12/29/15
Lessons 2: Operations on Algebraic expressions
The concept of unit
fraction is a special rule. Unit fraction is a fraction with 1 as numerator.
Unit fractions used unit fractions without repetition except 2/3.
Activity 1: Egyptian Fraction
Ancient Egyptians
used unit fraction. Now, be like an Ancient Egyptian. Give the unit fractions
in Ancient Egyptian way.
Activity 2: Anticipation Guide
There are sets of
rational algebraic expressions in the table below. Agree if the entries in
column I is equivalent to the entry in column II and check disagree if the
entries in the two columns are not equivalent.
x2 – xy x2 – y2 • x
+ y x2 – xy
|
x-1 – y -1
|
Agree
|
6y – 30 y2 + 2y
+ 1 ÷ 3y – 15 y2 + y
|
2y
y + 1
|
Agree
|
5 4x2 + 7 6x
|
15
+ 14x 12x2
|
Agree
|
a b – a – b a – b
|
a +
b b – a
|
Agree
|
a + b b – b a + b
/1b + 2a
|
a2
a + b
|
Agree
|
Activity 3: Picture Analysis
Picture construction
workers building a house. Each of them are assigned to different jobs. Analyse
what you pictured.
·
Work is accelerated
when more workers are added to help.
·
Work is
disrupted when one or more workers did not do their job.
·
The rate
is affected by efficiency, speed, and numbers.
Activity 4: MULTIPLYING RATIONAL ALGEBRAIC EXPRESSIONS
·
Express
the numerators and denominators into prime factors as possible.
·
Simplify
rational expression using laws of exponents.
·
The
product of two rational expressions is the product of the numerators divided by
the product of the denominators.
01/06/16
Exercises
Find the product of the following
rational algebraic expressions.
·
4x/u
·
aˆ2+ab/2b
·
x/x+5
·
x+1/y+1
·
a-b/a+1
Activity 5: What’s my Area?
Calculate the area of the equations
below:
·
b^2/2-b x (b-2/2b)^2
= -b/4
·
(s+3/s)^2 * (s^2/3s
+ 9s)^2 = 1/3
·
y^2 – 4/9 * (3y*1/y^2+2y)
Activity 6: Circle Process
·
Write the steps of multiplying
Rational Algebraic Expressions.
o Simplify using Law of Exponent.
o Factor the solution.
o Cancel out to simplify.
o Multiply the rest of the equation.
Activity 7: Dividing Rational Expressions
Observe the steps of dividing
Rational Algebraic Expressions
· Multiply the dividend and
the reciprocal of the divisor.
o Simplify using Law of Exponent.
o Factor the solution.
o Cancel out to simplify.
o Multiply the rest of the equation.
01/14/15
Activity 8: Missing Dimension
Answer the missing
length of each equation.
·
X^2 –
100/8 ÷ 2x^2 + 20/20
o
5x + 50/
4
·
21/3x-21
÷ x^2/35
o
2x^3 –
14x^2/245
Activity 9: Chain Reaction
Write the
instructions for dividing rational algebraic expressions.
·
Find the
reciprocal of the second fraction.
·
Simplify
the coefficient.
·
Multiply
the new coefficient.
·
Multiply
the variables.
·
Subtract
the largest variable by the smaller variables.
Activity 10: Adding and subtracting similar rational
algebraic expressions.
·
Combine
like terms in the numerator.
·
Factor
out the numerator and denominator.
01/27/16
Activity 11: Adding and subtracting dissimilar rational
algebraic expressions
In adding or
subtracting dissimilar rational expressions change the rational algebraic expressions
into similar rational algebraic expressions using the least common denominator
or LCD and proceed as in adding similar fractions.
Adding
and subtracting dissimilar rational algebraic expressions
|
Express the denominators as
prime factors.
|
Take the factors of the
denominators. When the same factor is present in more than one denominator,
take the factor with the highest exponent. The product of these factors is
the LCD.
|
Find a number equivalent to 1
that should be multiplied to the rational algebraic expressions so that the denominators
are the same with the LCD.
|
Activity 12: Flow Chart
Write the steps in
answering Adding and Subtracting Dissimilar Rational Algebraic Expressions and Adding
and Subtracting Similar Rational Algebraic Expressions.
Adding
and subtracting dissimilar rational algebraic expressions
|
Adding
and subtracting Similar rational algebraic expressions
|
|
Express the denominators as
prime factors.
|
Combine like terms in the numerator.
|
|
Take the factors of the
denominators.
|
Factor out the numerator and denominator.
|
|
When the same factor is present
in more than one denominator, take the factor with the highest exponent.
|
multiply the subtrahend by –
1 in the numerator
|
|
The product of these factors
is the LCD.
|
||
Find a number equivalent to 1
that should be multiplied to the rational algebraic expressions so that the denominators
are the same with the LCD.
|
Activity 13: What’s wrong with me?
For the solution in the first
box: The error in this item is the (6 – x) becomes (x – 6). The factor of (6
– x) is -1(x – 6).
|
For the solution in the
second box: The wrong concepts here are a – 5 (a) becomes a2 – 5a and the
numerator of subtrahend must be multiplied by -1.
So: a – 5 (a) is equal to a –
5a.
|
For the solution in the third
box: 3 must not be cancelled out. The concept of dividing out can be applied
to a common factor and not to the common variable or number in the numerator
and denominator.
|
For the solution in the
fourth box: b2 – 4b + 4 must be factored out as (b – 2) (b – 2). The concept
of factoring is essential in performing operations on rational algebraic
expressions.
|
Activity 14: Complex Rational Algebraic Expressions
Rational algebraic
expression is said to be in its simplest form when the numerator and
denominator are polynomials with no common factors other than 1. If the
numerator or denominator, or both numerator and denominator of a rational algebraic
expression is also a rational algebraic expression, it is called a complex
rational algebraic expression. To simplify the complex rational expression, it
means to transform it into simple rational expression. You need all the
concepts learned previously to simplify complex rational expressions.
02/10/16
Activity 15: Treasure Hunt
Answer the expressions below.
x^2-4/x^2 ÷ x+2/x
|
x^2 - 2
|
x^2/2 + x/3 ÷ 5x/3
|
5x/3
|
3/x + 3x + 2 ÷ x/x + 2
|
3/x^2+x
|
Activity 16:
List the steps in writing Complex Rational
Algebraic Expressions
Factor the denominator and
numerator.
|
Rewrite the expression in
division form.
|
Find the reciprocal and
multiply.
|
Cancel the like terms.
|
Simplify.
|
02/17/16
Activity 17: Reaction Guide
Observe the
equations. Agree or disagree their solutions.
(x^2-xy)/(x^2y^2)*(x+y)/(x^2-xy)
|
(y+x)/(x^2y^2)
|
Disagree
|
(6y – 30/y^2) + (2y + 1)
÷ (3y – 15) / (y^2+y)
|
2y/y+1
|
Agree
|
(5/4x^2) + (7/6x)
|
Agree
|
|
(a/(b-a)-(b/(a-b)
|
(b+a)/(b−a)
|
Disagree
|
Activity 18: Word Problem
Answer the problems
below
Two vehicles travelled (x +
4) kilometers. The first vehicle travelled for (x^2 – 16) hours while
the second travelled for 2x – 4 hours.
|
|||
Given
|
Distance
|
Time
|
Rate
|
A
|
(x + 4)
|
(x^2 – 16)
|
1/x-4
|
B
|
(x + 4)
|
2x – 4
|
x+4/2(x−2)
|
Jem Boy and Roger were asked
to fill the tank with water. Jem Boy can fill the tank in x minutes alone
while Roger is slower by 2 minutes compared to Jem Boy if working alone.
|
|||
Given
|
Time
|
Combine
two Expressions
|
|
A
|
1/1+x
|
2/3+x
|
|
B
|
1/2-x
|
||
02/24/16
·
The product of two rational
expressions is the product of the numerators divided by the product of the
denominators.
·
The quotient of two rational
algebraic expressions is the product of the dividend and the reciprocal of the
divisor.
·
In adding or subtracting similar
rational expressions, add or subtract the numerators and write the answer in
the numerator of the result over the common denominator.
·
In adding or subtracting dissimilar
rational expressions, change the rational algebraic expressions into similar
rational algebraic expressions using the least common denominator or LCD and proceed
as in adding similar fractions.
·
A rational algebraic expression is
said to be in its simplest form when the numerator and denominator are
polynomials with no common factors other than 1.
If the numerator or
denominator, or both numerator and denominator of a rational algebraic
expression is also a rational algebraic expression, it is called a complex rational
algebraic expression.
Simplifying complex rational expressions is transforming
it into a simple rational expression. You need all the concepts learned
previously to simplify complex rational expressions.
Activity
19:
Write the steps in calculating work rate.
·
Work = Time x Rate
·
Time = Rate / Work
·
Rate = Work / Time
Activity
20:
Answer the problem below
|
(x+2/x)+
(x+1/x)
|
(x+1/x)+(x)
|
(2x-1/x)+(x+2/2)
|
|
(2x+3)/x
|
(2x+1)/x
|
(3x+1)/x
|
11/04/15
Ever wondered how many people are needed to complete a job? An example
is constructing buildings? What are the bases of their wages? These questions
will be answered through Rational Algebraic Expressions.
Rational
Algebraic Expressions
Activity 1: Match it
to me
Match the following
phrases to the appropriate mathematical expressions.
·
Ratio of x and 4 added to 2
o
x/4 + 2
·
The product of the square root of 3 and y.
o
√3y
·
Sum of b and 2 less than the square of b.
o
b^2 / b + 2
·
Product of p and q divided by 3.
o
pq/3
·
One third the square of c
o
3/c^2
·
Ten times the number of y increased 6
o
10y+6
·
Cube of z decreased by 9
o
Z^2-9
Activity 2:
How
fast
Supposed you are to print
a 40 page research paper. You observed that printer (A) finished printing in 2
minutes.
·
How long for the printer to print 100 pages?
o
5 minutes
·
How many pages for printer A to print x pages?
o
Printer A: 20 pages = 100 minute
Activity 3:
KWH
Write your ideas about
the knowledge you gained during the last activities.
What I know?
|
What I want
to know
|
Rational
Algebraic Expressions appear to be like polynomials. They can calculate area
and speed of an object.
|
I want to
know more about the application in Rational Algebraic Expressions.
|
Activity 4:
Match
it to me –Revisited
List the polynomials from activity 1 and classify them as
polynomials and Expressions.
Polynomials
|
Expressions
|
A^2
+ 2a
|
X:4 + 2
|
10y + 6
|
√3y
|
Z^3
– 9
|
Pq/3
|
B^2/b
+ 2
|
3/c^2
|
Activity 5: Compare and Contrast
Write the similarities between
polynomials and expressions.
Similarities
|
They both have terms.
|
They both calculate various
things.
|
They both have variables and
numbers.
|
They are both expressions.
|
Difference
|
|
Polynomials
|
Expressions
|
They are easily factored.
|
They appear to be polynomials
but they are not.
|
Simple Structure
|
Complex Structures
|
Can calculate very easily.
|
Can calculate more things but
not easily.
|
Activity 6: My Definition Chart
Write your own initial definition
below:
Initial Definition
|
Rational
Algebraic Expressions is used to calculate and distribute numbers accurately.
|
Activity 7:
Classify
me
Classify each expression below
Rational
Algebraic Expressions
|
Expressions
|
Y+2/y-2
|
C^4/m-m
|
c/n-2
|
1 – m/-m
|
a/y^2-x^9
|
k/3k-6k
|
Activity 8:
My
Definition Chart
Put the final definition below:
Definition
|
Rational algebraic expression
is a ratio of two polynomials where the denominator is not equal to zero.
|
Activity 9: Let the
pattern answer it:
A
|
B
|
C
|
A
|
B
|
C
|
A
|
B
|
C
|
|
2*2*2*2*2
|
2^5
|
32
|
3*3*3*3*3
|
3^5
|
243
|
4*4*4*4*4
|
4^5
|
1024
|
|
2*2*2*2
|
2^4
|
16
|
3*3*3*3
|
3^4
|
81
|
4*4*4*4
|
4^4
|
256
|
|
2*2*2
|
2^3
|
8
|
3*3*3
|
3^3
|
27
|
4*4*4
|
4^3
|
61
|
|
2*2
|
2^2
|
4
|
3*3
|
3^2
|
9
|
4*4
|
4^2
|
16
|
|
2
|
2^1
|
2
|
3
|
3^1
|
3
|
4
|
4^1
|
4
|
11/11/15
Activity 10:
3 things I found out
|
Rational Algebraic Expressions involve fractions
|
Rational Algebraic Expressions don’t equal to zero, especially
the denominator
|
|
They calculate better than polynomials
|
|
2 interesting things
|
They have similar structures to non-rational algebraic
expressions
|
They are different from polynomials in terms of solving
methods
|
|
1 question I still have
|
Can Rational Algebraic Expressions solve unknown geometric
properties?
|
Activity 11: Who is
right?
Simplify n^3/n^4, there are two
solutions below. Guess which one is correct:
Solution 1
|
Solution 2
|
n^4/n^3 = n^-3+4 = n^(3)+4
= n^7
|
n^3/n^4 = n^3/ 1/n-^4 =
n^3 n^4/1
|
Quotient Law
|
Polynomial Rule
|
·
Answer: There
is no wrong answer. Both of them have the correct solution. Rational Algebraic Expressions can be solved in many different
methods.
Activity 12: Speedy
Mars
Situation: Mars finished a 15 meter
dash in 3 seconds.
·
How fast can Mars run?
o 15m / 3s = 5m/s
·
How fast can Mars run in 6
seconds?
o 30m / 6s = 5 m/s
·
How fast can Mars run in 60
meters?
o 60m/3m = 20m/s
Activity 13: My Value
Find the value of each expression
Expression
|
Value A
|
Value B
|
Solution
|
Total
|
a^2 + b^3
|
2
|
3
|
a^3 + b3 =
2^3 + 3^2 + + 4+27 = 31
|
31
|
a^-2/b^-3
|
-2
|
3
|
a^-2/b^3 =
-2^2/3^3 * 3^3/2^2
|
24/4
|
The following values show patterns of numbers to solve.
|
||||
11/18/15
Activity 16: Connect to equivalent
Match column A with
column B
Column A
|
Column B
|
5/20
|
1/4
|
8/12
|
2/3
|
4/8
|
1/4
|
5/15
|
1/3
|
6/8
|
3/4
|
Simplifying Rational Algebraic Expressions
·
9a+8b/12
o
4(a+2b)/4*3
– Factor the expression
§
a + 2b/3
·
15c^3
d^4e/12c^2d^5w
o
3*5c^2e/3*4c^2d^4dw
Exercises: Simplify the following
·
y^2+5x+4/y^2-3x-4
o
=
y+4/y-4
·
-21a^2b^2/28a^3b^3
o
= 3/4ab
·
xˆ2-9/x^2-7x+12
o
=
x+3/x-4
·
X^2-5x-14/x^2+4x+4
o
= x-7/x+2
Activity 17: Match it down
Match the following
equivalents to its expressions
- -1
- a-1/1-a
- a-8/-9+8
- 1
- 3a+1/1+3a
- a+5
- a^2+10a+25/a+5
- a^2+6a+5/a+1
Activity 18: Circle Process
Write the steps on
simplifying rational algebraic expressions.
Steps
|
Find the LCD of the expressions
|
Cancel out the expressions
using the LCD
|
Factor the expressions
|
Simplify
|
12/09/15
Activity 19: How fast 2
Analyse the table
below:
Printer
|
Rate
|
Time
|
Work
|
A
|
10 ppm
|
2 minutes
|
40 Pages
|
22.5 ppm
|
45 Pages
|
||
75 ppm
|
150 Pages
|
||
B
|
3x ppm
|
30 Pages
|
|
3x + 5 ppm
|
35 Pages
|
||
20 ppm
|
2 minutes
|
40 Pages
|
Work-related problems
usually deal with things that work at different speeds. Determining how much work
is done, is called rate.
Learned
|
Affirmed
|
Challenged
|
I learned that rate of work
could be dealt using Rational Algebraic Expressions
|
Rational Algebraic
Expressions is connected to special products and factoring.
|
I find it a little difficult
at first, to calculate work rate.
|
Activity 20: Hours and Rates
Analyse the table
below:
Printer
|
Work
|
Time
|
Rate
|
A
|
1B
|
3H
|
3ppm
|
30B
|
10H
|
||
75B
|
25H
|
||
195B
|
65H
|
||
B
|
30x^2 + 40x
|
30x^2 + 40x + 1ppm
|
|
120x + 160
|
120(x) + 160ppm
|
||
10
|
10ppm
|
||
1B
|
3x + 20
|
3(BX) + 20ppm
|
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