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Operations and Whole Operations and Whole Numbers: Developing MeaningNumbers: Developing MeaningModel by beginning with word problems

Real-world setting or problem

ModelsConcretePictorialMental

Language

Mathematical World(symbols)

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Understanding Understanding Addition and SubtractionAddition and Subtraction

Problem Type

Description Quantities Involved

JoinElements are being added or joined in a set

Starting amount, the change amount, and resulting amount

Separate Elements are being removed from a set

Starting amount, the change amount, and resulting amount

Part-Part-Whole

These problems focus on the relationship between a set and its two subsets

Two parts and the whole

Compare Involve comparisons between two different sets

Two wholes and the difference

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Eleven Addition and Eleven Addition and Subtraction Problem TypesSubtraction Problem Types

http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_mhttp://www.mathplayground.com/ThinkingBlocks/thinking_blocks_modeling%20_tool.htmlodeling%20_tool.html

Join Result Unknown

Peter had 4 cookies. Erika gave him 7 more cookies. How many cookies does Peter have now?

Change UnknownPeter had 4 cookies. Erika gave him some more cookies. Now Peter has 11 cookies. How many cookies did Erika give him?

Start UnknownPeter 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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SeparateSeparate Result Unknown

Peter had 11 cookies. He gave 7 cookies to Erika. How many cookies does Peter have now?

Change UnknownPeter had 11 cookies. He gave some cookies to Erika. Now Peter has 4 cookies. How many cookies did Peter give to Erika?

Start UnknownPeter 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-WholePart-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 UnknownPeter 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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CompareCompare Difference Unknown

– Peter has 11 cookies and Erika has 7 cookies. How many more cookies does Peter have than Erika?

Larger UnknownErika has 7 cookies. Peter has 4 more cookies than Erika. How many cookies does Peter have?

Smaller UnknownPeter has 11 cookies. Peter has 4 more cookies than Erika. How many cookies does Erika have?

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Using Models to Solve Using Models to Solve Addition and Subtraction Addition and Subtraction

ProblemsProblemsDirect 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 Direct Modeling for Join and SeparateJoin 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 Modeling part-part-whole and compare Problemscompare 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 Writing Number Sentences for Addition and SubtractionAddition 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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Remember the Opposite-Remember the Opposite-Change RuleChange Rule

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 85

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The Opposite-Change RuleThe Opposite-Change Rule

Another way. Subtract and add 4. 59 (subtract 4) 55

+ 26 (add 4) + 30

85

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Same-Change Rule for Same-Change Rule for SubtractionSubtraction

92 –36 = ?One way add 4 92 (add 4) 96 - 36 (add 4) – 40 56Another way subtract 6 92 (subtract 6) 86 - 36 (subtract 6) - 30 56

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Multiplication AlgorithmsMultiplication Algorithms

Multiplying in Columns (Standard Algorithm)

Lattice Method

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Multiplying in Columns Multiplying in Columns MethodMethod

4 * 236 = ? 1 2

2 3 6

* 4

9 4 4

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Lattice MethodLattice Method6 * 815 = ? (4890)The box with cells and diagonals is called a

lattice. 8 1 5

4

8

0

6

3

06

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Types of Multiplication and Types of Multiplication and Division ProblemsDivision 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?

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Types of Multiplication and Types of Multiplication and Division ProblemsDivision Problems

RatePartitive Divison – size of group is unknownExample:

On 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 Types of Multiplication and Division ProblemsDivision 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 Types of Multiplication and Division ProblemsDivision Problems

Number of equal groups is unknownQuotative DivisionExample:Jasmine spent $100 on some new CDs.

Each CD cost $20. How many did she buy?

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The EndThe End

www.math.ccsu.edu/mitchell/numbersandoperations2.ppt