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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.
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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 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.
Kriz: Dominika: 4x+2y2(2x+y)
Who is correct? {"type":"choice","form":{"alts":["Neither","Dominika","Kriz","Both"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
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.
a+b=b+a
4+5+1✓10=1+5+4⇓=10✓
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.
(a+b)+c=a+(b+c)
(3+9)+412+4✓16=3+(9+4)⇓=3+13⇓=16✓
Since the Associative Property of Addition is an axiom, it does not need a proof.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 ✓
It has been shown that a+0=a. By the Commutative Property of Addition, 0+a is also equal to a.
Example
Rolling For Speed
Dominika and her friends are playing the tabletop role playing game that they created.
50+d
In addition to this, Dominika has an upgrade item that adds 15 feet to the total distance.
(50+d)+15
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.
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b Write the expression from Part A.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["d"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["d+65"]}}
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.
(50+d)+15⇓(d+50)+15
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.
(d+50)+15⇓d+(50+15)
The final step is to complete the addition.
d+(50+15)⇓d+65
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.
d+65
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.
a⋅b=b⋅a
3⋅2⋅6✓36=2⋅6⋅3⇓=36✓
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.
(a⋅b)⋅c=a⋅(b⋅c)
(2⋅4)⋅68⋅6✓48=2⋅(4⋅6)⇓=2⋅24⇓=48✓
Since the Associative Property of Multiplication is an axiom, it does not need a proof.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=n timesa+a+…+a
If n=1, the sum has only one term.
a⋅1=1 timea
Therefore, a⋅1 is equal to a. Also, by the Commutative Property of Multiplication, 1⋅a=a. Discussion
Zero Property of Multiplication
The result of multiplying any number by 0 is always 0.
a⋅0=0
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, the zero can be rewritten 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.
Example
Time to Attack
Dominika and Zain are preparing a cooperative attack to take down a strong enemy together.
a Zain's character currently has an effect that diminishes their attack points. If Zain's character has an attack of z, the resulting points can be written as an expression.
z⋅54
Another player gives Zain a boost that multiplies Zain's attack points by 45.
(z⋅54)⋅45
Write the expression for Zain's character's attack points as a single term with a coefficient. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["z"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["z","1z"]}}
b Dominika's character has d attack points. Dominika also has some items that increase attack points. Her total attack points can be written as an expression.
1.2⋅(d⋅1.5)
Write this expression as a term with a single coefficient. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["d"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["1.8d"]}}
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.
(z⋅54)⋅45⇓z⋅(54⋅45)
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⋅(54⋅45)⇓z⋅1
Finally, this expression can be simplified to z by using the Identity Property of Multiplication.
z⋅1=z
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 1.5 can be changed using the Commutative Property of Multiplication.
1.2⋅(d⋅1.5)⇓1.2⋅(1.5⋅d)
Similar to Part A, the Associative Property of Multiplication can be used to group the constants.
1.2⋅(1.5⋅d)⇓(1.2⋅1.5)⋅d
Finally, the product can be found. The expression can be written as a unique term with the variable d and its coefficient.
(1.2⋅1.5)⋅d=1.8d
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.
5(2x+y)
Every health potion has the same price and every magic potion has the same price. a What is the price of a health potion?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"copper","answer":{"text":["10"]}}
b What is the price of a magic potion?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"copper","answer":{"text":["5"]}}
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 x health potions is the product of x and the price of a single health potion.
Price of x Health Potions:price of a health potion⋅x
Similarly, the price of y magic potions is the product of y and the price of one magic potion.
Price of y Magic Potions: price of a magic potion⋅y
The total price of x health potions and y magic potions is the sum of these products. price of a health potion⋅x+price of a magic potion⋅y
However, the given expression is written differently.
5(2x+y)
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(b+c)=ab+ac
A similar procedure can be performed on the given expression to rewrite it.
Now that the expression is rewritten, the x-term can be compared with the price of x health potions.
price of a health potion⋅x⇕10x
As previously stated, the total for the x health potions is the product of x by the price of a single health potion, which is of 10 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.
10x+5y
This time the focus should be on the term with the variable y. This term can be compared with the price of y magic potions.
price of a magic potion⋅y⇕5y
This indicates the total price paid for the y magic potions, which is the result of multiplying y by the price of one magic potion, 5 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.
12−62⋅3=6=6
Since both 12−6 and 2⋅3 equal 6, 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.
y+42+y+2
In both of these expressions, any value for y results in the same number. For example, substitute 5 for y. y+4⇓5+4=92+y+2⇓2+5+2=9
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"copper","answer":{"text":["130"]}}
Hint
Rewrite each price as a multiple of 13.
Solution
The 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. In this case, the given numbers are all multiples of 13. This means that they can be rewritten as the product of some number and 13.
13⋅5+13⋅3+13⋅2
Next, the Distributive Property can be applied to separate common factor 13 from each term.
13(5+3+2)
This result is an expression with an addition of small numbers and a multiplication. This expression can be solved using the order of operations.
Therefore, Dominika's party needs 130 copper coins to buy all three items.
Example
A Dragon with a Quiz
Dominika and her party encountered a dragon!
14(2z+71)
Select all expressions that are equivalent to the expression above. {"type":"multichoice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">7<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.8359999999999999em;vertical-align:-0.686em;\"><\/span><span class=\"mord\">7<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mord\">4<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">4<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,3,4]}
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 (2z+71)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.
Expression 7(14z+1)
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 14.14(2z+71)
Distr
Distribute 14
14⋅2z+14⋅71
MultII
Multiply 14 by 2z
28z+14⋅71
MultII
Multiply 14 by 71
28z+714
ReduceFrac
ba=b/7a/7
28z+12
DivByOne
1a=a
28z+2
7(14z+1)
Distr
Distribute 7
7⋅14z+7⋅1
IdPropMult
Identity Property of Multiplication
7⋅14z+7
MultII
Multiply 7 by 14z
147z+7
ReduceFrac
ba=b/7a/7
2z+7
Expression 2z+28
Consider the simplification of the dragon's expression.28z+2
This expression is different than the expression 2z+28, so these expressions are not equivalent. Expression 2+28z
Consider the simplification of the dragon's expression again.28z+2
Applying the Commutative Property of Addition to this expression results in the next possible equivalent expression.
28z+2=2+28z
The expressions are equivalent. Expression 2(1+14z)
The previous expression was shown to be equivalent to the dragon's expression, so it can be compared to the final potential equivalent expression.2+28z
The constant term is a multiple of 2 and the coefficient of the variable term is also a multiple of 2. Knowing this, both terms can be rewritten.
2⋅1+2⋅14z
The 2 can be separated from both terms by using the Distributive Property.
2⋅1+2⋅14z=2(1+14z)
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.
Kriz: Dominika: 4x+2y2(2x+y)
The total 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 corresponding copper coins from x gold coins are the result of multiplying the number of gold coins x by the number of copper coins that one gold coin is worth, 4.
Gold Coins:4x
In a similar way, the number of copper coins equivalent to y silver coins is the product of 2 and y. Gold Coins:Silver Coins:4x2y
The total copper coins from both types of coins is the sum of these expressions.
4x+2y
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 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)
Algebraic Properties
Algebraic Properties
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