Regrouping when subtracting decimals
is a common source of confusion for
students.
This video has illustrations
for helping students understand
regrouping.
Eddie plans to make a tapered
steel shaft on a lathe in the school's machine
shop.
The top diameter is eight hundred
thirty-four thousandth of an inch,
and the bottom diameter is five hundred
forty -seven thousandth of an inch.
How many thousandth of an inch greater is
the top diameter
then the bottom diameter. This square
has 834
shaded parts out of a thousand and represents the decimal
for the top diameter, and the decimal for
this square
is the bottom diameter, 547
shaded pasts out of a thousand. To
see the difference
in the shaded amount of the blue square
and  the yellow square, we will use a
transparency.
This difference
is the yellow shaded amount that is left.
And this difference
is 834
thousandths minus
547 thousandths. There are two
full tenths columns, now we have 2
tenths,
there are eight,
1, 2, 3, 4, 5, 6, 7, 8 small
hundredths squares, and there are 1,
2, 3, 4, 5, 6, 7 tiny
thousandths part's. So the difference
between the top diameter and the bottom
diameter is 287
thousandths of an inch. Let's compute
the difference of
the decimals we just found by using a
place value table.
Starting with the thousandths digits, we
need to subtract 7 thousandths,
but we only have 4 thousandths. Now
we know that one hundredth
is equal to 10 thousandths. So we'll
regroup
1 hundredth to obtain 10
more thousandths and that leaves us
with 2 hundredths in hundredths column. Now
10 thousandths plus 4 thousandths is 14
thousandths
minus 7 thousandths is 7 thousandths.
Next we need to subtract 4 hundredths
but we only have 2 hundredths, so we need
more hundredths, and
we know that 1 tenth
is equal to 10 hundredths. So we'll regroup
1 tenth
to obtain 10 more hundredths, and that leaves us with
7 tenths in the tenths column. Now 12 hundredths
minus 4 hundredths is 8 hundredths,
and 7 tenths minus 5 tenths is 2 tenths.
Place value tables are immediate steps between using Decimal Squares
to subtract decimals and using
subtraction algorithms or calculators.
The middle school
science club scale showed that a block
of ice
weighed 3 and 65 hundredths pounds. These squares represent 3
and 65 hundredths pounds,
the original weight
of the ice. One hour later
the block of ice had lost 1 and 78 hundredths
pounds.
We'll indicate that by subtraction. What
was the new weight of the ice?
We'll start by removing 78 hundredths of a pound,
but this square is only 65 hundredths.
So we replace one whole square
by a square with 100 hundredths.
Now we have a hundred and 65 hundredths.
Then we remove or take away 78 hundredths
which we can show by blocking out
78 hundredths. And this leaves 22
hundredths and 65 hundredths, so a total of 87 hundredths
remain. Then to subtract the 1 in 1 and
78 hundredths,
we remove one unit square. So 1
and 87 hundredths remain.
So the new weight of the ice is 1 and 87
hundredths pounds.
The algorithm
for computing this difference
is similar to the steps with the Decimal Squares.
We want to subtract 8 hundredths, but we only have 5 hundredths,
so we regroup 1 tenth
to gain 10 more hundredths, and this leaves
us with 5 tenths
in the tenths column. 10 and 5 is 15, and 15
minus
8 hundredths is 7 hundredths. Then we need to subtract 7 tenths, but
we only have 5 tenths, so we we regroup 1
unit
to gain 10 more tenths, and this leaves us with
2 units in the units column.
15 tenths minus 7 tenths is 8 tenths,
and 2 minus 1 is 1.
So 1 and 87 hundredths
is the new weight of the ice. Notice in the algorithm that
we subtracted the hundredths digits and then
the tenths digits,
but in using the squares, we subtracted the
78 hundredths
in one step. Blaine walked
8 tenths of a mile along a river. If the length
from 0 and 1
on the number line represents 1 mile, we
can represent the distance
Blaine walk by drawing an arrow,
from 0 to 8 tenths.
Then Blaine returned 5 tenths of a
mile along the same path, stopping at a
bridge.
Starting at 8 tenths on the number
line, a second arrow can be drawn back
toward zero 5 tenths. How far
is Blaine from the starting point? The
difference between these distances
is 3 tenths. So
8 tenths minus
5 tenths equals
3 tenths, and Blaine is 3 tenths of a mile
from the starting point. Moving forward
on a number line to represent one
amount
and then moving back toward zero to
represent
a second amount is one method showing
subtraction of decimals on the number line.
Next we look at a different method
of showing subtraction of decimals
on a number line. The circular opening
of a bluebird's house has a diameter
1 and 56 hundredths inches. The opening
for a sparrow's house is 1 and 18 hundredths inches.
What is the difference
in these dimensions? Let's draw an arc
to represent the opening of the bluebird's house,
and an arc to represent the opening for
the sparrow's house.
The difference in these mixed decimals can
be found by counting the hundredths
between these arrows. We can either count back,
or we can count forward, using the Add-Up
method.
Starting at 1.18, it's 2 hundredths
to 1.2.
From 1.2 to 1.5
its 3 tenths or 30 hundredths.
And from 1.5 to 1.56
is 6 hundredths. So the total distance
is 38 hundredths. And 1 and
56 hundredths minus
1 and 18 hundredths
equals 38
hundredths. So the difference in the
dimensions of
the openings of the two bird houses is
38
hundredths of an inch. You have seen
place value tables and number lines
for illustrating subtraction of decimals
and mixed decimals, and special attention
was given for understanding regrouping.
A demonstration of a game,
shooting laser beams at asteroids,
from decimalsquares.com. The
object of
this game is to shoot laser beams at asteroids. We'll
choose the Intermediate Level which
involves rounding decimals and
computing sums and differences. NIce shot.
This takes some concentration. Here we have a difference,
good shot. Six out of 7
successful shots.
