Transcript Operations and Whole Numbers: Developing Meaning


Operations and Whole
Numbers: Developing
Meaning
Model by beginning with word problems
Real-world setting or problem
Models
Concrete
Pictorial
Mental
Language
Mathematical World
(symbols)
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Understanding Addition and
Subtraction

There are four types of addition and
subtraction problems
– Join
action
– Separate
action
– Part-part-whole
relationships of quantities
– Compare
relationships of quantities
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Eleven Addition and
Subtraction Problem Types

Join
Result Unknown
Peter had 4 cookies. Erika gave him 7 more cookies. How many
cookies does Peter have now?
Change Unknown
Peter had 4 cookies. Erika gave him some more cookies. Now
Peter has 11 cookies. How many cookies did Erika give him?
Start Unknown
Peter had some cookies. Erika gave him 7 more cookies. Now
Peter has 11 cookies. How many cookies did Peter have to start
with?
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Separate
Result Unknown
Peter had 11 cookies. He gave 7 cookies to Erika. How
many cookies does Peter have now?
Change Unknown
Peter had 11 cookies. He gave some cookies to Erika.
Now Peter has 4 cookies. How many cookies did Peter
give to Erika?
Start Unknown
Peter had some cookies. He gave 7 cookies to Erika.
Now Peter has 4 cookies. How many cookies did Peter
have to start with?
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Part-Part-Whole
Whole Unknown
Peter had some cookies. Four are chocolate
chip cookies and 7 are peanut butter cookies.
How many cookies does Peter have?
Part Unknown
Peter has 11 cookies. Four are chocolate chip
cookies and the rest are peanut butter cookies.
How many peanut butter cookies does Peter
have?
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Compare
Difference Unknown
– Peter has 11 cookies and Erika has 7 cookies. How
many more cookies does Peter have than Erika?
Larger Unknown
Erika has 7 cookies. Peter has 4 more cookies than
Erika. How many cookies does Peter have?
Smaller Unknown
Peter has 11 cookies. Peter has 4 more cookies than
Erika. How many cookies does Erika have?
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
Using Models to Solve
Addition and Subtraction
Problems
Direct modeling refers to the process of
children using concrete materials to exactly
represent the problem as it is written.
 Join and Separate (problems involving
action) work best with Direct Modeling
 For example, John had 4 cookies. Jennifer
gave him 7 more cookies. How many
cookies does John have?(join)
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Direct Modeling for Join and
Separate

David had 10 cookies. He gave 7 cookies to
Sarah. How many cookies does David have
now? (separate)
 Brian had 10 cookies. He gave some
cookies to Tina. Now Brian has 4 cookies.
How many cookies did Brian give to
Tina?(separate)
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Modeling part-part-whole and
compare Problems

Michelle had 7 cookies and Katie had 3
cookies. How many more cookies does
Michelle have than Katie? (compare)
 Meghan has some cookies. Four are
chocolate chip cookies and 7 are peanut
butter cookies. How many cookies does
Meghan have? (part-part-whole)
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Writing Number Sentences for
Addition and Subtraction

Once the children have had many
experiences modeling and talking about real
life problems, the teacher should encourage
children to write mathematical symbols for
problems.
 A number sentence could look like this
2+5=?
Or
2 + ? =7
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Addition Algorithms

The Partial-Sums Method is used to find sums
mentally or with paper and pencil.
 The Column-Addition Method can be used to find
sums with paper and pencil, but is not a good
method for finding sums mentally.
 The Short Method adds one column from right to
left without displaying the partial sums(the way
most adults learned how to add)
 The Opposite-Change Rule can be used to subtract
a number from one addend, and add the same
number to the other addend, the sum is the same. 11
Partial-Sums Method

Example: 348 + 177=?
•
•
•
•
•
•
•
100s
3
+1
4
1
5
10s
4
7
0
1
1
2
1s
8
7
0 Add the 100s (300 + 100)
0 Add the 10s (40 + 70)
5 Add the 1s ( 8 + 7)
5 Add the partial sums (400+110+15)
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Column –Addition Method
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Example: 359 + 298=?
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100s 10s 1s
3
5 9
+2
9 8
5 14 17 Add the numbers in each column
5 15 7 Adjust the 1s and 10s: 17 ones = 1 ten
and 7 ones
Trade the 1 ten into the tens column.
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5 7 Adjust the 10s and 100s: 15 tens = 1
hundred and 5 tens. Trade the 1 hundred into the
hundreds column.
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A Short Method
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248 + 187=?
1 1
2 4 8
+1 8 7
4 3 5
8 ones + 7 ones = 15 ones = 1 ten + 5 ones
1 ten + 4 tens + 8 tens = 13 tens = 1 hundred + 3
tens
1 hundred + 2 hundreds + 1 hundred = 4 hundreds
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The Opposite-Change Rule
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Addends are numbers that are added.
In 8 + 4 = 12, the numbers 8 and 4 are addends.
If you subtract a number from one addend, and
add the same number to the other addend, the sum
is the same. You can use this rule to make a
problem easier by changing either of the addends
to a number that has zero in the ones place.
One way: Add and subtract
59 (add 1)
60
+26 (subtract 1) +25
15
85
The Opposite-Change Rule

Another way. Subtract and add 4.
 59 (subtract 4) 55
 + 26 (add 4)
+ 30

85
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Subtraction Algorithms

The Trade-First Subtraction Method is
similar to the method that most adults were
taught
 Left-to-Right Subtraction Method
 Partial-Differences Method
 Same-Change Rule
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The Trade-First Method

If each digit in the top number is greater
than or equal to the digit below it , subtract
separately in each column.
 If any digit in the top number is less than
the digit below it, adjust the top number
before doing any subtracting. Adjust the
top number by “trading”
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The Trade-First Method
Example
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Subtract 275 from 463 using the trade-first
method
 100s 10s 1s

4 6 3
- 2
7 5
 Look at the 1s place. You cannot subtract
5 ones from 3 ones
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The Trade-First Method
Example

100s 10s 1s
Subtract 463 - 275

5 13
 4
6 3
- 2
7 5
 So trade 1 ten for ten ones. Look at the tens
place. You cannot remove 7 tens from 5
tens.
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The Trade-First Method
Example
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Subtract 463 – 275
100s 10s 1s
15
3
5 13
4
6 3
- 2
7 5
1
8 8
So trade 1 hundred for 10 tens. Now subtract in
each column.
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Left to Right Subtraction
Method
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Starting at the left, subtract column by column.
9 3 2
-3 5 6
Subtract the 100s 932
- 300
Subtract the 10s 632
- 50
Subtract the 1s
582
- 6
576
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Partial-Differences Method

Subtract from left to right, one column at a
time. Always subtract the larger number
from the smaller number.
 If the smaller number is on the bottom, the
difference is added to the answer.
 If the smaller number is on top, the
difference is subtracted from the number.
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Partial-Differences Method
Example

8 4 6

-3 6 3
 Subtract the 100s 800 – 300 +5 0 0
 Subtract the 10s
60 – 40 - 2 0
 Subtract the 1s
6- 3 + 3
4 8 3
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Same-Change Rule Example
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92 –36 = ?
One way add 4
92 (add 4) 96
- 36 (add 4) – 40
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Another way subtract 6
92 (subtract 6) 86
- 36 (subtract 6) - 30
56
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Multiplication Algorithms

Partial-Products Methods
 Lattice Method
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Partial-Products Method
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You must keep track of the place value of
each digit. Write 1s 10s 100s above the
columns.
 4 * 236 = ?
 Think of 236 as 200 + 30 + 6
 Multiply each part of 236 by 4
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Partial – Products Method
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4 * 236 = ?
100s
2
*
4 * 200
8
4 * 30
1
4*6
0
Add these three
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partial products
10s 1s
3
6
4
0
0
2
0
2
4
4
4
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Lattice Method
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6 * 815 = ?
 The box with cells and diagonals is called a
lattice.
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8
1
5
4
0
3
6
8
6
0
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Types of Multiplication and
Division Problems
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Equal Grouping
Partitive Division – Size of group is unknown
 Example:
 Twenty four apples need to be placed into eight
paper bags. How many apples will you put in each
bag if you want the same number in each bag?
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Types of Multiplication and
Division Problems

Rate
 Partitive Divison – size of group is
unknown
 Example:
 On the Mitchell’s trip to NYC, they drove
400 miles and used 12 gallons of gasoline.
How many miles per gallon did they
average?
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Types of Multiplication and
Division Problems

Number of equal groups is unknown
 Quotative Division
 Example:
 I have 24 apples. How many paper bags will
I be able to fill if I put 3 apples in each bag?
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Types of Multiplication and
Division Problems

Number of equal groups is unknown
 Quotative Division
 Example:
 Jasmine spent $100 on some new CDs.
Each CD cost $20. How many did she buy?
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Partial Quotients Method

The Partial Quotients Method, the Everyday
Mathematics focus algorithm for division, might
be described as successive approximation. It is
suggested that a pupil will find it helpful to
prepare first a table of some easy multiples of the
divisor; say twice and five times the divisor. Then
we work up towards the answer from below. In
the example at right, 1220 divided by 16, we may
have made a note first that 2*16=32 and 5*16=80.
Then we work up towards 1220. 50*16=800
subtract from 1220, leaves 420; 20*16=320; etc..
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The End
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