Basic Algebra/Working with Numbers/Distributive Property

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Vocabulary

Sum The resulting quantity obtained by the addition of two or more terms.

Real Number: An element of the set of all rational and irrational numbers. All of these numbers can be expressed as decimals.

Term: A term is a number or a variable or the product of a number and a variable(s).

Monomial: An algebraic expression consisting of one term.

Binomial: An algebraic expression consisting of two terms.

Trinomial: An algebraic expression consisting of three terms.

Polynomial: An algebraic expression consisting of two or more terms.

Like Terms: Like terms are expressions that have the same variable(s) and the same exponent on the variable(s). Remember that constant terms are all like terms. This follows from the definition because all constant terms can be seen to have a variable with an exponent of zero.

Lesson

The distributive property is the short name for "the distributive property of multiplication over addition", although you will be using it to distribute multiplication over subtraction as well. When you are simplifying or evaluating you follow the order of operations. Sometimes you are unable to simplify any further because you cannot combine like terms. This is when the distributive property comes in handy.

Natural Language

When you first learned about multiplication it was described as grouping. You used multiplication as a way to condense the multiple addition of the same quantity. If you wanted to add $3+3+3+3$ you could think about it as four groups of three items.

|ooo| + |ooo| + |ooo| + |ooo|

You have 12 items. This is where $4\cdot 3$ comes in. So as you moved on you took this idea to incorporate variables as well. $3x$ is three groups of x.

$\underbrace {x+x+x} _{\text{three groups of x}}=3x$

And $3(2x)$ is three groups of $2x$ and $2x$ is $x+x$

$\underbrace {(x+x)+(x+x)+(x+x)} _{\text{three groups of 2x}}=3(2x)$

This gives you six x's or 6x. Now we need to take this idea and extend it even further. If you have $3(x+1)$ you might try to simplify using the order of operations first. This would have you do the addition inside the parentheses first. However, x and 1 are not like terms so the addition is impossible. We need to look at this expression differently if we are going to simplify it. What you have is $3(x+1)$ or in other words you have three groups of $(x+1)$

$\underbrace {(x+1)+(x+1)+(x+1)} _{\text{three groups of (x+1)}}=3(x+1)$

Here you can collect like terms. You have three x's and three 1's.

$\underbrace {x+x+x} _{\text{3x}}+\underbrace {1+1+1} _{\text{3}}=3x+3$

So you started with $3(x+1)$ and ended with $3x+3$

$3(x+1)=3x+3$

The last equation might make it easier to see what the distributive property says to do.

$3(x+1)=3(x)+3(1)=3x+3$

You are taking the multiplication by 3 and distributing that operation across the terms being added in the parentheses. You multiply the x by 3 and you multiply the 1 by 3. Then you just have to simplify using the order of operations.

What Is Coming Next

After you learn about the distributive property you will know how to multiply a monomial by a polynomial. Next, you can use this information to understand how to multiply a polynomial by a polynomial. You will probably move on to multiplying a binomial times a binomial. This will show up in something like (x+2)(3x+5). You can think of a problem like this as x(3x+5) + 2(3x+5). Breaking up the first binomial like this allows you to use your knowledge of the distributive property. Once you understand this use of the distributive property you can extend this understanding even further to justify the multiplication of any polynomial with any polynomial.

Sometimes while you are attempting to isolate a variable in an equation or inequality you will need to use the distributive property. You already know that you use inverse operations to isolate your desired variable, but before you do that you need to combine like terms that are on the same side of the equation (or inequality). Now there might be a step even before that. You will need to see if the distributive property needs to be used before you can combine like terms then proceed to use inverse operations to isolate a variable.

Word to the Wise

Remember that you still have the order of operations. If you can evaluate operations in a straightforward manner it is usually in your best interest to do so. The distributive property is like a back door to the order of operations for when you get stuck because you do not have like terms. Of course when you are dealing with only constant terms everything you encounter is like terms. The trouble happens when you introduce variables. This means that some terms cannot be combined. Remember that variables take the place of real numbers (at least in Algebra 1) so the same rules that govern real numbers will also govern the variables that hold their place and vice versa. You can use the distributive property even when you do not need to.

Example Problems

Example Problem #1:

Simplify $2(x+4)$

Solution to Example Problem #1:

Normally, to follow the order of operations you would add the two terms in the parenthesis first, then do the multiplication by. This does not work for this expression because x and 4 are unlike terms so you cannot combine them. We use the distributive property to help us find a way around the order of operations while still being sure that we keep the value of the express.

We distribute the multiplication by 2 across the addition. We will have 2 multiplied by x and 2 multiplied by 4.

$2(x)+2(4)$

Now we just need to finish the multiplication. $2(4)$ is equal to 8.

$2x+8$

We are done because we just have two terms being added and we cannot add them because they are not like terms.

Example Problem #2:

Simplify $3x(2x-4)$

Solution to Example Problem #2:

Since the terms inside the parentheses are not like terms we cannot combine them. We can use the distributive property to multiply by $3x$ .

$3x(2x-4)=3x(2x)-3x(4)$

This is the first example with subtraction in it. You keep this operation between the two terms just like we kept the addition between the two terms in the previous example. The next step is to multiply

$3x(2x)-3x(4)=6x^{2}-12x$

In order to complete the previous step you will already need to know how to multiply monomials.

To summarize all the steps...

$3x(2x-4)=3x(2x)-3x(4)=6x^{2}-12x$

Example Problem #3:

Solve for $x$ in $2(x+10)=60$

Solution to Example Problem #3:

To solve for a variable you must isolate it on one side of the equation. We need to get the $x$ out of the parentheses. Since we cannot go through the order of operations and just add x plus 10 then multiply by 2, we will have to use the distributive property. First, distribute the multiplication by 2 across the addition inside the parentheses.

$2(x+10)=60$

$2(x)+2(10)=60$

Now you can multiply

$2(x)+2(10)=60$

$2(x)+20=60$

Now we can work on getting the $x$ on one side by itself. You need to do the order of operations backwards so we can "undo" what is "being done to" $x$ . To get rid of adding 20 you need to subtract 20. And remember that an equation sets up a relationship that we need to preserve. If you subtract 20 from one side you need to subtract 20 from the other side as well to keep the balance.

$2(x)+20=60$

$2(x)+20-20=60-20$

$2(x)+0=40$

$2(x)=40$

Now we need to "undo" the multiplication by 2, so we divide by 2. Whatever you do to one side must be done to the other. So divide both sides by 2.

${\frac {2(x)}{2}}={\frac {40}{2}}$

$x=20$

This is it. You know you are done when the variable $x$ is by itself on one side, and it is.

Practice Games

http://www.quia.com/ba/15357.html

http://www.studystack.com/matching-1870

http://www.slideshare.net/rfant/distributive-property-algebra-1/ (slide show)

http://www.phschool.com/atschool/academy123/html/bbapplet_wl-problem-430723.html ( video explanation)

Practice Problems

(Note: solutions are in red)

Use the distributive property to rewrite the expression

$2(x+7)$ 2x+14

$5(y+6)$ 5y+30

$4(1+b)$ 4+4b

$x(a+b)$ xa+xb

$(3+x)6$ 18+6x

Notes for Educators

It is obvious to most educators in the classroom that students must have a good number sense to comprehend mathematics in a useful way. A critical part to have number sense is understanding multiplication of real numbers and variables that stand in the place of real numbers.

Students also need as much practice as possible with counting principles. Explaining multiplication and the distributive property as above helps to solidify some counting principles knowledge in the minds of the students.

In order to teach the distributive property an educator might be interested in how students first perceive knowledge of this kind. The better we understand how the brain obtains knowledge the more responsibly we can guide it.

Piaget model of cognitive development sets up level of understanding that the students minds passes through.

According to this chart, the distributive property would sit in sensory-motor or perhaps the pre-operational stages. Piaget's work has been largely criticized, but few doubt that it is a good starting place to think about how the brain acquires mathematical understanding.

Annette Karmiloff-Smith was students of Piaget and many believe that she brings his ideas forward. She believes that human brains are born with some preset modules that have the innate ability to learn and as you have experiences you create more independent modules. Eventually these modules start working together to create a deeper understanding and more applicable knowledge. The person moves from implicit to a more explicit knowledge which helps to create verbal knowledge.

Education, and specifically mathematics education plays a role during the process of moving from the instictually implicit stages to the more verbal explicit understanding. A student acquires procedural methods then learning they theory behind the procedure. This runs parallel to mathematics education. If you accept this model of how the mind comes to understand a concept, it would be critical to teach the students the procedural methods and mechanics of how the distributive property must be carried out. It would then be just as important to show them why this works out the way it does, or at least provide them with the educational opportunities to explore why it works out.

This exploration should take three stages. First the students needs to master the mechanics of the distributive property. In math ed terms, this might be considered drill and kill. The next step would be asking the students to reflect on why they think the distributive property has such a behavior. This could be related to encouraging metacognition with your students. Have them reflect not only on the procedure of the distributive property but also on why they think that. Hopefully the third and final step would be a the last two steps coming together in the students' minds as a solid understand of the distributive property.

Since this knowledge would probably first be link in the students mind as a procedure only helpful in a math classroom, it might also be beneficial to encourage the students to stretch this concept across domains. After all, one of the main purposes of a public mathematics education is to encourage logicality among the populous.

One of the most common errors for students to make is to just multiply the first number in the parentheses by the number outside. For example

$2(x+1)=2x+1$

This could initially be remedied by explaining the distributive property as taking 2 groups of (x+1) and adding them, like multiplication means to do.

This might lead to another misunderstand though. It might be confusing to think about things like $.5(x+1)$ or ${\frac {2}{3}}(x+4)$ because it is hard to think about .5 groups of (x+1) or ${\frac {2}{3}}$ of $(x+4)$ . When a student first learns about multiplication they are told that it like grouping things together to simplify the addition of the same number multiple times. Once they have mastered this concept multiplication is extended to all rational number. Now multiplication is better thought of as a scaling process. You are taking one number and scaling it by a factor of another. This same mental leap is needed to think about distributing a rational number because the distributive property is still just multiplication.

An effective method to explain multiplication as a scale factor is to have two number lines, one right above the other. If you are multiplying by ${\frac {1}{2}}$ then the scale factor is ${\frac {1}{2}}$ and you can draw guide lines from the top number line to the bottom number line that scale every number down by one half. So a line will be draw from 2 on the top number line to 1 on the bottom number line. Another line will be drawn from 3 on the top number line to 1.5 on the bottom number line and so on. Of course this method is easier to use if you have an interactive applet or program of some kind that allows you to update the scale factor immediately. Without this instant gratification the students may find this explanation too cumbersome to follow.

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