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.
Wednesday, September 7, 2011
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"
Sunday, September 4, 2011
Mr. W. the numbers man: Subtraction, Minus, Less Than, or Adding in the Op...
Mr. W. the numbers man: Subtraction, Minus, Less Than, or Adding in the Op...: Big Topic or perhaps not. In all the years I taught Algebra, this operation caused more problems for my students then any other. You wo...
Subtraction, Minus, Less Than, or Adding in the Opposite Direction
Big Topic or perhaps not. In all the years I taught Algebra, this operation
caused more problems for my students then any other. You would think that the "–"
sign would be easily understood by high school. If I could recall some of the comments
made by my students I would list them for you. But I cannot. What I can do is give you
an idea of what some were like.
1. Some called it a minus, while others argued that the "–" sign meant negative.
2. Those that declared it to mean negative said that it was not minus.
3. They would not read it as an opposite of a positive.
4. They would question the operation of a problem like 5 – 3 = ?
Lets try something: 1. We are going to put the integers –20 to +100 on the wall starting
in KINDERGARTEN.
2. We will call –11, "the opposite of 11"
If you teach 15 + 11 = ? You can go to the integers on your wall, start at 15 and
go 11 to the right. The students will get a great visual as you move to
the right, and your words will help the audio learners. Then you
have individuals show the class like you did it.
If you teach 15 – 11 = ? This is 15 + (–11) so you can again go to your integers
on your wall, start at 15 and go 11 to the left. Here again, the
power of your moving to the left will be of great benefit. And
of course all those future actors can do like you did for the class.
We are, "adding in the opposite direction."
So 5 – 3 = 2 The "–" sign means I am going to my left from where ever I am.
5 + 3 = 8 The "+" sign means I am going to my right from where ever I am.
I seldom see any the integers in the primary grades, but they should be there
for more reasons than just what I am speaking to here. When students are first
told about the integers, in 5th grade, they become confused. They now see a
sign "–" in front of a number like "–5" and if it is added to 8, what are talking
about? –5 + 8 = ?
Now think about what we did above. Start at –5 and go to the right 8. We
get to the number 3. We start at the "opposite of 5".
What about –5 – 8 = ?
We start at –5 and we go to our left –5 – 8 = –13.
+(–5) +(–8) = – 13
I contend that the more familiar your students are with numbers the easier
it will be for them to work with them. In that last problem, we are simply
adding to our left as we have added to our right.
WE ARE ADDING IN THE OPPOSITE
DIRECTION!
caused more problems for my students then any other. You would think that the "–"
sign would be easily understood by high school. If I could recall some of the comments
made by my students I would list them for you. But I cannot. What I can do is give you
an idea of what some were like.
1. Some called it a minus, while others argued that the "–" sign meant negative.
2. Those that declared it to mean negative said that it was not minus.
3. They would not read it as an opposite of a positive.
4. They would question the operation of a problem like 5 – 3 = ?
Lets try something: 1. We are going to put the integers –20 to +100 on the wall starting
in KINDERGARTEN.
2. We will call –11, "the opposite of 11"
If you teach 15 + 11 = ? You can go to the integers on your wall, start at 15 and
go 11 to the right. The students will get a great visual as you move to
the right, and your words will help the audio learners. Then you
have individuals show the class like you did it.
If you teach 15 – 11 = ? This is 15 + (–11) so you can again go to your integers
on your wall, start at 15 and go 11 to the left. Here again, the
power of your moving to the left will be of great benefit. And
of course all those future actors can do like you did for the class.
We are, "adding in the opposite direction."
So 5 – 3 = 2 The "–" sign means I am going to my left from where ever I am.
5 + 3 = 8 The "+" sign means I am going to my right from where ever I am.
I seldom see any the integers in the primary grades, but they should be there
for more reasons than just what I am speaking to here. When students are first
told about the integers, in 5th grade, they become confused. They now see a
sign "–" in front of a number like "–5" and if it is added to 8, what are talking
about? –5 + 8 = ?
Now think about what we did above. Start at –5 and go to the right 8. We
get to the number 3. We start at the "opposite of 5".
What about –5 – 8 = ?
We start at –5 and we go to our left –5 – 8 = –13.
+(–5) +(–8) = – 13
I contend that the more familiar your students are with numbers the easier
it will be for them to work with them. In that last problem, we are simply
adding to our left as we have added to our right.
WE ARE ADDING IN THE OPPOSITE
DIRECTION!
Saturday, September 3, 2011
Mr. W. the numbers man: The story Problem
Mr. W. the numbers man: The story Problem: When the teacher says open your text to page 236, and all you see is word problems, or story problems as they are often called, how do yo...
Subscribe to:
Posts (Atom)