Download Common Algebra Mistakes

Top Ten
Common Algebra Mistakes
Presentation by:
Valerie Beaman-Hackle
Associate Professor of Mathematics
Macon State College
http://facultyweb.maconstate.edu/valerie.beamanhackle/
For the Georgia Tutoring Association (GATA)
10th Annual Conference
Saturday, February 20, 2010
Macon State College
Macon, GA
#1 Multiplication before Division?
106  25


60  2  5
30 5
150


There are only 4 levels
in the order of
operations agreement!
On the Multiplication and
Division level, we must
work left to right.
What’s logical and easy
to do mentally may not
follow the order of
operations agreement.
Grits Eaten Daily Makes
Students Achieve.
#2 Inside vs. Outside
7  2(5  1)


7  2  ( 4)
7 8

1

There are only 4 levels in the
order of operations
agreement!
Doing what’s inside the
parentheses is on the top
level, but there is no “do
what’s outside the grouping
symbol.”
It may be helpful to write in
any understood multiplication
symbols.
What’s logical and easy to do
mentally may not follow the
order of operations
agreement.
#3 What’s negative?

5
2
 5  5  25
(5)  (5)(5)  25
2
5
2
1
1
 2 
5
25
There are 3 places a negative can
be in an exponential expression—
hanging out front, part of the base,
or part of the exponent.
 If the negative is not in
parentheses but instead hanging
out front of the base, then just
bring it down as part of your final
answer and proceed to evaluate
the exponential expression.
 The base is negative only if the
negative is inside the parentheses
and the exponent is outside the
parentheses.
 With a negative exponent, the base
is moved to the denominator and
the exponent becomes positive.
#4
Solving equations vs. simplifying expressions
4x  4  3x  2x  9
x  4  2x  9



4  x9
5  x

Keep equation
balanced!
Order of steps matter
Never combine terms
over the equals sign.
Think of it like a fence.
No jumping over the
fence
No sliding =
#5 Negative coefficients vs. negative terms
 6x  30
 6 x 30

6 6
x  5




-6 + x is different than
-6 ● x
Inverse (reverse) of
multiplication is division
Inverse (reverse) of
subtraction is addition.
May be helpful to show
solution to x – 6 = 30
along side of solution to
-6x = 30 to emphasize
the difference in the
problems
#6 Subtracting Polynomials

(3x  2)  (7 x  6)
3x  2  7x  6
 4x  4


The first set of
parentheses don’t
require any special
consideration, but the
second set does
Think of subtraction rule:
change subtraction to
addition, take opposite
of every term in ( )
Or think of distributing a
-1 to each term
#7 Squaring a binomial
3x  4

(3x  4)(3x  4)

2
9 x  12 x  12 x  16
2

9 x  24 x  16
2
3xy 
4 2
Squaring a binomial is
different that raising a
product to a power.
Rewrite in expanded
notation and FOIL.
May want to show an
example of raising a
product to a power to
emphasize difference
 (3) ( x) ( y )
2
2
4 2
 9x y
2
8
#8 Simplifying Rational Expressions
y  25
y 2  10 y  25
2



( y  5)( y  5)
( y  5)( y  5)
y 5
y5


Differentiate between
Factors and Terms
NEVER cancel TERMS.
NEVER cancel parts of
terms.
Divide out (cancel)
FACTORS!
Only MONOMIALS can
be factored using
expanded notation
#9 Solving Rational Equations
3x
1
 2x 
5
2
 3x

1
10  2 x   10 
 5

2
2
5 1
 3x 
 
10   10 2 x   10 
 5 
2
23x   102 x   51

Distribute LCD to all
terms (even those that
aren’t fractions) before
canceling denominators!

Also applies to simplifying
complex rational expressions.
6 x  20 x  5
14 x  5
x
5
14
#10 Solving Rational Equations
2x  3  4  7


2 x  3  11
2 x  3  121
2 x  118
x  59

Order of the steps
matters!
Isolate radical, THEN
square both SIDES (not
individual terms)!.
Also remember not to
combine terms inside
radical with those
outside.