Transcript SIM Math I - WordPress.com

x
2
2x
3
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?
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.
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.
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.
Were you able to find the area of each
rectangle?
How did you do it?
2 cm Rectangle 1
3 cm
Rectangle 2
4 cm
5 cm
4 cm Rectangle 3
2 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.
Rectangle 4
5 cm
Were you able to make a bigger rectangle out of the
two rectangle cut-outs?
How did you come up with the figure?
3 cm
What things did you consider?
5 cm
Rectangle 6
4 cm Rectangle 5
4 cm
6 cm
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.
Find the total area of each figure below. Express
your answer as an equation.
a)
8 cm
4 cm
5 cm
4 cm
6 cm
3 cm
6 cm
b)
a
b
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.
c
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
a)
x
c)
1
x
2 cm
3 cm
2
1 cm
Can you find now the total area of Jennylyn’s
land including the area of each rectangular lot?
Answer: __________________________
a
x
b
2
b)
c
d
2x
3
Answer:________________
Answer: __________________________
1
2
3
4
5
6
7
Find the products of the following binomials. Write
the letter corresponding to your answer in the boxes
above.
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?
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
Are you having difficulty?
Below are some important mathematical
ideas which may help you to find the
product of binomials.
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.
Area of a Rectangle
1) (3x + 2y)(2x + 3y)
2) (x3 + 3)(2x3 – 4)
3)  2a  2  5a 

 5


5

3
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.
Its area is:
4) (3m3 – 2n)(5m3 + 3n)
Consider the rectangle at the right.
A = 6 cm x 10 cm
A = 60 cm2
5) [2(x + 2y) + 3][3(x + 2y) - 4]
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
The area of the bigger rectangle formed on the
other hand is:
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.
AreaBigger Rectangle = 7 cm x (4 + 9) cm
= 7 cm x 13 cm
AreaBigger Rectangle = 91 cm2
4 cm
7 cm
9 cm
7 cm
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:
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
and
AreaSecond Rectangle = 7 cm x 9 cm = 63 cm2
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:
Consider the measures of the four rectangles below.
A1, A2, A3, and A4 represent the areas of each rectangle.
a) 4 and (2 + 3)
The area of each small rectangle is:
b) x and (x + 4)
A1 = 3 x 4
= 12
A3 = 1 x 4
=4
A2 = 3 x 2
=6
A4 = 1 x 2
=2
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
The sum of the areas of the four rectangles is
A1 + A2 + A3 + A4 = 12 + 6 + 4 + 2
c) a(b + c) = ab + ac
=24
To find the area of the rectangle formed, we need to
find first the measures of its sides. Thus,
A bigger rectangle can also be formed out of four
rectangles as shown below.
One side = 3 + 1
=4
Other 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.