### Video Transcript

The diagram shows the number π
plus π. By dividing it into four parts, as
shown by the dashed lines, one gets π plus π divided by four. Using the diagram, find another
expression for π plus π divided by four. Given that π plus π times π is a
multiplication equivalent to this division, what is the multiplication factor
π? What property of multiplication was
used to rewrite π plus π times one-fourth as one-fourth times π plus π? What property of multiplication has
been used to rewrite one-fourth times π plus π as π over four plus π over
four?

Letβs begin with the diagram. It says the diagram shows the
number π plus π. So, again, this entire diagram
represents π plus π. And then, it says by dividing it
into four parts, as shown by the dashed line here, here, and here, we get π plus
π, so the entire diagram divided by four because there are one, two, three, four
parts now. So, π plus π divided by four.

So, the first question says, using
the diagram, find another expression for π plus π divided by four. Well, instead of saying the entire
diagram has been divided by four, we could say the π portion was divided into four
pieces. And the π portion was divided into
four pieces, and together makes the whole diagram. So, we have the π-divided-by-four
portion plus the π-divided-by-four portion. So, here weβve used the diagram to
find another expression for π plus π divided by four.

Now we could double-check our
answer. Because if we would expand this
dividing by four, we would need to take π divided by four and π divided by
four. So, we would still get π divided
by four plus π divided by four.

Our next question states: given
that π plus π times π is a multiplication equivalent to this division, what is
the multiplication factor π? So, itβs saying that π plus π
times π is a multiplication equivalent to this division. So, here weβve written the
division, the divided by four, and we wanna know an equivalent multiplication to
this. So, writing π plus π times π, we
have the π plus πs and we wanna know what can we multiply by to make it the exact
same as dividing by four. Well, dividing something by four is
the same as multiplying by one-fourth. So, the multiplication equivalent
to this division would be the multiplication factor π of one-fourth.

Next, weβre asked what property of
multiplication was used to rewrite π plus π times one-fourth as one-fourth times
π plus π. So, we chose to rewrite these. So, theyβre equivalent to each
other. And thereβs a property that tells
us that this is okay to do. Well, notice on both sides the π
and the π are still being added together. And on both sides, weβre still
multiplying the π plus π times a one-fourth. However, the order at which weβre
writing these has switched. So, say we called π plus π π₯ and
one-fourth π¦. π₯ times π¦ is equal to π¦ times
π₯. And this is true because of the
property called the commutative property.

Lastly, it says what property of
multiplication has been used to rewrite one-fourth times π plus π as π over four
plus π over four. So, what exactly happened here? Taking one-fourth times π, we get
π over four, or π divided by four. And taking one-fourth times π, we
get π over four, or π divided by four. And then we keep the addition sign
between. This is called distributing. So, we use the distributive
property. And now weβve completed this
question.