Transcript 4.N.3 - youngachieversmath

Fractions &
Decimals
3rd Grade Standards for Fractions and Decimals:
4.N.3 Demonstrate an understanding of fractions as parts of unit wholes,
as parts of a collection, and as locations on the number line.
4.N.4 Select, use, and explain models to relate common fractions and
mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, and 1-1/2), find
equivalent fractions, mixed numbers, and decimals, and order fractions.
4.N.5 Identify and generate equivalent forms of common decimals and
fractions less than one whole (halves, quarters, fifths, and tenths).
4.N.6 Exhibit an understanding of the base ten number system by reading,
naming, and writing decimals between 0 and 1 up to the hundredths.
Taken from 2000 Massachusetts Curriculum Frameworks
What is a fraction?
What is a fraction?
One-fourth of
the whole is
shaded in.
A fraction is part of an
entire object or unit
whole.
Two-fourths of
the whole is
shaded in.
Three-fourths of
the whole is
shaded in.
Here the whole is
divided into four
equal parts called
fourths. Four-fourths
are shaded in here.
A fraction is part of a collection of objects.
This collection has 12 pieces, and fourtwelfths of the pieces are yellow.
A fraction is part of a collection
of objects.
A fraction is a place on the number line.
A fraction is a place on the
number line.
0
1
2
Here, there is a point at one-half.
1
How do fractions get their names?
1
2
3
4
4
12
Here the numerator is
1, and the denominator
is 2. When we say ½,
we mean that we have
one piece out of two
pieces.
Here the numerator is
3, and the denominator
is 4. When we say ¾,
we mean that we have
three pieces out of
four pieces.
Here the numerator is
4, and the denominator
is 12. When we say
4/12, we mean that
we have four pieces
out of twelve pieces.
The bottom number of a fraction is
the denominator. It tells us how
many equal pieces the whole has
been divided into, how many pieces
are in the collection, or how many
equal pieces the number line has
been divided into between 0 and 1.
The denominator is usually named
after the number in the
denominator: four becomes fourths,
12 becomes twelfths, etc.
The top number of a fraction is the
numerator. It tells us how many
pieces of that size we have.
Comparing Fractions
Just like whole
numbers, we can
compare the size of
different fractions to
put them in order.
Take a look at this
chart. Which
fractions are bigger
than other fractions?
Which fractions seem
to line up?
Comparing Fractions with the Same
Denominator
When you are trying to
compare fractions with the
same denominator, just
compare the numerator.
(Remember that the
numerator tells us how
many pieces there are.
How would you tell which
one is bigger?) Whichever
fraction has the larger
numerator is the larger
fraction.
¼ < 2/4
2/3>1/3
3/5>1/5
What happens when the fractions
have different denominators?
If they have the same
numerator, sometimes
you can just compare
them based on their
denominator. The
larger the denominator,
the smaller the pieces
will be because the
whole has to be cut into
more pieces.
¾___3/5
Three big pieces is
larger than 3 small
pieces, so
¾>3/5.
Finding a Common Denominator
Often times, you will have
two fractions that have
different numerators and
different denominators. In
these cases, you need to find a
common denominator. You
are trying to get the whole
divided into the same number
of pieces, so you can compare
the number of pieces.
Which one is smaller?
1/2
1.
2/5
First list multiples of 2 and 5:
5/10 > 4/10
2: 2, 4, 6, 8, 10, 12, 14…
2.
5: 5, 10, 15, 20…
So
Circle the one they have in common.
2/5 < 1/2.
Multiply each fraction by a fraction equal to 1:
1/1, 2/2, 3/3, etc., to get the common multiple
in the denominator. This is the same as
multiplying the numerator and the denominator
by the same number.
1x5
5
2x2
=
2x5
3.
4
=
10
5x2
10
Then compare the “new” fractions.
Fractions that are
equal to each other
are called
equivalent
fractions.
Adding and Subtracting with
Fractions
What does it mean to add
fractions? You are trying
to add to pieces of a
whole to see how much
you have. What happens
if the pieces are different
sizes?
Before adding and
subtracting fractions, the
fractions must have the
same denominator.
1/3
+
1/2
These pieces don’t
line up very well, so
we will need to find
pieces that line up
with both ½ and
1/3.
1. Make sure your fractions have a common denominator. If they
don’t have one, find one.
1/5 + 2/5—These fractions are ready to add because they have
the same denominator.
½ +2/3—You need to find a common denominator before you
can add these.
2. Then add (or subtract) across the numerators and keep the
same denominator.
1+2
5
3
=
5
So 1/5+2/5=3/5.
What happens when the
numerator is bigger than the
denominator?
A fraction with a
numerator that is larger
than the denominator
is called an improper
fraction. These are
fractions that are larger
than one. When might
you get a fraction
larger than one?
Moving from Improper
Fractions to Mixed Numbers
Often, the improper
fraction will be easier to
think about if you change it
to a mixed number. A
mixed number is a
combination of a whole
number and a proper
fraction: 1 1/3, 2 ½, etc.
1.
Subtract a fraction equal to 1
from the improper fraction:
3/2-2/2=1/2
2.
Sometimes you will have to do
this more than once. Keep track
of how many times you subtract
one because this will become
your whole number in the mixed
number.
3.
Then rewrite the fraction as the
whole number you subtracted
with the remaining fraction
beside it.
3/2= 1 ½
What is a Decimal?
A decimal is like a fraction because it is a number
between 0 and 1.
0.1
0.25
0.5
0.95
Decimals are often added to whole numbers by
joining them to the whole number with a decimal
point.
1.1
2.25
4.5
6.95
Decimals are fractions.
Decimals are fractions with denominators of
10, 100, 1000, and other powers of 10.
0.1=1/10
0.25= 25/100
0.5= 5/10
0.95=95/100
To write a decimal as a fraction find the
place value of the last digit of the decimal.
That number will be the denominator. Then
write the digits in the decimal over the
denominator.
Place Value with Decimals
Thousands Hundreds Tens
Ones
Tenths
Hundredths Thousandths
1
4
5
6
2
3
7
This number would be read:
One thousand two hundred thirty four and five
hundred sixty seven thousandths.
Rules for Reading Decimals
1. Always use “and” between the whole
number and the decimal to show where the
decimal point is. (Never say “and” when
you are reading a number without a
decimal.)
2. The decimal always has the name of the
last digit’s place value even if there are
non-zero digits in the other places. For
example, if the last digit is in the tenths
place, the decimal is in tenths. If the last
digit is in the thousandths place, the
decimal is in thousandths.
Every Fraction Can Be Written
As a Decimal
Do you remember what we
did to find common
denominators of
fractions?
To write a fraction as a decimal, most of
the time you can find a common
denominator that is a power of 10 (10,
100, 1000, 10,000, etc.) and multiply
each fraction by a fraction equivalent to
1 to get the new fraction with the new
denominator.
Then write the new fraction as a
decimal.
How do we write ½ as a decimal?
1.
List multiples of 2 and find one
that is a power of 10.
2, 4, 6, 8, 10
2.
3.
Multiply the numerator and
denominator of ½ each by 5 to
get the fraction in the new
denominator.
1x5
5
2x5
10
=
Then write the numerator of the
fraction in the appropriate place
values in the decimal.
5/10=0.5
Writing Fractions as Decimals
Sometimes, the denominator of a fraction
will never have a multiple that is also a
power of 10. This happens with 1/3. In
these cases, you have to divide the numerator
by the denominator to find the decimal.
3
Writing zeros after the decimal
point doesn’t, change the
number, but it makes it so that
we can divide 3 into what
looks like 10. Act as if the
numbers are whole numbers
and complete the division, but
remember to raise the decimal
point to the answer line.
0.333
1.000
-9
10
-9
10
-9
1
Repeating Decimals
As we saw when we tried to write 1/3 as a
decimal, some decimals keep repeating forever.
These decimals are called repeating decimals.
Because we can’t keep writing the pattern
forever, we write a bar over the part of the
decimal that repeats to show that it is a
repeating decimal.
1/3=0.33
Summary
•A fraction is part of a whole, part of a collection, and a
point on a number line.
•Before adding, subtracting, or comparing fractions, you
must have a common denominator.
•Decimals are fractions with denominators that are powers
of 10: 10, 100, 1000, etc.
•All fractions can be written as decimals.