1. Integers

United States of America

Grade 7

When we manipulate expressions with numbers, we follow a set of properties that apply to all numbers. Knowing these properties helps us to be able to rewrite expressions in a variety of different ways, and might make evaluating them easier.

The box below summarizes some of the properties of real numbers and gives an example of each.

Property | Symbols | Example |
---|---|---|

\text{Commutative property of addition} | a+b=b+a | -3+6=6+\left(-3\right) |

\text{Commutative property of multiplication} | a \times b=b \times a | 6 \times (-3)=(-3) \times 6 |

\text{Associative property of addition} | a+\left(b+c\right) \\ =\left(a+b\right)+c | 6+\left(\left(-3\right)+2\right)=\\\left(6+\left(-3\right)\right)+2 |

\text{Associative property of multiplication} | a\times(b\times c)=(a\times b)\times c | 6 \times \left(-3 \times 2\right)=\\ \left(6 \times (-3\right)) \times 2 |

\text{Distributive property} | a\times\left(b+c\right)= \\ a\times b+a\times c \\ \text{or} \\ a\times\left(b-c\right)= \\a\times b-a\times c | 4\left(-3+5\right)= \\ 4 \times \left(-3\right)+4 \times 5 \\ \text{or} \\ 4 \left(\left(-3\right)-5\right)= \\ 4 \times\left(-3\right)-4 \times 5 |

The **commutative property** is the reason that we can add numbers in any order or multiply numbers in any order. While it applies to multiplication and addition, it does not apply to expressions that are written as subtraction or division.

The applet below uses two sliders that you can use to adjust the number of rows of columns of the array. A multiplication expression is shown that represents the array. Use the bottom slider to generate the rotated version of the left array.

What do you notice when you rotated the array?

If we rotate the array in the applet, we can see that the rectangles are the same size when the length and width are switched. This demonstrates the commutative property for multiplication.

The **associative property** is the reason that we can group sums of numbers differently and the result remains the same. The same is true for products of numbers. However, it's not true for subtraction and division. That's why we say they are not associative.

The **distributive property** shows us how the product of a number and a sum or difference is applied to each term in the sum or difference.

We know that adding zero to a number gives us the same number. That's the **identity property for addition**. In a similar way, multiplying a number by one doesn't change the number. That's the **identity property for multiplication**.

We also know that opposites add to zero. That's the **inverse property of addition**.

All of these properties can be applied to help us evaluate expressions more easily.

Find the value of: 6+\left(-5\right)+5

Worked Solution

Use the commutative property of addition to fill in the missing number:

19 + \left(-15\right) = \left(-15\right) + ⬚

Worked Solution

Which property is demonstrated by the following statement?

4 \times (9 \times (-5)) = (4 \times 9) \times (-5)

A

Commutative property of multiplication

B

Associative property of multiplication

C

Distributive property

D

Associative property of addition

E

Commutative property of addition

Worked Solution

Using the distributive property, rewrite -11\times\left(7-3\right) as the sum or difference of two integers.

Worked Solution

Idea summary

Property | Symbols |
---|---|

\text{Commutative property of addition} | a+b=b+a |

\text{Commutative property of multiplication} | a \times b=b \times a |

\text{Associative property of addition} | a+\left(b+c\right) =\left(a+b\right)+c |

\text{Associative property of multiplication} | a\times(b\times c)=(a\times b)\times c |

\text{Distributive property} | a\times\left(b+c\right)= a\times b+a\times c \\ \text{or} \\ a\times\left(b-c\right)=a\times b-a\times c |