2. Math reform plan: Hammer away at basics
Inability to handle fractions is hurting, says presidential panel
updated 1:03 p.m. ET, Thurs., March. 13, 2008
WASHINGTON - Schools could improve students' sluggish math scores by
hammering home the basics, such as addition and multiplication, and then
increasing the focus on fractions and geometry, a presidential panel
recommended Thursday.
"Difficulty with fractions (including decimals and percents) is pervasive and is a
major obstacle to further progress in mathematics, including algebra," the panel,
appointed by President Bush two years ago, said in a report.
Because success in algebra is linked to higher graduation rates and college
enrollment, the panel focused on improving areas that form the foundation for
algebra. Average U.S. math scores on a variety of tests drop around middle school,
when algebra coursework typically begins. That trend led the panel to focus on
what's happening before kids take algebra.
A major goal for students should be mastery of fractions, since that is a "severely
underdeveloped" area and one that's important to later algebra success, the
report states.
It goes on to say that other critical topics — such as whole numbers and aspects of
geometry and measurement — should be studied in a more in-depth way.
3. History
• In the 1950’s a large scale stress on national
testing began. This caused non-math
elementary teachers to focus on students
getting test answers rather than proper
methodology for solving problems.
4. History
• Elementary Teachers observed that when
dividing fractions if they multiplied the right
denominator times the left numerator and the
right numerator times the left denominator,
from right to left, and then simplify would get
a correct answer as illustrated below.
7 ÷ 3
9 5
5. History
This was the origin of “Cross Multiply”. This method
was successful in helping students get a correct
test answer. But it does not follow the rules or
laws of math.
The rules of math are in the Order of Operations
which says PEMDAS from left to right. Cross
multiply only works from right to left.
6. History
• Then teachers noticed that when adding or
subtracting fractions with prime number
denominators, they could just multiply the
denominators to get the LCD.
• Then to change the numerator, just multiply
the right denominator times the left
numerator and left denominator times the
right numerator. This correctly change the
numerators and makes a new application for
the term “Cross Multiply”.
7. History
• Middle School teachers picked up on this term
and used it to help students learn to solve
proportions, just cross multiply and divide.
3 = x
5 5
• So now “Cross Multiply” was taught to add,
subtract, divide, and proportions.
8. History
• But, the same “Cross Multiply” cannot be used
for adding, subtracting, dividing and
proportions. Each is a slight variation and
none follows the Order of Operations,
PEMDAS from left to right.
• In addition, students remember the term
“Cross Multiply” and try and use it to multiply
fractions, the one time it cannot be used.
9. History
• So, students were promoted to higher math
classes with a technique to solve fractions that
does not follow the properties of math and
therefore will not work at higher math levels.
• To address this problem, in the 1960’s “New
Math” was developed to explain WHY steps
were taken. The problem was the teacher
needed to be a “Math Person” to understand
how to teach this method. So it failed.
10. History
• Since then different methods of math
instruction have been utilized to add and
subtract fractions that include such methods
to find the LCD as “Try and Figure” what
number both denominators will go into
evenly. However, “Try and Figure” is not a
property or law of math. The student is
expected to “Conjure” a number. Conjuring a
number is mysticism not mathematics.
11. History
• Another method tried recently is to take
multiples of the denominators and find the
first one that is the same. For example the
denominators of 2 & 3
2 4 (6) 8 10 12 14 16 18
3 (6) 9 12 15 18 21 24 27
12. Problems in Teaching
Methodology
• So, now try the denominators 2a and 3b using
this method.
2a 4a 6a 8a 10a 12a 14a 16a 18a
3b 6b 9b 12b 15b 18b 21b 24b 27b
As you can see, this method does not work with
this level of problem. Imagine students trying
to use this method to find the LCD of x2
– 1
and x2
+ 2x + 1. It cannot be done!
13. How to Fix the Problem
• Any real math education fix must be
– mathematically correct
– simple for students to learn
– simple enough for non-math teachers to be able
to understand as well as teach.
• I developed such a method which is in my
book and the fraction section information
follows.
14. Problems in Teaching
Methodology
• Take the problem
3 x 2 =
4 9
Current method is
3 x 2 = 6
4 9 36
Then simplify to
3 x 2 = 6 =1
4 9 36 6
This method is multiply, then simplify. There is nothing wrong
with this method. However, it cannot be used in higher
math.
15. Problems in Teaching
Methodology
• For example, if the student is working on a
quadratic fraction like this one
(3x2
+ 5x + 2) x (4x2
– 9)
(2x2
+ 9x + 9) (2x2
-3x -3)
they definitely do NOT want to multiply, then
simplify. They want to factor and simplify,
then multiply. So factor and simplify, and
then multiply is what I use in my method
16. How is your school doing?
Ask one of your elementary teachers how they
would teach students to work the problem on
the left and note the steps they take, probably
multiply, then simplify. Then ask them to use
the same steps to work the problem on the
right.
3 x 8 (3x2
+ 5x + 2) x (4x2
– 9)
4 9 (2x2
+ 9x + 9) (2x2
-x -3)
17. Solution: Multiply & Divide Fractions
• I give my students the following steps to
multiply or divide fractions:
1. Make it a fraction. If a mixed number,
make it into an improper fraction.
2 3 ÷ 1 1 (3x2
+ 5x + 2) x (4x2
– 9)
4 8 (2x2
+ 9x + 9) (2x2
-x -3)
11 ÷ 9 Already in fraction form
4 8
18. Solution: Multiply & Divide Fractions
2. Determine if it is multiply or divide. If
multiply go to the next step. If divide
inverse, reciprocal, or “flip” the second
fraction and only the second fraction.
11 x 8 (3x2
+ 5x + 2) x (4x2
– 9)
4 9 (2x2
+ 9x + 9) (2x2
-x -3)
These are both multiplication. Nothing needs to
be done.
19. Solution: Multiply & Divide Fractions
3. Factor into prime factors the numerators and
denominators and make them one fraction.
11 x 8 (3x2
+ 5x + 2) x (4x2
– 9)
4 9 (2x2
+ 9x + 9) (2x2
- x - 3)
11•1•2•2•2 (3x+2)(x+1)(2x+3)(2x-3)
2•2•3•3 (2x+3)(x+3)(x+1)(2x-3)
22. Solution: Add & Subtract Fractions
The key here is to teach students how to
CALCULATE the LCD and then change the
numerator. Here are the steps I teach for
these two fractions. They could be addition or
subtraction.
1 + 2 3 + 2___
15 9 x2
+ 2x + 1 2x2
+x - 1
23. Solution: Add & Subtract Fractions
1. Make it a fraction.
These are both already fractions, so no
changes are needed. If mixed numbers,
make them improper fractions.
1 + 2 3 + 2___
15 9 x2
+ 2x + 1 2x2
+x - 1
24. Solution: Add & Subtract Fractions
2. Factoring is the key to all fractions. The LCD
is looking for the least common
denominator, so factor the denominators.
1 + 2 3 + 2___
15 9 x2
+ 2x + 1 2x2
+x – 1
1 + 2 3 + 2____
3•5 3•3 (x+1)(x+1) (2x – 1)(x+1)
25. Solution: Add & Subtract Fractions
2. Circle the factors that are the same in both
denominators.
1 + 2 3 + 2____
(3)•5 3•(3) ((x+1))(x+1) (2x – 1)((x+1))
26. Solution: Add & Subtract Fractions
3. Next, put a box around all the other factors.
1 + 2 3 + 2______
(3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]((x+1))
The purpose is to identify the parts that will be
used.
27. Solution: Add & Subtract Fractions
4. Then multiply the box(es) on the right times
the numerator on the left.
1[3] + 2 3[(2x-1)] + 2___
(3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]((x+1))
5. Then multiply the box(es) on the left times
the numerator on the right.
1[3] + 2[5] 3(2x-1) + 2[(x+1)]___
28. Solution: Add & Subtract Fractions
6. Find the LCD
1[3] + 2[5] 3[(2x-1)] + 2[(x+1)]__
(3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]((x+1))
The LCD is ALWAYS the circle(s) on the left times
ALL box or boxes
(3)[5][3] ((x+1))[(x+1)][(2x-1)]
30. Solution: Add & Subtract Fractions
8. Combine Like terms, if possible, then check
to see if it can be simplified.
3 ± 10 6x – 3 ± 2x ± 2
45 2x3
+ 3x2
– 1
13 or -7 8x – 1____ or 4x – 5___
45 45 2x3
+ 3x2
– 1 2x3
+ 3x2
– 1
These cannot be simplified.
31. Summary
The Common Core by its very design, is
requiring schools to systemize their math
teaching methodology from K to 12 grades.
My system is the ONLY system, of which I am
aware, that can be used and taught from k-12
WITHOUT CHANGE. In addition, it is very
visual and when color markers are included in
instruction, greatly improves student and
teacher understanding.
