GAC004 AE2
Tom
GAC004 Assessment Event 2 Project 1
Factorization
Chinese name in pinyin: ChengYuhao
English name: Tom
Department, grade, class, program: ICC S1C9 AP
Word Count: 1756
ChengYuhao
GAC004 AE2
Tom
ChengYuhao
Different squares and stripes that represent different function during the process of
calculating
-Squares and stripes that represent the function of addition(+)
x
1
1
1
x
x
x
X^2
1
-Squares and stripes that represent the function of subtraction(-)
1
1
-x
GAC004 AE2
Tom
ChengYuhao
-1
1
x
a) 3x-6
Step 1: list the squares and stripes
1
1
1
-1
x
x
x
x
x
-1
-1
-1
-1
x
-1
GAC004 AE2
Tom
ChengYuhao
Step 2 Combine them into two new groups
-1
-1
-1
-1
-1
-1
x
x
x
6 squares of width 1 that each represent -1 can combined into a
groups that represent -6
3 stripes with length x and width 1 can represent 3x
Step3 Assemble these two groups
GAC004 AE2
Tom
ChengYuhao
3
Finally we will have the formation
that 3x-6=3(x-2).
the area of squares and stripes that
represent 3x-6 and 3(x-2) is equal
Hence that the two function are
equivalent.
x-2
b)3x+6
Step 1: list the squares and stripes
1
x
x
x
1
1
GAC004 AE2
Tom
ChengYuhao
1
1
1
Step 2 Combine them into two new groups
x
x
1
1
1
1
1
1
x
6 squares of width 1 that each represent 1 can combined into a groups that represent 6
GAC004 AE2
Tom
ChengYuhao
3 stripes with length x and width 1 can represent 3x .
Step3 Assemble these two groups
3
x
x
x
x+2
1
1
1
Finally we will have the formation that
1
1
1
3x-6=3(x-2).The area of squares and
stripes that represent 3x-6 and 3(x-2) is equal .Hence that the two functions are
equivalent.
b) 4x+8
Step 1: list the squares and stripes
GAC004 AE2
Tom
ChengYuhao
1
1
1
x
x
x
1
x
1
1
1
1
Step 2 Combine them into two new groups
x
x
1
x
1
x
GAC004 AE2
Tom
1
1
ChengYuhao
1
1
1
1
8 squares of width 1 that each represent 1 can combined into a groups that represent 8
4 stripes with length x and width 1 can represent 4x .
Step3 Assemble these two groups
4
x+2
x
x
1
1
x
1
1
1
x
1
1
1
GAC004 AE2
Tom
ChengYuhao
Finally we will have the formation that 4x+8=4(x+2).The area of squares and stripes
that represent 4x+8 and 4(x+2) is equal .Hence that the two functions are equivalent.
c)8x+4
Step 1: list the squares and stripes
x
x
x
1
x
1
x
1
Step 2 Combine them into two new groups
1
1
1
1
x
1
x
x
GAC004 AE2
x
Tom
x
x
x
x
x
Step3 Assemble these two groups
4
x
x
x
x
2x+1
x
x
1
1
x
x
1
1
ChengYuhao
x
x
GAC004 AE2
Tom
ChengYuhao
Finally we will have the formation that 8x+4=4(2x+1).The area of squares and stripes
that represent 8x+4and 4(2x+1) is equal .Hence that the two functions are equivalent.
e) x^2+7x
Step 1: list the squares and stripes
x^2
x
x
x
x
x
x
x
GAC004 AE2
Tom
ChengYuhao
x^2
x
x
x
x
x
x
x
Step 2 Combine them into two new groups
Step3 Assemble these two groups
x^2
x
x+7
x
x
x
x
x
x
GAC004 AE2
Tom
ChengYuhao
Finally we will have the formation that 8x+4=4(2x+1).The area of squares and stripes
that represent 8x+4and 4(2x+1) is equal .Hence that the two functions are equivalent.
f）x^2-16
Step 1: list the squares and stripes
x
x^2
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Step 2 Combine them into two new groups
GAC004 AE2
Tom
x^2
Step3 Assemble these two groups
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
x^2
x-4
ChengYuhao
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
GAC004 AE2
Tom
ChengYuhao
By covering the square that represent x^2 with 1square combined by 16 squares that
each represent -1,we will have two different rectangle one is x-4 long and x wide the
another
one is x-4 wide and x-4 long.The area of these two rectangle equals to the
formation x^2-16
Step 4 Assemble the two rectangles we got into one bigger rectangle.
x-4
x+4
As we assemble the two rectangle into one bigger rectangle , we can see that the
width of the rectangle is x-4 and the length of the rectangle is x+4. Therefore the area
of the rectangle equals to the formation (x-4)(x+4).Because the two rectangle is made
by covering the squares of length x with 16 squares of length that each represent -1,so
the area of the two rectangles equals to the area of the square of length x minus the
area of 16 squares of length 1 which also equals to the formation x^2-16,hence the
formations (x-4)(x+4) and x^2-16 are identical.
Step 5 list two groups that one represent the formation x(x+4) and the another
represent the formation 4(x+4).
GAC004 AE2
Tom
ChengYuhao
x
x+4
4
x+4
As shown above,now we have two rectangles the one is x long and x+4 wide
another one is 4 long and x+4 wide which means subtraction.
the
GAC004 AE2
Tom
ChengYuhao
Step 6 combine these two rectangle
4
x
4x
x^2
x-4
x+4
As we can see, after combine these two rectangles ,we will have a rectangle x-4 long
and x+4 wide which is equals to the rectangle we have in step4 which area equals the
formation (x+4)(x-4) and the formation x^2-16.So. in general we can come to a
conclusion that the formation x^2-16 and the formation x(x+4) - 4(x+4) are identical.
g)x^2+2x+1
Step 1: list the squares and stripes
GAC004 AE2
Tom
ChengYuhao
x^2
1
x
x
Step 2 combine the squares and stripes
GAC004 AE2
x+1
Tom
ChengYuhao
x^2
x
x
1
x+1
As the module shown above, the length and the width of the module are all
x+1.Hence the module is a perfect square of x+1 and it’s area can be represent as the
formation (x+1)(x+1) ,and the formation x^2+2x+1 is identical to the formation
(x+1)(x+1).
Step3 list two rectangle that one is x wide and x+1 long the another one is 1 wide and
x+1 long which represent subtraction.
GAC004 AE2
Tom
ChengYuhao
x
1
x+1
The two rectangles are shown above , one is combined by a square of x long and a
rectangle that is x long and 1 wide ,and the area of the rectangle can be represent as
the formation x(x+1).The another one is combined by a square of 1 long and a
rectangle that is x wide and 1 long, and the area of the rectangle can be represent as
the formation 1(x+1).
Step4 Combine these two rectangle
GAC004 AE2
x
Tom
x^2
ChengYuhao
x
x+1
1
x
1
x+1
As the module shown above ,the module is both x+1 wide and long ,Hence, it is a
square and the area of this square can be represent as (x+1)(x+1) which is identical to
the formation x^2+2x+1.Therefore the formation x(x+1)+1(x+1) is identical to the
formation x^2+2x+1.
Step 5: Check the answer by Excel
GAC004 AE2
Tom
ChengYuhao
56+8=64
As the picture shown above,the result of two formations are equal and the figures are
the same , therefore the two formations are identical.
h) x^2+3x+2
Step1 list the squares and stripes
GAC004 AE2
Tom
ChengYuhao
x^2
1
1
Step 2 Try to assemble the squares and stripes into
two groups one is the rectangles that is x+1 wide and
x long the another one is the rectangle that is x+1
wide and 2 long.
x
x
x
GAC004 AE2
Tom
x^2
ChengYuhao
x+1
x
x
As the modules shown above, we can see that the
squares and strips that represent the formation
x^2+3x+2,can be assembled into two groups one is the
x
x
x+1
rectangles that is x+1 wide and x long the another one is
the rectangle that is x+1 wide and 2 long.Hence the
formation x^2+3x+2 and the formation x(x+1)+2(x+1)
is identical.
1
1
2
GAC004 AE2
Tom
ChengYuhao
Step 5: Check the answer by Excel
16+56=72
GAC004 AE2
Tom
ChengYuhao
As the picture shown above,the result of two formations are equal and the figures are
the same , therefore the two formations are identical.
i) x2 +5x+6 Show the sections of these that demonstrate factorization by group
in pairs.
Step 1 list the squares and stripes
x^2
x
x
x
x
Step2 combine these squares and stripes into groups
x^2
x
1
1
1
1
1
1
GAC004 AE2
x
x
1
Tom
x
x
x
1
1
1
1
1
Step3 Assemble them into one group
ChengYuhao
GAC004 AE2
Tom
ChengYuhao
x+2
x+3
As the module shown above, the squares and stripes are combined into a big rectangle
which is x+3 wide and x+2 long.The area of this big rectangle can be represented into
the formation (x+3)(x+2).Because the sum of the area of those squares and stripes are
x^2+5x+6.Therefore the formation (x+2)(x+3) is identical to the formation x^2+5x+6.
Step 4 Divide the big rectangle into two
section
x+2
x+2
x
3
GAC004 AE2
Tom
ChengYuhao
The first one is a rectangle which is x wide and x+2 long, the area of this rectangle is
x(x+2)The second one is a rectangle which is 3 wide and x+2 long, the area of this
rectangle is 3(x+2)
As the two modules shown above , the formation x(x+2)+3(x+2),because it also
represent the area of these squares and stripes so the formation x(x+2)+3(x+2) is
identical to the formation x^2+5x+6.Hence, the factorization of x2 +5x+6 can be done
by grouping in pairs.
Step 5: Check the answer by Excel
GAC004 AE2
Tom
ChengYuhao
As the picture shown above,the result of two formations are equal and the figures are
the same , therefore the two formations are identical.Thus, x2 +3x+2 can be grouped
in pair by x(x+2) + 3(x+2).
j)x^2+6x-40
Step 1 list the squares and stripes
x^2
x
1
x
x
x
x
x
GAC004 AE2
Tom
Step2 combine these squares and stripes into groups
Step3 try to assemble them into one group
ChengYuhao
GAC004 AE2
Tom
ChengYuhao
As we trying to assemble them into a big rectangle ,we find out that these modules
can’t form a regular module
Therefore we can try to divide the formation 6x into 10x-4x
Step4 try to assemble them into one group another time
GAC004 AE2
Tom
ChengYuhao
x+10
x-4
As the module shown above, the squares and stripes are combined into a big rectangle
which is x+10 wide and x-4 long.The area of this big rectangle can be represented
into the formation (x+10)(x+4).Because the sum of the area of those squares and
stripes are
x^2+6x-40.Therefore the formation (x+10)(x-4) is identical to the
formation x^2+6x-40.
Step 4 Divide the big rectangle into two section
x+10
GAC004 AE2
Tom
x
x+10
ChengYuhao
GAC004 AE2
Tom
ChengYuhao
-4
The first one is a rectangle which is x wide and x+10 long, the area of this rectangle is
x(x+10)The second one is a rectangle which is -3 wide and x+10 long, the area of this
rectangle is -4(x+10)
As the two modules shown above , the formation x(x+10)-4(x+10),because it also
represent the area of these squares and stripes so the formation x(x+10)-4(x+10) is
identical to the formation x^2+6x-40.Hence, the factorization of x^2+6x-40 can be
done by grouping in pairs.