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Expressions represent mathematical ideas by using multiple operations, numbers, and variables. However, sometimes there may be more than one way to write the same idea. This lesson teaches how to manipulate an expression to rewrite it different ways and how to determine if several expressions represent the same idea.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Who Is Correct?

Dominika and her friends are playing a tabletop role playing game that they created. There are three different type of coins in the game — copper, silver, and gold.

Copper, Silver, and Gold Coins
Each type of coin has a different value. The least valuable coin is the copper one. A gold coin is worth copper coins and a silver coin is worth copper coins. Wondering how many copper coins gold coins and silver coins are worth, Kriz and Dominika came up with different expressions.
Who is correct?
Discussion

Properties of Addition

The addition operation has some properties that allow to rewrite an algebraic expression without changing the values when evaluating. The first one has to do with the order of the addition.

Rule

Commutative Property of Addition

The order in which two or more terms are added does not affect the value of the sum. In other words, the addends can be written in any order.

For example, adding to produces the same result as adding to In both cases, the sum is This property also applies to the sum of more than two terms.
Since the Commutative Property of Addition is an axiom, it does not need a proof.
Discussion

Associative Property of Addition

The way three or more terms are grouped when added does not affect the value of the sum.

For example, consider the sum Grouping and adding it to produces the same result as grouping and adding it to
Since the Associative Property of Addition is an axiom, it does not need a proof.
Discussion

Identity Property of Addition

Adding to any number always results in the number itself.

Because of this, is called the Additive Identity.

Proof

 Informal Justification
Consider a number By the Reflexive Property of Equality, is equal to itself.
Let be another number. If is added to and subtracted from the left-hand side of the above equation, the equality still holds true.
Finally, is equal to
It has been shown that By the Commutative Property of Addition, is also equal to
Example

Rolling For Speed

Dominika and her friends are playing the tabletop role playing game that they created.

Tabletop-RPG-dice-set-on-map.jpg

Dominika's character needs to move very far as quickly as possible. It is good that she has upgrades to move faster! Her character can move feet in one round, but she can roll a die to add additional distance. This additional distance can be written as
In addition to this, Dominika has an upgrade item that adds feet to the total distance.
Help Dominika do the following.
a Performing a few steps in the correct order allows to rewrite the expression so that it has two terms. Write the correct order.
b Write the expression from Part A.

Hint
a Use the properties of addition to rewrite the expression.
b Write the expression found by following the steps from Part A.

Solution
a The given expression can be rewritten by combining the two constant terms. The first step to rewrite it is to arrange the terms so the constant terms are close together. This is done applying the Commutative Property of Addition.
Looking at the expression above, the next step would be to group the constant terms together. This can be done with the Associative Property of Addition.
The final step is to complete the addition.
Doing this process, the expression has been rewritten as an expression with only two terms. The steps to get to this point can be summarized as follows.
Step Operation
First Commutative Property of Addition
Second Associative Property of Addition
Third Evaluate the Addition
b The resulting expression was written by following the steps from Part A.
When this expression is evaluated, the result is the same as evaluating the given original expression.
Discussion

Properties of Multiplication

Similar to the Properties of Addition, the multiplication operation has some properties that allow to rewrite an algebraic expressions without modifying the results when evaluating.

Rule

Commutative Property of Multiplication

The order in which two or more factors are multiplied does not affect the value of the product. That is, the multiplicands can be written in any order.

For example, multiplying by produces the same result as multiplying by In both cases, the product is This property also applies to the product of more than two terms.
Since the Commutative Property of Multiplication is an axiom, it does not need a proof.
Discussion

Associative Property of Multiplication

The way three or more factors are grouped when multiplied does not affect the value of the product.

For example, consider the product Grouping and multiplying it by produces the same result as grouping and multiplying it by
Since the Associative Property of Multiplication is an axiom, it does not need a proof.
Discussion

Identity Property of Multiplication

Any number multiplied by is equal to the number itself.

Because of this, the number is called the Multiplicative Identity.

Proof

 Informal Justification
Consider a number By the definition of multiplication, multiplied by another number can be written as times the addition of
If the sum has only one term.
Therefore, is equal to Also, by the Commutative Property of Multiplication,
Discussion

Zero Property of Multiplication

The result of multiplying any number by  is always

As a consequence of this, if an entire expression is multiplied by the result is It does not matter the number of terms that the expression has.

Proof
Consider a number multiplied by zero.
The number zero can be rewritten as the subtraction of any number from itself. For simplicity, the zero can be rewritten as
Then the number can be distributed to simplify the expression on the right-hand side of the equation.
Distr

Distribute

DiffZero

It has been shown that By the Commutative Property of Multiplication, is also equal to
Example

Time to Attack

Dominika and Zain are preparing a cooperative attack to take down a strong enemy together.

Tabletop-RPG-characters.jpg

a Zain's character currently has an effect that diminishes their attack points. If Zain's character has an attack of the resulting points can be written as an expression.
Another player gives Zain a boost that multiplies Zain's attack points by
Write the expression for Zain's character's attack points as a single term with a coefficient.
b Dominika's character has attack points. Dominika also has some items that increase attack points. Her total attack points can be written as an expression.
Write this expression as a term with a single coefficient.

Hint
a Which property would allow the constant terms to be grouped together?
b Which property allows the order of the terms to be changed?

Solution
a An expression is given and needs to be rewritten as a single term with a coefficient. The first thing to notice is that there are two multiplications by a constant in the expression. By using the Associative Property of Multiplication, these constant numbers can be grouped together.
Now the fractions can be multiplied directly. Notice that the grouped fractions are reciprocals of one another. This means that multiplying them results in
Finally, this expression can be simplified to by using the Identity Property of Multiplication.
After the attack modifiers, Zain's character has their original attack points.
b Now it is time to simplify another expression. The constants can be grouped together to multiply them. The order of the variable and the number can be changed using the Commutative Property of Multiplication.
Similar to Part A, the Associative Property of Multiplication can be used to group the constants.
Finally, the product can be found. The expression can be written as a unique term with the variable and its coefficient.
Dominika's character has almost double their original attack power!
Example

Buying Potions

After a hard battle, Dominika and her friends went to buy some potions for health and magic.

Health potion and magic potion
They decided to buy health potions and magic potions. Dominika wrote an expression for the total amount of copper coins that they need to buy the potions.
Every health potion has the same price and every magic potion has the same price.
a What is the price of a health potion?
b What is the price of a magic potion?

Hint
a Use the Distributive Property to rewrite the given expression.
b Use the expression found in Part A.

Solution
a It is important to understand what the expression represents. The price of health potions is the product of and the price of a single health potion.
Similarly, the price of magic potions is the product of and the price of one magic potion.
The total price of health potions and magic potions is the sum of these products.
However, the given expression is written differently.
The good thing is that the expression can be rewritten using the Distributive Property. This property shows how the multiplication of a number and a sum can be rewritten as a sum of products.
A similar procedure can be performed on the given expression to rewrite it.
Distr

Distribute

MultII

Multiply by

Now that the expression is rewritten, the term can be compared with the price of health potions.
As previously stated, the total for the health potions is the product of by the price of a single health potion, which is of copper.
b In Part A, the given expression was rewritten to identify the price of a health potion. The same expression can be used to find the price of a magic potion.
This time the focus should be on the term with the variable This term can be compared with the price of magic potions.
This indicates the total price paid for the magic potions, which is the result of multiplying by the price of one magic potion, copper.
Discussion

Expressions that Represent the Same Quantity

In this lesson, different properties of operations were used to rewrite expressions into different expressions that evaluate to the same results. These expressions may be different, but they represent the same quantity and they are called equivalent expressions.

Concept

Equivalent Expression

Two or more expressions are equivalent expressions if they have the same result when evaluated. In the case of numerical expressions, two equivalent expressions result in the same number when evaluating the operations.
Since both and equal these expressions are equivalent. In a similar way, two algebraic expressions are equivalent if both expressions result in the same number for every value of the variables.
In both of these expressions, any value for results in the same number. For example, substitute for
It is important to remember that, when the properties of addition, multiplication, and the Distributive Property are used to rewrite an expression, the resulting expression is always equivalent to the initial one.
Example

Doing Mental Math When Buying Equipment

With the loot from their quest, Dominika and her party go to a town to buy some useful equipment. The shop gives prices in copper coins.

Magic Staff with a price of 65, Sword with a price of 39, and Shield with a price of 26
Dominika wants to know the total that they need to buy the equipment, but she does not have a calculator nearby. Help Dominika to find the total using mental math.

Hint

Rewrite each price as a multiple of

Solution
The sum of the prices can be written as a numerical expression.
When evaluating an expression with mental math, sometimes it is easier to find an equivalent expression with operations that are easier. In this case, the given numbers are all multiples of This means that they can be rewritten as the product of some number and
Next, the Distributive Property can be applied to separate common factor from each term.
This result is an expression with an addition of small numbers and a multiplication. This expression can be solved using the order of operations.
AddTerms

Add terms

MultII

Multiply by

Therefore, Dominika's party needs copper coins to buy all three items.
Example

A Dragon with a Quiz

Dominika and her party encountered a dragon!

Professor Emoji
However, instead of a battle, the dragon presented a quiz and asked for equivalent expressions of an algebraic expression.
Select all expressions that are equivalent to the expression above.

Hint

Think about the different properties of operations.

Solution

Equivalent expressions have the same result for every value of the variables. These expressions are written by applying properties of operations. The properties can be applied to the given expressions to determine which are equivalent to the expression given by the dragon.

Expression

The given expression can be seen as the product of two numbers. One way to get an equivalent expression is to use the Commutative Property of Multiplication.
(2z+1/7)14
This means that this expression is equivalent to the given expression.

Expression

The Distributive Property can be used to compare this expression to the given expression. When simplified, both expressions should be equal to each other if they are equivalent. The given expression can be simplified by distributing
Distr

Distribute

MultII

Multiply by

MultII

Multiply by

ReduceFrac

DivByOne

Now simplify the questionable expression by distributing
Distr

Distribute

IdPropMult

Identity Property of Multiplication

MultII

Multiply by

ReduceFrac

This simplifed expression is different from the one given by the dragon. Therefore, these expressions are not equivalent.

Expression

Consider the simplification of the dragon's expression.
This expression is different than the expression so these expressions are not equivalent.

Expression

Consider the simplification of the dragon's expression again.
Applying the Commutative Property of Addition to this expression results in the next possible equivalent expression.
The expressions are equivalent.

Expression

The previous expression was shown to be equivalent to the dragon's expression, so it can be compared to the final potential equivalent expression.
The constant term is a multiple of and the coefficient of the variable term is also a multiple of Knowing this, both terms can be rewritten.
The can be separated from both terms by using the Distributive Property.
The expression is equivalent to the previous expression, so it is also equivalent to the expression given by the dragon!
Closure

Multiple Ways of Being Right

At the beginning of this lesson, it was mentioned that in the game there are three different types of coins with different values.

Copper, Silver, and Gold Coins
The copper coins are the least valuable. The gold coins are worth copper coins and the silver coins are worth copper coins. Wondering how many copper coins gold coins and silver coins are worth, Dominika and Kriz proposed two different expressions.
The total copper coins can be found by adding the copper coins that correspond to gold coins to the copper coins from the silver coins. The corresponding copper coins from gold coins are the result of multiplying the number of gold coins by the number of copper coins that one gold coin is worth,
In a similar way, the number of copper coins equivalent to silver coins is the product of and
The total copper coins from both types of coins is the sum of these expressions.
This means that Kriz is right. This is great, but there is something important to note. Both coefficients of the expression can be rewritten as multiples of This factor can then be separated from the addition by using the Distributive Property.
Rewrite

Rewrite as

Rewrite

Rewrite as

FactorOut

Factor out

IdPropMult

Identity Property of Multiplication

This is the result that Dominika wrote. Since this was done using the properties of operations, these expressions are equivalent. Therefore, both Kriz and Dominika are correct. There are multiple ways of being right!
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