Operations and Whole Numbers: Developing Meaning

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

3. 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?

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. 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.

12. Partial-Sums Method • Example: 348 + 177=? • 100s 10s 1s • 3 4 8 • +1 7 7 • 4 0 0 Add the 100s (300 + 100) • 1 1 0 Add the 10s (40 + 70) • 1 5 Add the 1s ( 8 + 7) • 5 2 5 Add the partial sums (400+110+15)

13. Column –Addition Method • Example: 359 + 298=? • 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. • 6 5 7 Adjust the 10s and 100s: 15 tens = 1 hundred and 5 tens. Trade the 1 hundred into the hundreds column.

14. A Short Method • 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

15. 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

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

17. 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

18. 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”

19. The Trade-First Method Example • 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

20. 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.

21. The Trade-First Method Example • 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.

22. Left to Right Subtraction Method • 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

23. 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.

24. 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

25. 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) 86 • - 36 (subtract 6) - 30 • 56

26. Multiplication Algorithms • Partial-Products Methods • Lattice Method

27. 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 200 + 30 + 6 • Multiply each part of 236 by 4

28. Partial – Products Method • 4 * 236 = ? 100s 10s 1s • 2 3 6 • * 4 • 4 * 200 8 0 0 • 4 * 30 1 2 0 • 4 * 6 0 2 4 • Add these three 9 4 4 • partial products

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

30. 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?

31. 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?

32. 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?

33. 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?

34. 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..

35. The End