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Addition and subtraction Multiplication and division Order of operations ExponentsHere is everything you need to know about negative numbers, including how to add, subtract, multiply, and divide negative numbers and how to order negative numbers.

Students will first learn about negative numbers as a part of the number system in 6 th grade.

**Negative numbers** are any numbers less than zero. They have a negative or minus sign (-) in front of them.

Numbers greater than zero are referred to as **positive numbers.** If there is no sign in front of a number, the number is positive.

On the **number line** below, you can see some positive and negative integers (whole numbers):

The numbers in orange are negative values and the blue numbers are positive values. Zero is neither positive nor negative.

Just like you can add, subtract, multiply and divide positive numbers, you can do the same with negative numbers whether they are integers, decimals or fractions.

When adding and subtracting negative numbers, use a number line.

**If you are adding, move to the right of the number line,** or the positive direction.

**If you are subtracting, move to the left of the number line,** or the negative direction.

Sometimes a question might have two operations next to each other.

For example, 4+(-2).

**If the signs are the same, replace them with a positive sign.**

**If the signs are different, replace them with a negative sign.**

The chart below summarizes this:

As a rule of thumb: Same signs add, different signs subtract.

For example,

Solve: -4+13.

Solve 4+-7.

There are two **different** signs written next to each other. These become **negative.**

So, 4+-7 = 4-7

Similar rules apply for multiplying and dividing negative numbers.

**If the signs are the same, the answer is positive.**

**If the signs are different, the answer is negative.**

When multiplying negative numbers:

The same rules apply for dividing negative numbers:

For example,

Solve: -4 \times-3.

4 \times 3=12Both numbers have the **same** sign, so the answer will be **positive.**

Solve: 12 \div-2.

12 \div 2=6The numbers have **different** signs, so the answer will be **negative.**

12 \div-2=-6

Integers (whole numbers) are ordered on a number line based on **positive** integers and **negative **integers. The center of the number line is marked as 0.

The integers that are** greater than** 0 are **positive** integers. The integers that are **less than** 0 are **negative** integers.

The more an integer is **negative**, the smaller the value is. The more an integer is **positive**, the bigger the value is.

For example,

Put these numbers in ascending order.

4, \; -2, \; 0, \; 3, \; -3, \; -5, \; 1, \; 2Remember: ascending means smallest to biggest.

-5 is the **most negative** number, so this is the smallest value.

4 is the **most positive** number, so this is the biggest value.

So, putting them in ascending order you get, -5, \; -3, \; -2, \; 0, \; 1, \; 2, \; 3, \; 4

For example,

Put these numbers in descending order.

-1, \; -6, \; -12, \; -4, \; -3, \; -8Remember, descending means biggest to smallest.

-1 is the **least negative** number, so this is the biggest value.

-12 is the **most negative** number, so this is the smallest value.

So, putting them in descending order you get, -1, \; -3, \; -4, \; -6, \; -8, \; -12

How does this relate to 6 th grade math?

**Grade 6: The Number System (6.NS.C.5)**Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., 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 6: The Number System (6.NS.C.6)**

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

**Grade 6: The Number System (6.NS.C.7)**

Understand the ordering and absolute value of rational numbers.

Use this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 15+ questions with answers covering a range of 4th, 5th and 7th grade multiplication and division topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 15+ questions with answers covering a range of 4th, 5th and 7th grade multiplication and division topics to identify areas of strength and support!

DOWNLOAD FREEIn order to add and subtract negative numbers:

**If you have two signs next to each other, change them to a single sign.****Circle the first number on the number line.****Use the number line to add or subtract your numbers.****Write your final answer.**

Solve: -5+8.

**If you have two signs next to each other, change them to a single sign.**

- If the signs are the same in the middle of the calculation, replace them with a positive sign (+).
- If the signs are different, replace them with a negative sign (-).

In this case you do not have two signs next to each other.

2**Circle the first number on the number line.**

The first number in the question is (-5).

3**Use the number line to add or subtract your numbers.**

- If you are adding, move to the right of the number in step 2 \; (\rightarrow).
- If you are subtracting, move to the left of the number in step 2 \; (\leftarrow).

In this case you are adding the 8 so move 8 spaces right from the (-5) on the number line:

4**Write your final answer.**

Solve: -7-(-5).

**If you have two signs next to each other, change them to a single sign.**

- If the signs are the same in the middle of the calculation, replace with a positive sign (+).
- If the signs are different, replace with a negative sign (-).

In this case you have a minus and a minus next to each other. Since the signs are the same, replace with a plus sign (+).

-7 + 5

**Circle the first number on the number line.**

The first number in the question is (-7).

**Use the number line to add or subtract your numbers.**

- If you are adding, move to the right of the number in step 2 \; (\rightarrow).
- If you are subtracting, move to the left of the number in step 2 \; (\leftarrow).

In this case you are adding 5 so move 5 spaces right from the (-7) on the number line:

**Write your final answer.**

-7-(-5)=-2

In order to multiply and divide negative numbers:

**Multiply or divide the numbers normally.****Change the sign using the rules of multiplying and dividing negative numbers.**

Solve: -6 \times (-7).

**Multiply or divide the numbers normally.**

6 \times 7=42

**Change the sign using the rules of multiplying and dividing negative numbers.**

- If the signs are the same at the beginning of each number, the answer is positive.
- If the signs are different, the answer is negative.

-6 \times -7

In this case we have a negative number times a negative number.

The signs are the same so we must have a positive answer.

-6 \times(-7)=42

Solve: -25 \div 5.

**Multiply or divide the numbers normally.**

25 \div 5=5

**Change the sign using the rules of multiplying and dividing negative numbers.**

- If the signs are the same, the answer is positive.
- If the signs are different, the answer is negative.

-25 \div 5

In this case we have a negative number divided by a positive number.

The signs are different so we must have a negative quotient.

-25 \div 5=-5

The table below shows the temperatures recorded in Chicago at different times of the day.

a) What is the product of the highest and lowest temperatures?

b) What is the difference between the temperatures at 2 am and 1 pm?

**Part \textbf{a)} What is the product of the highest and lowest temperatures?**

The highest temperature is 3^{\circ} \mathrm{F}.

The lowest temperature is -8^{\circ} \mathrm{F}.

**Multiply or divide the numbers normally.**

8 \times 3=24

**Change the sign using the rules of multiplying and dividing negative numbers.**

- If the signs are the same, the answer is positive.
- If the signs are different, the answer is negative.

3 \times -8

In this case, you have a positive number times a negative number.

The signs are different so it must have a negative answer.

3 \times-12=-24^{\circ}\mathrm{F}

** \textbf{b)} What is the difference between the temperatures at \bf{1} pm and \bf{2} am?**

The temperature at 2 am was -8^{\circ}\mathrm{F} and the temperature at 1 pm was 3^{\circ}\mathrm{F}.

To find the difference we must work out -8 - 3.

**If you have two signs next to each other, change them to a single sign.**

- If the signs are the same, replace them with a positive sign (+).
- If the signs are different, replace them with a negative sign (-).

In this case you do not have two signs next to each other.

**Circle the first number on the number line.**

The first number in the question is (-8)

**Use the number line to add or subtract your numbers.**

- If you are adding, move to the right of the number in step 2 \; (\rightarrow).
- If you are subtracting, move to the left of the number in step 2 \; (\leftarrow).

In this case, you are subtracting the 3 , so move 3 spaces right from the (-8) on the number line:

**Write your final answer**.

-8-3=-11^{\circ}\mathrm{F}

In order to order negative numbers:

**Plot the given numbers on a number line.****List the numbers in ascending or descending order.**

Put these numbers in ascending order.

5, \; -2, \; 0, \; -4, \; -8, \; 3, \; 1**Plot the given numbers on a number line.**

**List the numbers in ascending or descending order.**

The question asked for the numbers to be placed in ascending order, or from least to greatest.

Looking at the number line, the smallest number is –8. You can then list the rest of the numbers based on their location on the number line, moving from left to right.

The numbers listed in ascending order are:

–8, \; –4, \; –2, \; 0, \; 1, \; 3, \; 5

Put these numbers in descending order.

-5, \; 3, \; -3, \; -7, \; 1, \; 5**Plot the given numbers on a number line.**

**List the numbers in ascending or descending order.**

The question asked for the numbers to be placed in descending order, or from greatest to least.

Looking at the number line, the largest number is 5. You can then list the rest of the numbers based on their location on the number line, moving right to left.

The numbers listed in descending order are:

5, \; 3, \; 1, \; –3, \; –5, \; –7.

- Use real-world examples when discussing negative numbers. Use a bank account as an example and explain when you spend money, the amount of money spent is usually shown as a negative, and the amount in your account decreases.

- Teach students about opposites, explaining that every negative number has a corresponding positive number with the same magnitude.

- Present students with common misconceptions about negative numbers and have students work through why the example presented is incorrect.

**Thinking a greater negative means a larger number**

Students sometimes assume that the larger a negative number the greater it is. For example, students might incorrectly assume -3 is greater than 2 because 3 is a larger number.

**Raising a negative number to a power greater than one**

Remember, when raising a negative number to a power greater than 1 , the resulting answer could be positive or negative. When you raise a negative number to an odd exponent, the resulting answer is negative; when you raise a negative number to an even exponent, the resulting answer is positive.

**Confusing when to change the signs**

When adding or subtracting with negative numbers, the signs change only if they are next to each other in the middle of the calculation and are different.

- Multiplication
- Multiplicative comparison
- Multiplying multi digit numbers
- Division
- Dividing multi digit numbers
- Long division
- Negative times negative
- Multiplying and dividing integers
- Multiplying and dividing rational numbers

1. Solve: -8+10.

-2

2

-18

18

Find the first number (–8) on the number line and move to the right the number of places as the second number (10).

2. Solve: -11-(-8).

-3

3

-19

19

There are two negative signs (-) together, so change it to a positive (+).

-11+8

3. Solve: -8 \times (-9).

-72

72

17

-17

8 \times 9 = 72

The signs are the same so the answer is positive.

-8 \times (-9)=72

4. Solve: -144 \div 3.

147

-147

-48

48

144 \div 3 = 48

The signs are different so the answer will be negative.

-144 \div 3= -48

5. The table shows the temperature in Buffalo, NY at different times of the day in degrees Fahrenheit.

What is the difference between the highest and lowest temperatures?

1^{\circ} \mathrm{F}

7^{\circ} \mathrm{F}

10^{\circ} \mathrm{F}

17^{\circ} \mathrm{F}

The calculation you need to solve here is 9-\,-8. There are two negative signs (-) together so this becomes a positive (+).

The signs are different so the answer will be negative.

9-\,-8=9+8=17

6. Place the following numbers in ascending order:

-2, \; 5, \; -6, \; 0, \; 4, \; -3

5, \; 4, \; 0, \; -2, \; -3, \; -6

-6, \; -3, \; -2, \; 0, \; 4, \; 5

0, \; -2, \; -3, \; 4, \; 5, \; -6

-6, \; 5, \; 4, \; -3, \; -2, \; 0

Plot the numbers on a number line.

Ascending order means from least to greatest, so find the smallest number (left side of the number line) and list numbers moving from left to right on the number line.

Negative numbers are not natural numbers. Natural numbers are all positive integers from 1 to infinity, often referred to as counting numbers.

When you compare negative numbers, you compare the magnitude of each number. The farther left a number is on the number line, the smaller it is.

Negative numbers can be rational or irrational depending on whether it is expressed as fractions or not. For example, –3 is rational because it can be expressed as \cfrac{-3}{1}, while the square root of –2 is irrational.

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