What you should learn

To use the distributive property to simplify expressions

NCTM Curriculm Standards 2 - 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Distributive Property

Term

Like Terms

Equivalent Expressions

Coefficient

Simplest form

Introduction: In the school cafeteria, each student can pick one hot or cold entree and one salad from the following in order to get a "Meal Deal" lunch at a special low price.

Jenine wants to know how many different lunches are possible. She can make a table to represent all the possible combinations.

There are 2 *4 or 8 possible hot lunches.

There are 2 * 3 or 6 possible cold lunches.

There are (2 * 4) + (2 * 3) possible lunches from which students can choose.

Note that (2 * 4) + (2 * 3) = 8 + 6 or 14.

Jenine can also represent this problem by using the following table.

According to this table, there are 2 types of salads times the total number of entrees, 4 + 3, or 7.

Either way you look at it, there are 14 possible lunches. That's because the following is true.

This is an example of the distributive property.

Distributive Property: For any numbers a, b, and c

a(b+c)=ab+ac

(b+c)a=ba+ca

a(b-c)=ab-ac

(b-c)a=ba-ca

Notice that it doesn't matter whether a is placed on the right or the left of the expression in parentheses.

The symmetric property of equality allows the distributive property to be written as follows.

Exercise 1: The Oak Grove High School Spirit Club wants to make a banner for the championship football game. The students have two large pieces of paper to use for hte banner. One piece is 5 feet by 13 feet, and the other is 5 feet by 10 feet. They plan to use both pieces for the banner. Find the total area of the banner.

The toatl area of the banner can be found in two ways.

Method 1: Add the areas of hte smaller rectangles.

A = wL1 + wL2

= 5(13) + 5(10)

= 65 + 50

= 115

Method 2: Multiply the width by the length.

A = wl

= 5 (13 + 10)

= 5 (23)

= 115

The area is 115 square feet.

You can use the distributive property to multiply in your head.

Exercise 2: Use the distributive property to find each product.

a. 7 * 98

7 * 98 = 7(100 - 2) = 700 - 14 = 686

b. 8(6.5)

A term is a number, a variable, or a product or quotient of numbers and variables. Some examples of terms are , (1/4)a, and 4y. The expression 9y + 13y + 3 has three terms.

Like terms are terms that contain the same variables, with corresponding variables having the same power. In the expression 8x + 2x + 5a + a, 8x and 2x are like terms, and 5a and a are also like terms.

We can use the distributive property and properties of equality to show that 3x + 8x = 11x. In this expression, 3x and 8x are like terms.

The expression 3x + 8x and 11x are called equilvalent expressions because they denote the same number. An expression is in simplest form when it is replaced by an equivalent expression having no like terms and no parentheses.

Exercise 3: Simplify (1/4)x + 2x + (11/4)x

The coefficient of a term is the numerical factor. For example, in 23ab, the coefficient is 23. In xy, the coefficient is 1 since, by the multiplicative identity property, 1 * xy = xy. Like terms may also be defined as terms that are the same or that differ only in their coefficients.

Activity: Do the following two examples...

1. Name the coefficient in each term

a. 145xy

b. ab

c.

2. Simplify each expression.

a.

b.

Further Application: Modeling Mathematics: The Distributive Property

Materials: algebra tiles and product mat

Throughtout your study of mathematics, you have used rectangles to model multiplication. For example, the figure below shows the multiplication 2(3 + 1) as a rectangle that is 2 units wide and 3 + 1 units long. The model shows that the expression 2(3 + 1) is equal to 2 * 3 + 2 * 1. The sentence 2(3 + 1) = 2 * 3 + 2 * 1 illustrates the distributive property.

You can use special tiles called algebra tiles to form rectangles that model multiplication. A 1-tile is a square that is 1 unit long and 1 unit wide. Its area is 1 square unit. An x-tile is a rectangle that is 1 unit wide and x units long. Its area is x square units.

Activity: Find the product 3(x + 2) by using algebra tiles. First, model 3 (x + 2) as the area of a rectangle.

Step 1: The rectangle has a width of 3 units and a length of x + 2 units. Use your algebra tiles to mark off the dimensions on a product mat.

Step 2: Using the marks as a guide, make the rectangle with algebra tiles. The rectangle has 3 x-tiles and 6 1-tiles. The area of the rectangle is x + 1 + 1 + x + 1 + 1 + x + 1 + 1, or 3x + 6. Thus, 3(x + 2) = 3x + 6.

 

Model: Find each product by using algebra tiles.

1. 2 (x = 1)

2. 5 (x + 2)

3. 2 (2x + 1)

4. 2 (3x + 3)

Draw: Tell whether each statement is true or false. Justify your answer with algebra tiles and a drawing.

5. 3(x + 3) = 3x + 3

6. x(3 + 2) = 3x + 2x

Write: 7. You Decide...Helen says that 3(x + 4) = 3x + 4, but Adita says that 3(x + 4) = 3x + 12.

a. Which classmate is correct?

b. Use words and/or models to give an explanation that demonstrates the correct equation.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 25 - 51 odd, 52 - 60

Alternative Homework: Enriched: 24 - 48 even, 49 - 60

Extra Practice: Students book page 758 Lesson 1-7

Extra Practice Worksheet: Click Here.

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