# Operations and Whole Numbers: Developing Meaning

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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)

There are four types of addition and subtraction problems Join action Separate action Part-part-whole relationships of quantities Compare relationships of quantities

Eleven Addition and Subtraction Problem Types

Separate Result Unknown

Part-Part-Whole Whole Unknown

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?

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)

Direct Modeling for Join and Separate

Modeling part-part-whole and compare Problems

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 ? =7

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.

Partial-Sums Method Example: 348 + 177=? 100s 10s 1s 3 4 8 +1 7 7
Add the 100s ( ) Add the 10s ( ) Add the 1s ( 8 + 7) Add the partial sums ( )

Example: =? 100s 10s 1s Add the numbers in each column Adjust the 1s and 10s: 17 ones = 1 ten and 7 ones Trade the 1 ten into the tens column. Adjust the 10s and 100s: 15 tens = hundred and 5 tens. Trade the 1 hundred into the hundreds column.

A Short Method =? 1 1 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

The Opposite-Change Rule
Addends are numbers that are added. In = 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) +26 (subtract 1) +25 85

The Opposite-Change Rule
Another way. Subtract and add 4. 59 (subtract 4) 55 (add 4) 85

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

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”

Subtract 275 from 463 using the trade-first method 100s 10s 1s Look at the 1s place. You cannot subtract 5 ones from 3 ones

100s 10s 1s Subtract So trade 1 ten for ten ones. Look at the tens place. You cannot remove 7 tens from 5 tens.

Subtract 463 – 275 100s 10s 1s 15 So trade 1 hundred for 10 tens. Now subtract in each column.

Left to Right Subtraction Method
Starting at the left, subtract column by column. Subtract the 100s 932 - 300 Subtract the 10s - 50 Subtract the 1s - 6 576

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.

Partial-Differences Method Example
Subtract the 100s – Subtract the 10s – Subtract the 1s

Same-Change Rule Example
92 –36 = ? One way add 4 92 (add 4) 96 - 36 (add 4) – 40 56 Another way subtract 6 92 (subtract 6) (subtract 6) - 30

Multiplication Algorithms
Partial-Products Methods Lattice Method

Partial-Products Method
You must keep track of the place value of each digit. Write 1s 10s 100s above the columns. 4 * 236 = ? Think of 236 as Multiply each part of 236 by 4

Partial – Products Method
4 * 236 = ? s 10s 1s * 4 * 4 * 4 * Add these three partial products

Lattice Method 6 * 815 = ? The box with cells and diagonals is called a lattice. 4 8 6 3 6

Types of Multiplication and Division Problems
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?

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?

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?

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?

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 *16=800 subtract from 1220, leaves 420; 20*16=320; etc..

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