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Let's begin with a question,
is it necessary to teach fractions
before teaching decimals. For many years
fractions have been taught first in schools and then used to
teach decimals, and this is still being done.
The main reason for this practice maybe the fractions have
been around for over 5,000 years, while decimals have
been around for less than five hundred years. The first lesson
in this series videos will help you
answer that question. Here is a model
for decimals and percents. Each square with red shading
has ten equal parts, so we can describe this square by saying,
it has 3 parts that are shaded and 10 equal parts, or
3 parts out of 10 are shaded. The green squares each have 100 parts.
There are 10 little parts in each column. So there are 35 parts out of 100, or
we can say it has 100 parts and 35 parts are shaded. Now the yellow
squares
have a thousand tiny parts, and to help see those parts,
we will use a larger square.
Now this block down here has 10, here's five and five, so we will
identify that block as 10 parts.
And there are 10 of these blocks in a column,
so each column has 100 parts.
So, there's 100, 200, 3, 4, 5, 6, 700,
and then we've got 10, 20, 30, 40, 50, 60, 75,
so let's just fill those in there,
So this square can be described by saying it has a thousand equal parts,
and 775 are shaded, or
775 parts out of 1000
are shaded. Notice that we can describe
these three types of squares by using just
whole numbers. This provides a non-threatening
introduction. In the early grades, it's best to start with just the red and
green squares.
Students can sort these types of squares and
look for patterns in the shaded amounts. They will find there's
9 different red squares,
19 different green squares, and if they use the yellow squares, there are
29 squares for total
of 57 decimal squares in a deck. After students are familiar with
describing shaded amounts with members,
we can bring in decimals. The decimal for 3 parts out of 10
is .3, and the name of this decimal is three tenths.
The decimal for 35 parts out of 100
is .35,
and the name of this decimal is thirty-five hundredths. The decimal for
675 parts out of 1000 is
point 675,
and the name of this decimal is six hundred and seventy-five
thousandths. One activity
would be having students select decimal squares
and write the decimals for the shaded amount. Conversely,
for a given decimal, student can shade
a blank decimal square.
For this square,
we will shade four columns
for the decimal .4.
And a square for hundreds
Let's shade
the first column.
There's 10,
10, 20, 3,
23 parts out of a hundred,
for the decimal .23.
And this square with a thousand parts, and there are 100 tiny parts in this first column,
and there are 10 in each block here, so 10, 20,
30, 35, 37.
So that is the shading
for the square for a hundred and thirty-five thousandths.
The shading of blank squares can be creative.
Point 26,
or 26 hundredths have been shaded for this square.
In this example,
.4 or 4 parts out of 10
have been shaded. In one final example,
four hundredths
or 4 parts out of 100 have been shaded.
Notice that we cannot use .4
for four parts out of 100, because .4
is the decimal for the square with 10 parts and 4 parts shaded.
Once decimals are introduced,
they can be used to talk about parts of things and applications.
The first and second graders, this square
could represent a cake with seven tenths
left. Students can make up stories,
such as three parts of the cake have been eaten. For older students,
who have seen tenths of a pound in bathroom scales,
if one whole square represents one pound,
then this square represents seven tenths of a pound. Suppose it costs
65 cents to mail an over-size envelop.
If one whole square
represents one dollar, then this square represents sixty-five hundredths of a dollar.
In track meets
time is measured thousandths of a second. If this whole square
represents one second, then the shaded amount of this square
represents one hundred and twenty-five thousandths of a second. There are eight games
on decimalsquares.com. We will look at the first game,
Beat the Clock. This game
can be played against the clock or against another player.
We will select the clock. Maria
enters her name, selects the Beginner Level,
selects forty seconds for the clock time,
and clicks the square
to start.
Types the decimal,
clicks the clock to check the answer. Then clicks the clock again
for the next square.
Maria
is doing very well. She knows
her decimals. This video
has used decimal squares, a region model, for
illustrating decimal symbols
with understanding, and we did not use fractions.
In fact, fractions is so difficult to teach,
it might make sense to teach decimals
before teaching fractions.