Distributive Property

Distributive Property exercise

• Describe what the Distributive Property means for Bella and Anton.

  Hints

  Don't worry about capital and small letters. They can be used the same way. But remember, you should always be consistent when naming variables.

  When mathematical expressions are equal to each other, it does not matter whether you write $a=b$ or $b=a$.

  Take a look at both of the longer terms. What do they describe? Do they have a different result?

  Solution

  The Distributive Property is very useful for calculating things in everyday life because it helps you simplify terms.

  Let's take a look at how Bella and Anton have applied the Distributive Property. As we have seen in previous examples, it does not matter if they buy their ice cream and then meet or first meet and buy the ice cream together.

  What does this mean? We can express with the following mathematical expression:

  $3 \times 1 + 3 \times 1 = 3 \times (1 + 1) = 6$

  In general, we know that the Distributive Property says that distributing a multiplier over a sum of numbers will result in the same answer as multiplying each addend separately and then summing.

• Explain how to use the Distributive Property for scenarios with more than 3 scoops of ice cream.

  Hints

  First check whether it works with more than $3$ scoops of ice cream.

  Finally there should be a general formula for the Distributive Property.

  Solution

  Bella has recognized that her equation does not depend on the number of scoops. So she tries to increase the number of scoops to $4$. Let's take a look at what she finds:

  • $4 \times (1 + 1) = 4 \times 1 + 4 \times 1 = 8$
  • $4 \times (1 + 1) = 4 \times 2 = 8$

  Both equations have the same result. As we can see, distributing a multiplier over a sum of numbers results in the same answer as multiplying each addend separately and then summing.

  Generalizing this, we just have to substitute the numbers with variables. Therefore, we substitute the number of scoops with $c$, Anton with $a$ and Bella with $b$. The following formula describes our situation using the Distributive Property.

  • $c \times (a + b) = c \times a + c \times b$

• Describe how the Distributive Property changes when a third kid buys ice cream.

  Hints

  The Distributive Property is not limited to a certain number of addends inside the parentheses.

  Introducing the variable $d$, which stands for the Dave, allows us to extend the formula to $c \times (a + b + d)=c \times a + c \times b + c\times d$.

  Solution

  The Distributive Property is not limited to a certain number of addends inside the parentheses. This enables us to evaluate the number of ice cream scoops even when Dave joins Bella and Anton.

  Therefore, the total number of scoops is: $4 \times (1 + 1 + 1)$.

  Again, we can solve this in two different ways:

  • We could distribute the multiplier over the sum inside the parentheses first. This gives us $4 \times 3 = 12$.
  • Alternatively, we could multiply each addend separately and then sum. Using this method we get $4 \times 1 + 4 \times 1 + 4 \times 1 = 4 + 4 + 4 = 12$.

  The result is always the same. As you can see, we can always increase the number of addends: $c \times (a + b + d) = c \times a + c \times b + c \times d$. The variable $d$ stands for Dave. In contrast, increasing the number of multipliers is not possible and would not make sense at all. If you want to verify this fact, try it out for yourself!

• Use the Distributive Property to demonstrate what happens when Mrs. Harper bakes fewer pancakes.

  Hints

  The Distributive Property works with subtraction, too.

  We can express that with variables: $c \times (a - b) = c \times a - c \times b$

  In general we can say $c \times \left( a \pm b \right) = c\times a \pm c \times b$. The sign $\pm$ stands for either $+$ or $-$.

  Solution

  The Harper family really loves pancakes. Every Sunday morning, each member of the Harper family eats $3$ pancakes. Fortunately, Mrs. Harper knows how to use the Distributive Property. Her normal calculation goes like this:

  • $3\times ( 1 + 1 + 1 +1 + 1+ 1) = 3 \times 6 = 18$.

  But today, Franny and Sammy are visiting their grandma. Mrs. Harper has an answer to that:

  • $3 \times ( 1 + 1 + 1 +1 + 1+ 1 - 1 - 1) = 3\times (6-2)$.

  She still has an example of the Distributive Property, but this time, it includes subtraction. As we will see, this is no problem at all. Mrs. Harper has two possibilities:

  • She could calculate the parentheses first. Solving it this way, she gets $3 \times 4$, giving her $12$.
  • Alternatively, she can multiply each item inside the parentheses separately. This also gives her $3 \times 6 - 3 \times 2 = 18 - 6 = 12$.

  The result is always the same. She only has to be aware that the operator inside the parentheses changes. Instead of $+$ she has to use $-$.

• Use the Distributive Property to solve the expression: $3 \times (4 + 5)$.

  Hints

  The Distributive Property says that distributing a multiplier over a sum of numbers will result in the same answer as multiplying each addend separately and then summing.

  As a formula, this idea is expressed as follows: $c \times (a + b) = c \times a + c \times b$.

  Solution

  There are infinite examples where you can use the Distributive Property. Let's take a look at one of them. The general formula is: $c \times (a + b) = c \times a + c \times b$. So we can replace the $c$ with $3$, $a$ with $4$ and $b$ with $5$. What we're left with is $3 \times (4 + 5)$.

  We always have two different possibilities when solving these types of problems:

  • We could calculate the sum of $4+5$ first. We get $9$, which has to be multiplied by $3$. This leaves us with $27$.
  • The second way is to multiply each addend separately and then sum the resulting numbers, giving us $3 \times 4 + 3 \times 5$. As we can see, the result is the same: $12 + 15 = 27$.

• Simplify the expression $(27 + 12 - 21) \div 3$ using the Distributive Property.

  Hints

  The Distributive Property also works with division.

  Pay attention to the single operator inside the parentheses.

  Solution

  The Distributive Property can include a divisor, too. The difference is that we do not multiply the content of the parentheses, but divide it. As you may have learned in school, you can divide a dividend by multiplying by the reciprocal of the divisor. With a divisor, we also have a multiplier. So the rules for the Distributive Property have not really changed.

  Let us take a look at the expression $(27 + 12 - 21) \div 3$. We have two options for solving it:

  • First, we can evaluate the parentheses, which equals $18$. Then we divide by $3$ and get $6$.
  • On the other hand, we can divide each term inside the parentheses separately, paying attention to each operator. That gives us $27 \div 3 + 12 \div 3 - 21 \div 3 = 9 + 4 - 7$. In the end, we get $6$, too.

  As you can see, the Distributive Property has several variations. It can include both addition and subtraction inside the parentheses as well as multiplication and division outside the parentheses. In the case of division, the Distributive Property only works when a single number stands behind the parentheses.