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Intro to distributive property

Practice decomposing the factors in multiplication problems and see how it affects the product.

Breaking up multiplication

This array is made up of 3 rows with 6 dots in each row. The dots show 3, times, 6, equals, 18.
If we add a line dividing the dots into two groups, the total number of dots does not change.
The top group has 1 row with 6 dots. The dots show 1, times, 6.
The bottom group has 2 rows with 6 dots in each row. The dots show 2, times, 6.
We still have a total of 18 dots.

Distributive property

The math rule that allows us to break up multiplication problems is called the distributive property.
The distributive property says that in a multiplication problem, when one of the factors is rewritten as the sum of two numbers, the product does not change.
Using the distributive property allows us to solve two simpler multiplication problems.
In the example with the dots we started with start color #1fab54, 3, end color #1fab54, times, start color #7854ab, 6, end color #7854ab.
We broke the start color #1fab54, 3, end color #1fab54 down into start color #1fab54, 1, plus, 2, end color #1fab54. We can do this because start color #1fab54, 1, plus, 2, equals, 3, end color #1fab54
We used the distributive property to change the problem from start color #1fab54, 3, end color #1fab54, times, start color #7854ab, 6, end color #7854ab to left parenthesis, start color #1fab54, 1, plus, 2, end color #1fab54, right parenthesis, times, start color #7854ab, 6, end color #7854ab.
The start color #7854ab, 6, end color #7854ab gets distributed to the start color #1fab54, 1, end color #1fab54 and start color #1fab54, 2, end color #1fab54 and the problem changes to:
left parenthesis, start color #1fab54, 1, end color #1fab54, times, start color #7854ab, 6, end color #7854ab, right parenthesis, plus, left parenthesis, start color #1fab54, 2, end color #1fab54, times, start color #7854ab, 6, end color #7854ab, right parenthesis
Now we need to find the two products:
6, plus, 12
And finally, the sum:
6, plus, 12, equals, 18
start color #1fab54, 3, end color #1fab54, times, start color #7854ab, 6, end color #7854ab, equals, 18 and
left parenthesis, start color #1fab54, 1, plus, 2, end color #1fab54, right parenthesis, times, start color #7854ab, 6, end color #7854ab, equals, 18
Practice problem 1
Which expressions are the same as 4, times, 9?
Choose all answers that apply:

Small numbers

Some numbers like 1, comma, 2, comma, 5, and 10 are easier to multiply. The distributive property allows us to change a multiplication problem so that we can use these numbers as one of the factors.
For example, we can change 4, times, 12 into 4, times, left parenthesis, start color #01a995, 10, end color #01a995, plus, start color #74cf70, 2, end color #74cf70, right parenthesis.
The array of dots on the left shows left parenthesis, start color #01a995, 4, times, 10, end color #01a995, right parenthesis. The array of dots on the right shows left parenthesis, start color #74cf70, 4, times, 2, end color #74cf70, right parenthesis.
Now we can add the expressions to find the total.
left parenthesis, start color #01a995, 4, times, 10, end color #01a995, right parenthesis, plus, left parenthesis, start color #74cf70, 4, times, 2, end color #74cf70, right parenthesis
equals, start color #01a995, 40, end color #01a995, plus, start color #74cf70, 8, end color #74cf70
equals, 48
Since 10 and 2 are both easy to multiply, using the distributive property for this problem made finding the product easier.

Practice problem 2

The dots represent 9, times, 4.
Problem 2, part A
Which expression shows the dots above the dotted line?
Choose 1 answer:

Problem 2, Part B
Which expression shows the dots below the dotted line?
Choose 1 answer:

Problem 2, Part C
left parenthesis, 5, times, 4, right parenthesis
left parenthesis, 4, times, 4, right parenthesis, equals, start text, space, t, o, t, a, l, space, n, u, m, b, e, r, space, o, f, space, d, o, t, s, end text

More practice

Problem 3A
  • Current
The dots represent 3, times, 8.
Which expression can we use to calculate the total number of dots?
Choose 1 answer:

Working with large numbers

The distributive property is very helpful when multiplying larger numbers. Look at how we can use the distributive property to simplify 15, times, 8.
We will start by breaking start color #11accd, 15, end color #11accd into start color #11accd, 10, plus, 5, end color #11accd. Then we will distribute the 8 to both of these numbers.
start color #11accd, 15, end color #11accd, times, 8, equals, left parenthesis, start color #11accd, 10, end color #11accd, times, 8, right parenthesis, plus, left parenthesis, start color #11accd, 5, end color #11accd, times, 8, right parenthesis
empty spaceequals, space80, plus, 40
empty spaceequals, space120
Problem 4
Use the distributive property to find the product.
start color #11accd, 18, end color #11accd, times, 3, equals, left parenthesis, start color #11accd, 10, end color #11accd, times, 3, right parenthesis, plus, left parenthesis, space
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
times, 3, right parenthesis
empty space, equals, space, 30, plus
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
empty space, equals, space
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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