Algebra I
Coordination
1.
3  15  3  5  2 23
 3  15  3  5  2 5
 3  15  3  5  32
 3  5  160
 8  160
 168
This problem falls under the coordination problem since we have to use the
properties of order of operations in order to solve it. We first have to deal with the
exponents, where we add 2 and 3 together first, and then evaluate it with respect to 2.
We then have to deal with the multiplications/divisions, and finally the
additions/subtractions. If we were to do it any other way, our answers would not be right.


2. Solve 2 3a 2  5ab  b  3ba  4bb  1 when a  1 and b  2 .
3. If the perimeter of a rectangle is 10 inches, and one side is longer than the other, how
long are the sides?
Generalization
1. Given 4 x 2  7 x  15 , where a  4 , b  7 , and c  15 , factor.
4 x 2  7 x  15
Find the product ac :
4 15  60
Think of two factors that add up to 7:  5 and 12
Write 7 x as  5x and  12x :
4 x 2  5 x  12 x  15
Group the two pairs of terms:
4x
Remove common factors:
x4 x  5  34x  5
2

 5x  12 x  15
Factor out a common factor:
x  34x  5
Although this procedure is done for this specific problem, we can generalize it to
solve other problems. We can replace the numbers with variables and list the steps for a
general case.
2. Given a  Z , we have
a  a  2a
a  a  a  3a
a  a  a  a  4a
.
.
.
.
a  a  a  a  a  a  .......  a  na
If a  2 , show that the above generalization is true for n  10.
3. Derive the formula for a triangle given the fact that the formula for the area of a
rectangle is A  bh.
Reverse/Inverse
1. Find the sides of a rectangle, given the fact that the area is 36 cm, and the length is
twice the width.
A  36  lw, l  2 w
36  2 w  w
18  w 2
w3 2
l6 2
This problem is the reverse type since we are asked to solve for the length and
width given the area of the rectangle. The process reverses the step we usually take in
finding the area of the rectangle.
2. Given 2,1 and 4,5 , find the equation of the line passing through these points.
3. Given factors x  2, x  3 , find an appropriate polynomial.
Verification
1. Verify the fact that the solutions of a polynomial can be found by factoring or by
using the quadratic formula with x 2  5 x  6.
x 2  5x  6  0
x  3x  2  0
x  2, x  3
and
5  24  4  6  1 5  1

2
2
5 1
5 1
x
 3, x 
 2.
2
2
x
Algebra II
Coordination
1. The sum of Laurie’s and Peggy’s age is 34 years. In 5 years, twice Laurie’s age
increased by Peggy’s age will be 67 years. How old is each now?
Define a variable
Let n = Laurie’s age now
34  n = Peggy’s age now
n  5 = Laurie’s age in 5 years
34  n  5 = Peggy’s age in 5 years
Write an equation
2n  5  34  n  5  67
Solving
2n  10  34  n  5  67
n  49  67
n  18
Answer the problem
Laurie’s age is 18,and Peggy’s age is 16.
This problem requires coordination since we first have to understand the problem
and use the skill of translating verbal expressions into algebraic expressions to write an
equation. Next, we have to use the skill of additions/subtractions along with
multiplications/divisions along with the properties of equality to solve the equation.
Finally, we have to take the algebraic answer and translate it back to the verbal
expression.
1. Find the slope, y-intercept, and x-intercept of the line whose equation is
5 x  3 y  9.
1. Solve the following system of equations
2 x  3 y  2 z  14
4 x  2 y  z  15 .
x  y  3z  8
Generalization
1. Find the sum of numbers 1 through 1000.
First, we consider a simpler problem. Find the sum from 1 through 10.
1  10  11
2  9  11
3  8  11
4  7  11
5  6  11
Now, extend to the original problem
1  1000  1001
2  999  1001
.
.
499  502  1001
500  501  1001
Thus, the sum of 1 through 1000 is 500  1001  500,5000.
In this problem, we have to apply a simple scheme to solve a simple problem and
extend it to solve a more complicated problem. We can further generalize this scheme to
solve other similar problems.
1. Solve x 4  13x 2  36  0.
1. Consider the series form by the first n positive odd integers
1  3  5  7  ...  (2n  1)
Find the sum of the series.
Reverse/Inverse
1. Write the simplest polynomial equation with integral coefficients whose roots are
4 and 2  3i.
If 2  3i is root, so is 2  3i
x  2  3i x  2  3i x  4 = x  2  3ix  2  3ix  4


=  x  2  9i 2 x  4
2


= x 2  4 x  13 x  4
= x 3  8 x 2  29 x  52.
In this problem, instead of finding the roots of a polynomial, we are given the
roots and we need to find the equation. Thus, we are working in reverse.
2. Solve the equation 2 x  27.
3. Factor 16a 2  4.
Verification
1. Verify that 1  cot 4 x  2 csc 2 x  csc 4 x.
1  cot 4 x  2 csc 2 x  csc 4 x
1  cot x 1  cot x   2 csc x  csc x
1  1  csc x  1csc x  2 csc x  csc
2  csc xcsc x  2 csc x  csc x
2
2
2
2
2
2
2
4
2
2
4
x
4
2 csc 2 x  csc 4 x  2 csc 2 x  csc 4 x
This problem combines generalization, coordination, and inverse all together.
The coordination part combines the different concepts such as substitution, factoring,
equality, etc. The inverse part is that once we have verified the equality, we can reverse
the process and go back to our original equation. Lastly, this process works in general,
and not just for this equation.
Pre-Calculus
Coordination
1. Two airplanes leave Detroit at the same time, with the Boeing 737 flying due east
toward New York, and the DC9 flying due south toward Atlanta. The two aircraft travel
at constant speeds with the Boeing 737 flying 50 mph faster than the DC9. After flying
for 1.5 h, the airplanes are 870 mi apart. How fast are the two airplanes flying?
Detroit
New York
d=(r+50)t
d=rt
d
Atlanta
Applying the Pythagorean Theorem, we have
(distance south)2 + (distance east)2 = distance apart
r = rate of the DC9
r  50 = rate of the Boeing 737
1.5 = time travel
1.5r = distance of the southbound DC9
1.5r  50 = distance of the eastbound Boeing 737
1.5r 2  1.5r  502  870 2


2.25r 2  2.25 r 2  100r  5625  756,900
4.5r 2  225r  5625  767,900
4.5r 2  225r  751,275  0
Using the quadratic formula,
r
 225 
2252  44.5 756,275
24.5
r  384.359... or r  434.359...
r  50  434.359...
The DC9 is traveling at about 384 mph, and the Boeing 737 is traveling at about
434 mph.
This problem uses the idea of the Pythagorean Theorem to relate the distances of
the two airplanes. We also needed to use the distance = rate*time formula, along with the
ability to solve an algebraic equation. Lastly, we needed to use the quadratic formula to
find the rate at which the planes are traveling.
2. Solve x  3  2 x  4 and check for extraneous solution.
3. Find all the zeroes of f x   10 x 5  3x 2  x  6 , and identify them as rational or
irrational.
Generalization
1. Multiply 2  3i and 5  i . Write the answer in standard form. Then develop a general
rule for multiplying two complex numbers.
2  3i 5  i  = 25  2 i   3i 5  3i  i 
= 10  2i  15i  3i 2
= 10  2i  15i  3
= 13  13i
In general, we have
a  bi c  di  = ac  di   bi c  di 
= ac  adi  bci  bdi 2
= ac  adi  bci  bd
= ac  bd   ad  bci.
In this problem, we first multiply two complex numbers together with the
appropriate operations to obtain another complex number. We then generalize our
scheme so that it will work for any two complex numbers.
2. Solve 2 x 1  3.
3. Solve triangle ABC given that a  11, b  5,   20 .
Reverse/Inverse

2
.
1. Find the exact value of the expression cos 1  

 2 

2
2
   if and only if cos   
cos 1  
and 0     . Since cos is

2
2


negative,  must be in quadrant II, and  
3
is a solution. Therefore,
4

2  3

cos 1  
.

 2  4
This problem falls under the reverse/inverse category since we are asked to find
the inverse of some value of a trigonometric function. This requires us to have a
knowledge of the trigonometric function, and also the signs of the function in a specific
quadrant in order to arrive at the right answer. We could have reverse this problem and
find the value of the trigonometric function given an angle measurement.
2. Given that x  0, x   , x 

3
5
, find the corresponding equation containing
3
,x 
these four values.
3. Convert the rectangular form x  3   y  2  13 to polar form.
2
2
Verification
1. Verify that the area of triangle ABC is given by A  ss  a s  bs  c  , where
s
1
a  b  c  is the semiperimeter (half the full perimeter).
2
1
ab sin 
2
4 A  2ab sin 
A
16 A 2  4a 2 b 2 sin 2 

 4a 2 b 2 1  cos 2 

 4a 2 b 2  4a 2 b 2 cos 2 
 4a 2 b 2  2ab cos  

2
 4a 2 b 2  a 2  b 2  c 2


 2ab  a 2  b 2  c 2


 c 2  a 2  2ab  b 2



2
  2ab  a  b  c  
 a  2ab  b   c 
2
2
2
 c 2  a  b   a  b   c 2
2
2
2
2

2 2
2
 c  a  b c  a  b   a  b   c a  b   c 
 c  a  b c  a  b a  b  c a  b  c 
 2 s  2a 2 s  2b 2 s  2c 2 s 
16 A 2  16ss  a s  bs  c 
A 2  ss  a s  b s  c 
A  ss  a s  bs  c  .
This problem combines generalization, coordination, and inverse all together. The
coordination part combines the different concepts such as the Pythagorean identity, the
Law of cosines, the differences of squares, etc. The inverse part is that once we have
verified the equality, we can reverse the process and go back to our original equation.
Lastly, this process works in general, and not just for this equation.