Download 2.1 Using Properties to Form Equivalent Ex

2.1. Using Properties to Form Equivalent Expressions 7.EE.1, 7.EE.2
www.ck12.org
2.1 Using Properties to Form Equivalent Expressions 7.EE.1, 7.EE.2
Students will know and understand the basic properties in math as applied to rational numbers and algebraic
expressions. Factoring and expanding expressions are used as multiple representations in word problems.
Sophie decided to make cupcakes and a cake for dessert. She put half the batter in a cake pan and the other half
in cupcake tins. Which tasted better? They both tasted the same because they used the same batter, the same
ingredients. That is exactly like equivalent expressions. They look different, but when you break them down, they
are the same variables and numbers.
Just like cake and cupcakes from the same batter, we will first look at different expressions and learn how to make
equivalent forms. Then we will use this to make understanding word problems easier.
Examining Expressions for Equivalence Using Numbers
What is an expression in math? An expression is simply a math problem that has not been solved. It has numbers
and symbols, but no equal sign. An algebraic expression is a math expression that uses numbers, symbols, and
5
are
variables. Equivalent means that the expressions have the same value, but don’t look the same. 24 and 10
1
equivalent fractions. They look different, but both represent 2 .
Does 3 + (4 + 5) have the same value as (3 + 4) + 5?
If we follow the order of operations in math, operations in parentheses are done first.
3 + (9) and (7) + 5 are the two expressions after the first operation. Now we add in each expression. Are the final
values the same? They both simplify to 12. The expressions are equivalent. This is also a way to prove a property
of math. It is called the associative property of addition. This property says that if all signs are addition, it does
not matter how you group the numbers with the parentheses, the answer will come out the same.This property also
works if you use multiplication signs instead of addition signs.
Does -3(4 - 10) have the same value as (-3 • 4) + (-3 • -10)?
Again we follow the order of operations in math, completing the work in parentheses first.
38
www.ck12.org
Chapter 2. Unit 2: Expressions and Equations
−3 (4 − 10)
(−3 · 4) + (−3 · −10)
−3 (4 + −10)
−12 + 30
−3 (−6)
18
18
18
The two expressions are equivalent. Here again, another property of math is proved. It is called the distributive
property. It says that multiplying the outside number by each term inside the parentheses gives the same value as
doing the parentheses first, and then multiplying by the number outside.
Does 5 • 42 have the same value as (5 • 4)2 ?
5 · 42
(5 · 4)2
5 · 16
(20)2
80
400
In this case, the expressions are not equivalent. The same numbers and exponent were used in each, but the
parentheses in the second equation made the answer very different.
Examining Algebraic Expressions for Equivalence
When looking at algebraic expressions, it is sometimes harder to see if they have the same value. One way to help
is by expanding the expression. Expanding the expression means that we can use the distributive property to get
rid of the parentheses to see what the expression looks like. Expanding also means that we can separate 4a into two
parts, 2a + 2a just like we can separate 4 into 2 + 2.
Another way is by combining like terms. Variables are just like numbers. We can add, subtract, multiply, and divide
with variables that are the same.
Is 3a + 4b equivalent to a + 3b + 2a + b?
Looking at the second expression, we can switch the order of the 3b + 2a so that the a’s and b’s are next to each
other. This is using the commutative property.
a + 3b + 2a + b → a + 2a + 3b + b
Now combine like terms.
a + 2a + 3b + b → 3a + 4b
The two expressions are exactly the same now. This means they are equivalent.
Is 5(6a - 3b) + 14b equivalent to 30a - b?
The first expression can be expanded. We use the distributive property to expand.
5(6a - 3b) +14b → 5(6a) + 5(-3b) +14b → 30a + -15b + 14b
Combine the like terms.
30a - b
The two expressions are exactly the same now. They are equivalent.
The substitution principle can also be used to see if two algebraic expressions are equivalent. This means we pick a
number to put in where the variable is in both expressions and see if the answers are the same. Let’s look at a model
to help understand this process.
39
2.1. Using Properties to Form Equivalent Expressions 7.EE.1, 7.EE.2
www.ck12.org
Are 4x + 9 and 2(2x + 5) - 1 equivalent expressions? (Reminder: equivalent means look different but have the
same value.)
To use the substitution principle we first pick a number to replace x. Pick an easy number to work with. We will
use 3. So where there is an x, put in a three. Then just follow order of operations and get a final answer for each.
4x + 9
2 (2x + 5) − 1
4 (3) + 9
2 (2 (3) + 5) − 1
12 + 9
2 (6 + 5) − 1
21
2 (11) − 1
22 − 1
21
Since both expressions ended up as 21, it means that they are equivalent.
Are (3x - 2)2 and 3x2 - 4 equivalent expressions?
Let’s try the substitution principle again. Pick a number (a simple one). We will use 4. Replace the variable x with
4. Then use order of operations to find the answer. This is also called evaluating the expression. We have to be
careful with these expressions because of the exponent. It is important to complete parentheses first, then exponents,
then multiplication in each expression.
(3x − 2)2
3x2 − 4
(3 (4) − 2)2
3 (4)2 − 4
(12 − 2)2
3 (16) − 4
2
48 − 4
100
44
(10)
So even though the two expression look similar, by using the substitution principle, we prove that they are not
equivalent.
Using Properties to Create Equivalent Expressions
When we create equivalent expressions, we use properties in math. The properties are like the rules we follow when
moving and changing the way the expressions look. The table below lists the basic properties of math and how they
look using numbers and variables.
40
www.ck12.org
Chapter 2. Unit 2: Expressions and Equations
Sometimes creating a new, equivalent expression is actually simplifying the expression. Simplifying means to do
all the math operations you can in an expression. You make it as simple as possible.
If we start out with the expression 10 - 3(x + 2), we can create an equivalent expression by simplifying. Follow the
steps to see how this is done.
10 − 3 (x + 2)
10 − 3 (x) − 3 (2)
distributive property
10 − 3x − 6
multiply
10 + −3x + −6
additive inverse
10 + −6 + −3x
commutative property
4 + −3x
Notice that we used three properties to simplify. Naming the properties is called justifying or explaining your work.
Notice also that we did NOT do 10 - 3 first. Order of operations says that multiplication comes before subtracting.
Other times we start out with an expression that we cannot simplify and are asked to make an equivalent expression. When this happens there can be more than one answer.
The expression 5m - 10 cannot be simplified any more. We can still make equivalent expressions. You need to
think, "What expressions can I make that can become 5m - 10?". Break the expression down. 5 is equal to 3 +
2. So 5m can be broken down into 3m + 2m.
5m - 10 = 3m + 2m - 10
You could break down the - 10 as well.
5m - 10 = 5m - 6 - 4
There are lots of ways to do this when you break down with addition or subtraction. There is also a way to break
down some expressions by a process called factoring. It is actually the opposite of doing the distributive property.
Let’s use our example of 5m - 10. If we look at the numbers 5 and -10, there is a common factor for both. (Remember
that a factor is something that can divide evenly into a number.) 5 can divide into both 5 and -10 evenly. So 5m 10 can also be 5(m - 2). Here are some examples to help you understand:
Example 1
Write two equivalent expressions to the following: 16x − 12
Since this expression is already in simplest form (we can’t combine anything), we make equivalent forms by
breaking the expression down.
The first is by factoring.
41
2.1. Using Properties to Form Equivalent Expressions 7.EE.1, 7.EE.2
www.ck12.org
16x − 12
16x ⇒ 2, 2, 2, 2, x
−12 ⇒ −1, 2, 2, 3
4 (4x − 3)
The second is by breaking down the 16x using addition.
16x − 12
10x + 6x − 12
The two equivalend expressions are 4(4x - 3) and 10x + 6x - 12.
These four websites help you really understand the power of properties in mathematics: http://mathmaster.org/video/commutative
property-for-addition/?id=931
http://mathmaster.org/video/associative-property-for-multiplication/?id=932
http://mathmaster.org/video/distributive-property/?id=933
http://learnzillion.com/lessons/809-factor-an-expression shows an alternate way to factor an algebraic expression.
Word Problems and Equivalent Expressions
Which is simpler to understand: a math problem in words or a math problem in numbers and symbols?
A math problem gives us an answer. A word problem tells us why the answer makes sense.
Word problems are looked at in different ways by different people. You may come up with one math problem and
someone else has a different math problem. Both may be right! By using properties of numbers, it is possible to
have different math expressions that actually are the same.
Geometry problems are a good way to see how "different" is still the same. Given a rectangle where the length is
twice as long as the width, there are different ways to make an expression to show perimeter. Perimeter is adding
up all the sides of a rectangle.
It is always a good idea to draw a diagram in geometry so we can picture the problem. We need to label the length
and width. Twice means to multiply by two. That is why the width is x and the length is 2x.
42
www.ck12.org
Chapter 2. Unit 2: Expressions and Equations
Opposite sides of a rectangle are the same length. So now we label all four sides. To find the perimeter we put
addition signs between all four side.
What other ways can we write this expression?
We could combine like terms: x + 2x + x + 2x → 6x
We could expand even more: x + 2x + x + 2x → x + x + x + x + x + x
We could use the commutative property and then show the factored form: x + 2x + x + 2x → x + x + 2x + 2x → 2x
+ 2(2x)
All four ways of expressing the perimeter are correct. There actually is more than one right answer.
Example 2
The perimeter of a triangle is 6x. The triangle is equilateral. Represent the perimeter of the triangle using algebraic
expressions. Show or explain the procedure you used.
Again we draw a diagram. We state what makes a triangle equilataral, and give an example using numbers.
If the lengths of the sides are 5, then to find the perimeter we would choose one of two ways:
5 + 5 + 5 or 3 (5)
Is there a way to make 6x look like one of the two ways? Can we break 6x up into three equal parts like the
three 5’s? We make a diagram to think.
If we use 2, then 2x + 2x + 2x =6x. Also if we use 2 in the second equation, 3(2x) = 6x. The same length, 2x,
can be each side of the triangle.
So if all three sides can be the same expression, 2x, then 6x can be the perimeter of an equliateral triangle.
http://learnzillion.com/lessons/815-rewrite-an-expression-by-expanding-it is a video that shows how to create equivalent expressions in geometry.
Now let’s look at writing different math problems from a single word problem.
Jamie and Zoe made cookies for a bake sale. Jamie made 15 more than Zoe. Each cookie sold for 25¢. Using c to
represent the number of cookies Zoe made, write an expression to represent the total money earned for selling all
the cookies.
Zoe made c cookies. Jamie made 15 more. More means to add. Jamie made c + 15 cookies.
43
2.1. Using Properties to Form Equivalent Expressions 7.EE.1, 7.EE.2
www.ck12.org
Each cookie was 25¢. So to find the money earned, we multiply .25(c) for Zoe and .25(c + 15) for Jamie. Total
means to add, so we show the addition of both Jamie’s and Zoe’s cookies.
.25c + .25 (c + 15)
WAIT!
This is not the only right answer. What if we thought of all the cookies together rather than Jamie’s and Zoe’s
separately. To find total cookies, we would first add c and c + 15 to get the total cookies.
c + c + 15 ⇒ 2c + 15
.25 (2c + 15)
The way we think about a word problem can set up different, but equivalent expressions.
http://www.youtube.com/watch?v=nsQ_tHSe4qU is a video tutorial on making expressions from word problems.
associative property, commutative property, distributive property, algebraic expression, factoring, expanding an
expression, equivalent, combine like terms
44