数学加减法方法

1、Operations and Whole Numbers: Developing MeaningModel by beginning with word problemsReal-world setting or problemModelsConcretePictorialMentalLanguageMathematical World(symbols)1Understanding Addition and SubtractionThere are four types of addition and subtraction problemsJoinactionSeparateactionPa

2、rt-part-whole relationships of quantitiesComparerelationships of quantities2Eleven Addition and Subtraction Problem TypesJoin 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. N

3、ow 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?3Separate Result Unknown Peter had 11 cookies. He gave 7 cookies to Erika. How many cookies does Pe

4、ter 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?4Part-Part-Whole W

5、hole 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?5Compare Dif

6、ference UnknownPeter 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 co

7、okies does Erika have?6Using Models to Solve Addition and Subtraction ProblemsDirect 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 ModelingFor example, John had

8、 4 cookies. Jennifer gave him 7 more cookies. How many cookies does John have?(join)7Direct Modeling for Join and SeparateDavid 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

9、 many cookies did Brian give to Tina?(separate)8Modeling part-part-whole and compare ProblemsMichelle 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 man

10、y cookies does Meghan have? (part-part-whole)9Writing Number Sentences for Addition and SubtractionOnce 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

11、 like this2 + 5 = ? Or 2 + ? =710Addition AlgorithmsThe 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

12、 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.11Partial-Sums MethodExample: 348 + 177=?100s 10s1s 3 48+1 7 7 4 0 0 Add the

13、 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)12Column Addition MethodExample: 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 te

14、ns column. 6 5 7 Adjust the 10s and 100s: 15 tens = 1 hundred and 5 tens. Trade the 1 hundred into the hundreds column.13A Short Method248 + 187=? 1 1 2 4 8+ 1 8 7 4 3 58 ones + 7 ones = 15 ones = 1 ten + 5 ones1 ten + 4 tens + 8 tens = 13 tens = 1 hundred + 3 tens1 hundred + 2 hundreds + 1 hundred

15、= 4 hundreds14The Opposite-Change RuleAddends 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 th

16、e addends to a number that has zero in the ones place.One way: Add and subtract 59 (add 1) 60+26 (subtract 1) +25 8515The Opposite-Change RuleAnother way. Subtract and add 4. 59 (subtract 4) 55+ 26 (add 4) + 30 8516Subtraction AlgorithmsThe Trade-First Subtraction Method is similar to the method tha

17、t most adults were taughtLeft-to-Right Subtraction Method Partial-Differences MethodSame-Change Rule17The Trade-First MethodIf 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

18、it, adjust the top number before doing any subtracting. Adjust the top number by “trading”18The Trade-First Method ExampleSubtract 275 from 463 using the trade-first method100s 10s 1s 4 6 3- 2 7 5Look at the 1s place. You cannot subtract 5 ones from 3 ones19The Trade-First Method Example100s 10s 1s

19、Subtract 463 - 275 5 13 4 6 3 - 2 7 5So trade 1 ten for ten ones. Look at the tens place. You cannot remove 7 tens from 5 tens.20The Trade-First Method ExampleSubtract 463 275100s 10s 1s 15 3 5 13 4 6 3- 2 7 5 1 8 8So trade 1 hundred for 10 tens. Now subtract in each column.21Left to Right Subtracti

20、on MethodStarting at the left, subtract column by column. 9 3 2 - 3 5 6Subtract the 100s 932 - 300Subtract the 10s 632 - 50Subtract the 1s 582 - 6 57622Partial-Differences MethodSubtract from left to right, one column at a time. Always subtract the larger number from the smaller number.If the smalle

21、r 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.23Partial-Differences Method Example 8 4 6 - 3 6 3Subtract the 100s 800 300 +5 0 0 Subtract the 10s 60 40 - 2 0Subtract the 1s 6 - 3 + 3 4 8 324Same-Change R

22、ule Example92 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 5625Multiplication AlgorithmsPartial-Products MethodsLattice Method26Partial-Products MethodYou must keep track of the place value of each digit. Write 1s 10s 100s above t

23、he columns. 4 * 236 = ?Think of 236 as 200 + 30 + 6Multiply each part of 236 by 427Partial Products Method4 * 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 products28Lattice Method6 * 815 = ?The box with cells and diagonals is called a lattice. 8

24、1 54 80 63 0629Types of Multiplication and Division ProblemsEqual GroupingPartitive Division Size of group is unknownExample: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?30Types of Multiplication and Divi

25、sion ProblemsRatePartitive Divison size of group is unknownExample: On the Mitchells trip to NYC, they drove 400 miles and used 12 gallons of gasoline. How many miles per gallon did they average?31Types of Multiplication and Division ProblemsNumber of equal groups is unknownQuotative DivisionExample