Why Johnny Can`t Do Mathematics

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Why Johnny Can’t Do
Mathematics
DON CZARCINSKI, PH. D.
LOURDES UNIVERSITY
“Why Johnny Can't Read”
by Rudolf Flesch
 Reading is Fundamental
 to Mathematics also.
 APPLICATIONS OF LINEAR
EQUATIONS
 Word Problems
I want you to estimate the answer to this problem;
you do NOT have to find the exact answer. If you do,
however, it will be worth an extra 2 points. Joe could
paint a garage in 8 hours. Bill could paint the same
garage in 6 hours. If they work together on that
garage, how long would it take the two of them to
paint it?
 72 students
 None added the two rates
 33 made reasonable guesses
 28 averaged the two rates even though I told them
they did not have to compute an answer.
When a number is decreased by 12 the result is 15.
Find the number.
 The unknown quantity is ordinarily denoted by a




letter, x.
“A number is decreased by 12” implies the operation
of subtraction.
The phrase “the result is” gives rise to an equal sign.
So the problem can be translated into the equation:
x – 12 = 15.
John has a total of $1.60 in nickels and dimes. If he
has twice as many nickels as dimes, how many of
each does he have?
Here the word “total” gives rise to an equation
How does one arrive at this total?
Number of nickels + the number of dimes
There is also a relation between the number of
nickels and the number of dimes.
 Do nothing until you finish reading the
instructions completely
 Clap your hands
 Stand up
 If you are the first one here, announce that you are
the king of following instructions.
Do nothing until you finish reading the
instructions completely
 READ
 REACT
 RESPOND
 Find the Greatest Common Factor of Two Numbers.
 Find the Least Common Multiple of Two Numbers.
The Greatest Common Factor of two numbers
is a divisor of both numbers.
That means the GCF of two numbers must be
SMALLER THAN BOTH
The Least Common Multiple of two numbers is a
multiple of both numbers.
That means the LCM of two numbers must be
LARGER THAN BOTH
I teach two methods for finding the Greatest Common Factor
 The Prime Factors Method
 The Euclidean Algorithm
To find the Greatest Common Factor for 42 and 70
by using the Prime Factors Method
 Get Prime Factorization of the two numbers
 42 = 2 · 3 · 7
 70 = 2 · 5 · 7
 Match (and multiply) the prime factors and
 GCF(42, 70) = 14





To find the Greatest Common Factor for 42 and 70 by
using the Euclidean Algorithm
70 ÷ 42 = 1 with remainder 28
42 ÷ 28 = 1 with remainder 14
28 ÷ 14 = 2 with remainder 0
So GCF(42, 70) is the last divisor,
14
1) Use the prime factors method to find the Greatest
Common Factor of the pair of numbers 48 and 90
2) Use the Euclidean Algorithm to find the Greatest
Common Factor of the pair of numbers, 35 and 42.
3) Find the Least Common Multiple of the pair of numbers
18 and 48
What is the noun in the phrase
“Least Common Multiple?”
 Multiple
 “Common” and “Least” are modifying the noun
9) Find a factor common to 6 and 8.
Then I ask:
10) Find the Greatest Common Factor of the following
pair of numbers:
30 and 42
11) Find a common multiple of 6 and 8.
Then I ask:
12) Find the Least Common Multiple of the following
pair of numbers
30 and 42
Find the following: (2 points)
24 and 18
 * Don’t ask, because I won’t tell. Figure it out on your
own.
 16 found the Greatest Common Factor
 7 found the Least Common Multiple
 11 found both
 5 found neither
 2 realized that the only mathematical interpretation
of the word “and” is
addition, so
24 and 18 equals 42
What are the multiples of 6?
Several answer:
1, 2, 3, 6
I now ask
Find five multiples of 4.
Which is a better assessment of a student’s
knowledge of finding Least Common Multiple?
1) Find the Least Common Multiple of 4 and 6.
Or
2) One waitress has every fourth day off; her friend has
every sixth day off. They are both off today. How long
will it be before they have the same day off?
Why do you want students to know how to find the
Least Common Multiple?
You leave your campsite and travel 3 miles south.
You see a bear in the distance in front of you so you
travel 1 mile east
You then travel 3 miles north and arrive at your
campsite.
What color was the bear?
“Mathematical Ideas” 12th Edition
Miller, Hereen and Hornsby
Addison Wesley, 2011
Chapter 1 The Art of Problem Solving
The Art of Reading Mathematics Problems
Today is your first day driving a city bus. When you
leave downtown you have twenty-three passengers.
At the first stop, three people exit and five get on the
bus. At the second stop, eleven people exit and eight
people get on the bus. At the third stop, five people
exit and three get on the bus.
 23 – 3 + 5 = 25
 25 – 11 + 8 = 22
 22 – 5 + 3 = 20
 How old is the bus driver?
 READ
 REACT
 RESPOND
What is the Least Common Multiple of 4 and 6?
READ
What is the noun in the phrase
Least Common Multiple?
Answer: Multiple
REACT
 What are the multiples of 4 and of 6?
 Multiples of 4 are: 4, 8, 12, 16, 20, 24
 Multiples of 6 are: 6, 12, 18, 24, 30, 36
 What are the common multiples of 4 and 6?
 12, 24, 36, . . .
RESPOND
 What is the smallest of these Common Multiples of 4
and 6?
 12
“Zero product property of real numbers,”
That is if the product of two real numbers was 0,
then one or both of them had to be 0.
Solve the equation
(x + 2)(x – 3) = 0
Some students
a) performed the implied multiplication
b) factored the resulting quadratic
(often differently than was given)
c) set the two linear polynomials equal to 0 and solved
those equations to obtain the solutions of the given
quadratic equation.
 See . . . Do
 See
Think
Do
See × . . . Do ×
 x2 – x – 6 = 0
 x(x – 1) = 6
 x = 6, 7
 Six-product Property
Solve the equation
x2 = x
Quadratic equations have two solutions.
They may be different
They may be the same
They may not be real numbers
But there are TWO.
Here, 0 and 1
PEMDAS
Perform the indicated operations:
5 – 3[4 – (2 – 9)]
See subtraction Do subtraction
2[4 – (2 – 9)]
2[4 – (-7)]
2[11] = 22
Scorpio’s head is given by the equation
9x2 + 4y2 =36
Dirty Harry fires his .44 Magnum along the line
y=½x+3
Does Dirty Harry blow Scorpio’s head clean off?
Harry’s shot miss him completely,
that is, the line does not intersect the ellipse
Harry’s shot grazes him,
that is, the line is tangent to the ellipse
Or Harry blows Scorpio’s head clean off
the line intersected the ellipse
.44
The Toledo Blade
Wednesday June 13, 2012
Services will be held at the funeral home on
Saturday, June 23rd.
Guess when yours truly went to the funeral home.
Saturday, June 16th.
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