This lesson well illustrate
relationships between
tenths, hundredths, and thousandths in place value tables
that are important for understanding 
regrouping
when adding decimals. It will also
illustrate addition of
decimals on tenths and hundreds
number lines. This place value table
will be used for adding decimals. This
decimal square
represent 675 thousandths.
There are 1, 2, 3, 4, 5, 6
full columns, 6 tenths,
1, 2, 3, 4, 5, 6, 7 hundredths,
and five
tiny thousandths. The next square
represents 748 thousandths.
There are 7 full columns,
7 tenths, 1, 2, 3, 4 hundredths,
and
5, 6, 7, 8 tiny thousandths.
Before adding
these decimals, we will use this large
unit square to review the relationships
between tenths,
hundredths, and thousandths. This large
square represents one unit, and we will
record the number of
units in the ones column. This red shaded column
is one of ten equal parts, so one of these parts
is called 1 tenth. This small green
shaded square is one of 100
equal parts, so we call one of these
one hundredth. And 10 of these small green
squares equal
1 tenth. This tiny orange part is one of a 1000 equal parts in the unit,
so we will call one of these
one thousandth. And 10 of these thousandths
equals 1 hundredth. 5 thousandths plus
8 thousandths is 13 thousandths, and we know that whenever we have 10
thousandths we can replace that by one
hundredth. So we leave
3 thousandths here,
we'll regroup 10 thousandths
to the hundredths column, and indicate that by placing 1 in the hundreds column.
Now we have 1 hundredth, 7 and 4, so that's
12 hundredths.
Now whenever we have 10 hundredths, we know that we have 1 tenth,
so we'll leave the 2 hundredths
here and regroup 10 hundredths
to 1 tenth, and record that by putting a 1
in the tenths column.
1, 6, and 7 tenths are
14 tenths, and we know that whenever we have 10
tenths we have 1 unit, so we leave the 4
here,
record the 4 tenths, and
regroup 10 of the tenths by writing a 1 in the ones column.
And so we bring that down, and the sum of these two decimals
is 1 and 423 thousandths.
You can see why we usually begin
by adding the digits in the thousandths column. Because if there are 10 or more,
we can replace 10 thousandths by 1 hundredth
and regroup to the hundredths column.
The number line is another model
of teaching addition of decimals. Six tenths
of this decimal square is shaded,  and it
corresponds
to 6 tenths on the number line. Suppose Jay
hikes 6 tenths of a mile and stops to
rest.
If the length from 0 to 1 represents one
mile,
the line shows that Jay has hiked 6 tenths of a mile, and
we can mark his progress
by an arrow. If Jay continues
7 tenths of a mile further and stops at an
abandoned mine,
we can place a square representing
7 tenths next to the shaded amount of the 6 tenths,
and mark Jay's progress on the number
line.
Or, we can count off 2, 3, 4, 5, 6, 7, seven more tenths on the
number line
to the point 1 and 3 tenths.
We can represent the total length
that Jay has hiked by an addition equation.
Six tenths plus 7 tenths
equals 1 and 3 tenths miles.
The hundredths number line
has 10 equal spaces for tenths
between 0 and 1, and each of these spaces
is divided into 10 equal parts for
hundredths.
Similarly, the decimal square
for hundredths has 10 equal
columns or parts, and each column
is divided into ten equal parts for
hundredths.
The shaded amount of this decimal square
represents 65 hundredths.
And we can match this shaded amount up to the decimal
on the number line for 65 hundredths.
Now this has six full columns and
half a column,
so we go to the decimal 6 tenths and 7 tenths, we go halfway between,
and that number that's halfway between
is the decimal for 65 hundredths.
Suppose Jade is able to decrease
her time in the 400-meter freestyle
swimming event
by 65 hundredths of a second. If the line from
zero
to 1 represents one second, we can mark her
decrease in time
on the number line, 65 hundredths of a second.
By further training, Jade is able to
decrease her time by another
70 hundredths of a second. You can
illustrate this by
a decimal square with 70 parts out of a hundred shaded.
We'll place the shaded amount of this square, 
begin it at that point for the decimal
65 hundredths. And we can see that the shaded amount goes
about halfway between 1 and 3 tenths
and 1 and 4 tenths.
Or, starting at 65 hundredths,
we can count ten hundredths at the time, so
65, 75
85, 95, 105, 115, 125, 135.
And we can mark Jade's
decrease in time on the number line,
70 hundredths of the second. The number
line shows that Jade's total decrease in
time
is 1 and 35 hundredths of a second. We can record this by
an addition equation, 65 hundredths
plus 70 hundredths
equals 1 and 35
hundredths seconds. The decimal square
is a region model showing the whole unit
and equal parts of
the unit. The number line is a linear
model
and the unit from 0 to 1 is divided into
10 equal parts,
and this case for the hundredths line
is divided into a 100 equal parts. You
have seen
the visual the decimal squares model
connected to both place value tables
and number lines
for adding decimals. All too often, connections between different models
for adding decimals are not shown. Let's go
to decimalsquares.com and look at
the game Decimal Darts.
If a dart misses a balloon,
an adjustment can be made by adding or
subtracting decimals.
In this game, we will throw darts and try to break balloons.
We click "Start Game", click "1 Player".
The player enters a name, "Elayna",
selects a difficulty level of 1, and clicks "CONTINUE".
The dart board is like a number line
going from a low of 2.7
to a high of 3.21. The player
selects a number, clicks
"Throw Dart"  Great shot!
We broke one balloon with one dart.
The player types in another number, clicks "Throw Dart".
That was a miss, but it gives us some
good information.
Beautiful, two balloons gone with three
darts.
Nice job!
