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Think About It!

Goal: Define the properties of operations and recognize how to use them for addition and multiplication.

Daniel with two plates of celery each plate has a group of 3 celery sticks and one of four celery sticks.

Daniel is making “ants on a log” snacks. He will use celery sticks, raisins, and nut butter. Daniel has 2 plates of celery sticks. Each plate has a group of 3 sticks and a group of 4 sticks. Daniel wants to know how many sticks he has in all.

Daniel remembers a new way to write an equation. When we have a situation like the plates of celery, we can create an equation that fits what we need to find.

The number of plates pointing to the 2 in the equation.

\({2 \times (3 + 4) = ?}\)

3 and 4 are the groups of celery on each plate and points to the label.

How can we help Daniel find the total number of celery sticks with this new type of equation? We can use the properties of operations! The properties of operations are rules about adding or multiplying that help us understand and solve equations. These properties are the identity property, the commutative property, the associative property, and the distributive property. We will use these properties to help us solve different equations.

Identity

Identity Property

2 celery sticks

\({ 2 + 0 = 2 }\)

adding to celery sticks to zero equals 2.

Commutative

Commutative Property

2 celery sticks

3 celery sticks

\({ 2 + 3 = 5 }\)

3 celery sticks

2 celery sticks

\({3 + 2 = 5}\)

\({2 + 3 = 3 + 2}\)

Both equations equal 5. Both equations are equal to each other.

Associative

Associative Property

2 celery sticks grouped with 2 celery sticks to make 4

1 celery stick

\({(2+ 2) + 1 = 5}\)

2 celery sticks

2 celery sticks grouped with 1 celery stick to make 3

\({2 + (2+1) = 5}\)

\({(2+2) + 1 = 2 + (2+1)} \)

Both equations equal 5 so both equations are equal to each other.

Distributive

Distributive Property

2 groups of 3 celery sticks make 6 celery sticks

1 celery sticks

\({2 \times (2 + 1) = 6}\)

How can we use the properties of operations to solve addition equations? The properties help us understand numbers and how they work together when we add or multiply. We can use what we have learned from each property to solve other addition equations.

Click each row to solve addition equations using the properties of operations.

\({\Large64 + 0 = ?}\)	\({\Large64 + 0 = ?}\) We know that any number plus 0 is the same number. This is the identity property. 64 plus 0 equals 64.
\({\Large23 + 41 = 64}\)    \({\Large41 + 23 = ?}\)	\({\Large23 + 41 = 64}\) \({\Large41 + 23 = ?}\) We know that changing the order of the addends does not change the sum. This is the commutative property. If 23 plus 41 equals 64, so does 41 plus 23
\({\Large(13 + 30) + 21 = 64}\) \({\Large13 + (30 + 21) = ?}\)	\({13+(30+21)=?}\) arrow pointing to \({43+21 = 64}\) or \({13+(30+21)=?}\) arrow point to \({13+51=64}\) We know that we can add the addends in any order that works for us. This is the associative property. We can put parentheses around the two addends we want to add first. Both addition sentences equal 64.
\({8 \times (4+4)=?}\)	\({8 \times (4 + 4 = ?)}\) arrow pointing to \({8 times (8) = 64}\) This is the distributive property. We distribute the 8 by multiplying it by the first 4, then the second 4 and add the products together.

Let’s use the properties of operations to solve Daniel’s math sentence!

a plate of celery with 1 group of 3 celery sticks and a 1 group of 4 celery sticks

another plate of celery with 1 group of 3 celery sticks and a 1 group of 4 celery sticks

\({2 \times (3+4)=?}\)

Look at Daniel’s math sentence. What do you notice? Click the Show Me button to see!

Show Me

Start by solving the equation in parentheses.

There are two numbers inside the parentheses. There is an addition sign between them. There is a number outside the parentheses.

Start by solving the equation in parentheses.

\({(3+4)}\)

We noticed that there are two numbers inside the parentheses, an addition sign between them, and a number outside the parentheses.

Question

Which property can we use to help us solve this equation?

Answer

We can use the distributive property to help us solve this equation!

2 times 3 plus 4 equals 2 times 3 plus 2 times 4

We know we can distribute the 2 through the parentheses. That means you can multiply 2 by 3 and then by 4.

Question

What is \({2 \times 3}\)? What is \({2 \times 4}\)? 

Answer

\({2 \times 3 = 6}\) and \({2 \times 4 = 8}\)

\({6 + 8 = 14}\)

Now that we know the products inside the parentheses, we can add the products together.

Question

How many celery sticks does Daniel have?

Answer

Daniel has 14 celery sticks.

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Slide:

We can use what we know about the properties of operations to solve addition and multiplication equations!