Arithmetic

Here you will learn about arithmetic, including key terminology and mathematical symbols, using the four operations with positive and negative integers, and inverse operations.

Students will first learn about arithmetic as part of number and operations in 4th and 5th grade and continue to build on this knowledge in the number system in 6th and 7th grade.

What is arithmetic?

Arithmetic is the study of numbers and the operations between them. It is an elementary branch of mathematics.

The definition of arithmetic comes from the Greek word “arithmos”, meaning number, or the art of counting.

To solve problems using basic arithmetic, you need to understand and use the four operations.

The four operations you will be using are: addition, subtraction, multiplication, and division.

Arithmetic with four operations

Below is a table showing the four operations of arithmetic with their associated symbols.

Addition	Subtraction	Multiplication	Division

Each operation has a different function that you should be confident with using.

Addition

Addition is the operation of combining two or more numbers together.

The two numbers can be represented by a and b, and c will represent the sum of a and b.

This would be written as a+b=c and pronounced a plus b is equal to c.

Addition is commutative, which means that the order in which addition is carried out does not matter.

For example,

Addition can be done with positive and negative integers, fractions, and decimals.

Addition represents a movement up the number line. Here are some examples:

In order to solve addition problems with larger numbers, you can use the standard algorithm.

For example, 347+21 :

It is important to consider the place value of each digit and line up the corresponding digits in each number.

Subtraction

Subtraction is the operation of finding the difference between two numbers.

The number remaining when the number b is subtracted from the number c would equal a, or the answer to a subtraction problem.

This would be written as c-b=a and pronounced c take away b is equal to a.

Subtraction is not commutative. If the order of the numbers within the calculation changes, the result will change.

For example,

7-4 ≠ 4-7

Subtraction can also be done with positive and negative integers, fractions, and decimals. Subtraction represents a movement down the number line. Here are some examples:

In order to solve subtraction problems with larger numbers, you can use the standard algorithm.

For example,

It is important to consider the place value of each digit and line up the corresponding digits and decimal place in each number.

Multiplication

Multiplication is essentially repeated addition.

If you have n copies of a, you multiply a by n to find how many are in the new set, m.

This is the same as calculating a+a+a+… \, n times.

This would be written as n\times{a}=m and pronounced n times a is equal to m.

The product is the answer you get when multiplying one number by another. The multiplicand is the quantity to be multiplied by the multiplier, which will give you a product.

multiplicand \times multiplier = product

For example,

In this calculation,

6 is the multiplicand, 3 is the multiplier and the answer, 18 is the product.

The product will be 0 if either the multiplicand or multiplier is 0.

Multiplication is commutative. The order in which the calculation is performed does not matter.

For example,

Multiplication can be done with positive and negative integers, fractions, and decimals. When multiplying positive and negative numbers, the following rules apply:

To solve multiplication problems with larger numbers, you can use an area model.

For example,

Division

Division shares or breaks a number into equal sized numbers of groups.

If the number m can be shared equally between n groups, with no remainder, then this is written as m\div{n}=a and pronounced m divided by n is equal to a.

The quotient is the answer you get when dividing one number by another.

The word quotient comes from Latin and means ‘how many times’. When dividing, you are finding out ‘how many times’ a number goes into another number.

dividend \div divisor = quotient

For example,

In this calculation,

8 is the dividend, 4 is the divisor and the answer, 2 is the quotient.

The quotient will only be 0 if the dividend is 0 but the divisor is not.

For example,

Unlike multiplication, division is not commutative. If the order of the numbers within the calculation changes, the result will change.

12\div{4} ≠ {4}\div{12}

Division can also be done with positive and negative integers, fractions, and decimals. When dividing positive and negative numbers, the following rules apply:

To perform division problems with larger numbers, you can use long division.

For example,

Inverse operations

Inverse operations reverse an operation that has been carried out. Below is a table outlining some operations along with their inverse operations.

Note that you can switch the columns so the inverse operation of subtraction is addition and the inverse operation of division is multiplication, etc.

Using the inverse operations helps us see relationships between numbers and are often referred to as fact families. Fact families show the relationship between the same set of numbers, just as inverse operations do.

Step-by-step guide: Inverse operations

PEMDAS

When solving problems involving basic arithmetic operations, it is important to apply the order of operations, or PEMDAS. PEMDAS tells us what order to perform the operations in.

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

In the calculation

multiplication should be done before the addition.

What is arithmetic?

Common Core State Standards

How does this relate to 4th, 5th, and 6th grade math?

• Grade 4: Number and Operations in Base 10 (4.NBT.4)
  Fluently add and subtract multi-digit whole numbers using the standard algorithm.

• Grade 5: Number and Operations in Base Ten (5.NBT.5)
  Fluently multiply multi-digit whole numbers using the standard algorithm.

• Grade 5: Number and Operations – Fractions (5.NF.1)
  Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

• Grade 5: Number and Operations – Fractions (5.NF.4)
  Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

• Grade 6: The Number System (6.NS.1)
  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, example, by using visual fraction models and equations to represent the problem.

• Grade 6: The Number System (6.NS.2)
  Fluently divide multi-digit numbers using the standard algorithm.

• Grade 6: The Number System (6.NS.3)
  Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

• Grade 6: The Number System (6.NS.5)
  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (example, temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

• Grade 7: The Number System (7.NS.A.1)
  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

• Grade 7: The Number System (7.NS.A.2)
  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

How to use arithmetic with positive and negative numbers

In order to use arithmetic with positive and negative numbers:

1. Determine which operation you will use.
2. Perform the chosen operation.

Arithmetic examples

Example 1: addition with decimals

Calculate the value of 10.9+34.3.

1. Determine which operation you will use.

Here you will use addition.

2Perform the chosen operation.

Using the standard algorithm, you have:

Starting on the right, the tenths place, you add 9+3=12. The tenths column value of 2 is placed below the answer line, and the 1 is carried above the addition problem, so that you can add it to the next column total (the units column).

Adding 0, 4, and the 1 that was carried from the tenths column, you have 0+4+1=5 and so 5 goes into the answer bar for the units column.

The final column requires us to add 1 and 3, \, 1+3=4 and so you put a 4 in the tens column.

So 10.9+34.3=45.2.

Example 2: subtraction (word problem)

96 mathematicians attended a celebration party at a hotel. 38 mathematicians stayed overnight. How many mathematicians did not stay overnight?

Perform the chosen operation.

Writing subtraction using the standard algorithm, you have:

Starting from the units column, 6-8 is a negative number, so you have to borrow from the tens column.

16-8=8 and so the value for the units column is 8.

Continuing with the tens column, you have 8-3=5.

58 guests did not stay overnight.

Example 3: multiplication (word problem)

A sack can hold 36 potatoes. A farmer packs 24 sacks of potatoes with no leftover potatoes. How many potatoes does the farmer pack?

Perform the chosen operation.

Using the area model, you have:

By multiplying each row value by each column value, you get:

Adding the new values in the area model, you have:

The farmer packs 864 potatoes.

Example 4: division (word problem)

A bar of chocolate is made up of 84 individual cubes. The bar is 6 cubes wide. How many rows does the chocolate bar have?

Perform the chosen operation.

Using partial quotients, you have:

You will think of a multiple of 6 that will get you as close to 84, without going over.

6 \times 10=60

84-60=24

You will think of another multiple of 6 that will get you as close to 24 as possible, without going over.

\begin{aligned} & 6 \times 4=24 \\\\ & 24-24=0 \end{aligned}

You will add your two partial quotients together 10 + 4 = 14.

The chocolate bar has 14 rows.

Example 5: addition, with negative numbers

Calculate (-3)+(-2).

Example 6: subtraction, with negative numbers

Calculate 5-(-3).

Example 7: multiplication, with negative numbers

Calculate 8\times(-2).

Example 8: division, with negative numbers

Calculate \cfrac{-120}{-3}.

Teaching tips for arithmetic

• If a student is struggling with using algorithms with the given operation, they need more experience with concrete models.

• When teaching students to use the algorithm for division, or long division, provide graph paper for students to use to keep their work neat and legible.

• Be sure to practice a problem solving method when working with word problems. Students may be able to solve a given equation, but struggle when having to pull information out of and comprehend a word problem.

• The use of a number line can be very impactful when working with positive and negative numbers.

Easy mistakes to make

• Parentheses are not an arithmetic operation
  Operations allow us to perform calculations between two or more numbers. In the associative property, parentheses dictate the order in which this should be completed and so they are technically not an operation. However, parentheses are sometimes used in place of multiplication symbols, for example, 6(7-3).

• Multiplying by \bf{0} by any number equals \bf{0}
  It is important to remember that multiplying anything by 0 gives the answer 0.

• Multiplying, instead of dividing, when working with fractions
  Mistakes can easily be made when dividing fractions. When dividing a number by a fraction, a commonly seen error is that the value is multiplied by the fraction instead. For example, 12\div\frac{1}{2}=6 is incorrect. Instead, the correct answer is 12\div\frac{1}{2}=12\times{2}=24.

• Knowing the value of decimal places are important
  It’s important that students remember the concept of place value when solving with decimals. Knowing the value of the numbers you are performing an operation with is crucial.

The next lessons are

• Properties of equality
• Addition and subtraction
• Multiplication and division
• Calculator skills
• Money word problems
• Skip counting
• Number sense
• Two step word problems

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