2^{3+4} = 1 + 2^7 = 1 + 128 = 129 (the superscript groups 3 + 4) - sqrt(1 + 3) + 5 = sqrt(4) + 5 = 2 + 5 = 7 (the root bar groups 1 + 3) ### Brackets Use brackets ( ) to override or emphasise precedence. Nested brackets may use different shapes, e.g. [2 × (3 + 4)] − 5 = [2 × 7] − 5 = 14 − 5 = 9. ### Inverse operations and checking - Addition and subtraction are **inverses**; multiplication and division are **inverses**. - Check an answer by doing the **inverse operation** (e.g. check 7 × 8 = 56 with 56 ÷ 8 = 7), or by estimating first. ### Key points - Higher **precedence** means performed first. - Brackets can be treated as a single expression. - These conventions exist to avoid ambiguity.

Order of operations: The conventions deciding which arithmetic operations are performed first when evaluating an expression.
Precedence: The rank of an operation; an operation with higher precedence is performed before one with lower precedence.
BIDMAS/BODMAS: Brackets, Indices (Orders), Division and Multiplication, Addition and Subtraction — the conventional order of operations.
Brackets (parentheses): Grouping symbols used to override or emphasise precedence; the sub-expression inside is evaluated first.
Indices (powers): A superscript showing repeated multiplication of a base, e.g. 3^4 = 81; evaluated before multiplication and addition.
Root: The inverse of a power; the radical symbol √ with a bar (vinculum) groups the radicand, e.g. sqrt(1 + 3) = 2.
Inverse operation: The operation that undoes another: addition and subtraction are inverses; multiplication and division are inverses.
Long multiplication: A formal written method that multiplies by each digit in turn and adds the partial products.
Long division: A formal written method that repeatedly divides, multiplies, subtracts and brings down digits.
Short division: A compact written division method working digit by digit, carrying remainders.

フラッシュカード

What does BIDMAS stand for, and what does it tell you to do?

Brackets, Indices, Division and Multiplication, Addition and Subtraction. It gives the order in which operations in an expression are performed: brackets first, then powers and roots, then division and multiplication, then addition and subtraction.

What is the value of 1 + 2 × 3?

7. Multiplication is done before addition, so 1 + (2 × 3) = 1 + 6 = 7.

What is the value of (1 + 2) × 3?

9. The brackets are evaluated first, so (1 + 2) × 3 = 3 × 3 = 9.

What is the value of 1 − 2 × 3⁴?

−161. The power is done first, then the multiplication, then the subtraction: 1 − 2 × 81 = 1 − 162 = −161.

What is the value of 1 + 2³⁺⁴?

129. The whole superscript is grouped by its position, so 2³⁺⁴ = 2⁷ = 128, then 1 + 128 = 129.

What is the value of √(1 + 3) + 5?

7. The bar of the root sign covers 1 + 3, so √(4) + 5 = 2 + 5 = 7.

What is the value of [2 × (3 + 4)] − 5?

9. Work from the innermost brackets outwards: 2 × 7 = 14, then 14 − 5 = 9.

In an expression with only multiplication and division, in what order do you work?

From left to right. Multiplication and division have equal precedence, so they are evaluated in the order they appear.

In an expression with only addition and subtraction, in what order do you work?

From left to right. Addition and subtraction have equal precedence, so they are evaluated in the order they appear.

What is the precedence of an operation?

The rank of an operation. An operation with higher precedence is performed before one with lower precedence.

Why do we need an agreed order of operations?

To avoid ambiguity: without fixed conventions, an expression such as 1 + 2 × 3 could have more than one value. The conventions let notation stay brief while having a single clear meaning.

How can you override the usual order of operations in an expression?

By putting part of the expression in brackets (parentheses). Brackets are evaluated first, so (2 + 3) × 4 = 20 instead of 2 + 3 × 4 = 14.

What is an inverse operation? Give the inverse of addition and of multiplication.

An operation that undoes another. The inverse of addition is subtraction; the inverse of multiplication is division.

How can inverse operations be used to check an answer?

Apply the inverse operation to your result and see whether you return to the starting number. For example, if 47 + 68 = 115, then 115 − 68 should give 47.

What is the value of 3 + 5²?

28. The power is evaluated before the addition: 3 + 25 = 28.

What is the value of 3 × 5²?

75. The power is evaluated before the multiplication: 3 × 25 = 75.

What is the value of (3 + 5)²?

64. The brackets are evaluated first, so 8² = 64.

Which written method would you use to multiply two 2-digit numbers such as 47 × 68?

Long multiplication: multiply by each digit in turn, lining up the place values, then add the partial products.

What is the difference between short division and long division?

Short division is a compact method used when dividing by a single-digit number. Long division sets out each step and is used when dividing by a larger number (two or more digits).

How do you divide a decimal by a whole number using short division?

Set out the division as usual and keep the decimal point in the answer lined up with the decimal point in the number being divided, then divide digit by digit.

What does the radical symbol √() mean, and what is the bar over the number for?

It means square root. The bar (the vinculum) shows exactly which expression is the radicand, so no extra brackets are needed: √(1 + 3) means the square root of 4.

Evaluate 20 − 3 × 4 + 2.

10. Multiplication first: 20 − 12 + 2. Then work left to right: 8 + 2 = 10.

Evaluate 36 ÷ 6 × 2.

12. Division and multiplication have equal precedence, so work left to right: 36 ÷ 6 = 6, then 6 × 2 = 12.

Evaluate 2 + 3 × (8 − 5)².

29. Brackets first: 8 − 5 = 3. Then the power: 3² = 9. Then multiplication: 3 × 9 = 27. Then addition: 2 + 27 = 29.

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