4. Algebraic Properties
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Chapter 4
4.
Algebraic Properties
This lesson delves into the fundamental rules that govern the manipulation of mathematical expressions, known as algebraic properties. These include the commutative property, which allows you to rearrange the order of numbers when adding or multiplying; the associative property, which lets you change the grouping of numbers; and the distributive property, which explains how to expand expressions involving both addition and multiplication. Through real-world examples like calculating costs in a role-playing game, the material illustrates how these properties can simplify complex calculations and help solve everyday problems. Understanding these properties is crucial for anyone looking to excel in algebra and beyond.
Show less Show more expand_more| Student Learning Objectives: |
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| | 16 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Algebraic Properties
Slide of 16
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 explores how to manipulate an expression to rewrite it in different ways and how to determine if several expressions represent the same quantity.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
How Many Copper Coins?
A board game has three types of coins: copper, silver, and gold.
Each type of coin has a different value. The least valuable coin is the copper one. A gold coin is worth 4 copper coins and a silver coin is worth 2 copper coins. Which algebraic expression represents how many copper coins x gold coins and y silver coins are worth? rl Expression A: & 4x + 2y Expression B: & 2(2x+y)
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Discussion
Properties of Addition
Addition has some properties that allow us to rewrite an algebraic expression without changing the final value when evaluating. The first property 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.
a+ b= b+ a
For example, adding 3 to 6 produces the same result as adding 6 to 3. In both cases, the sum is 9. This property also applies to the sum of more than two terms. 4+ 5+ 1 &= 1+ 5+ 4 &⇓ 10 &= 10 ✓
Since the Commutative Property of Addition is an axiom, it does not need a proof.
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Discussion
Associative Property of Addition
The way three or more terms are grouped when they are added does not affect the value of the sum.
( a+ b)+ c= a+( b+ c)
Consider the sum 3+9+4. Grouping 3+9 and adding it to 4 produces the same result as grouping 9+4 and adding it to 3. ( 3+ 9)+ 4&= 3+( 9+ 4) &⇓ 12 + 4 &= 3 + 13 &⇓ 16 &= 16 ✓
Since the Associative Property of Addition is an axiom, it does not need a proof.
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Discussion
Identity Property of Addition
Adding 0 to any number always results in the number itself.
a+0=a
Because of this, 0 is called the Additive Identity.
Proof
Informal JustificationConsider a number a. By the Reflexive Property of Equality, a is equal to itself.
a=a
Let b be another number. If b is added to and subtracted from the left-hand side of the above equation, the equality still holds true.
a=a ⇔ a+b-b=a
Finally, b-b is equal to 0.
a+b-b=a ⇔ a+0=a ✓
We have shown that a+0=a. By the Commutative Property of Addition, 0+a is also equal to a.
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Example
Rolling For Speed
Dominika and her friends are playing a tabletop role playing game. 
Dominika's character can move 50 feet in one round, but she can roll a die to add additional distance. This additional distance can be written as d.
50+d
She also has an upgrade item that adds 15 feet to the total distance.
(50+d) + 15
a
Arrange the steps needed to simplify the expression in the correct.
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b
Write the expression from Part A.
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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 simplified by combining the two constant terms. The first step to rewrite it is to arrange the terms so the constant terms are close together. Let's use the Commutative Property of Addition.
( 50+ d) + 15 ⇓ ( d+ 50)+15 The next step would be to group the constant terms together. This can be done using the Associative Property of Addition. (d+50 )+15 ⇓ d+ (50+15 ) The final step is to complete the addition. d+( 50+15) ⇓ d+ 65 The expression has now been rewritten as an expression with only two terms. Let's review the steps we took to get to this point.
| Step | Operation |
|---|---|
| First | Commutative Property of Addition |
| Second | Associative Property of Addition |
| Third | Evaluate the Addition |
b
We wrote this expression in Part A.
d+65 When this expression is evaluated, the result is the same as evaluating the given original expression.
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Discussion
Properties of Multiplication
Multiplication also 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.
a* b = b* a
For example, multiplying 5 by 4 produces the same result as multiplying 4 by 5. The product for both is 20. This property also applies to the product of more than two terms. 3* 2* 6 &= 2* 6 * 3 &⇓ 36 &= 36 ✓
Since the Commutative Property of Multiplication is an axiom, it does not need a proof.
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Discussion
Associative Property of Multiplication
The way three or more factors are grouped when they are multiplied does not affect the value of the product.
( a* b)* c= a*( b* c)
For example, consider the product 2* 4* 6. Grouping 2* 4 and multiplying that product by 6 produces the same result as grouping 4* 6 and multiplying that product by 2. ( 2* 4)* 6&= 2*( 4* 6) &⇓ 8* 6&= 2* 24 &⇓ 48 &= 48 ✓
Since the Associative Property of Multiplication is an axiom, it does not need a proof.
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Discussion
Identity Property of Multiplication
Any number multiplied by 1 is equal to the number itself.
a *1=a
Because of this, the number 1 is called the Multiplicative Identity.
Proof
Informal JustificationConsider a number a. By the definition of multiplication, a multiplied by another number n can be written as n times the addition of a.
a* n =a+a+... +a_(ntimes)
If n=1, the sum has only one term.
a* 1 =a_(1time)
Therefore, a* 1 is equal to a. By the Commutative Property of Multiplication, 1* a=a.
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Discussion
Zero Property of Multiplication
The result of multiplying any number by 0 is always 0.
a*0=0
If an entire expression is multiplied by 0, the result is 0, no matter how many terms the expression has. (a+29-3b+12* xy0.24)*0=0 endgathered
Proof
Consider a number a multiplied by zero.
a* 0
The number zero can be rewritten as the subtraction of any number from itself. For simplicity, we will rewrite the zero as 1-1. a* 0 = a* (1-1)
Then the number a can be distributed to simplify the expression on the right-hand side of the equation.
It has been shown that a*0=0. By the Commutative Property of Multiplication, 0* a is also equal to 0.
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Example
Photo Resizing
Emily is working on a photo layout for the yearbook.
a
The original width of one photo is z. Emily first multiplies the width by 45 to make it smaller.
z* 4/5 She makes the photo larger by multiplying it by 54. ( z* 4/5 ) * 5/4 Write the final width of the photo as a single term with a coefficient.
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b
On the cover of the yearbook, the school's logo design starts with a length d. Emily makes a few adjustments to the size.
1.2* (d* 1.5) Write this expression as a term with a single coefficient.
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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
We want to rewrite the given expression as a single term with a coefficient. The first thing to notice is that there are two multiplications by a constant in the expression. Let's use the Associative Property of Multiplication to group the constants.
( z* 4/5 ) * 5/4 [0.3em] ⇓ [0.3em] z* (4/5 * 5/4) Now the fractions can be multiplied directly. Notice that the grouped fractions are reciprocals of one another. This means that multiplying them results in 1. z* (4/5 * 5/4) [0.3em] ⇓ z* 1 Finally, this expression can be simplified to z by using the Identity Property of Multiplication. z* 1 = z After the size modifications, the photo is returned to its original size.
b
For the second expression, let's group the constants together again so we can multiply them. We can use the Commutative Property of Multiplication to change the order of the variable and the constant 1.5.
1.2* ( d* 1.5) ⇓ 1.2* ( 1.5 * d) Use the Associative Property of Multiplication to group the constants. 1.2* (1.5 * d) ⇓ (1.2* 1.5) * d Now we can find the product. The expression can be written as a unique term with the variable d and its coefficient. ( 1.2* 1.5) * d = 1.8d the final logo design is 1.8 times its original size.
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Example
Fruit for Sale
Jordan and Tadeo stopped by a fruit stand. They decided to buy x watermelons and y bags of peaches.
The expression shows the total amount of money they spent on the fruit. 5(2x + y)
a
What is the price of a watermelon?
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b
What is the price of a bag of peaches?
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Hint
a
Use the Distributive Property to rewrite the given expression.
b
Use the expression from Part A.
Solution
a
It is important to understand what the expression represents. The cost of x watermelons is the product of x and the price of a single watermelon.
Cost ofxWatermelons: price of a watermelon* x Similarly, the cost of y bags of peaches is the product of y and the price of one bag of peaches. Cost ofyBags of Peaches: price of a bag of peaches* y The total cost of x watermelons and y bags of peaches is the sum of these products. price of a watermelon * x + price of a bag of peaches * y However, the given expression is written differently. 5(2x + y) We can rewrite either expression 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( b + c) = a b + a c Let's distribute the 2 in the given expression.
Now that the expression is rewritten, the x-term can be compared with the cost of x watermelons. price of a watermelon* x ⇕ 10 x The watermelons cost $ 10 each.
b
In Part A we rewrote the given expression to find the price of a watermelon. We can use the same expression to find the price of a bag of peaches.
10 x + 5 y This time we will compare the y-term with the cost of y bags of peaches. price of a bag of peaches* y ⇕ 5 y Each bag of peaches y cost $5.
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Discussion
Expressions that Represent the Same Quantity
Properties of operations can be used to rewrite expressions without changing their value. Expressions that have the same value are called equivalent expressions.
Concept
Equivalent Expression
Two or more expressions are equivalent expressions if they have the same result when evaluated. Equivalent numerical expressions result in the same number when evaluating the operations. 12 - 6 &= 6 2*3 &= 6 Since both 12-6 and 2*3 equal 6, these expressions are equivalent. Two or more algebraic expressions are equivalent if all expressions result in the same number for every value of the variables. r y + 4 2+y + 2 In both of these expressions, any value for y results in the same number. For example, substitute 5 for y. ccc y + 4 & & 2+y+2 ⇓ & & ⇓ 5 + 4 = 9 & & 2+ 5+2=9
When the properties of operations and the Distributive Property are used to rewrite an expression, the resulting expression is always equivalent to the initial expression.
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Example
Doing Mental Math When Buying Equipment
While playing a video game, Dominika goes to a shop to buy some useful equipment. The shop gives prices in gold coins.
Dominika wants to know the cost of the equipment, but she does not have a calculator nearby. Help her find the total using mental math.
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Hint
Rewrite each price as a multiple of 13.
Solution
We can write sum of the prices can be written as a numerical expression.
65 + 39 +26
When evaluating an expression with mental math, sometimes it is easier to find an equivalent expression with operations that are easier to manage. The given numbers are all multiples of 13, so they can be rewritten as the product of some number and 13.
13*5 + 13*3 +13*2
Next, let's apply the Distributive Property to separate the common factor 13 from each term.
13(5 + 3 +2)
This result is an expression with an addition of small numbers and a multiplication. We can solve this expression using the order of operations.
Therefore, Dominika needs 130 gold coins to buy all three items.
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Example
Equivalent Expressions
Select all expressions that are equivalent to the expression below. 14( 2z+1/7)
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Hint
Think about the different properties of operations.
Solution
Equivalent expressions have the same result for every value of the variables. We can apply the properties of operations to the given expressions to determine which are equivalent to the given one.
Expression ( 2z+ 17)14
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 to change the order of the factors.
This means that this expression is equivalent to the given expression.
Expression 7( z14+1)
Let's use the Distributive Property to compare this expression to the given expression. We will start by distributing 14 in the given expression.
14( 2z+1/7)
Distr
Distribute 14
14*2z+14* 1/7
MultII
Multiply 14 by 2z
28z+14* 1/7
MultII
Multiply 14 by 1/7
28z+14/7
ReduceFrac
a/b=.a /7./.b /7.
28z+2/1
DivByOne
a/1=a
28z+2
Now let's simplify the questionable expression by distributing 7.
7(z/14+1)
Distr
Distribute 7
7* z/14+7* 1
IdPropMult
Identity Property of Multiplication
7* z/14+7
MultII
Multiply 7 by z/14
7z/14+7
ReduceFrac
a/b=.a /7./.b /7.
z/2+7
This simplifed expression does not match the given one, so these expressions are not equivalent.
Expression 2z+28
We simplified the given expression by distributing 14. 28z+2 This expression is different than the expression 2z+28, so these expressions are not equivalent.
Expression 2+28z
We simplified the given expression by distributing 14. 28z+2 If we apply the Commutative Property of Addition to this expression, we get the next possible equivalent expression. 28z+2 = 2+28z The expressions are equivalent.
Expression 2(1+14z)
Let's start with our rearranged simplified expression from the previous part. 2+28z Both the constant term and the coefficient of the variable term are multiples of 2. Let's separate this factor out. 2*1+2*14z Now we can factor out 2 from both terms by using the Distributive Property. 2*1+2*14z = 2(1+14z) The expression is also equivalent to the original expression.
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Closure
Multiple Ways of Being Right
A board game has three types of coins: copper, silver, and gold.
The copper coins are the least valuable. The gold coins are worth 4 copper coins and the silver coins are worth 2 copper coins. Earlier we considered two expressions that could represent how many copper coins x gold coins and y silver coins are worth. rl Expression A: & 4x + 2y Expression B: & 2(2x+y) The total number of copper coins can be found by adding the copper coins that correspond to x gold coins to the copper coins from the y silver coins. The copper coins equal to x gold coins is the result of multiplying the number of gold coins x by the number of copper coins that one gold coin is worth, 4. rl Gold Coins: & 4x The number of copper coins equivalent to y silver coins is the product of 2 and y. rl Gold Coins: & 4x Silver Coins: & 2y The total copper coins from both types of coins is the sum of these expressions. 4x + 2y This means that Expression A is correct. However, notice that both coefficients of the expression can be rewritten as multiples of 2. This factor can then be separated from the addition by using the Distributive Property.
4x + 2y
Rewrite
Rewrite 4x as 2*2x
2*2x + 2y
Rewrite
Rewrite 2y as 2*1y
2*2x + 2*1y
FactorOut
Factor out 2
2(2x+1y)
IdPropMult
Identity Property of Multiplication
2(2x+y)
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Algebraic Properties
Exercises
Which of the following expressions is equivalent to 4+x+4? A.& x * 4 B.& x(4* x) C.& x+ 4 +4 D.& (4+x)+x
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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Consider the given options that might be an equivalent expression for 4+x+4. A.& x * 4 B.& x(4* x) C.& x+ 4 +4 D.& (4+x)+x We want to know which of these is equivalent to the given expression. Equivalent expressions are equal no matter what number is substituted for the same variable. With this in mind, we can evaluate each option for a value of the variable and see if they are equal. Let's start by evaluating the given expression for x= 1.
The value of the given expression is 9. We can do the same for the given options to see which of them is an equivalent expression for x= 1. Let's see the results in a table!
| Expression | Substitution | Value | Equivalent Expression? |
|---|---|---|---|
| x * 4 | 1* 4 | 4 | No |
| x (4* x) | 1(4* 1) | 4 | No |
| x +4+4 | 1+4+4 | 9 | Yes |
| (4+x)+x | (4+ 1)+ 1 | 6 | No |
As we can see, only x+4+4 has the same value as the given expression. This means that this expression might be equivalent! To make sure that the expression is indeed equivalent, let's see if there are any properties that we can apply to modify the given expression to get the expression in C. Let's compare them first! Given Expression:& 4+x+4 Expression C:& x+4 +4 The terms in these expressions are the same but the order is different. The good thing is that we can reorder the given expression by using the Commutative Property of Addition to get expression C. 4+x+4 = x+4+4 This means that the correct answer is C.
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Solution
A property is being demonstrated. 5* (y* 9) = (5* y) * 9 Which property is it?
{"type":"choice","form":{"alts":["Commutative Property of Addition","Commutative Property of Multiplication","Associative Property of Addition","Associative Property of Multiplication"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":3}
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We want to identify the operation being performed to identify which of the Properties of Operations is being demonstrated. 5* (y* 9) = (5* y) * 9 In this case, the terms are being multiplied. The property being used shows that changing the grouping of the factors being multiplied does not change the product. This relates to the Associative Property of Multiplication.
In this exercise, we saw an example of the Associative Property of Multiplication. Now, let's review some other Properties of Operations.
- Commutative Property of Addition: This property shows that changing the order of the factors being added does not change the sum.
- Commutative Property of Multiplication: This property shows that changing the order of the factors being multiplied does not change the product.
- Associative Property of Addition: This property shows that changing the grouping of the terms being added does not change the sum.
- Distributive Property: This property shows that the product of a number and a parenthetical factor is equal to the sum of the products of the number and each term inside the parentheses.
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Solution
Simplify the following expression. 10+(14+a)
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We need to examine the given expression to simplify it. First, let's look at the operations used in the expression. 10+(14+a) The expression only has additions. According to the order of operations, a constant should be added to a variable first. However, we cannot complete this operation. Instead, let's simplify the expression by adding the two constants. We can use the Associative Property of Addition to group the constants. 10+(14+a) = (10+14)+a Now we can add the constants.
Note that we can also use the Commutative Property of Addition to have the variable term before the constant term. a+24=24+a Since we cannot add the variable and the constant, we have simplified the expression. Good job!
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Solution
Simplify the following expression. 7(n+2)
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We want to find an equivalent expression with the least number of terms to simplify the given expression. Consider the given expression. 7(n+2) We can see it as the product of two factors. Let's use the Distributive Property to find an equivalent expression. 7(n+2)&=7(n)+7(2) &=7n+14 The variables and constants cannot be added without evaluating the variable. Because of this, the expression is simplified. We did it!
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Consider the following expression. 4(3a+2) Which of the following is an equivalent expression? Select all that apply.
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We are asked to determine which of the given expressions are equivalent to the expression 4(3a+2).
| Expression | Is It Equivalent to 4(3a+2)? |
|---|---|
| 12a+8 | |
| 6a+8+6a | |
| 12a+6 |
Two expressions are equivalent if one expression can be rewritten into the other one by using Properties of Operations. First, let's try to rewrite the expression 4(3a+2) using the Distributive Property.
We found that 4(3a+2) and 12a+8 are equivalent expressions. This is one of our options!
| Expression | Is It Equivalent to 4(3a+2)? |
|---|---|
| 12a+8 | Yes |
| 6a+8+6a | |
| 12a+6 |
Now let's use the Properties of Operations to rewrite the next expression.
We can see that 6a+8+6a and 12a+8 are equivalent expressions. Therefore, the expression is also equivalent to 4(3a+2)!
| Expression | Is It Equivalent to 4(3a+2)? |
|---|---|
| 12a+8 | Yes |
| 6a+8+6a | Yes |
| 12a+6 |
Now let's determine if the last expression is equivalent to the others. Earlier, we found that 4(3a+2) is equivalent to 12a+8. 4(3a+2)=12a+8 Notice that the expression 12a+8 is not equivalent to 12a+6 because we cannot turn 8 into 6 using the Properties of Operations. 12a+8 ≠ 12a+6 ⇕ 4(3a+2) ≠ 12a+6 This means that 4(3a+2) is not equivalent to 12a+6.
| Expression | Is It Equivalent to 4(3a+2)? |
|---|---|
| 12a+8 | Yes |
| 6a+8+6a | Yes |
| 12a+6 | No |
Now we know which expressions are equivalent. Great job!
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Algebraic Properties
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