Math 310 - Dr. Miller - Self-Paced Lesson: Decimal Arithmetic - Spring 2015
We all know that decimal numbers are ones that are written with a decimal point and digits
to the right and/or left of the point, numbers such as 3.14, 0.598, or 6.0. In daily speech
we sometimes use the word ”decimal” to refer either to a decimal number or to the decimal
point - as in ”3.14 is a decimal” or ”there are 2 digits after the decimal” - but in teaching,
it’s important to be clear with our language, so I’ll try to use the complete phrases ”decimal
number” and ”decimal point.” Also, in a number such as 0.598, showing the 0 in front makes
it less likely that a child will overlook the decimal point and misread the number as just 598.
The first thing we all learn in our early encounters with decimal numbers is what the new
place values (the positions to the right of the decimal point) stand for. Just as in MATH
210, you spent time discussing the names of the positions to the left - ones, tens, hundreds,
thousands, and so on - in MATH 310, we will continue to use those positions as well as those
to the left - tenths, hundredths, thousandths, etc. A basic exercise in learning to understand
place values whether for MATH 210 whole numbers or MATH 310 decimal numbers is to
write the numbers in so-called expanded form. This specifically means taking each individual
digit in the number and showing how it contributes to the overall value of the entire number
by multiplying the digit by the value of its position as well, then adding these all up. For
positions to the right of the decimal point, it’s important that you name them with their
fractional names 1/10, 1/100, and so on, because simply saying they mean 0.1, 0.01, etc.
has not clarified AT ALL to a child what it means to have a position to the right. Here are
some examples of rewriting decimal numbers in expanded form:
5962.8 = 5 · 1000 + 9 · 100 + 6 · 10 + 2 · 1 + 8 ·
1
10
1
1
1
1
1
+ 7·
+ 1·
+ 0·
+ 3·
10
100
1000
10000
100000
1
In the second example, we as adults know that the ”0 · 10000 ” portion doesn’t contribute to
the value and so are inclined to skip over it. However, to help children learn the place value
names consistently, I encourage you NOT to skip over a position in your expanded forms
just because there’s a 0 in the position.
2.97103 = 2 · 1 + 9 ·
When it comes to expanded form, we learn in MATH 210 that we can more conveniently
just use exponents on the number 10 to name the positions, because those exponents count
up in a completely predictable pattern. This pattern actually extends to the left as well.
Think of it this way: if
the ten thousands position (10, 000s) means 104 and
the thousands position (1, 000s) means 103 and
the hundreds position (100s) means 102 and
the tens position (10s) means 101
with the exponents counting DOWN by one each time we move one position to the right,
then it makes sense to believe that they KEEP counting down, so that
the ones position (1s) means 100
the tenths position (
1
s) means 10−1
10
1
s) means 10−2 and so on
100
This pattern shows up even more clearly if we label the positions in a ”blank” number with
their exponent names:
the hundredths position (
104
103
102
101
100
•
10−1
10−2
10−3
Two things to help children understand are (1) that there’s no ”symmetry” to the left and
right in that they should NOT write 100 again as the position just to the right of the decimal
point, and (2) the reason 100 goes on the LEFT is because 101 is only the tens place, and
we do still need an exponent name for the ones place before we hit the decimal point.
These exponent-showing ways of naming the positions mean that the following are also
correct expanded forms:
5962.8 = 5 · 103 + 9 · 102 + 6 · 101 + 2 · 100 + 8 · 10−1
2.97103 = 2 · 100 + 9 · 10−1 + 7 · 10−2 + 1 · 10−3 + 0 · 10−4 + 3 · 10−5
Notice again how NOT skipping over that zero digit makes it easy to get the position names
right. On assignments and exams, I’ll always let you know whether I want expanded form
WITH exponents (including on 101 and 100 ), like these immediately above, or expanded
form WITHOUT exponents, like the ones on the front page.
Teachers can strengthen student understanding of place value names and meanings by having children solve ”decimal clue” problems. I have provided you with part of a fourth grade
worksheet entitled ”Decimal Number Logic” that features such problems. Take a few minutes to work those children’s level problems now.
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗(worksheet) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
I too like to ask decimal clue problems, but obviously, I must increase the level of difficulty to
make them more appropriate for the skill set you all must develop. Typically, my problems
will not tell you which digits appear in advance, so you should initially expect that any of
the digits 0-9 can go in any positions, with the exception that a number like 00578.2 doesn’t
make sense and instead should be written as plain 578.2. I will also sometimes tell you that
a number does not have ”trailing 0s,” by which I just mean that you don’t get to add extra
0s onto the right end either: 5.6000 has trailing 0s, but writing it as 5.6 gets rid of them.
A good strategy to use in solving my decimal clue problems is to make a series of blank
positions for your number - as many as you think you need (past students have often labeled
them underneath with the exponent names for the positions too) - and record in each blank
the digits that you rule in or out for going there. It’s also handy to keep digits that have
to go ”together” lined up horizontally. I do sometimes include information about rounding,
so if you don’t remember how to round numbers from MATH 210, you’ll want to review
that; I’ve attached a 5th grade rounding explanation to the 4th grade worksheet you just
completed. Finally, my decimal clue problems sometimes have several correct answers, and
I may ask you to find some or all of them. Here’s a sample problem and solution:
Find all Hindu-Arabic (that’s just the name for our base ten system of writing numbers, as
you learned in MATH 210) that simultaneously satisfy all of these clues:
The digit 7 is not used. There are no trailing 0s.
There are no digits in the 10−3 and smaller positions.
Rounded to the nearest ten, the number is 20.
The digit in the 100 position is one more than the digit in the hundredths position.
The sum of the ones, tens, and hundredths digits is 15.
Solution: I’ll make my blanks and put Xs in the ones I realize I don’t need. It’s also ok just
to not draw those positions in the first place.
X
X
101
X
100
•
10−1
10−2
X
X
How did I know to cross out the hundreds position and beyond on the LEFT? Because if
the number only rounds to 20, it can’t be big enough to need those positions. Let’s use the
rounding clue next - you would typically have just one drawing that evolves as you use each
clue, but in this explanation, I’ll have to show you each change to the diagram separately. So
here’s what it looks like after I write the possible digits for the tens and ones columns, based
on the rounding clue:
2
2
2
2
2
1
1
61
1
1
4
3
2
1
0
9
8
67
6
5
I get all these options
because ”15 point something”
up to ”24 point something”
are the only numbers
that would round to 20
when rounded to the nearest ten
However I also have to
cross out 17
because of the 7 clue
•
10−1
10−2
Now let’s interpret the clue about the 100 and the hundredths positions. The digit in the 100
position - that’s the ones position - is BIGGER: it’s 1 more than the other position. So now
I know what I can write in the hundredths position, but I still won’t know about the tenths
position yet:
2
2
2
2
62
1
61
61
1
1
4
3
2
1
60
9
68
67
6
5
??
??
??
??
??
??
??
cross out
??
•
10−1
3
2
1
0
not possible to have DIGIT thats one less than 0
8
6 7 − no good : 7s not allowed
17
5
4
Notice how a few more options have been ruled out. (When writing by hand, I just cross
out the whole line, but my typesetting software cna only show you one digit crossed out at a
time.) Finally, we’re ready for the clue about the ones, tenths, and hundredths digits added
together: we need them to make a total of 15, and I can already see that several options won’t
be able to do that when I am only allowed to write a single-digit value in the tenths position.
Here goes:
2
62
62
62
62
61
61
61
1
1
4
63
62
61
60
69
68
67
6
5
•
8
3
??
6 2 − no good : can0 t reach 15 with one more digit
??
6 1 − no good : can0 t reach 15 with one more digit
??
6 0 − no good : can0 t reach 15 with one more digit
ruled out earlier
??
6 8 − no good : already over 15
??
6 7 − ruled out earlier
ruled out earlier
4
5
6
4
So in the end, it looks like we are left with just three possible answers: 24.83, 16.45, and
15.64. Attached is an activity with two decimal clue problems on it. Work the activity
together in your groups, and then hand it in to me today as a 5-point score.
The assignment for this material is posted on the projector screen.