The order in which we evaluate expressions can be ambiguous. If we do the addition first, then

4+3 · 2=7 · 2

= 14.

On the other hand, if we do the multiplication first, then

4+3 · 2=4+6

= 10.

So, what are we to do? Of course, grouping symbols can remove the ambiguity

Grouping Symbols

Parentheses, brackets, or curly braces can be used to group parts of an expression. Each of the following are equivalent:

(4 + 3) · 2 or [4 + 3] · 2 or {4+3} · 2

In each case, the rule is “evaluate the expression inside the grouping symbols first.” If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.

Thus, for example,

(4 + 3) · 2=7 · 2

= 14.

Note how the expression contained in the parentheses was evaluated first. Another way to avoid ambiguities in evaluating expressions is to establish an order in which operations should be performed. The following guidelines should always be strictly enforced when evaluating expressions.

Rules Guiding Order of Operations

When evaluating expressions, proceed in the following order.

- Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
- Evaluate all exponents that appear in the expression.
- Perform all multiplications and divisions in the order that they appear in the expression, moving left to right.
- Perform all additions and subtractions in

Example 1

Evaluate 4 + 3 · 2.

**Solution**

Because of the established *Rules Guiding Order of Operations*, this expression is no longer ambiguous. There are no grouping symbols or exponents, so we immediately go to rule three, evaluate all multiplications and divisions in the order that they appear, moving left to right. After that we invoke rule four, performing all additions and subtractions in the order that they appear, moving left to right.

[ egin{aligned} 4+3 dot 2=4+6 = 10 end{aligned} onumber ]

Thus, 4 + 3 · 2 = 10.

Exercise

Simplify: 8 + 2 · 5.

**Answer**18

Example 2

Evaluate 18 − 2 + 3.

**Solution**

Follow the *Rules Guiding Order of Operations*. Addition has no precedence over subtraction, nor does subtraction have precedence over addition. We are to perform additions and subtractions as they occur, moving left to right.

[ egin{aligned} 18 − 2 + 3 = 16 + 3 & extcolor{red}{ ext{ Subtract: 18 − 2 = 16.}} = 19 & extcolor{red}{ ext{ Add: 16 + 3 = 19. }} end{aligned} onumber ]

Thus, 18 − 2 + 3 = 19.

Exercise

Simplify: 17 − 8 + 2.

**Answer**11

Example 3

Evaluate 54 ÷ 9 · 2.

**Solution**

Follow the *Rules Guiding Order of Operations*. Division has no precedence over multiplication, nor does multiplication have precedence over division. We are to perform divisions and multiplications as they occur, moving left to right.

[ egin{aligned} 54 div 9 cdot 2=6 dot 2 & extcolor{red}{ ext{ Divide: 54 } div ext{ 9 = 6. }} = 12 & extcolor{red}{ ext{ Multiply: 6 } cdot ext{ 2 = 12. }} end{aligned} onumber ]

Thus, 54 ÷ 9 · 2 = 12.

Exercise

Simplify: 72 ÷ 9 · 2.

**Answer**16

Example 4

Evaluate 2 · 3^{2} − 12.

**Solution**

Follow the *Rules Guiding Order of Operations*, exponents first, then multiplication, then subtraction.

[ egin{aligned} 2 cdot 3^2 - 12 = 2 dot 9 - 12 & extcolor{red}{ ext{ Evaluate the exponent: 3^2 = 9. }} = 18 - 12 & extcolor{red}{ ext{ Perform the multiplication: } 2 cdot 9 = 18. } = 6 & extcolor{red}{ ext{ Perform the subtraction: } 18 - 12 = 6.} end{aligned} onumber ]

Thus, 2 · 3^{2} − 12 = 6.

Exercise

Simplify: 14 + 3 · 4^{2}

**Answer**62

Example 5

Evaluate 12 + 2(3 + 2 · 5)^{2}.

**Solution**

Follow the Rules Guiding Order of Operations, evaluate the expression inside the parentheses first, then exponents, then multiplication, then addition.

[ egin{aligned} 12 + 2(3 + 5 cdot 5 )^2 = 12 + 2(3 + 10)^2 ~ & extcolor{red}{ ext{ Multiply inside parentheses: 2 } cdot 5 = 10.} = 12 + 2(13)^2 ~ & extcolor{red}{ ext{ Add inside parentheses: } 3 + 10 = 13.} = 12 + 2(169) ~ & extcolor{red}{ ext{ Exponents are next: } (13)^2 = 169.} = 12 + 338 ~ & extcolor{red}{ ext{ Multiplication is next: } 2(169) = 338.} = 350 ~ & extcolor{red}{ ext{ Time to add: } 12 + 338 = 350.} end{aligned} onumber ]

Thus, 12 + 2(3 + 2 · 5) 2 = 350.

Exercise

Simplify: 3(2 + 3 · 4)^{2} − 11.

**Answer**577

Example 6

Evaluate 2{2 + 2[2 + 2]}.

**Solution**

When grouping symbols are nested, evaluate the expression between the pair of innermost grouping symbols first.

[ egin{aligned} 2( 2 + 2[2 + 2]) = 2(2 + 2[4]) ~ & extcolor{red}{ ext{ Innermost grouping first: } 2 + 2 = 4.} = 2(2+8) ~ & extcolor{red}{ ext{ Multiply next: } 2[4] = 8.} = 2(10) ~ & extcolor{red}{ ext{ Add inside braces: } 2 + 8 = 10.} = 20 ~ & extcolor{red}{ ext{ Multiply: } 2(10) = 20} end{aligned} onumber ]

Thus, 2(2 + 2[2 + 2]) = 20.

Exercise

Simplify: 2{3 + 2[3 + 2]}.

**Answer**26

## Fraction Bars

Consider the expression

[ frac{6^{2}+8^{2}}{(2+3)^{2}} onumber ]

Because a fraction bar means division, the above expression is equivalent to

[left(6^{2}+8^{2} ight) div(2+3)^{2} onumber ]

The position of the grouping symbols signals how we should proceed. We should simplify the numerator, then the denominator, then divide.

Fractional Expressions

If a fractional expression is present, evaluate the numerator and denominator first, then divide.

Example 7

Evaluate the expression

[ frac{6^{2}+8^{2}}{(2+3)^{2}}. onumber ]

**Solution**

Simplify the numerator and denominator first, then divide.

[ egin{aligned} frac{6^{2}+8^{2}}{(2+3)^{2}}=frac{6^{2}+8^{2}}{(5)^{2}} ~ & extcolor{red}{ ext{ Parentheses in denominator first: } 2 + 3 = 5} = frac{36+64}{25} ~ & extcolor{red}{ ext{Exponents are next: } 6^2 = 36,~ 8^2 = 64,~ 5^2 = 25.} = frac{100}{25} ~ & extcolor{red}{ ext{ Add in numerator: } 36 + 64 = 100} = 4 ~ & extcolor{red}{ ext{ Divide: } 100 div 25 = 4.} end{aligned} onumber ]

Thus, (frac{6^{2}+8^{2}}{(2+3)^{2}}=4).

Exercise

Simplify: (frac{12+3 cdot 2}{6})**Answer**

3

## The Distributive Property

Consider the expression 2 · (3 + 4). If we follow the “Rules Guiding Order of Operations,” we would evaluate the expression inside the parentheses first. 2 · (3 + 4) = 2 · 7 Parentheses first: 3 + 4 = 7. = 14 Multiply: 2 · 7 = 14.

However, we could also choose to “distribute” the 2, first multiplying 2 times each addend in the parentheses.

[ egin{aligned} 2 cdot (3 + 4) = 2 cdot 3 + 2 cdot 4 ~ & extcolor{red}{ ext{ Multiply 2 times both 3 and 4.}} = 6 + 8 ~ & extcolor{red}{ ext{ Multiply: } 2 cdot 3 = 6 ext{ and } 2 cdot 4 = 8.} = 14 ~ & extcolor{red}{ ext{ Add: } 6 + 8 = 14.} end{aligned} onumber ]

The fact that we get the same answer in the second approach is an illustration of an important property of whole numbers.^{1}

The Distributive Property

Let *a*, *b*, and *c* be any whole numbers. Then,

*a* · (*b* + *c*) = *a* · *b* + *a* · *c*.

We say that “multiplication is distributive with respect to addition.”

Multiplication is distributive with respect to addition. If you are not computing the product of a number and a sum of numbers, the distributive property does not apply.

Caution! Wrong Answer Ahead!

If you are calculating the product of a number and the product of two numbers, the distributive property must not be used. For example, here is a common misapplication of the distributive property.

[ egin{aligned} 2 cdot (3 cdot 4) = (2 cdot 3) cdot (2 cdot 4) = 6 cdot 8 = 48 end{aligned} onumber ]

This result is quite distant from the correct answer, which is found by computing the product within the parentheses first.

[ egin{aligned} 2 cdot (3 cdot 4) = 2 cdot 12 = 24. end{aligned} onumber ]

In order to apply the distributive property, you must be multiplying times a sum.

Example 8

Use the distributive property to calculate 4 · (5 + 11).

**Solution**

This is the product of a number and a sum, so the distributive property may be applied.

[ egin{aligned} 4 cdot (5 + 11) = 4 cdot 5 + 4 cdot 11 ~ & extcolor{red}{ ext{ Distribute the 4 times addend in the sum.}} = 20 + 44 ~ & extcolor{red}{ ext{ Multiply: } 4 cdot 5 = 20 ext{ and } 4 cdot 11 = 44.} = 64 ~ & extcolor{red}{ ext{ Add: } 20 + 44 = 64.} end{aligned} onumber ]

Readers should check that the same answer is found by computing the sum within the parentheses first.

Exercise

Distribute: 5 · (11 + 8).

**Answer**95

The distributive property is the underpinning of the multiplication algorithm learned in our childhood years.

Example 9

Multiply: 6 · 43.

**Solution**

We’ll express 43 as sum, then use the distributive property.

[ egin{aligned} 6 cdot 43 = 6 cdot (40 + 3) ~ & extcolor{red}{ ext{ Express 43 as a sum: } 43 = 40 + 3} = 6 cdot 40 + 6 cdot 3 ~ & extcolor{red}{ ext{ Distribute the 6.}} = 240 + 18 ~ & extcolor{red}{ ext{ Multiply: } 6 cdot 40 = 240 ext{ and } 6 cdot 3 = 18.} = 258 ~ & extcolor{red}{ ext{ Add: } 240 + 18 = 258.} end{aligned} onumber ]

Readers should be able to see this application of the distributive property in the more familiar algorithmic form:

( egin{array}{r}{43} { imes 6} hline 18 {frac{240}{258}}end{array})

Or in the even more condensed form with “carrying:”

( egin{array}{r}{^{1} 43} {frac{ imes 6}{258}}end{array})

Exercise

Use the distributive property to evaluate 8 · 92.

**Answer**736

Multiplication is also distributive with respect to subtraction.

The Distributive Property (Subtraction)

Let *a*, *b*, and *c* be any whole numbers. Then,

*a* · (*b* − *c*) = *a* · *b* − *a* · *c*.

We say the multiplication is “distributive with respect to subtraction.”

Example 10

Use the distributive property to simplify: 3 · (12 − 8).

**Solution**

This is the product of a number and a difference, so the distributive property may be applied.

[ egin{aligned} 3 cdot (12 - 8) = 3 cdot 12 - 3 cdot 8 ~ & extcolor{red}{ ext{ Distribute the 3 times each term in the difference.}} = 36 - 24 ~ & extcolor{red}{ ext{Multiply: } 3 cdot 12 = 36 ext{ and } 3 cdot 8 = 24.} = 12 ~ & extcolor{red}{ ext{Subtract: } 36 - 24 = 12.} end{aligned} onumber ]

**Alternate solution **

Note what happens if we use the usual “order of operations” to evaluate the expression.

[ egin{aligned} 3 cdot (12 - 8) = 3 cdot 4 ~ & extcolor{red}{ ext{ Parentheses first: } 12 - 8 = 4.} = 12 ~ & extcolor{red}{ ext{ Multiply: } 3 cdot 4 = 12.} end{aligned} onumber ]

Same answer.

Exercise

Distribute: 8 · (9 − 2).

**Answer**56

## Exercises

In Exercises 1-12, simplify the given expression.

1. 5+2 · 2

2. 5+2 · 8

3. 23 − 7 · 2

4. 37 − 3 · 7

5. 4 · 3+2 · 5

6. 2 · 5+9 · 7

7. 6 · 5+4 · 3

8. 5 · 2+9 · 8

9. 9+2 · 3

10. 3+6 · 6

11. 32 − 8 · 2

12. 24 − 2 · 5

In Exercises 13-28, simplify the given expression.

13. 45 ÷ 3 · 5

14. 20 ÷ 1 · 4

15. 2 · 9 ÷ 3 · 18

16. 19 · 20 ÷ 4 · 16

17. 30 ÷ 2 · 3

18. 27 ÷ 3 · 3

19. 8 − 6+1

20. 15 − 5 + 10

21. 14 · 16 ÷ 16 · 19

22. 20 · 17 ÷ 17 · 14

23. 15 · 17 + 10 ÷ 10 − 12 · 4

24. 14 · 18 + 9 ÷ 3 − 7 · 13

25. 22 − 10 + 7

26. 29 − 11 + 1

27. 20 · 10 + 15 ÷ 5 − 7 · 6

28. 18 · 19 + 18 ÷ 18 − 6 · 7

In Exercises 29-40, simplify the given expression.

29. 9+8 ÷ {4+4}

30. 10 + 20 ÷ {2+2}

31. 7 · [8 − 5] − 10

32. 11 · [12 − 4] − 10

33. (18 + 10) ÷ (2 + 2)

34. (14 + 7) ÷ (2 + 5)

35. 9 · (10 + 7) − 3 · (4 + 10)

36. 9 · (7 + 7) − 8 · (3 + 8)

37. 2 · {8 + 12} ÷ 4

38. 4 · {8+7} ÷ 3

39. 9+6 · (12 + 3)

40. 3+5 · (10 + 12)

In Exercises 41-56, simplify the given expression.

41. 2+9 · [7 + 3 · (9 + 5)]

42. 6+3 · [4 + 4 · (5 + 8)]

43. 7+3 · [8 + 8 · (5 + 9)]

44. 4+9 · [7 + 6 · (3 + 3)]

45. 6 − 5[11 − (2 + 8)]

46. 15 − 1[19 − (7 + 3)]

47. 11 − 1[19 − (2 + 15)]

48. 9 − 8[6 − (2 + 3)]

49. 4{7[9 + 3] − 2[3 + 2]}

50. 4{8[3 + 9] − 4[6 + 2]}

51. 9 · [3 + 4 · (5 + 2)]

52. 3 · [4 + 9 · (8 + 5)]

53. 3{8[6 + 5] − 8[7 + 3]}

54. 2{4[6 + 9] − 2[3 + 4]}

55. 3 · [2 + 4 · (9 + 6)]

56. 8 · [3 + 9 · (5 + 2)]

In Exercises 57-68, simplfiy the given expression.

57. (5 − 2)^{2}

58. (5 − 3)^{4}

59. (4 + 2)^{2 }

60. (3 + 5)^{2}

61. 2^{3} + 3^{3}

62. 5^{4} + 2^{4}

63. 2^{3} − 1^{3}

64. 3^{2} − 1^{2}

65. 12 · 5^{2} + 8 · 9+4

66. 6 · 3^{2} + 7 · 5 + 12

67. 9 − 3 · 2 + 12 · 10^{2}

68. 11 − 2 · 3 + 12 · 4^{2}

In Exercises 69-80, simplify the given expression.

69. 4^{2} − (13 + 2)

70. 3^{3} − (7 + 6)

71. 3^{3} − (7 + 12)

72. 4^{3} − (6 + 5)

73. 19 + 3[12 − (2^{3} + 1)]

74. 13 + 12[14 − (2^{2} + 1)]

75. 17 + 7[13 − (2^{2} + 6)]

76. 10 + 1[16 − (2^{2} + 9)]

77. 4^{3} − (12 + 1)

78. 5^{3} − (17 + 15)

79. 5 + 7[11 − (2^{2} + 1)]

80. 10 + 11[20 − (2^{2} + 1)]

In Exercises 81-92, simplify the given expression.

81. ( frac{13+35}{3(4)})

82. ( frac{35+28}{7(3)})

83. ( frac{64-(8 cdot 6-3)}{4 cdot 7-9})

84. ( frac{19-(4 cdot 3-2)}{6 cdot 3-9})

85. (frac{2+13}{4-1})

86. ( frac{7+1}{8-4})

87. ( frac{17+14}{9-8})

88. ( frac{16+2}{13-11})

89. ( frac{37+27}{8(2)})

90. ( frac{16+38}{6(3)})

91. ( frac{40-(3 cdot 7-9)}{8 cdot 2-2})

92. ( frac{60-(8 cdot 6-3)}{5 cdot 4-5})

In Exercises 93-100, use the distributive property to evaluate the given expression.

93. 5 · (8 + 4)

94. 8 · (4 + 2)

95. 7 · (8 − 3)

96. 8 · (9 − 7)

97. 6 · (7 − 2)

98. 4 · (8 − 6)

99. 4 · (3 + 2)

100. 4 · (9 + 6)

In Exercises 101-104, use the distributive property to evaluate the given expression using the technique shown in Example 9.

101. 9 · 62

102. 3 · 76

103. 3 · 58

104. 7 · 57

## Answers

1. 9

3. 9

5. 22

7. 42

9. 15

11. 16

13. 75

15. 108

17. 45

19. 3

21. 266

23. 208

25. 19

27. 161

29. 10

31. 11

33. 7

35. 111

37. 10

39. 99

41. 443

43. 367

45. 1

47. 9

49. 296

51. 279

53. 24

55. 186

57. 9

59. 36

61. 35

63. 7

65. 376

67. 1203

69. 1

71. 8

73. 28

75. 38

77. 51

79. 47

81. 4

83. 1

85. 5

87. 31

89. 4

91. 2

93. 60

95. 35

97. 30

99. 20

101. 558

103. 174

^{1}Later, we’ll see that this property applies to all numbers, not just whole numbers

## 1.5: Order of Operations - Mathematics

In math problems it's important to do the operations in the right order. If you don't, you may end up with the wrong answer. In math, there can be only one correct answer, so mathematicians came up with rules to follow so we can all come up with the same correct answer. The correct order in math is called the "**order of operations**". The basic idea is that you do some things, like multiplication, before others, like addition.

For example, if you have 3 x 2 + 7 = ?

This problem could be solved two different ways. If you did the addition first you would get:

If you do the multiplication first, you get:

The second way is correct as you should do the multiplication first.

- Do everything inside of brackets first.
- Next, any exponents or roots (if you don't know what these are, don't worry about them for now).
- Multiplication and division, performing them left to right
- Addition and subtraction, performing them left to right

**Let's do a few examples:**

Now we do the multiplication and division, left to right:

Now addition and subtraction, left to right:

**Note:** even on the last step if we had added 35 + 1 first then we would have done 41 - 36 = 5. This is the wrong answer. So we need to do the operations in order and left to right.

**Another order of operations example:**

6 x 12 - (12 x 7 - 10) + 2 x 30 ÷ 5

We do the math inside the brackets first. We do the multiplication in the brackets first:

6 x 12 - (84 - 10) + 2 x 30 ÷ 5

Multiplication and division next:

**How to remember the order?**

There are different ways to remember the order. One way is to use the word PEMDAS. This can be remembered by the phrase "Please Excuse My Dear Aunt Sally". What it means in the Order of Operations is "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". When using this you must remember that multiplication and division are together, multiplication doesn't come before division. The same rule applies to addition and subtraction.

## Related Resources

The various resources listed below are aligned to the same standard, (5OA02) taken from the CCSM (Common Core Standards For Mathematics) as the Expressions and equations Worksheet shown above.

*Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.*

### Example/Guidance

### Worksheet

Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:

## PEMDAS

PEMDAS is an acronym that stands for " P lease e xcuse m y d ear a unt s ally," which is a mnemonic device intended to help with memorizing the order of operations:

This tells us the order in which we need to perform the respective operations. Multiplication and division can be grouped together because they are inverses, so the order they are performed in doesn't matter. When trying to decide whether to multiply or divide first (assuming all higher priority operations have already been taken into account) compute the operations in order from left to right. The same process is used for addition and subtraction. In some cases PEMDAS is written as PE(MD)(AS), to indicate this relationship.

Also, whenever a number, or group of numbers, is next to another number or group of numbers that are in parenthesis, if there is no explicit operation written between them, the operation is multiplication.

This problem is slightly tricky because if we were to have multiplied the 2 × 10 instead of dividing the 5 ÷ 2, we would get an incorrect answer of 0.25.

Another way to practice order of operations is to construct a "maze" where progress through the maze is tied to successfully completing order of operations problems. There are many ways this can be done. Below is one example.

Solve the problem in the rectangle marked "Start," and follow the arrow for the solution that you get. If the solution that you get isn't available, that means that your solution is incorrect. However, just because the solution you acquire is available, doesn't necessarily mean that it is correct. Solve your way through the maze until you arrive at the "Finish" rectangle. Once you do, check the solution below to see if the path you followed to get to the solution was the correct one.

It is also possible to construct more complicated mazes. The point is just to test your understanding of order of operations since some of the solutions in the maze are possible to acquire by making certain mistakes with the order of operations.

If you progressed through the maze correctly, you would have moved from rectangle one through two, four, five, six, and then to the finish. The solutions to each of the problems in the rectangles are listed below the numbering in the list corresponds to those in the top of each rectangle.

## 1.5: Order of Operations - Mathematics

Order of operation comes into play when a mathematical expression has more than one arithmetical operation.

Order of operations refers to the precedence of performing one arithmetical operation over another while working on a mathematical expression.

**Here are the Rules:**

1. Evaluate expressions inside parentheses.

2. Evaluate all powers

3. Perform all multiplications and/or divisions from left to right

4. Perform all additions and/or subtractions from left to right.

### More About Order of Operations

Order of operations if not rigidly followed can lead to two different solutions to the same expression.

PEMDAS or BEDMAS help you remember order of operations.

PEMDAS - Please Excuse My Dear Aunt Sally

P - Parentheses

E - Exponents

M - Multiplication

D - Division

A - Addition

S - Subtraction

### Video Examples: Order Of Operations

**BEDMAS**

B - Brackets

E - Exponents

D - Division

M - Multiplication

A - Addition

S - Subtraction

### Examples of Order of Operations

2 + (25 - 4) × 20 ÷ 2

First do all operations inside parentheses

2 + (21) × 20 ÷ 2

Perform all multiplications and divisions, from left to right.

2 + 420 ÷ 2

2 + 210

Perform all additions and subtractions from left to right.

212

### Solved Example on Order of Operations

#### Ques: Evaluate the variable expression 5x 4 + 4 when x = 3 using order of operations.

##### Choices:

A. 419

B. 404

C. 409

D. 414

Correct Answer: C

#### Solution:

Step 1: 5*x* 4 + 4[Original expression.]

Step 2: = 5 × (3) 4 + 4 [Substitute *x* = 3.]

Step 3: = 5 × 81 + 4 [Evaluate power.]

Step 4: = 405 + 4[Multiply 5 with 81.]

Step 5: = 409 [Add.]

Step 6: So, the value of 5*x* 4 + 4 for *x* = 3 is 409.

## Published by

### PRESH TALWALKAR

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By way of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. It's been quite a journey! I thank everyone that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.

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## The order in which Excel performs operations in formulas

In some cases, the order in which a calculation is performed can affect the return value of the formula, so it's important to understand how the order is determined and how you can change the order to obtain the results you want.

Formulas calculate values in a specific order. A formula in Excel always begins with an equal sign (**=**). Excel interprets the characters that follow the equal sign as a formula. Following the equal sign are the elements to be calculated (the operands), such as constants or cell references. These are separated by calculation operators. Excel calculates the formula from left to right, according to a specific order for each operator in the formula.

**Operator precedence in Excel formulas**

If you combine several operators in a single formula, Excel performs the operations in the order shown in the following table. If a formula contains operators with the same precedence—for example, if a formula contains both a multiplication and division operator—Excel evaluates the operators from left to right.

## Probability and Statistics

The abstract branch of mathematics, probability and statistics use mathematical concepts to predict events that are likely to happen and organize, analyze, and interpret a collection of data. Amongst the relatively newer branches of mathematics, it has become indispensable because of its use in both natural and social sciences. The scope of this branch involves studying the laws and principles governing numerical data and random events. Presenting an interesting study, statistics, and probability is a branch full of surprises.

### Planning for learning and teaching

"One of the teaching approaches which contribute particularly well to successful learning in mathematics is - well planned opportunities for children and young people to learn through investigate, active approaches" Learning Together: Mathematics - HMIE

The following questions may provide a stimulus for discussion:

- When planning learning and teaching what type of activities provide opportunities for learners to work independently as well as collaboratively?
- What steps are planned to review, improve and sustain these types of activities? See: Skills in Practice - Developing Thinking Skills

Well planned activities, incorporating Bloom’s Revised Taxonomy for Learning, are a useful tool for developing learners’ understanding and skills in numeracy and mathematics. The following activities have been developed to support staff to adopt the use of Bloom’s Revised Taxonomy in their planning:

- Bloom’s Revised Taxonomy planning tool for numeracy and mathematics can be used to support quality questioning.
- Bloom’s Higher Order Fans provide: Plenary questions to promote higher order thinking in the numeracy and mathematics classroom exemplar activities which can be used to develop higher order thinking in numeracy and mathematics from early to fourth level in number and number process, fractions, decimal fractions and percentages and measurement.

### Downloads

### Activities to support learning and teaching

This section is designed to support staff and learners by providing practical activities for the numeracy and mathematics classroom:

#### Practical activity 1 - Hinge questions

The Mathematics Excellence Group advocates strongly the planning of questions into lesson preparation. Such questions have been called 'hinge questions'. The idea is that the teacher plans every lesson with a 'hinge' a point in the lesson when the teacher can check on student understanding, and then decide what to do next. 'Hinge' questions are typically designed to test learners' understanding of one important concept in a lesson—one that is critical for pupils to comprehend before the teacher moves on in the lesson.

#### Practical activity 2 - Starter and stand-alone activities

Putting a different ‘spin’ on lesson starters is one way to stimulate thinking and problem solving and also generates some very interesting discussions between learners and staff. Longer starters could be used as stand-alone activities during lessons.

#### Practical activity 3 - Self and peer assessment

Peer assessment makes greater demands on dialogue between learners. It encourages learners to externalise their thinking, explaining their understanding to others. In endeavouring to support others in their understanding, the learner is involved in utilising higher order thinking skills.

#### Practical activity 4 - Using incorrect answers

Through their use of effective questioning and discussion, teachers will use misconceptions and wrong answers as opportunities to improve and deepen children’s understanding of mathematical concepts.

#### Practical activity 5 - Using summative assessment formatively

Using summative assessments in a meaningful way to raise learners’ awareness of their strengths and development needs is vital in promoting understanding in mathematics. High quality discussion and debate from analysing summative tests provides an opportunity for learners to further develop higher order thinking and questioning skills.

PDF file: Building the Curriculum 5

### Reflective questions

- What kind of techniques and activities do you find are useful and effective for evaluating learners' progress informally?
- What kind of opportunities do you already provide for learners to discuss their progress?

### About the author(s)

This resource was created within Education Scotland’s Numeracy team in conjunction with Scottish Government.

## Four Fours (Order of operations)

2017-11-26 2020-08-11 http://calc.amsi.org.au/wp-content/uploads/sites/15/2016/02/amsi-calculate.png Calculate 200px 200pxThis lesson has been designed for students in Upper Primary. It builds on the problem “Four Goodness Sake” found on the NRICH website (https://nrich.maths.org/1081). It is aimed at helping students develop an increased understanding into the order of operations and how the use of brackets in an equation can modify the solution.

The lesson activity requires students to use up to four fours and any operation to make the numbers from 0 to 100. For example, 4 + 4 + 4 + 4 = 16.

Initially, students will use the more familiar operations, +, −, ×, ÷, along with brackets to create their equations. For example, (4 × 4 + 4) ÷ 4 = 5. Gradually, finding solutions for the remaining numbers in the list becomes more challenging. Here, it is a good opportunity to introduce some other operations to the students, including √4 = 2 and 4! = 24. In my experience, at least one student in the class will be familiar with the square root operation √*n*, which will make it easier to introduce. The other operation *n*!, known as *factorial*, will be less familiar to students.

*n*! is the product of all positive integers less than or equal to *n f*or example, 4! = 4 × 3 × 2 × 1 = 24, or 3! = 3 × 2 × 1 = 6

With the introduction of these two less familiar operations, the number of solutions that students can find to the initial problem increases.

Recently, in a 5/6 class in NSW the students were quick to understand the initial problem. After naming some of the more obvious solutions, the challenge soon came to find solutions for the numbers from 1 to 10. We decided to use the large classroom blackboard to record the solutions as they were suggested. On the side of the board a “working out” space was included so problems where the “order of operations” was a factor could be tested before being added to the main list.

Although several solutions larger than 10 were also discovered, the number 10 itself became a bit of a sticking point. It was here that I prompted students to think of using different operations. One student suggested , but soon realised that this would not be allowed as it involved using a number other than 4. Another student said that the opposite to “squared” was “square root” and soon a new operation was available to the students. This “new” knowledge was then used by students to provide multiple solutions for the number 10, including:

After another five to ten minutes of working, when the sharing of possible solutions had again slowed down I introduced the factorial operation (. Not only did this help students find solutions to some different numbers, it also helped them to develop less complicated solutions to some of previous numbers. For example:

With this new information in place, over the course of one hour, the class of students were able to develop solutions to more than 30 numbers. As the lesson was finishing, students were still eagerly sharing potential solutions.

For more information, the attached lesson plan shows the relevant links to the Australian Curriculum and highlights the related content and outcomes as listed in the NSW Mathematics K-10 syllabus.