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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). Understanding Addition and Subtraction.

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Operations and Whole Numbers: Developing Meaning

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  1. 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)

  2. Understanding Addition and Subtraction

  3. Eleven Addition and Subtraction Problem Typeshttp://www.mathplayground.com/ThinkingBlocks/thinking_blocks_modeling%20_tool.html • 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?

  4. 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?

  5. 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?

  6. 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?

  7. 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)

  8. 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)

  9. 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)

  10. 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

  11. Remember the Opposite-Change 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

  12. The Opposite-Change Rule • Another way. Subtract and add 4. 59 (subtract 4) 55 + 26 (add 4) + 30 85

  13. Same-Change Rule for Subtraction 92 –36 = ? One way add 4 92 (add 4) 96 - 36 (add 4) – 40 56 Another way subtract 6 92 (subtract 6) 86 - 36 (subtract 6) - 30 56

  14. Multiplication Algorithms • Multiplying in Columns (Standard Algorithm) • Lattice Method

  15. Multiplying in Columns Method 4 * 236 = ? 1 2 2 3 6 * 4 9 4 4

  16. Lattice Method • 6 * 815 = ? (4890) • The box with cells and diagonals is called a lattice. • 8 1 5 6

  17. 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?

  18. Types of Multiplication and Division Problems • Rate • Partitive Divison – size of group is unknown Example: On Mitchell’s trip to NYC, they drove 400 miles and used 12 gallons of gasoline. How many miles per gallon did they average?

  19. 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?

  20. 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?

  21. The End www.math.ccsu.edu/mitchell/numbersandoperations2.ppt

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