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Table
of Contents
Chapters:
Introduction
1. What
does this mean to me?
2. Fractions:
Pieces of what?
3. Decimals:
What’s your point?
4. Percentage:
How much of what?
INTRODUCTION
“I hate math!!!!”
How many times have you heard this? How many times have you said this? I think
we all have, way too many times. We are programmed to think of math as scary,
as too hard, as confusing, and to be honest it is; until you are taught that it
can be interesting and fun. No, I’m not lying! In school I absolutely hated
math, I despised it, I pretty much almost failed it. I squeaked by with a D to
get out of high school, but then in college learned I had to have at least a B
to pass. That is when I decided the only way I was going to pass was by
changing how I looked at it. Once I did that I found out I could teach myself
what I needed to know with a little research and some great websites.
When I started
teaching 6th grade I had to teach all subjects, it never failed that
the first day of school when I handed out text books the math book got shoved
into the desk and ignored until it was required. The other books were leafed
through, examined, comments made, even the science books got more respect than
the math books. I would ask students to number a card 1 – 4 then put the
subjects in order with the one they liked the best being number 1, next best
number 2, etc. One year out of 32 students not one single student put math at
number 1, two students put it at number 2, and 30 students put it at number 4.
I vowed that by Christmas at least half of that class would have math at number
1! I was wrong,, it only took the 1st quarter and three fourths of
the class had it at number 1! Let me tell how it happened, and hopefully it
will help you put the fear of math behind you too.
Chapter
1
What
Does This Mean to Me?
Math needs to mean
something to the person trying to learn it; it can’t be just numbers and
formulas thrown at them. If you can make a connection to what you are learning,
you have a better chance of learning it and a much better chance of teaching
it. Ask questions before you start, find out what your audience already knows.
Depending on the grade level, students know different levels of the area of
math you are working on. For example; if you are working on fractions the
questions you ask might start with:
·
What are fractions?
·
How do you use fractions?
·
Where can you find fractions?
·
Give me an example of fractions in everyday
use?
If you are working
on decimals or percentages, just replace the word fractions with the subject
you are working with. Using a chart tablet, write the questions on the chart
and then in different colors fill in the answers students give you. Post the
chart you make in an obvious place in the classroom, or if you are
homeschooling place it somewhere you will see it regularly. Invite the students
to add to the list as they come up with more answers.
After you fill in
your chart and post it, answer the questions yourself, explaining to your
students why the answers are what they are. Go into detail, don’t just say “A
fraction is a piece of a number”, that makes no sense at all, it’s not just a
piece of a number, it is a piece of a “WHOLE” unit. I will go into detail more
in the section on fractions, I am just making the point here that you have to
be detailed in your explanation, but you have to put it at the student’s level
you are working with. You will have students at more than one level in your
class, or in your family, and you have to make the explanation pertinent to all
of them. Your lowest might need
clarification that ¼ means divide a circle into 4 pieces and color one piece.
But wait! Does that student understand what the 1 means and what the 4 signifies?
Probably not, you haven’t told them yet! You can’t jump into fractions without
explaining the Numerator and the Denominator. This goes back to knowing your
audience. What can you use to make this mean something to your students?
I like to use examples
from my own life, it not only makes it easier on me, it gives the students
something in common with me, and believe it or not it does not hurt to identify
with your students. If you are using fractions, decimals, or percentages talk
about sales in the newspaper, or signs in stores. When you are learning
measurements think of building something, cooking something, even trying on
clothing. There are so many ways to make a math lesson relevant to the learner.
I suggest using folders and as you come up with ideas of your own to use with
these lessons write them down and keep them for future use.
Chapter
2
FRACTIONS
What
is a fraction?
What
does it mean?
What
can I do with it?
Ok,
in order to know what a fraction is you have to know what makes up a fraction:
A numerator, a fraction bar, and a denominator are the parts of a fraction.
Numerator:
The top number in a fraction, it means the number of pieces of the whole unit
you are working with.
Fraction Bar:
Divides the numerator from the denominator
Denominator:
The bottom number, this is how many pieces the whole unit is divided into.
EXAMPLE:
½ = One out of two pieces.
Draw
a circle, draw a line through the middle dividing it into two pieces. Color one
piece, leave the other piece blank. Explain that the circle is divided into two
pieces that is the denominator, and one piece is colored, that is the
numerator. I like to say that the bar in the fraction can be said as “out of”.
So 1/2 can be said as “one half” or “one out of two” meaning one out of two
pieces is colored. That would be a very basic but very plain example of what a
numerator and a denominator is.
What
does a fraction mean? It means you have a taken one whole unit and divided it
into pieces. A unit could be one pizza, two pies, ten sodas, or even a hundred
people. The biggest misconception that confuses people is that a fraction is a
piece of the number 1, it isn’t, it can be a piece of anything. If the unit is
10 people, then the denominator is 10. If 1 out of those 10 people is wearing a
blue shirt, the numerator is 1. So the fraction would be 1/10 meaning 1 out of
10 people are wearing a blue shirt. If you are working with one pizza divided
into eight slices, then the denominator is 8. If you are serving 2 slices to
each person then the numerator is the number 2. This means each person would
get 2/8, or 2 out of 8 pieces, of the pizza.
What can I do with
fractions? You can do a lot with fractions, in fact I’ll bet if you think about
it you can come up with several times a week, sometimes a day, that you use
them. Remember being in kindergarten and your best buddy brought a candy bar in
their lunch? “Halfies!!!” Yep, you wanted half of that candy bar didn’t you?
You wanted a fraction of that candy bar, 1/2 of it to be exact. When you bake cookies, the recipe calls for
1/3 of a cup of butter, or 2 ½ cups of flour, you need to know what those
fractions mean.
There are some
fractions we use a lot more than others, those are called “benchmark
fractions”. These are fractions you hear all the time: 1/4, 1/2, 3/4, and 1/10
because of it’s relationship to decimals. Benchmark fractions are used to
compare other numbers to and are often used to estimate measurements.
The
figure below illustrates 1/10. The whole
unit, a rectangle, is divided into 10 pieces, so the denominator is 10. One
piece is shaded blue, so the numerator is 1, thus 1/10. One out of ten pieces
is shaded blue.
1/10
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2/10
|
3/10
|
4/10
|
5/10
|
6/10
|
7/10
|
8/10
|
9/10
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10/10 = 1
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The
figure below represents 2/8. The whole unit, a rectangle, is divided into 8
pieces, 2 of the pieces are shaded red, thus 2/8. Two out of eight pieces are
shaded red.
1/8
|
3/8
|
5/8
|
7/8
|
2/8
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4/8
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6/8
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8/8 = 1
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No matter how many
pieces the whole unit is divided into, when the numerator and the denominator
are both the same number, it equals 1. Whether it is 1 piece, or 1 whole unit
divided into multiple pieces, if those two numbers are the same, it equals 1.
EXAMPLES:
1/1
= 1÷1=1
2/2
= 2÷2=1
3/3
= 3÷3=1
Chapter
3
Decimals
What
is a decimal? A decimal is a small dot that separates a whole number on the
left of the decimal from the fraction of a number on the right side of the
decimal. When I start that first lesson using decimals I can hear the moans and
groans, the comments, the expression of fear, or confusion, and the inevitable
“I hate decimals!” Why? Because my audience doesn’t know how to work with them
yet. They have heard others complaining about them, showing fear of them, and
worse, giving up on them. The first thing you have to learn to be successful
with decimals is your place value. This is something students are supposed to
start learning in kindergarten, and move up each grade learning more and
understanding the meaning of what place value is.
I
like to go straight into decimals from fractions while they are still fresh in
their minds; this helps make the transition easier and the relationship between
fractions and decimals will be more obvious to most of them. The first thing I
do is either on the overhead viewer, white board, or chalkboard, is put a big
dot right in the middle of it. Ask the students what that dot means, use the
definition I have already given you if they don’t come up with it themselves.
To the right of the decimal put a 0 and ask what place that 0 is in:
.0 The answer would be the
tenths place. Make sure that they understand you are working with a fraction of
a number so it has the “th” on the end of ten, not an “s”. Numbers to the left
of the decimal have an “s” (ones, tens, etc.) to the right of the decimal they have “th” (tenths,
hundredths, etc.) on the end. Go over it again, make a big deal out of it.
Right now just 0’s to illustrate place value, we will add other numbers
shortly, you are making sure they know their place value right now. I always have students make an index card with
place values marked on it and tape it to the top of their desks until they are
positive they know place value. By making their own chart it helps retain that
information more than if they went out and bought one, it’s ownership of their
work.
I
make it a point to try and give at least two examples of the idea I am trying
to illustrate to them:
.0
Tenths, .00 Hundredths, .000 Thousandths, .0000 Ten Thousandths, .00000 Hundred
Thousandths, .000000 Millionths. In this example the place value of the last 0
is given.
Then
there is the traditional place value chart:
The
important thing here is that the student finds the method that helps them the
most. All learners are not the same, so all methods will not work on all
students the same either.
Graph
paper is a great learning tool for decimals; students can make their place
value chart and put examples underneath the labels.
When
you are working with decimals you must keep you decimals lined up, if they are
not lined up it is extremely easy to get the decimal in the wrong place in your
answer. There is a huge difference between 1.234 and 123.4!
Working
With Decimals Examples
Math
is visual, there are many ways to write problems down and everyone is different
in what works for them. When writing an addition problem with decimals it can
be written as 1.234 + 123.4. If you are experienced with decimals
that may work for you and you can add them together and your decimal in the
correct place; but if you aren’t familiar, or are just starting out, it may be
easier for you to write it in a stacked format:
1.234
or 1.234
+123.4__ +123.400
By stacking
the numbers you can make sure your decimal is in the correct space and lined up
perfectly. In the above examples I first used the numbers as written, in the
second example I put 00 behind the 4 in the tenths space on the bottom number.
Sometimes this will give a visual learner a better understanding of that number
and they can add 4 + 0 easier than just dropping the four down below the line.
It’s still adding 4 and 0 but many need that place value holder.
Whether you
are adding, subtracting, or multiplying decimals, as long as they are lined up
correctly, and you bring your decimal down into the proper position, you answer
will be set up correctly. Dividing decimals is a whole different issue though
and takes a lot more work. It can be made easier, and if you learn how to move
your decimals to make whole numbers you will be able to divide any decimal
using long division.
Dividing
Decimals
Let’s
divide a decimal by a whole number first, it is the easiest way to learn to
divide the “dreaded decimal”!
__ _____
Take
9.1 ÷ 7 and write in long division
format: 7 ) 9.1
Ok
this can be the confusing step but once you do it a few times it will be much
easier. You are going to remove your decimal from your equation completely and
do the math as in box 1. In box 2 you put the decimal back between 9 and 1,
then take it straight up between the 1 and 3. So 9.1 ÷ 7 = 1.3.
1. 2.
Dividing
a decimal by a decimal is a little more challenging but is done in a similar
manner, but you remove the decimal from your divisor completely:
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