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...
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 you feel. Does your stomach begin
to turn, do you brake out in hives, or do you just want to close the book. Sometimes
I wonder if your teachers felt the same way, when it came time for them to teach you
how to solve them.
Is there a magic way to solve them, NO!
However, there are some strategies to help you out.
1. READ THE PROBLEM
You will not know what the problem is about unless you read it. Math has to be read,
not stared at. It is up to teachers, from day one to force students to read everything in
their Math book. Do not try to make it easy by telling them they do not have to read.
Remember, life is a story, not a simple equation to solve.
2. DECIDE WHAT THE EXACT
QUESTION THAT IS BEING
ASKED. WRITE IT DOWN!
Here we have to read carefully. The question is not always at the end of the problem.
Sometimes the question is embedded in the problem, and often there is more than one
question. As you write it down you should be putting yourself into the problem,
becoming a part of the story. The closer we are to understanding the question, the
closer we are to the answer.
3. DRAW A PICTURE OR DIGRAM
This is often what makes it easy to solve. In problems that deal with distance, rate,
and time, after making the chart you will realize that it is all about the distance.
Similar charts can be made with chemistry, money, and age problems.
4. CHOSE AN OPERATION OR
MAKE AN EQUATION TO
SOLVE AND FOLLOW
THROUGH
Doing this step is all about your abilities in the basics. Once you are here
you are 80% done. Following through means that once you have a solution,
it needs to make sense. Just because you have an answer does not mean
you are correct. Reread the problem, and see that it fits.
5. WRITE YOUR ANSWER IN A
SENTENCE ANSWERING THE
QUESTION OR QUESTIONS
YOU WROTE AT THE BEGINNING
The question is a story, the answer deserves to be a sentence. If the question
is how many miles did Harvey travel, your answer needs to be; Harvey traveled
x number of miles. This way students answer questions in Math the same way
they do in History or Science.
or story problems as they are often called, how do you feel. Does your stomach begin
to turn, do you brake out in hives, or do you just want to close the book. Sometimes
I wonder if your teachers felt the same way, when it came time for them to teach you
how to solve them.
Is there a magic way to solve them, NO!
However, there are some strategies to help you out.
1. READ THE PROBLEM
You will not know what the problem is about unless you read it. Math has to be read,
not stared at. It is up to teachers, from day one to force students to read everything in
their Math book. Do not try to make it easy by telling them they do not have to read.
Remember, life is a story, not a simple equation to solve.
2. DECIDE WHAT THE EXACT
QUESTION THAT IS BEING
ASKED. WRITE IT DOWN!
Here we have to read carefully. The question is not always at the end of the problem.
Sometimes the question is embedded in the problem, and often there is more than one
question. As you write it down you should be putting yourself into the problem,
becoming a part of the story. The closer we are to understanding the question, the
closer we are to the answer.
3. DRAW A PICTURE OR DIGRAM
This is often what makes it easy to solve. In problems that deal with distance, rate,
and time, after making the chart you will realize that it is all about the distance.
Similar charts can be made with chemistry, money, and age problems.
4. CHOSE AN OPERATION OR
MAKE AN EQUATION TO
SOLVE AND FOLLOW
THROUGH
Doing this step is all about your abilities in the basics. Once you are here
you are 80% done. Following through means that once you have a solution,
it needs to make sense. Just because you have an answer does not mean
you are correct. Reread the problem, and see that it fits.
5. WRITE YOUR ANSWER IN A
SENTENCE ANSWERING THE
QUESTION OR QUESTIONS
YOU WROTE AT THE BEGINNING
The question is a story, the answer deserves to be a sentence. If the question
is how many miles did Harvey travel, your answer needs to be; Harvey traveled
x number of miles. This way students answer questions in Math the same way
they do in History or Science.
Thursday, September 1, 2011
Percents, as viewed as a PROPORTION
There are three basic kinds questions we need to answer when it comes to percents;
- What number is 40% of 24?
- 28 is what percent of 50?
- 15 is 24 percent of what number?
I recall my son coming home, and being perplexed by these problems. I ask him,
"If there were a way that you could solve all of these problems with one idea,
would you want to know about it?"As usual he said, "yes dad".
"If there were a way that you could solve all of these problems with one idea,
would you want to know about it?"As usual he said, "yes dad".
So I asked him what he knew about a percent, and he told me that it was part
of 100. I told him that is half of the story. A number written as a fraction with a
denominator of 100, we can then say that its NUMERATOR is a percent.
of 100. I told him that is half of the story. A number written as a fraction with a
denominator of 100, we can then say that its NUMERATOR is a percent.
43/100 = 43%
27/100 = 27%
Now what if I ask about a fraction say ¾. I said to my son, this is the second part of the story.
3 is what PERCENT of 4?
Therefore, 3/4 = %/100 or is/of = percent/100
Remember we are solving for the NUMBER in the % position. This is just
like changing a fraction to a common denominator. Divide 4 into 100, we
get 25. Now we multiply 25 times 3, and we get 75.
Thus ¾ = 75/100 or 75%.
3 is 75 PERCENT of 4.
In the previous problem 4 went into 100 evenly, but that will not always occur. But do not
worry. There is another way to solve for the percent.
3 is what PERCENT of 8?
3/8 = %/100
RETURN TO THE BASIC LAW OF PROPORTIONS
8 x % = 3 x 100
% = 300 ÷ 8
% = 37.5
3 is 37.5% of 8.
Next, to figure out a Percent of another number, we are wanting a piece of it.
That piece relates to the the number 100. That is 30% means 30/100 of an
amount or .30 of that amount. Given the following problem, we have two choices.
What number is 45% of 20?
Choice 1: 45% = 45/100 = .45 Now I can Multiply 20 x .45 = 9
Choice 2: Rely on my Proportion. ?/20 = 45/100
Here we learn another powerful solution method of proportions. Our "?" is the
unknown amount. And as we did above, 20 goes into 100, 5 times.
?/1 = 45/5 is left
But 5 goes into 45, 9 times. Thus our ? = 9
Now for our last part. What if the money in your pocket is 15% of your total
wealth. What is your total wealth? If it were me, not much. To start with we
need to know what is in your pocket. Lets say it is $10.00. So what do we
already understand.
15% = 15/100 You have $10.
10/x = 15/100 $10.00 "is" from "is/of = %/100"
Look at the numbers. None go in evenly as we go around , but we can
reduce a fraction.
15/100 = 3/20
Thus 10/x = 3/20
then 3x = 200
x = 66⅔
$ 66.67 is your total wealth?
With a single formula we can solve all three types of Percent problems. Our 100
is always under the percent and the is, is always over the of.
is ÷ of = % ÷ 100
Remember we are solving for the NUMBER in the % position. This is just
like changing a fraction to a common denominator. Divide 4 into 100, we
get 25. Now we multiply 25 times 3, and we get 75.
Thus ¾ = 75/100 or 75%.
3 is 75 PERCENT of 4.
In the previous problem 4 went into 100 evenly, but that will not always occur. But do not
worry. There is another way to solve for the percent.
3 is what PERCENT of 8?
3/8 = %/100
RETURN TO THE BASIC LAW OF PROPORTIONS
8 x % = 3 x 100
% = 300 ÷ 8
% = 37.5
3 is 37.5% of 8.
Next, to figure out a Percent of another number, we are wanting a piece of it.
That piece relates to the the number 100. That is 30% means 30/100 of an
amount or .30 of that amount. Given the following problem, we have two choices.
What number is 45% of 20?
Choice 1: 45% = 45/100 = .45 Now I can Multiply 20 x .45 = 9
Choice 2: Rely on my Proportion. ?/20 = 45/100
Here we learn another powerful solution method of proportions. Our "?" is the
unknown amount. And as we did above, 20 goes into 100, 5 times.
?/1 = 45/5 is left
But 5 goes into 45, 9 times. Thus our ? = 9
9 is 45% of 20.
Now for our last part. What if the money in your pocket is 15% of your total
wealth. What is your total wealth? If it were me, not much. To start with we
need to know what is in your pocket. Lets say it is $10.00. So what do we
already understand.
15% = 15/100 You have $10.
10/x = 15/100 $10.00 "is" from "is/of = %/100"
Look at the numbers. None go in evenly as we go around , but we can
reduce a fraction.
15/100 = 3/20
Thus 10/x = 3/20
then 3x = 200
x = 66⅔
$ 66.67 is your total wealth?
With a single formula we can solve all three types of Percent problems. Our 100
is always under the percent and the is, is always over the of.
is ÷ of = % ÷ 100
Sunday, August 28, 2011
Mr. W. the numbers man: The life of a PROPORTION
Mr. W. the numbers man: The life of a PROPORTION: Every year I would stand in front of my Algebra I class with a problem behind me that looked like the following: X ÷ 3 = (X + 1) ÷ 4 I wo...
Mr. W. the numbers man: Can you make 10
Mr. W. the numbers man: Can you make 10: The first operation a child learns is to ADD . There is much time spent in adding this number to that. I strongly believe learning how t...
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