The language of Math is used incorrectly in our society by most of us. Funny, we do not tolerate incorrect English, or improper grammar, but math seems to have its own set of rules which we seem to want to butcher. The butchering extends from us not all understanding the definitions, so we make some up. When we do, and it gets extended beyond us, whoops we have a bigger problem. Lets try to correct a couple.
"Two Negatives make a Positive"
Not quite, which of the following are we talking about?
1. The opposite of a negative integer is a positive integer
2. A negative integer multiplied by a negative integer = a positive integer
3. The sum of two negative integers is a larger negative integer
The statement in red above is confusing and incorrect. The 3 statements that follow
are correct, and keep us aware of the operation we are discussing. We can use a
pattern to see how this happens.
#1. – (–5) = 5 In this statement we are saying the opposite of negative 5
= positive 5. It is also saying "the opposite of", "the opposite
of 5" which is 5. So, we have gone from 5 to –5 and back
to 5, all in one statement.
#2. (–5) (–5) = 25 So, why is this true? Lets look at a PATTERN!
5 x 5 = 25 5 x (–1) = – 5
5 x 4 = 20 5 x (–2) = –10
5 x 3 = 15 5 x (–3) = –15
5 x 2 = 10 5 x (–4) = –20
5 x 1 = 5 5 x (–5) = –25
5 x 0 = 0 Since Multiplication is commutative!
(–5) x 5 = – 25 so (–5) x (–1) = 5
(–5) x 4 = – 20 (–5) x (–2) = 10
(–5) x 3 = –15 (–5) x (–3) = 15
(–5) x 2 = –10
(–5) x 1 = –5
(–5) x 0 = 0
As we see the product of a positive integer and a negative integer is a negative
integer. However, the product of 2 negative integers is
a positive integer.
#3. (–5) + (–5) = –10 Here we are adding to the left from 0. Thus if I add
2 negative integers I must get a larger negative integer.
We, as educators, must be clear to our students about
what is taking place, and say it correctly.
You might say: The sum of 2 negative integers is a
negative integer.
Clarify for them that a problem like, –3 – 4 = ? is a sum and not a product.
they could rewrite it as:
+(–3) + (–4) = –7
"Lets cross cancel first"
I do not know how this improper comment became a part of our lexicon, but
it has. People the correct word is Factor!!!
I would start a class out by writing:
8 ÷ 12 ⋄ 4 ÷ 5 = or 8/12 ⋄ 4/5 =
I would then reduce the fraction 8/12 to 2/3. A hand would go up, and when I call
on them they say, "you can't do that its not on a cross"
My response was always the same, "gee, is the reducing of fractions
now illegal, I did not get the memo".
The phrase "cross cancel" places into the minds of students the wrong information.
We should never use phrases that distort the true meaning of what we are intending
to teach.
Because of this phrase I have had students factor numbers across an = sign in a
proportion.
Example: They would factor the 3's in ⅓ = 3/x
Then they would say, "but its on a cross isn't it."
They had heard the wrong phrase so often, for them it was correct. If we are going
to teach Math as we teach any subject, use the correct vocabulary.
Tuesday, September 27, 2011
Thursday, September 22, 2011
Mr. W. the numbers man: Factoring in Algebra
Mr. W. the numbers man: Factoring in Algebra: FACTOR THE FOLLOWING: 3X² –10X – 8 This is a common problem in in the second semester of Algebra I. Every teacher has ...
Factoring in Algebra
FACTOR THE FOLLOWING:
3X² –10X – 8
This is a common problem in in the second semester of Algebra I. Every teacher
has their favorite way of teaching it, but for many students it brings tears to their
eyes and pains to their stomach.
O.M.G., 2 negatives and the coefficient of x² is not 1!!
What is that now, two negatives make a positive?
4 x 2 = 8, but 4 + 2 = 6, 3 is ½ of 6, oh darn, what
do I do anyway!!
So what's the question, "factor"?
Are you asking these questions, because you are in a Algebra I class, or you have
been there and now your children want your help. In my last blog I talked about
problems that could go through many grades, and as the student learned more they
would be able to answer the deeper question. This is one of those examples.
Most, if not all of us would agree that without knowing our multiplication tables
Math can be a struggle. The process for solving the problem above can begin
with a game in the 3rd grade. By the second part of the year most of the students
will have learned their multiplication tables through 12. As a warm–up teachers
could start playing a game I called "I'M LOOKING FOR TWO NUMBERS".
The teacher asks:
"I'M LOOKING FOR TWO NUMBERS WHOSE
PRODUCT IS 24 AND WHOSE SUM IS 10."
What is the student learning?
What words do I need to know the definitions of?
The student is learning to "factor", break down into its divisors.
They learn and remember the following terms:
Three knew questions without much thinking on the teachers part,
but great analysis for the students. The game does not have to change
in any way, but as your students get introduced to integers we can
add knew sums. For the above problem:
When our students get to Algebra I the problem at the top of the
blog should not scare anyone. We are looking for 2 numbers
whose product is –24, and whose difference is –10.
3x² – 10x – 8 factors to
(3x + 2)(x – 4)
3X² –10X – 8
This is a common problem in in the second semester of Algebra I. Every teacher
has their favorite way of teaching it, but for many students it brings tears to their
eyes and pains to their stomach.
O.M.G., 2 negatives and the coefficient of x² is not 1!!
What is that now, two negatives make a positive?
4 x 2 = 8, but 4 + 2 = 6, 3 is ½ of 6, oh darn, what
do I do anyway!!
So what's the question, "factor"?
Are you asking these questions, because you are in a Algebra I class, or you have
been there and now your children want your help. In my last blog I talked about
problems that could go through many grades, and as the student learned more they
would be able to answer the deeper question. This is one of those examples.
Most, if not all of us would agree that without knowing our multiplication tables
Math can be a struggle. The process for solving the problem above can begin
with a game in the 3rd grade. By the second part of the year most of the students
will have learned their multiplication tables through 12. As a warm–up teachers
could start playing a game I called "I'M LOOKING FOR TWO NUMBERS".
The teacher asks:
"I'M LOOKING FOR TWO NUMBERS WHOSE
PRODUCT IS 24 AND WHOSE SUM IS 10."
What is the student learning?
What words do I need to know the definitions of?
The student is learning to "factor", break down into its divisors.
They learn and remember the following terms:
- Sum: to Add
- Difference: to subtract
- Product: to multiply
They realize there is more then one way to make 24, or 36,
or 16, or many other numbers as a product of 2 numbers . This requires
them to further analyze the question:
them to further analyze the question:
6 x 4 = 24
3 x 8 = 24
2 x 12 = 24
1 x 24 = 24
But only 6 + 4 = 10
∴ the 2 numbers are 6 and 4
Now ask the same question, and end it by saying;
- The sum is 11
- The sum is 14
- The sum is 25
Three knew questions without much thinking on the teachers part,
but great analysis for the students. The game does not have to change
in any way, but as your students get introduced to integers we can
add knew sums. For the above problem:
- The sum is –10
- The difference is 3
- The difference is 10
When our students get to Algebra I the problem at the top of the
blog should not scare anyone. We are looking for 2 numbers
whose product is –24, and whose difference is –10.
3x² – 10x – 8 factors to
(3x + 2)(x – 4)
Wednesday, September 21, 2011
Teaching a Language in Math
Yesterday I was lucky to be a substitute at Salmon Creek School. I was taking over for
teachers who were being debriefed after observing a lecture to their students on a
writing program they have adopted. This program starts in 2nd grade and
goes through 8th grade. Each teacher uses the same terms and ideas to teach the
concepts they have always taught. The program has wonderful patterns and color
coding to help the students remember their writing skills.
I thought wouldn't this be great for Math. If all teachers defined words the same way.
If all phrases that take away from learning Math could be eliminated. If there were
problems that start in the 2nd grade and go to 8th grade. There are problems like this!
Now as students gain more knowledge in Math they can come closer to answering the
entire question, or answer deeper problems on the same question.
Example: The Tortoise and the Hare were having a race.
The race was 200', 100' forward and 100' back.
The Hare takes 2 steps of 3' in the same time that
the Tortoise takes 3 hops of 2'.
a) Who reached 100' first?
b) Who won the race?
c) How far back was the loser when
the winner crossed the finnish line?
d) If the loser was in the air when
the winner crossed the finnish line,
at what height was he?
We could start this problem in 2nd grade. Here we should only look
at part "a". But how do we do this? First of all, how far is 100'?
Our 2nd grade teacher can use the problem as an introduction to
measurement. Having measuring tapes of 25' they could find a
distance of 100' by simply adding 25', 4 times.
Next they could have one person be the Tortoise and one be the
Hare. They could then see how how each would move from the
start toward the 100' mark.
0_6'__12'__18'__24'___________________________96'_____100'
Both the Hare and the Tortoise arrive at 96' together!
What happens next?
How does the way they move effect what happens?
Are they both on the ground at 100'?
The tortoise in his next two hops lands on top of the 100' mark,
but the Hare's first step takes him to 99' and he would be in the
air going past the 100' mark as the Tortoise lands on the
100' mark.
They both get to 100' at the some time.
As our students are introduced to measurement and they find out about
feet and inches. They may also ask about the marks between the inch
marks.
It is up to our 3rd to 5th grades to discuss the "b" of the problem.
They must decide who wins the race and why. While they are doing
that they can work on the understanding of ½, ¼, or ¾ of an inch.
In the 2nd grade the use of the physical measuring tape will help students
to see how being at 2¾ then adding ¾ gets you to 3½. In the following
years they will see how 2 and 6/4 creates 3½.
They will understand how to solve "c" eventually, but we can leave "d"
to much latter.
As the students move through the grades it is up to the teachers to use
the same terminology when solving this problem, so there is no confusion
in the minds of our youngsters.
teachers who were being debriefed after observing a lecture to their students on a
writing program they have adopted. This program starts in 2nd grade and
goes through 8th grade. Each teacher uses the same terms and ideas to teach the
concepts they have always taught. The program has wonderful patterns and color
coding to help the students remember their writing skills.
I thought wouldn't this be great for Math. If all teachers defined words the same way.
If all phrases that take away from learning Math could be eliminated. If there were
problems that start in the 2nd grade and go to 8th grade. There are problems like this!
Now as students gain more knowledge in Math they can come closer to answering the
entire question, or answer deeper problems on the same question.
Example: The Tortoise and the Hare were having a race.
The race was 200', 100' forward and 100' back.
The Hare takes 2 steps of 3' in the same time that
the Tortoise takes 3 hops of 2'.
a) Who reached 100' first?
b) Who won the race?
c) How far back was the loser when
the winner crossed the finnish line?
d) If the loser was in the air when
the winner crossed the finnish line,
at what height was he?
We could start this problem in 2nd grade. Here we should only look
at part "a". But how do we do this? First of all, how far is 100'?
Our 2nd grade teacher can use the problem as an introduction to
measurement. Having measuring tapes of 25' they could find a
distance of 100' by simply adding 25', 4 times.
Next they could have one person be the Tortoise and one be the
Hare. They could then see how how each would move from the
start toward the 100' mark.
0_6'__12'__18'__24'___________________________96'_____100'
Both the Hare and the Tortoise arrive at 96' together!
What happens next?
How does the way they move effect what happens?
Are they both on the ground at 100'?
The tortoise in his next two hops lands on top of the 100' mark,
but the Hare's first step takes him to 99' and he would be in the
air going past the 100' mark as the Tortoise lands on the
100' mark.
They both get to 100' at the some time.
As our students are introduced to measurement and they find out about
feet and inches. They may also ask about the marks between the inch
marks.
It is up to our 3rd to 5th grades to discuss the "b" of the problem.
They must decide who wins the race and why. While they are doing
that they can work on the understanding of ½, ¼, or ¾ of an inch.
In the 2nd grade the use of the physical measuring tape will help students
to see how being at 2¾ then adding ¾ gets you to 3½. In the following
years they will see how 2 and 6/4 creates 3½.
They will understand how to solve "c" eventually, but we can leave "d"
to much latter.
As the students move through the grades it is up to the teachers to use
the same terminology when solving this problem, so there is no confusion
in the minds of our youngsters.
Monday, September 19, 2011
Probability
In the last 10 to 15 years PROBABILITY has shown up a lot more at the elementary
school level. As a matter of fact it happens in 6th grade for most and in some places
even earlier. It did for 2 reasons:
school level. As a matter of fact it happens in 6th grade for most and in some places
even earlier. It did for 2 reasons:
- It is a great example of a fraction in life.
- It is a fine way to look at percents.
For many of us as parents, when these questions come home we may be at a loss to
help little Johnny or Mia. But we can help if we just think of that coin being tossed
in the air. There are a total of 2 possibilities: HEADS OR TAILS. However, there
is only one winner.
The Probability of of getting heads or tails is ½, or 50% or .50
So we can think of Probability as
P(EVENT OCCURRING) =
Number of favorable outcomes
Total number of outcomes
Number of favorable outcomes
Total number of outcomes
We can write probabilities as a
- FRACTION
- DECIMAL
- PERCENT
Example: You have a box containing 10 marbles. 3 are RED,
4 are YELLOW, 2 are GREEN, and one in WHITE.
They are all the same size and feel the same. If you
reach in and randomly draw one out of the box, what
is the Probability that it will be YELLOW?
4 Favorable and 10 Total
∴ 4/10 = 2/5 You have 2 chances out of 5.
Example: Given the same box and marbles you have above, what is
the percentage chance of picking a GREEN marble?
2 Favorable and 10 Total
∴ 2/10 = 1/5 1/5 = .20 = 20%
You have a 20% chance of drawing out a GREEN marble.
What I have just discussed with you is called THEORETICAL PROBABILITY
because it is based on knowing all of the outcomes that are equally likely.
This needs to be separated from EXPERIMENTAL PROBABILITY which is
based on repeated trials of an experiment.
EXPERIMENTAL PROBABILITY OF AN EVENT
4 are YELLOW, 2 are GREEN, and one in WHITE.
They are all the same size and feel the same. If you
reach in and randomly draw one out of the box, what
is the Probability that it will be YELLOW?
4 Favorable and 10 Total
∴ 4/10 = 2/5 You have 2 chances out of 5.
Example: Given the same box and marbles you have above, what is
the percentage chance of picking a GREEN marble?
2 Favorable and 10 Total
∴ 2/10 = 1/5 1/5 = .20 = 20%
You have a 20% chance of drawing out a GREEN marble.
What I have just discussed with you is called THEORETICAL PROBABILITY
because it is based on knowing all of the outcomes that are equally likely.
This needs to be separated from EXPERIMENTAL PROBABILITY which is
based on repeated trials of an experiment.
EXPERIMENTAL PROBABILITY OF AN EVENT
= NUMBER OF SUCCESSES
NUMBER OF TRIALS
Example: You roll a six sided die. The chance that any one side
comes up is equally likely. You roll the die 40 times
and note all the results. What is the Experimental
Probability of getting an odd number on any role?
Here are your results:
RESULT 1 2 3 4 5 6
TIMES ROLED 6 7 7 5 10 5
23 out of 40 are odd! 57.5% are odd
Our P(odd) is 57.5% (Experimental)
20/40 or 50% are odd is the
Theoretical Probability.
This is only beginning of a long discussion. In my next Blog I will discuss
the ideas of independent and dependent events.
NUMBER OF TRIALS
Example: You roll a six sided die. The chance that any one side
comes up is equally likely. You roll the die 40 times
and note all the results. What is the Experimental
Probability of getting an odd number on any role?
Here are your results:
RESULT 1 2 3 4 5 6
TIMES ROLED 6 7 7 5 10 5
23 out of 40 are odd! 57.5% are odd
Our P(odd) is 57.5% (Experimental)
20/40 or 50% are odd is the
Theoretical Probability.
This is only beginning of a long discussion. In my next Blog I will discuss
the ideas of independent and dependent events.
Friday, September 16, 2011
Mr. W. the numbers man: The 3 M's of Statistics
Mr. W. the numbers man: The 3 M's of Statistics: If you are 50 or older and did not take Statistics the next 3 words were not part of your youth, and certainly not part of your every day d...
The 3 M's of Statistics
If you are 50 or older and did not take Statistics the next 3 words were not
part of your youth, and certainly not part of your every day discussions.
part of your youth, and certainly not part of your every day discussions.
- MEAN
- MEDIAN
- MODE
The first word described the "bully" in your class that just was not nice.
The second was the line you crossed when you passed someone on the road.
The last had to do with a musical scale if you were so inclined.
But not TODAY!
Open any 5th grade math book and you find these words with slightly
different definitions. And this is because we are now a part of the
DATA GENERATION!
Our children as ourselves are bombarded with data and statistics, day-in
and day-out. We are even supposed to understand what standard deviation
is all about. I don't know about you, but that was not in my 5th grade
curriculum.
curriculum.
However, this does not mean Johnny will not be coming home and asking
us, "what's a mode, is that what you get with apple pie?"
NO, not this time.
By giving you a set of data, we can try to understand.
The following is a list of the height in closest inches of the 11 boy in
your class this year.
50, 50, 52, 53, 54, 54, 54, 57, 59, 59, 60
MEAN This is the average of the numbers. So all we have to do
is add them up and then divide by 11.
The sum is 602/11 = 54.73 or ≈ 54¾ inches
Thus: Mean ➜Average
MEDIAN This is the man in the middle. Like the line in the two–lane
road that it is in the middle. We are talking about the data piece in the middle.
So if we count from both ends toward the middle we land on the 6th number, 54.
The man in the middle is 54
This is because we have an odd number of data pieces. If we had a even
number, when we got to the middle we would have our fingers on two
different numbers. We would then take the average of those 2 pieces.
Thus: Mean➜Man in the Middle
MODE This not the scoop of ice cream with the apple pie. No, it is the
pice of data that happens the most often. Look at the list above. There are
three boys that are 54 inches. Therefore it is the mode of this data. A list
of data like the one above can have more than one mode, or it may not have
any if all the pieces are different.
Thus: Mode➜The one that happens most often
NO, not this time.
By giving you a set of data, we can try to understand.
The following is a list of the height in closest inches of the 11 boy in
your class this year.
50, 50, 52, 53, 54, 54, 54, 57, 59, 59, 60
MEAN This is the average of the numbers. So all we have to do
is add them up and then divide by 11.
The sum is 602/11 = 54.73 or ≈ 54¾ inches
Thus: Mean ➜Average
MEDIAN This is the man in the middle. Like the line in the two–lane
road that it is in the middle. We are talking about the data piece in the middle.
So if we count from both ends toward the middle we land on the 6th number, 54.
The man in the middle is 54
This is because we have an odd number of data pieces. If we had a even
number, when we got to the middle we would have our fingers on two
different numbers. We would then take the average of those 2 pieces.
Thus: Mean➜Man in the Middle
MODE This not the scoop of ice cream with the apple pie. No, it is the
pice of data that happens the most often. Look at the list above. There are
three boys that are 54 inches. Therefore it is the mode of this data. A list
of data like the one above can have more than one mode, or it may not have
any if all the pieces are different.
Thus: Mode➜The one that happens most often
Mr. W. the numbers man: Computer or English Language/mixed numbers or impr...
Mr. W. the numbers man: Computer or English Language/mixed numbers or impr...: The last two days I have spent trying to put higher Math symbols into my blog easily. How difficult could this be I thought. I found a fre...
Computer or English Language/mixed numbers or improper fractions
The last two days I have spent trying to put higher Math symbols into my blog easily.
How difficult could this be I thought. I found a free down lode called Math Type.
I did not take me much time to learn how to create the equations that I wanted. Then
the problems began.
How do I get these equations to my blog?
Do I just move them to my desk top, open my
blog, drag and drop?
If only it was that easy. Here is where I ran into HTML, SEO, TeXLook.eqp,
Symbol palette, Template palette, Garamond and Times News Roman. Although,
a couple of these were familiar I felt like I was swimming in a an ocean I had
never learned about.
It is up to us to understand the new technologies, but where do I start. It was
recommended to me to get a book called "Blogging for Dummies". For as much
as Math is a language that is easy for me to understand, this language of
computers still gives me a start. I have great friends that are geeks when it comes
to this stuff. Soon it will be second nature, but at the moment, like many of
you, I struggle.
But this I can help you with
MIXED NUMBERS TO IMPROPER FRACTIONS
One of the first concepts we learn in fractions, is changing a mixed number into
improper fraction.
Example: 2¾ is read as 2 and ¾ = 2 + ¾
The Process is Simple 4 x 2 = 8 and 8 + 3 = 11 therefore 11/4
But why!
Here is why. We need to remember that 1 = 4/4, so 2 = 8/4 or 8 ÷ 4 = 2
Now 8/4 + 3/4 = 11/4
Lets try another: 6⅔ = what mixed number?
3 x 6 = 18 and 18 + 2 = 20, therefore 20/3
What I cannot stress enough is reading the number; 6⅔ = 6 + ⅔
The reason I mention the last item in bold is this. When the "+" is placed
between the 6 and the ⅔ students seem to think this will change the
process. But since both of the statements are equal then the process is
the same and the result is the same.
As a matter of fact, when this concept is first introduced, it should be
shown with and without the "+" sign. My reason is simple. Soon the
student will be faced with the following:
6 – ⅔ = ?
All we have to do is 3 x 6 = 18 and 18 – 2 = 16/3
Instead of adding 2 we subtract 2: The sign tells us to do.
Now we look forward to Algebra: x + ½ combine
We know that the x and the ½ cannot be added to make a larger number,
because we do not know what x is. However, that does not stop us from
rewriting the sum under one denominator using the same simple process
we have used for years.
2 ⋄ x = 2x + 1 = 2x + 1/2
The phrase 2x + 1 is now over one denominator, 2.
Now if you were given different values of x, it would change the value of
the phrase
x's value phrases value
0 ½
1 3/2
7 15/2
–3 –5/2
Example: ⅓ – y needs to be combined into one fraction.
3 ⋄ (–y) = –3y and –3y + 1/3
Same process, we now have a mixed number, that in Algebra helps us,
and we are not miles out of our comfort zone. In all cases it is important
that we read what it says, do not stare at it and hope to understand.
How difficult could this be I thought. I found a free down lode called Math Type.
I did not take me much time to learn how to create the equations that I wanted. Then
the problems began.
How do I get these equations to my blog?
Do I just move them to my desk top, open my
blog, drag and drop?
If only it was that easy. Here is where I ran into HTML, SEO, TeXLook.eqp,
Symbol palette, Template palette, Garamond and Times News Roman. Although,
a couple of these were familiar I felt like I was swimming in a an ocean I had
never learned about.
It is up to us to understand the new technologies, but where do I start. It was
recommended to me to get a book called "Blogging for Dummies". For as much
as Math is a language that is easy for me to understand, this language of
computers still gives me a start. I have great friends that are geeks when it comes
to this stuff. Soon it will be second nature, but at the moment, like many of
you, I struggle.
But this I can help you with
MIXED NUMBERS TO IMPROPER FRACTIONS
One of the first concepts we learn in fractions, is changing a mixed number into
improper fraction.
Example: 2¾ is read as 2 and ¾ = 2 + ¾
The Process is Simple 4 x 2 = 8 and 8 + 3 = 11 therefore 11/4
But why!
Here is why. We need to remember that 1 = 4/4, so 2 = 8/4 or 8 ÷ 4 = 2
Now 8/4 + 3/4 = 11/4
Lets try another: 6⅔ = what mixed number?
3 x 6 = 18 and 18 + 2 = 20, therefore 20/3
What I cannot stress enough is reading the number; 6⅔ = 6 + ⅔
The reason I mention the last item in bold is this. When the "+" is placed
between the 6 and the ⅔ students seem to think this will change the
process. But since both of the statements are equal then the process is
the same and the result is the same.
As a matter of fact, when this concept is first introduced, it should be
shown with and without the "+" sign. My reason is simple. Soon the
student will be faced with the following:
6 – ⅔ = ?
All we have to do is 3 x 6 = 18 and 18 – 2 = 16/3
Instead of adding 2 we subtract 2: The sign tells us to do.
Now we look forward to Algebra: x + ½ combine
We know that the x and the ½ cannot be added to make a larger number,
because we do not know what x is. However, that does not stop us from
rewriting the sum under one denominator using the same simple process
we have used for years.
2 ⋄ x = 2x + 1 = 2x + 1/2
The phrase 2x + 1 is now over one denominator, 2.
Now if you were given different values of x, it would change the value of
the phrase
x's value phrases value
0 ½
1 3/2
7 15/2
–3 –5/2
Example: ⅓ – y needs to be combined into one fraction.
3 ⋄ (–y) = –3y and –3y + 1/3
Same process, we now have a mixed number, that in Algebra helps us,
and we are not miles out of our comfort zone. In all cases it is important
that we read what it says, do not stare at it and hope to understand.
Wednesday, September 14, 2011
Friday, September 9, 2011
The 3 Levels of Math
I wish to extend my thanks to my friend Mike for what I am about to write. He and I
spend hours discussing math, physics, dark matter, string theory and alike. We both
have similar interests that most don't. We both even happen to be avid stamp
collectors. Mike once said to me, "I believe there are 3 levels of Math, 1 – all the
symbols we will ever learn, 2 – the equal sign, and 3 – well the rest are just recipes."
I thought for a moment, and as a retired Math teacher I did spend most of my days
surrounded by symbols, explaining what they meant to my students. I also had to
agree that there is no symbol more important than the equal sign, because it allows
things to be solved in equations and alike. Finally, we have our formulas
that solve problems in chemistry, economics, statistics, physics, etc..
Now lets take a close look at what I mean.
level 1 SYMBOLS
+ Here are the symbols we learn, add, subtract, multiply,
— and divide, in arithmetic. For a lot of us this is quite
× enough. But look at the key board I am touching.
÷ Every key has a symbol and by holding down the option key WOO!!!
=
These 5 only scratch the surface of all the symbols we need to learn and understand.
And some symbols mean the same thing, while others only hold position and do not
ask us to do anything. It is up to us as teachers of Math or any subject to first define
clearly what a symbol means. I believe that many of our students, when exposed to
a new symbol, become confused and often do not even ask about it.
× ⋄ ( ) ab All of these symbols can be used for the operation – multiplication.
The x is the times sign, but so is the raised dot. Two sets of items
in parenthesis (5+4)(6–3), or two variables with nothing between
them: rt = d. Rate times Time = Distance
a₁, a₂, a₃, a₄, . . . Now this is just a list. The numbers 1 - 4 are saying
these are the first four items in the pattern. They do
not tell us anything about the size of the items in the
pattern. The three dots at the end tells us that the pattern
continues as it did.
Example: 4, 7, 11, 14, 18, 21, 25, 28, 32, . . . add 3, add 4, add 3, add 4, . . .
< Less than, 5 < 8. Just when we thought we knew when things were the
> the same, now we have to know when they are smaller than something
≤ else. Then we have less than or equal to, but not greater than. Or we
≥ have greater than, which is not equal to. For heavens sake, which one
≠ do I use.
Sometimes symbols are a lot more than what they seem to be. It is as though those Math
people are just lazy. Why don't they just tell us with out the symbols? Oh well!
{ } {x: x ≥ 0} The set of numbers called x, such that all values of x are
greater than or equal to zero.
⎢ ⎥ ⎢ – 5 ⎥ the absolute value of –5. The absolute value is simply its distance
away from zero. The value here is 5.
Then there is GEOMETRY
∠ These are just a few of my favorites that show up in 5th grade. By this time, only
⊥ age 10 or 11, have been bombarded by hundreds of symbols. And we have to
≅ understand them all, and how they are used. Our first stands for angle, any angle
∼ in any figure; ∠A of triangle ABC. Our second means perpendicular, that is
∕∕ two lines that intersect at 90º (more symbols) ∠s. Our 3rd symbol ≅, means
congruent or the same size and shape. Our 4th ∼ means similar to, or same
shape, but not the same size. Our last symbol stands for parallel lines. Lines
in the same plane that do not intersect.
∩ This is the symbol for intersect.
I could go on for pages more, but you get the idea, and I have just started.
Can it be any wonder why learning is difficult if we do not repeat a symbol over and
over. Then we must build patterns for these symbols so that they make sense.
Level 2 = THE EQUAL SIGN
No sign is more important. It allows us to solve. It is the glue that holds all other
ides together. From Kindergarten on students should be writing boldly
5 + 4 = 9
They must write it down, not just say it or read it.
Where would we be today without E = mc², or V = ⅓Î’h, or C = Ï€d.
These formulas could not exist without the = sign. So with this sign we are able to
understand when things have the same value. And when they do we can sometimes
substitute one statement in for another to get a new idea.
Certainly, one of the more difficult concepts starts in Algebra, when so many
statements = 0. 5x³ – 4x² + 3x – 30 = 0 How could so much be nothing,
but that is what it says, and there must be a truth to it. It simply means if
we place the correct number in for the variable, if one exists, the phrase on the
left will have a value of 0. In this case it is 2. Far more difficult than 5 + 4 = 9,
but understanding the power of the = sign, and realizing why we use it will
make us more comfortable in our Math classes.
level 3 THE RECIPES OR FORMULAS
Above you saw some formulas that we all are aware of.
Remember this one:
a² + b² = c²
There are not to many adults, who can't tell you this one. If you ask them what
it is for you may not get the right answer, but a² + b² = c² they know for sure. Its
the Pythagorean Theorem, undoubtably the most elegant theorem of Math.
x = – b ± √ b² – 4ac ÷ 2a
The quadratic formula, the grandaddy of all Algebra I formulas. These are just 2
of so many of them that you learn in school. Next comes the confidence of working
with them. Now we have to place numbers in for the variables and get solutions.
spend hours discussing math, physics, dark matter, string theory and alike. We both
have similar interests that most don't. We both even happen to be avid stamp
collectors. Mike once said to me, "I believe there are 3 levels of Math, 1 – all the
symbols we will ever learn, 2 – the equal sign, and 3 – well the rest are just recipes."
I thought for a moment, and as a retired Math teacher I did spend most of my days
surrounded by symbols, explaining what they meant to my students. I also had to
agree that there is no symbol more important than the equal sign, because it allows
things to be solved in equations and alike. Finally, we have our formulas
that solve problems in chemistry, economics, statistics, physics, etc..
Now lets take a close look at what I mean.
level 1 SYMBOLS
+ Here are the symbols we learn, add, subtract, multiply,
— and divide, in arithmetic. For a lot of us this is quite
× enough. But look at the key board I am touching.
÷ Every key has a symbol and by holding down the option key WOO!!!
=
These 5 only scratch the surface of all the symbols we need to learn and understand.
And some symbols mean the same thing, while others only hold position and do not
ask us to do anything. It is up to us as teachers of Math or any subject to first define
clearly what a symbol means. I believe that many of our students, when exposed to
a new symbol, become confused and often do not even ask about it.
× ⋄ ( ) ab All of these symbols can be used for the operation – multiplication.
The x is the times sign, but so is the raised dot. Two sets of items
in parenthesis (5+4)(6–3), or two variables with nothing between
them: rt = d. Rate times Time = Distance
a₁, a₂, a₃, a₄, . . . Now this is just a list. The numbers 1 - 4 are saying
these are the first four items in the pattern. They do
not tell us anything about the size of the items in the
pattern. The three dots at the end tells us that the pattern
continues as it did.
Example: 4, 7, 11, 14, 18, 21, 25, 28, 32, . . . add 3, add 4, add 3, add 4, . . .
< Less than, 5 < 8. Just when we thought we knew when things were the
> the same, now we have to know when they are smaller than something
≤ else. Then we have less than or equal to, but not greater than. Or we
≥ have greater than, which is not equal to. For heavens sake, which one
≠ do I use.
Sometimes symbols are a lot more than what they seem to be. It is as though those Math
people are just lazy. Why don't they just tell us with out the symbols? Oh well!
{ } {x: x ≥ 0} The set of numbers called x, such that all values of x are
greater than or equal to zero.
⎢ ⎥ ⎢ – 5 ⎥ the absolute value of –5. The absolute value is simply its distance
away from zero. The value here is 5.
Then there is GEOMETRY
∠ These are just a few of my favorites that show up in 5th grade. By this time, only
⊥ age 10 or 11, have been bombarded by hundreds of symbols. And we have to
≅ understand them all, and how they are used. Our first stands for angle, any angle
∼ in any figure; ∠A of triangle ABC. Our second means perpendicular, that is
∕∕ two lines that intersect at 90º (more symbols) ∠s. Our 3rd symbol ≅, means
congruent or the same size and shape. Our 4th ∼ means similar to, or same
shape, but not the same size. Our last symbol stands for parallel lines. Lines
in the same plane that do not intersect.
∩ This is the symbol for intersect.
I could go on for pages more, but you get the idea, and I have just started.
Can it be any wonder why learning is difficult if we do not repeat a symbol over and
over. Then we must build patterns for these symbols so that they make sense.
Level 2 = THE EQUAL SIGN
No sign is more important. It allows us to solve. It is the glue that holds all other
ides together. From Kindergarten on students should be writing boldly
5 + 4 = 9
They must write it down, not just say it or read it.
Where would we be today without E = mc², or V = ⅓Î’h, or C = Ï€d.
These formulas could not exist without the = sign. So with this sign we are able to
understand when things have the same value. And when they do we can sometimes
substitute one statement in for another to get a new idea.
Certainly, one of the more difficult concepts starts in Algebra, when so many
statements = 0. 5x³ – 4x² + 3x – 30 = 0 How could so much be nothing,
but that is what it says, and there must be a truth to it. It simply means if
we place the correct number in for the variable, if one exists, the phrase on the
left will have a value of 0. In this case it is 2. Far more difficult than 5 + 4 = 9,
but understanding the power of the = sign, and realizing why we use it will
make us more comfortable in our Math classes.
level 3 THE RECIPES OR FORMULAS
Above you saw some formulas that we all are aware of.
Remember this one:
a² + b² = c²
There are not to many adults, who can't tell you this one. If you ask them what
it is for you may not get the right answer, but a² + b² = c² they know for sure. Its
the Pythagorean Theorem, undoubtably the most elegant theorem of Math.
x = – b ± √ b² – 4ac ÷ 2a
The quadratic formula, the grandaddy of all Algebra I formulas. These are just 2
of so many of them that you learn in school. Next comes the confidence of working
with them. Now we have to place numbers in for the variables and get solutions.
Wednesday, September 7, 2011
The Power of Formulas
If you are over 45, you can remember problem like this from Algebra I.
Dave can do a job in 5 hours and his
friend Moreno can do it in 4 hours.
How long would it take if they did
it together?
Not only do I remember doing them, but when I began my career as a teacher in
1969 the same text was being used, and I found myself sending my class home
with those same problems. However, realizing they were all the same, except
that they might have a twist, Dave got to work one hour late, or something.
I thought there must be a better way, and that is what I asked my class.
This is why. Think of a circle. Do we find the Area of one circle differently
then in another. NO! A circle is a circle.
We have a FORMULA: RT = 1
This means the RATE/HOUR x TIME WORKED = 1
1 MEANS THE ENTIRE JOB
DAVE above does 1/5 of the job per hour because he does the job in 5 hours.
MORENO does ¼ of the job per hour because he does the job in 4 hours.
1/5 x 5 = 1 and ¼ x 4 = 1
Now we are simply combining their work. R₁T + R₂T =1
The 1 and 2 in the formula are subscripts, and are just used as counters
As I stood before my classes that 1st year I said,
"Lets do it in general, using variables"
(1/a) (x) + (1/b)(x) = 1
Here a and b stand for the times each spent working alone, and x is the
combined timed. Now we solve.
x/a + x/b = 1 Our common denominator is ab
(bx + ax)/ab = 1/1 Now use the rules of proportions.
(bx + ax) = ab Factor x out of both terms on left
x (a + b) = ab Divide by sum (a + b)
x = ab/(a + b)
Now think of our friends Dave and Moreno
Dave = 5 hours
Moreno = 4 hours
Thus if we use our new formula: x = (5)(4)/(5 + 4)
x = 20/9 hours or 2 and 2/9 hours
With the same Algebraic tools we can create a formula for one of them
being 1 hour late. With these formulas we can solve a difficult Algebra I
problems as easily as we use A = bh for a Rectangle.
Add this formula to you list. But I suggest deriving it in an Algebra I
class can inspire students to feel better about story problems, and look for
new strategies to solve them.
Dave can do a job in 5 hours and his
friend Moreno can do it in 4 hours.
How long would it take if they did
it together?
Not only do I remember doing them, but when I began my career as a teacher in
1969 the same text was being used, and I found myself sending my class home
with those same problems. However, realizing they were all the same, except
that they might have a twist, Dave got to work one hour late, or something.
I thought there must be a better way, and that is what I asked my class.
This is why. Think of a circle. Do we find the Area of one circle differently
then in another. NO! A circle is a circle.
We have a FORMULA: RT = 1
This means the RATE/HOUR x TIME WORKED = 1
1 MEANS THE ENTIRE JOB
DAVE above does 1/5 of the job per hour because he does the job in 5 hours.
MORENO does ¼ of the job per hour because he does the job in 4 hours.
1/5 x 5 = 1 and ¼ x 4 = 1
Now we are simply combining their work. R₁T + R₂T =1
The 1 and 2 in the formula are subscripts, and are just used as counters
As I stood before my classes that 1st year I said,
"Lets do it in general, using variables"
(1/a) (x) + (1/b)(x) = 1
Here a and b stand for the times each spent working alone, and x is the
combined timed. Now we solve.
x/a + x/b = 1 Our common denominator is ab
(bx + ax)/ab = 1/1 Now use the rules of proportions.
(bx + ax) = ab Factor x out of both terms on left
x (a + b) = ab Divide by sum (a + b)
x = ab/(a + b)
Now think of our friends Dave and Moreno
Dave = 5 hours
Moreno = 4 hours
Thus if we use our new formula: x = (5)(4)/(5 + 4)
x = 20/9 hours or 2 and 2/9 hours
With the same Algebraic tools we can create a formula for one of them
being 1 hour late. With these formulas we can solve a difficult Algebra I
problems as easily as we use A = bh for a Rectangle.
Add this formula to you list. But I suggest deriving it in an Algebra I
class can inspire students to feel better about story problems, and look for
new strategies to solve them.
Tuesday, September 6, 2011
Size counts
In my tenure as a high school educator I had classes of 17, and classes of 42. The
room was the same size and I had an assistant with the class of , you guessed it,
the one with 17. This was due to the fact that 1 of the children needed extra help.
I was lucky that the 42 sized class was an honors class, so discipline problems
were not there, but helping everybody was impossible.
I tell you this story because without investment in our education, all of our classes
may have 42 in them, and they are not all honors classes. When you increase the
numbers in a class dramatically it changes the ability of the educator to teach and
the child to learn.
We can look at some numbers that will be helpful.
Recently Texas reduced their education budget enough to put 40,000
teachers out of work.
Lets say that there were 500,000 teachers, all were Elementary, and each
had 25 students.
That would mean we had 12,500,000 students to begin with.
If we lost 40,000 that would leave 460,000 still working
Now if we divide 12,5000,000 by 460,000 ≈ 27.77
Gee that's only 3 more students, not that much.
Hold on there old chopping block!
What if part of those teachers were high school, and second lets say there were
1200 students at the school taking Math and 10 teachers.
1200 ÷ 10 = 120 students per teacher
five classes per teacher: 120 ÷ 5 = 24
What if 3 of those teachers were part of the 40000? We still have 1200 students.
1200 ÷ 7 ≈ 171.42 students per teacher
five classes per teacher: 171.42 ÷ 5 ≈ 34.28
increase of 10 students per class
As you see the change can be amazing. And not all of your classes are =. So one
might have 28, while another has 40. When classes get this big it is the child that
is the big loser. All it takes is one or two students in a class that can destroy its
entire environment. Until 100% of our parents send us students who care and want
only to learn, not screw around, discipline will be part of the management of a class.
However, any class above 30 in High School, or 25 in Middle or Elementary School
is too much. We as teachers need to be able to to reach all of them, not only a few.
Also, at the same time, the no child left behind program is turning
educators into test teachers. This is another comment that I do not wish to discuss
at this time.
Educators want to inspire, but they can't if there is not enough time to reach the
students they have, because their classes are so big. If we believe in a great
education for our children, then we are going to have to invest more time and
money into it. With our investment we can use the modern technologies that
you and I use every day, and that our students are learning and working with
outside of school to help their school experience more meaningful.
room was the same size and I had an assistant with the class of , you guessed it,
the one with 17. This was due to the fact that 1 of the children needed extra help.
I was lucky that the 42 sized class was an honors class, so discipline problems
were not there, but helping everybody was impossible.
I tell you this story because without investment in our education, all of our classes
may have 42 in them, and they are not all honors classes. When you increase the
numbers in a class dramatically it changes the ability of the educator to teach and
the child to learn.
We can look at some numbers that will be helpful.
Recently Texas reduced their education budget enough to put 40,000
teachers out of work.
Lets say that there were 500,000 teachers, all were Elementary, and each
had 25 students.
That would mean we had 12,500,000 students to begin with.
If we lost 40,000 that would leave 460,000 still working
Now if we divide 12,5000,000 by 460,000 ≈ 27.77
Gee that's only 3 more students, not that much.
Hold on there old chopping block!
What if part of those teachers were high school, and second lets say there were
1200 students at the school taking Math and 10 teachers.
1200 ÷ 10 = 120 students per teacher
five classes per teacher: 120 ÷ 5 = 24
What if 3 of those teachers were part of the 40000? We still have 1200 students.
1200 ÷ 7 ≈ 171.42 students per teacher
five classes per teacher: 171.42 ÷ 5 ≈ 34.28
increase of 10 students per class
As you see the change can be amazing. And not all of your classes are =. So one
might have 28, while another has 40. When classes get this big it is the child that
is the big loser. All it takes is one or two students in a class that can destroy its
entire environment. Until 100% of our parents send us students who care and want
only to learn, not screw around, discipline will be part of the management of a class.
However, any class above 30 in High School, or 25 in Middle or Elementary School
is too much. We as teachers need to be able to to reach all of them, not only a few.
Also, at the same time, the no child left behind program is turning
educators into test teachers. This is another comment that I do not wish to discuss
at this time.
Educators want to inspire, but they can't if there is not enough time to reach the
students they have, because their classes are so big. If we believe in a great
education for our children, then we are going to have to invest more time and
money into it. With our investment we can use the modern technologies that
you and I use every day, and that our students are learning and working with
outside of school to help their school experience more meaningful.
Monday, September 5, 2011
Mr. W. the numbers man: What is wrong with an IMPROPER FRACTON
Mr. W. the numbers man: What is wrong with an IMPROPER FRACTON: Consider the following problem: ¾(30 ÷ 6) = ? Knowing ORDER RULES, we do what is in the PAREN...
What is wrong with an IMPROPER FRACTON
Consider the following problem:
¾(30 ÷ 6) = ?
Knowing ORDER RULES, we do what is in the PARENTHESIS first.
¾(5) = ?
Now ¾ x 5 = 15/4
15/4
This is the point at which I begin having disagreements with many of my
educational partners, (I endeavor to be politically correct at times).
Some of my entering freshman would tell me that you have to change it
to a smaller number, or mixed number, because the "elephant is on the mouse".
Some would say it need s to be a decimal, improper fractions are not allowed.
I would tell them improper fractions are my friends.
So, why the controversy. To me there are 2 things to consider
¾(30 ÷ 6) = ?
Knowing ORDER RULES, we do what is in the PARENTHESIS first.
¾(5) = ?
Now ¾ x 5 = 15/4
15/4
This is the point at which I begin having disagreements with many of my
educational partners, (I endeavor to be politically correct at times).
Some of my entering freshman would tell me that you have to change it
to a smaller number, or mixed number, because the "elephant is on the mouse".
Some would say it need s to be a decimal, improper fractions are not allowed.
I would tell them improper fractions are my friends.
So, why the controversy. To me there are 2 things to consider
- What is the problem about?
- Where are we going in Math?
The above problem is not a Story Problem, and since 4 does not divide evenly into
15, why spend the time. If it did, then I would change it to the integer it is equal to.
If it was a story problem: Mary is making waffles that requires ⅔ cup of milk.
Because Mary is having her family over she is making
a double recipe. How much milk should Mary use?
HOW MUCH MILK SHOULD MARY USE?
DOUBLE MEANS TO MULTIPLY BY 2
2(⅔) = 4/3 cups of milk
this is 1⅓ cups of milk
Mary needs 1⅓ cups of milk for her recipe.
In this case I changed the improper fraction to a mixed number because I
need an exact number of cups. Measuring cups do not have 4/3 on them.
need an exact number of cups. Measuring cups do not have 4/3 on them.
As I would say to my students:
"MIXED NUMBERS ARE FOR
MEASURING"
Another example might be: Mr. Romero has 15 grams of salt in a container.
He wishes to use only ¾ of the salt. How many
grams does Mr. Romero need?
HOW MANY GRAM OF SALT WILL MR. ROMERO NEED?
¾(15) = ?
45/4 = 11.25
Mr. Romero will need 11.25 grams of salt.
In science we need decimal accuracy, and our measuring equipment
is of marked that way.
As I say:
"DECIMALS ARE FOR SCIENCE"
However, being able to use improper fractions is of primary importance in Math.
Trigonometry is done completely in improper fractions. When the numerator is
greater than the denominator, our answer is greater than 1, and for Math,
because of where it takes us, that is what we need to know.
As I say:
"IMPROPER FRACTIONS ARE
OUR FRIENDS AND THEY ARE
FOR MATH"
He wishes to use only ¾ of the salt. How many
grams does Mr. Romero need?
HOW MANY GRAM OF SALT WILL MR. ROMERO NEED?
¾(15) = ?
45/4 = 11.25
Mr. Romero will need 11.25 grams of salt.
In science we need decimal accuracy, and our measuring equipment
is of marked that way.
As I say:
"DECIMALS ARE FOR SCIENCE"
However, being able to use improper fractions is of primary importance in Math.
Trigonometry is done completely in improper fractions. When the numerator is
greater than the denominator, our answer is greater than 1, and for Math,
because of where it takes us, that is what we need to know.
As I say:
"IMPROPER FRACTIONS ARE
OUR FRIENDS AND THEY ARE
FOR MATH"
Subscribe to:
Posts (Atom)