x 2

3

2x

Carlo is very much in love with Jennylyn. However, he is finding hard time winning Jennylyn’s heart. Jennylyn said that she could only accept Carlo’s love proposal if he could give the total area of the rectangular land she owns. The land is divided into four rectangular lots and Jennylyn also wanted to know the area of each lot. How would Carlo grant the wishes of Jennylyn? Could you help him?

Let us now explore the answer to Mark’s problem. I have prepared a learning kit that will ask you to find the areas of rectangles of different sizes and the area of a whole rectangle given the measures of its corresponding parts. It also contains activity that will lead you in understanding the “Distributive Property of Multiplication Over Addition” which you may apply in finding the products of binomials.

If you want to know if you have understood the lesson, try to answer the set of given exercises. After successfully answering these exercises, I’m sure that you will jump out of joy and excitement. Enhance further your skills by answering the enrichment exercises. If will challenge you a lot.

Along the way while learning this lesson, you may encounter some difficulties. Don’t worry because I got some tips for you which you may find at the Reference section of this kit.

Rectangle 3

3 cm

4 cm

Rectangle 1

4 cm

2 cm

Rectangle 54 cm

4 cm

Rectangle 4

5 cm

2 cm

Rectangle 23 cm

5 cm

Rectangle 6

6 cm

5 cm

Make rectangle cut-outs having the same measures as the rectangles previously presented. Select two rectangles from these cut-outs and use them to form a bigger rectangle.

Were you able to find the area of each rectangle?

How did you do it?

Were you able to make a bigger rectangle out of the two rectangle cut-outs?

How did you come up with the figure?What things did you consider?

Suppose the measures of one of the rectangles used to form a bigger rectangle are 3 cm and 6 cm and the other rectangle measures 4 cm and 6 cm, what will be the measure of the rectangle formed? Illustrate.

3 cm

6 cm

6 cm

4 cm

Find the area of each rectangle used to form the bigger rectangle. Also, find the area of the bigger rectangle formed. How is the area of the bigger rectangle formed related to the areas of the smaller rectangles? How are the measures of the bigger rectangle formed related to the measures of the two smaller rectangles? Express this as an equation.

Find the total area of each figure below. Express your answer as an equation.

5 cm

4 cm 8 cma)

c

b

ab)

Let us extend further the activity. This time, use four rectangle cut-outs to form a bigger

rectangle.What do you think would the figure look like?

Illustrate using the space below.

What is the area of each rectangle used in forming the bigger rectangle?

How about the area of the rectangle formed? How is the area of the bigger rectangle formed related

to the areas of the smaller rectangles?How are the measures of the bigger rectangle formed related to the measures of the smaller

rectangles?Express this as an equation.

Answers

Find the total area of each of the following figures. Express your answer as an equation.

3 cm

3 cm

1 cm

2 cm

Answer: __________________________

a)

a

d

b

c

b)

Answer: __________________________

Can you find now the total area of Jennylyn’s land including the area of each

rectangular lot?

x

2

1

x

c)

2x

3

2x

Answer:________________

Hello GUYS!!! I still want you to work on this. If you can give me all the correct answers, then

Carlo would be more joyful.

What do you think would Jennylyn say to make Carlo more happy?

4321 765

Find the products of the following binomials. Write the letter corresponding to your answer in the boxes above.

1) (x + 1) (x + 2) R - x2 – 4x + 21

2) (x + 5)(x – 1) M - 6x2 + 9x – 10

3) (x – 2)(x – 4) A - x2 + 4x - 5

4) (x – 7)(x + 3) Y - 2x2 + 9x + 4

5) (2x + 1)(x + 4) M - x2 + 3x + 2

6) (3x – 2)(2x + 5) E - 3x2 + 19x – 40

7) (x + 8)(3x – 5) R - x2 – 6x + 8

Now that Carlo and Jennylyn found themselves in each other’s arms, let yourself then enrich with skills in finding products of binomials. Do the following by getting the products.

35 a

5a 522

1) (3x + 2y)(2x + 3y)

2) (x3 + 3)(2x3 – 4)

3)

4) (3m3 – 2n)(5m3 + 3n)

5) [2(x + 2y) + 3][3(x + 2y) - 4]

Are you having difficulty?Below are some important mathematical

ideas which may help you to find the product of binomials.

Area of a Rectangle

The area of a rectangle is the number of square units contained in the rectangle. To find the area of the rectangle, we just get the product of its measures or multiply its length and width.

Consider the rectangle at the right.Its area is: A = 6 cm x 10 cm

A = 60 cm2

A more general way of getting the area of a rectangle is using the equation A = lw, where A is the area, l is the length and w is the width of the rectangle.

Forming Rectangle Out of Smaller Rectangles and the Distributive Property of Multiplication Over Addition

Using two rectangles, we can form a bigger rectangle as long as the two rectangles have a side having the same measures. For example, if one rectangle has measures 4 cm and 7 cm and another rectangle has measures 7 cm and 9 cm, then we can form a bigger rectangle out of these two rectangles as shown below.

Looking at the figure, we notice that the measure of one of its sides is the measure of the side common to both rectangles which is 7 cm. To find the measure of the other side, we just add the measures of the sides not common to both rectangles. Hence, the measure of the other side of the rectangle formed is 4 cm + 9 cm or 13 cm.

Comparing the areas of the two smaller rectangles and the area of the bigger rectangle formed, the areas of the two smaller rectangles are:

AreaFirst Rectangle = 7 cm x 4 cm = 28 cm2

andAreaSecond Rectangle = 7 cm x 9 cm = 63 cm2

9 cm4 cm

7 cm7 cm

The area of the bigger rectangle formed on the other hand is:

AreaBigger Rectangle = 7 cm x (4 + 9) cm = 7 cm x 13 cm AreaBigger Rectangle = 91 cm2

Notice that the sum of the area of the two smaller rectangles used is equal to the area of the bigger rectangle formed.

AreaBigger Rectangle = AreaFirst Rectangle + AreaSecond Rectangle

91 cm2 = 28 cm2 + 63 cm2

91 cm2 = 91 cm2

We can also express this as:

AreaBigger Rectangle = AreaFirst Rectangle + AreaSecond Rectangle

7 x (4 + 9) = (7 x 4) + (7 x 9) 7 x 13 = 28 + 63 91 cm2 = 91 cm2

In the equation 7 x (4 + 9) = (7 x 4) + (7 x 9), we applied the Distributive Property of Multiplication to get the product. The multiplier is multiplied to each addend in the multiplicand.

7 x (4 + 9) = (7 x 4) + (7 x 9)

Let’s have other examples. Suppose we wanted to get the product of the following:

a) 4 and (2 + 3)

b) x and (x + 4)

c) a and (b + c)

Then, a) 4 x (2 + 3) = (4 x 2) + (4 x 3) = 8 + 12 = 20

b) x(x + 4) = x2 + 4x

c) a(b + c) = ab + ac

A bigger rectangle can also be formed out of four rectangles as shown below.

Consider the measures of the four rectangles below. A1, A2, A3, and A4 represent the areas of each rectangle.

The area of each small rectangle is:

A1 = 3 x 4 A3 = 1 x 4 = 12 = 4

A2 = 3 x 2 A4 = 1 x 2 = 6 = 2

The sum of the areas of the four rectangles is

A1 + A2 + A3 + A4 = 12 + 6 + 4 + 2

=24

To find the area of the rectangle formed, we need to find first the measures of its sides. Thus,

One side = 3 + 1 = 4Other side = 4 + 2

= 6

The area then is : AreaRectangle = 4 x 6

= 24

Notice that the sum of the areas of the four small rectangles is equal to the area of the rectangle formed.

AreaRectangle = A1 + A2 + A3 + A4

24 = 12 + 6 + 4 + 2 24 = 24

Again, we can also express this as:

AreaRectangle = A1 + A2 + A3 + A4

(3 + 1)(4 + 2) = (3 x 4) + (3 x 2) + (1 x 4) + (1 x 2)(4)(6) = 12 + 6 + 4 + 2 24 = 24

The equation (3+1)(4+2) = (3 x 4) + (3 x 2) + (1 x 4) + (1 x 2) is an illustration of the Distributive Property of Multiplication Over Addition. Each addend in the multiplier is multiplied to each addend in the multiplicand.

(3 + 1)(4 + 2) = (3 x 4) + (3 x 2) + (1 x 4) + (1 x 2)

This property is further applied in the following examples.

a(5 + 2)(3 + 6) = (5 x 3) + (5 x 6) + (2 x 3) + (2 x 6) (7)(9) = 15 + 30 + 6 + 12 63 = 63

b(a + b)(c + d) = ac + ad + bc + bd

Suppose we wanted to get the product of (x + 2) and (x + 3). Again, we apply the Distributive Property of Multiplication Over Addition. To show this, we have:

(x + 2)(x + 3) = x2 + 3x + 2x + 6

Since there are terms which are similar, we combine them.Hence, (x + 2)(x + 3) = x2 + 3x + 2x + 6

= x2 + 5x + 6

(x + 2)(x + 3) = x2 + 5x + 6 is an example of getting products of binomials applying the Distributive Property of Multiplication Over Addition. The binomials are (x + 2) and (x + 3) and the product is x2 + 5x + 6.