Multiplication and Division

NZ Level 3

Distributive property for multiplication

Lesson

On the applet below

- create an array representing a multiplication of a single and double digit number
- hit the box to SPLIT
- adjust where you want to split the longer side
- Did the area of the whole shape change when we split it?
- What can we say about the sum of the 2 smaller areas when compared to the whole rectangle?

The distributive property of multiplication might be a long expression, but the meaning of it is really useful for us. Before we can use it though, we need to know what it means, but more importantly, trust that it works.

It's a good idea to refresh our memories of arrays for multiplication, as this helps us imagine what we are working out. Then, we can discover the distributive property of multiplication, and use it to solve multiplication number problems. In Video 1, we prove that the distributive property of multiplication works, so have a look at how we do this.

Once we trust that we can split our number problems into smaller problems, we can use this to solve other problems. Let's work through some examples in Video 2, including how you can use the distributive property of multiplication to solve some trickier multiplication problems.

Remember!

We can write an equation such as $5\times12$5×12 as $5\times10+5\times2$5×10+5×2. So that it works for any number, we could write this as:

$a\times bc=a\times b+a\times c$`a`×`b``c`=`a`×`b`+`a`×`c`

We want to use the distributive property to rewrite $2\times19$2×19 as easier multiplications.

This diagram shows how $2$2 groups of $19$19 objects can be split up.

Use the diagram to fill in the blank to make the statement true.

$2\times19=2\times\left(10+\editable{}\right)$2×19=2×(10+)

Fill in the blanks to show how $2$2 groups of $\left(10+9\right)$(10+9) can be split up into smaller multiplications.

$2\times\left(10+9\right)=2\times10+2\times\editable{}$2×(10+9)=2×10+2×

Fill in the missing number to make the equation true.

$2\times\editable{}=2\times\left(7+3\right)$2×=2×(7+3)

Choose the mathematical equation that matches the written statement:

$5$5 groups of $19$19 is the same as $5$5 groups of $($($10$10 and $9$9$)$).

$5\times19=5+\left(10+9\right)$5×19=5+(10+9)

A$5\times19=5+\left(10\times9\right)$5×19=5+(10×9)

B$5\times19=5\times\left(10+9\right)$5×19=5×(10+9)

C$5\times19=19\times\left(10+9\right)$5×19=19×(10+9)

D$5\times19=5+\left(10+9\right)$5×19=5+(10+9)

A$5\times19=5+\left(10\times9\right)$5×19=5+(10×9)

B$5\times19=5\times\left(10+9\right)$5×19=5×(10+9)

C$5\times19=19\times\left(10+9\right)$5×19=19×(10+9)

D

Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.

Generalise the properties of addition and subtraction with whole numbers