Students will simplify algebraic expressions using the identity properties of addition and multiplication, the commutative and associative properties of addition and multiplication, and the distributive property of multiplication over addition.
Background for Teachers
Enduring Understanding (Big Ideas):
Properties apply in both numeric and algebraic situations. Properties can expedite simplifying expressions
- What is the product of any number and 1? What is the sum of any number and 0?
- How does applying the commutative or associative properties affect the sum or product?
- How can I demonstrate the use of the distributive property of multiplication over addition?
- How do properties help me simplify algebraic expressions?
Apply properties in simplifying algebraic expressions
Commutative property, Associative property, Multiplicative Identity Property, Additive Identity Property, distributive property, algebraic expression, simplify
Ways to Gain/Maintain Attention (Primacy):
Contest, predicting, music, technology, stories, analogy, manipulative, writing, movement, cooperative discussion, journaling
Starter: Accessing prior knowledge
- Which of these representations does not tell us to multiply?
- 6 7
- 8 x 10
- Use Mental Math to compute.
- 3 + (17 + 138)
- 1( ½ + 4 + ½ )
- 5 x 26 x 2
- 2(13) + 2(7)
- 5(2 + 10)
- 231 8 0
Lesson Segment 1: (Accessing prior knowledge) What is the product of any number and 1? What is the sum of any number and 0? How does applying the commutative or associative properties affect the sum or product? How can I demonstrate the use of the distributive property of multiplication over addition?
Team Contest: Use the #1 question on the starter to review properties by asking students to look at property words on the board. Tell them you can compute much faster and easier by using these properties. Have students take out a paper for an assignment activity called Properties Guess, and number the paper a-f. As you mentally compute each starter problem, have students quietly discuss with their team and write which property or properties they think you applied. Ask students to respond after they have written the property they think you applied. Any team who correctly identified the property(s) earns a point. You may need correct their thinking as you go over each problem. After discussing an expression, have students write the correct property and how it was applied to simplify each expression on their paper.
- 3 + (17 + 138) Use associative property to regroup adding 3 and 17 first.
- 1( ½ + 4 + ½ ) Use the commutative property to reorder ½ + ½ + 4 in the parentheses, and then multiply by 1 using the identity property.
- 5 x 26 x 2 Use the commutative property to reorder 5 x 2 x 26.
- 2(13) + 2(7) Use the distributive property to multiply the sum of 13 and 7 (20) by 2.
- 5(2 + 10) Use the distributive property to multiply 5 x 2 (10) and 5 x 10 (50). Add 10 and 50.
- 231 8 0
Tell students these properties work for addition and multiplication with variables too. If they have their properties foldable from September Lesson 7, they could use it to compare. Make this foldable for properties with variables. Fold both edges toward the center. Clip on the dotted line to the fold to make four shutters. Inside students should write examples of the application of these properties using variables. Work with students to write simple algebraic examples such as:
a + b = b + a
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
1a = a
a(b+c) = ab + ac
Lesson Segment 2: How do properties help me simplify algebraic expressions?
Sing or say the Properties Song to review (attached).
Accessing and building background knowledge:
Tell the students when they were finding a simple answer for the operations in the contest, they were simplifying the expression. Give a brief explanation for the word, simplify such as, What we mean by simplifying an expression is to make the expression more simple to understand or look at without changing the value of the expression.
In our language we often simplify expressions. For example, we could say, Hi there. How are you doing? Or, we could say, Hey, Sup? The meaning is the same, but the second expression is much shorter and simpler than the original expression. In mathematics we want to write expressions as simply as possible, but do not want to change their meaning or value. We want the simplified expression to be equivalent to the original, longer expression.
Ask the following questions and have students record the examples on their Team Contest record paper.
Q. When we say two expressions are equivalent what does that mean? For example when we say 3 + 1 is equivalent to 4 (or 3 + 1 = 4), what does that mean? The equal sign tells us one expression is equivalent to the other or in other words, the expressions have the same value.
Q. If two expressions are equivalent, must they always look exactly the same? What makes you think so?
2 6 = 3 4
3(2 5) = (3 2)5
3(5 + 6) = 3 5 + 3 6
Q. How can we know whether two expressions are equivalent if they dont look alike? One way to verify that two expressions are equivalent, is to simplify each expression.
Example 1: 2 6 = 3 4
2 6 simplified is 12
3 4 simplified is 12
12 = 12.
So, 2 6 = 3 4
Example 2: 3(2 5) = (3 2)5
3(2 5) is 3(10) =30
(3 2)5 is (6)5 = 30
So, 3(2 5) = (3 2)5
Example 3: 3(5 + 6) = 3 5 + 3 6
3(11) = 33
3 5 + 3 6 is 15 + 18 = 33
33 = 33
So, 3(5 + 6) = 3 5 + 3 6
Tell students these ideas about equivalency and simplifying apply with variables as well as numbers. We use properties to simplify algebraic expressions. When we simplify an algebraic expression using properties, we can compare the original expression with the simplified expression to make sure they are equivalent. A simplified expression is always equivalent to the original. Students will be simplifying algebraic expressions, and then substituting values in the expressions to verify equivalency.
Work with the class to complete the Simplifying Algebraic Expressions worksheet.
Lesson Segment 3: How can the distributive property be applied to algebraic expressions?
Using Algeblocks, work through the attached Distributive Property Using Algeblocks investigation (attached). This is a powerful visualization for applying the distributive property with variables. Make sure students build, draw and represent as instructed.
Discuss their models.
Assign text practice as needed.
observation, questioning, writing, mental math, student response cards
This lesson plan was created by Linda Bolin.
Simplifying Using the Distributive Property Lesson
The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Take a look at the problem below.
2(3 + 6)
Because the binomial "3 + 6" is in a set of parentheses, when following the Order of Operations, you must first find the answer of 3 + 6, then multiply it by 2. This gives an answer of 18.
2(3 + 6)
! Incorrect Method !
It would be incorrect to remove the parentheses and multiply 2 and 3 then add 6, as this would give an incorrect answer of 12.
2(3 + 6)
2 * 3 + 6
6 + 6
Examine the expression below.
The two terms inside the parentheses cannot be added because they are not like terms. Therefore, 2 + 4x, the expression inside the parentheses, cannot be simplified any further. To simplify this multiplication, another method will be needed. This is where the Distributive Property comes in.
Distributing a Number
We continue with previous example.
6(2 + 4x)
The Distributive Property tells us that we can remove the parentheses if the term that the polynomial is being multiplied by is distributed to, or multiplied with each term inside the parentheses.
This definition is tough to understand without a good example, so observe the example below carefully.
6(2 + 4x)
now by applying the Distributive Propery
6 * 2 + 6 * 4x
The parentheses are removed and each term from inside is multiplied by the six.
Now we can simplify the multiplication of the individual terms:
12 + 24x
Distributing a Negative Sign
The next problem does not have a number outside the parentheses, only a negative sign.
-(3 + x2)
There are two easy ways to simplify this problem. The first and simplest way is to change each positive or negative sign of the terms that were inside the parentheses. Negative or minus signs become positive or plus signs. Similarly, positive or plus signs become negative or minus signs. Recall that in the case of 3, no positive or negative sign is shown, so a positive sign is assumed.
-3 - x2
We will now work through this problem again, but using a different method.
-(3 + x2)
Recall that any term that does not have a coefficient has an implied coefficient of 1. Because of the negative sign on the parentheses, we instead assume a coefficient of negative one. Thus, we can rewrite the problem as
-1(3 + x2)
Now the -1 can be distributed to each term inside the parentheses as in the first example in this lesson.
-1 * 3 + -1 * x2
-3 + -x2
A variable can be distributed into a set of parentheses just as we distributed a negative sign or a number. Consider the following example.
x(y + 1)
We can now apply the distributive property to the expression by multiplying each term inside the parentheses by x.
x * y + x * 1
Now simplifying the multiplication, we get a final answer of
xy + x
The same is true when a problem consists of a number, variables, and parentheses:
4x(x2 + 9)
Again, multiply each term inside the parentheses by the multiplier outside the parentheses.
4x * x2 + 4x * 9
4x3 + 36x
Quiz on the Distributive Property
Distributive Property Resources
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