Transcript Decimals

EMSE 3123
Math and Science in Education
Not Whole Numbers II:
Decimals
Presented by
Frank H. Osborne, Ph. D.
© 2015
1
Rational Numbers as Decimals
The decimal system should be a natural
outgrowth of the place value ideas covered
earlier a well as fractions.
We have already prepared children for
decimals when we introduced denominators
of 10 or 100.
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Rational Numbers as Decimals
Preparation for understanding tenths.
3
Rational Numbers as Decimals
Preparation for understanding hundredths.
4
Rational Numbers as Decimals
Examples: we indicate decimal numbers .15
and .23 on the grid by shading.
Students should be able to order any set of
decimals from smallest to largest by
shading them on the grid.
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Rational Numbers as Decimals
Each grid is a unit square. For each grid,
a. What fraction is shaded?
b. What decimal part is shaded?
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Rational Numbers as Decimals
Each grid is a unit square. For each grid,
a. What fraction is shaded?
b. What decimal part is shaded?
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Rational Numbers as Decimals
Decimals permit the place-value notation to
be extended to rational numbers.
Remember that we expressed whole
numbers such as 138 with manipulatives.
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Rational Numbers as Decimals
• Now we can add in some of the smaller parts
(say 5 of the 1/10 to the 138 we already have).
• To indicate we are moving into rational
numbers we use a dot (.) (decimal point) as an
indicator.
This is 138.5
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Rational Numbers as Decimals
We can also add hundredths. If we add 2 of
the hundredths pieces to the 138.5, we get
138.52.
It can continue on indefinitely, with the next
pieces being 1/1000, then 1/10,000, etc.
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Rational Numbers as Decimals
An alternative is to use Dienes Blocks. When
we started with these, the smallest equaled
one unit but any block can be used to
represent the whole. For example:
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Rational Numbers as Decimals
Let us use the large Dienes block cube as one.
How would you represent 4.326?
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Rational Numbers as Decimals
Let us use the large Dienes block cube as one.
How would you represent 4.326?
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Decimal Addition and Subtraction
• We use manipulatives for addition and
subtraction of decimals in the same way
that was used for whole numbers.
• Start with expressing each term using
manipulatives, then combine and regroup if
adding, or take away with borrowing if
necessary for subtraction.
• Students should be comfortable with this
and easily use play money, trading chips,
Dienes bocks, or Cuisenaire rods.
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Decimal Addition and Subtraction
With play money use only pennies (1/100 of a
dollar), dimes (1/10 of a dollar), dollars,
tens or hundreds. Money is ideal for
decimal addition or subtraction. We could
express $14.57 as
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Decimal Addition and Subtraction
With Dienes blocks, we could do the same
presentation.
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Decimal Addition and Subtraction
Let us add $14.57 and $25.64. We combine
and exchange for the next higher block
when we get ten.
In the cents column, the 7 hundredths plus
the 4 hundredths give us one tenth and one
hundredth left over.
In the tenths column we have 5 tenths plus 6
tenths plus the 1 tenth carried over for a
total of 12 tenths. We make 1 whole with 2
tenths left over.
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Decimal Addition and Subtraction
Let us add $14.57 and $25.64. We combine
and exchange for the next higher block
when we get ten.
The 1 whole is carried and added to the 4 and
5 to make 10 giving us 1 ten to carry.
Finally we have 1 ten plus 2 tens plus the 1
ten that was carried for a total of 4 tens.
The result is $40.11.
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Decimal Addition and Subtraction
Result:
Conclusion: Adding and subtracting decimals
is not any different from adding and
subtracting whole numbers.
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Decimal Addition and Subtraction
We can also use trading chips as an
illustration. We will start by making the
yellow chip to be one. We can add whole
numbers such as 2475 + 3566 = 6011.
+
=
We combine the chips and regroup.
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Decimal Addition and Subtraction
+
=

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Decimal Addition and Subtraction
Result:
• Instead of calling the yellow chip one we
could have made the green chip to be one.
• This would put a decimal point between
green and blue.
• We would have added $24.75 and $35.66 to
make the total to be $60.41.
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Decimal Addition and Subtraction
Subtraction of decimal numbers can be
performed in a similar fashion. This
example is 24.75 from 35.66.
35.66 – 24.75 =
-
=
23
Decimal Addition and Subtraction
Here are the steps. Take away 5 yellow.
Then, exchange 1 green for 10 blue—take
away 7 blue. Take away 4 green. Finally,
take away 2 reds.

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Decimal Addition and Subtraction

So, our answer to the problem is
35.66 – 24.75 = 10.91
25
Multiplying Decimals
Let us multiply 3.2 x 2. To assist us, we will let
a flat = 1, a rod will = 0.1 and a unit = 0.01.
A way to visualize multiplying decimals is to
build an array with these items.
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Multiplying Decimals
• An array is a rectangular arrangement of a
quantity in rows and columns.
• Use the pattern of directions in multiplying.
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Multiplying Decimals
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Multiplying Decimals
Each
product is
found on
the array.
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Multiplying Decimals
Here is another example: 4.6 x 1.3 =
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Multiplying Decimals
Here is another example: 4.6 x 1.3 = 5.98
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Dividing Decimals
• We have seen how we can apply whole
number techniques of addition and
subtraction to decimals.
• Similarly, we can apply the techniques of
multiplying and dividing of whole numbers
to decimals.
• We start by multiplying 3 x 7 = 21.
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Dividing Decimals
However, if the 3 x 7 is part of a whole unit
grid, then the numbers inside are actually
.3 and .7 as shown.
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Dividing Decimals
Each tiny square is 1/100 of the whole so we
can move six parts and show that we have
21/100.
Also, we can see that 21/100 is the same as
2/10 + 1/100.
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Dividing Decimals
• By the time children get this far they should
be proficient in whole number operations,
including the use of algorithms.
• They should also have some experience in
using manipulatives as applied to decimal
operations.
• Ultimately, they will realize that there is
really no difference between decimal
operations and whole number operations.
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Dividing Decimals
• As each fraction has a decimal equivalent,
division of decimals proceeds in the same
way as division of fractions and whole
numbers.
• When children begin division of decimals,
they should already be quite familiar with the
meaning of division.
• How much is 0.2 ÷ 0.05?
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Dividing Decimals
How much is 0.2 ÷ 0.05?
We know that 0.2 ÷ 0.05 is the same as
2/10 ÷ 5/100 or, “How many .05’s fit into .2?”
We see that .05 fits into .2 four times,
therefore 0.2 ÷ 0.05 = 4.
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The End
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