Understanding Addition and Subtraction with Whole Numbers

1. Understanding Addition and Subtraction with Whole Numbers Adapted from: Guiding Children's Learning of Mathematicsby Leonard M. Kennedy (Author), Steve Tipps (Author), Art Johnson (Author)

2. Understanding Addition and Subtraction with Whole Numbers The Goal is Computational Fluency When students leave grade 5, they should be able to solve problems involving whole-number computation and should recognize that each operation will help them solve many different types of problems. They should be able to solve many problems mentally, to estimate a reasonable result for a problem, to efficiently recall or derive the basic number combinations for each operation, and to compute fluently with multi-digit numbers. (NCTM, 2000, 149)

3. Understanding Addition and Subtraction with Whole Numbers Conceptual Understanding • Real-life problem-solving stories provide a meaningful context for addition and subtraction • Children learn about addition and subtraction facts and rules behind them • by modeling and acting out with physical objects • by representing the operations with pictures • through naming and labeling the results of their actions

4. Understanding Addition and Subtraction with Whole Numbers Addition The action of joining or combining sets • Juan has four red pencils and three blue pencils. • Heather has 37 books in her library. She receives 5 books as presents for her birthday. • The school collection for earthquake victims was $149 on Friday and $126 on Saturday.

5. Understanding Addition and Subtraction with Whole Numbers Subtraction - Take Away Part of an original set is moved, lost, eaten, or spent • Jeff collected wheelie cars. He had 16, but traded 3 of them for a new track. • Janyce had $94 saved from birthday money. She spent $33 on new shoes, $14 for a new top, and $27 for a new skirt. • Jamal had 36 cookies. He gave two cookies to twelve friends.

6. Understanding Addition and Subtraction with Whole Numbers Subtraction - Whole-Part-Part The whole group has a common characteristic, but subgroups have distinct characteristics • The class has 25 children. Fourteen are girls. • Deidra has 17 stuffed animals. Four are bears. • Sally has 48 snapdragons. Thirty of them are white. The rest are yellow. • Darius counted 114 people at the family reunion. Forty-nine had his last name

7. Chapter 7Understanding Addition and Subtraction with Whole Numbers Subtraction - Comparison Compare the size of two sets or two objects • Damon was the shortest student at 5’1” and Shawna was the tallest at 5’9”. • The collection for earthquake victims was $149 on Friday and $126 on Saturday. • The circus put on two performances. The matinee was attended by 8,958 people and the evening by 9,348 people.

8. Chapter 7Understanding Addition and Subtraction with Whole Numbers Subtraction - Completion Compares an existing set with a desired, incomplete, or imaginary set • Saundra has saved $9. The CD she wants costs $16. • Sean is collecting state quarters. He already has 23. • The recipe called for 3 cups of water for each can of concentrate. Mr. Lopez put 6 cans of concentrate in the punch bowl. He has put in 10 cups of water so far.

9. Understanding Addition and Subtraction with Whole Numbers Knowing basic number combinations—the single-digit addition and multiplication pairs and their counterparts for subtraction and division—is essential. (NCTM, 2000, p 32)

10. Understanding Addition and Subtraction with Whole Numbers Strategic Learning of Basic Facts • Children invent and learn strategies for the basic facts. • counting on with one, with two, with three • commutative property (flip flop facts) • identity property • doubles and near doubles (neighbors) • sums of ten • associative property • Children practice facts with the strategies to develop accuracy and reasonable speed.

11. Understanding Addition and Subtraction with Whole Numbers Guidelines for Practice • Accuracy is the primary goal. Speed is secondary. • Practice strategies and facts students understand. • Keep practice sessions brief – 5 minutes. • Use a variety of materials and procedures such as games, computer activities, and flash cards. • Emphasize individual improvement rather than same goal for all children. • Review and maintain basic facts to keep the skill active.

12. Understanding Addition and Subtraction with Whole Numbers Equally essential is computational fluency—having and using efficient and accurate methods of computing… (NCTM, 2000, p 32)

13. Understanding Addition and Subtraction with Whole Numbers Transition to Larger Numbers • The same addition and subtraction situations introduced with number 1-100 apply to situations with larger numbers. • Addition and subtraction with larger numbers requires • understanding of the operations and basic facts • understanding place value representation of larger numbers

14. Understanding Addition and Subtraction with Whole Numbers Real life situations provide the context for adding and subtracting with larger numbers. • Populations of cities and countries • Areas of states and countries • Attendance at sports events • Miles on vacation, miles to the moon and planets • Weight and volume of wheat, corn, oil

15. Understanding Addition and Subtraction with Whole Numbers Working with Numbers to 1000 and Beyond • Problem-solving stories extend learning addition and subtraction with larger numbers • by modeling and acting out with base ten models up to 100, 500, and 1000 • by representing the operations with pictures of place value materials • through recording the results of their actions and pictures with numerals and place value.

16. Understanding Addition and Subtraction with Whole Numbers Developing Algorithms • After actions with materials and recording results, students invent and/or learn alternative and standard algorithms as step by step procedures for adding and subtraction numbers to 1000 and larger than 1000. • At the same time, students learn when and how to use other computational processes. • Estimation • Mental computation • Technology- calculator and computer

17. Understanding Addition and Subtraction with Whole Numbers Traditional and Alternative Addition Algorithms • In addition, regrouping and renaming is the traditional algorithm. • Low stress algorithm start with the largest place value and works left to right. • Compensation algorithm allows students to adjust the addends to make a hard addition problem into an easy problem.

18. Understanding Addition and Subtraction with Whole Numbers Traditional Low-stress Compensation 1 1 574 574 574 -11 563 +289+289+289 +11 + 300 863 700 863 150 13 863

19. Understanding Addition and Subtraction with Whole Numbers Traditional and Alternative Subtraction Algorithms • Decomposing is the traditional subtraction algorithm. • Equal addition adjusts the problem to make the subtraction easier. • Compensation is similar to compensation in addition in adjusting the problem to make an easier subtraction problem. • Low stress starts in the largest place value.

20. Understanding Addition and Subtraction with Whole Numbers Traditional Low-stress Equal Addition 416 14 5 7 4 574 574+11 585 -2 8 9-289-289+11 - 300 2 8 5 300 285 -10 -5 285

21. Understanding Addition and Subtraction with Whole Numbers Number Sense, Reasonableness, and Estimation • Throughout development of addition and subtraction concepts and procedures, number sense is developed. • In problem situations, students ask themselves, “Does this answer make sense?” • Estimation procedures are developed in situations that do not require exact answers. • Front-end estimation • Rounding estimation

22. Understanding Addition and Subtraction with Whole Numbers Addition Estimation Addition Front-End Rounding 574 574 600 +289+ 289+300 700 900 150 850

23. Understanding Addition and Subtraction with Whole Numbers Subtraction Estimation Subtraction Front End Rounding 574 574600 -289-289 - 300 300 300 -10 290

24. Understanding Addition and Subtraction with Whole Numbers Mental Computation • Alternative algorithm and estimation procedures build confidence in mental computation. • Compatible numbers ( number combinations of 10, 100, 1000) are useful in algorithms, estimation, and mental computation.

25. Understanding Addition and Subtraction with Whole Numbers Regardless of the particular [computation] method used, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general. Students also need to be able to estimate and judge the reasonableness of results. (NCTM, 2000, p 32)