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Numeration and Addition/Subtraction Errors

Numeration and Addition/Subtraction Errors. Addition and Subtraction Tasks. Solve each problem. How are they similar? Different? Monica has six pens. Tyrisha gives her five more. How many does Monica now have?

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Numeration and Addition/Subtraction Errors

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  1. Numeration and Addition/Subtraction Errors

  2. Addition and Subtraction Tasks Solve each problem. How are they similar? Different? Monica has six pens. Tyrisha gives her five more. How many does Monica now have? Pedro has 11 pieces of candy. He only had six before going to the store. How many did he buy? Kobe’s mom gave him five dollars. He now has eleven dollars. How much money did he have before?

  3. Kobe’s mom gave him five dollars. He now has eleven dollars. How much money did he have before? • How much do I end up with? • What do I start with? • What changes?

  4. Structures of Addition Problems How are these problems similar? How are these problems different?

  5. Initial- the starting value Result- the “sum” or “total” Change- the amount added to the initial value Structures of Addition Problems 5 + 4 = 9

  6. Structures of Addition Tasks • Join tasks • Problems where items or groups are combined • Result unknown • Traditional addition problem, e.g. 4 + 5 = 9 • Change unknown • The second addend is missing, e.g. 4 + = 9 • Initial unknown • The first addend is missing, e.g. + 5 = 9 • Research shows that Join are easiest and Initial Unknown are the hardest.

  7. Structures of Addition Problems • Write four addition/subtraction story tasks. Include at least one of each type of problem. • Pair up and exchange tasks • Identify the type of task • Model each task in two of these ways • Cuisenaire rods • Base-10 blocks • Drawing a picture • Jot down how you modeled the task.

  8. Structures of Addition Tasks • Which was the most difficult to write? • Which was the most difficult to model? • Which manipulatives or approaches were most appropriate?

  9. Addition and Subtracting- Concrete Errors • Misuse of the manipulatives • Miscounting the manipulatives • Counting 1-to-1correspondence • Counting from one • Counting on • Misapplication of Action Language • I had four cookies. Today my Mom gave me more. Now I have nine. • What do you think students’ most common error is?

  10. Addressing Concrete Errors • Act out the problem • I had four cookies. Today my Mom gave me more. Now I have nine. • What do we start with? • What does the nine represent? • How should we act this out? • Check work when using manipulatives • Avoid double counting or omitting

  11. Addition- Symbolic Errors • Mike, p. 102 • What types of addition problems would Mike answer correctly? • Based on his error pattern, how would he solve? 58 538 +51+272

  12. Addition- Symbolic Errors • Mike, p. 102 • Work making a group of 10 • Bundled sticks, base-10 blocks • Place value frames • Chip-trading games • Different colored chips Proportional manipulatives Non-proportional manipulatives

  13. Proportional vs Non-proportional • Proportional • Base-10 blocks • Bundled straws • Popsicle sticks • Students see the relationship– 10 ones equal 1 ten • Non-proportional • Colored chips • Money (e.g., 10 pennies = 1 dime) • Students struggle to see the relationship since size of manipulatives are not proportional

  14. Mary (p. 103) What error is occurring? Are any of them correct? How would she solve? 254 618 +535+782 Possible ways to help her?

  15. Addition- Symbolic Errors • Mary, p. 103 • Estimate sums • Base-10 blocks • Pattern Chart/Place Value Chart

  16. In the Standards- K • Count objects in a set • Joining sets • Separating sets • Comparing sets

  17. In the Standards- 1st Grade • Develop fluency with single-digit addition and corresponding differences using strategies such as modeling, composing and decomposing quantities, using doubles, and making tens.

  18. In the Standards: 2nd Grade • Develop fluency with multi-digit addition and subtraction through 999 using multiple strategies. • Strategies for adding and subtracting numbers. • Estimation of sums and differences in appropriate situations. • Relationships between operations

  19. In the Standards: 3rd Grade • Develop fluency with multi-digit addition and subtraction through 9,999 using: • Strategies for adding and subtracting numbers. • Estimation of sums and differences in appropriate situations. • Relationships between operations.

  20. In the Standards: 4th and 5th Grade • Skills with whole numbers are maintained • Work with decimals and fractions is the focus with addition and subtraction

  21. Teaching the Basic Facts • Should the basic computational facts be taught in school? • How much focus should be on the basic facts in the various grade levels? • What strategies do you think are most effective at teaching the basic facts? • Readings…we’ll come back to this in November

  22. Symbolic Errors- Gary (p. 102) 7 + 8 = 14 7 + 6 = 12 8 + 6 = 13 What would he do for… 7 + 7 = 8 + 5 = What is Gary doing incorrectly? Suggested modifications?

  23. Subtraction- Errors • Misuse of Action Language The shuttle astronauts have been aloft for 16 hours. How many more hours will it take to complete one full 24 hour day in space? 16 + __ = 24 16 + 24 = 40. Here, students will say the answer is 40. Operative- knowing the action, “what to do” Figurative- interpreting and reading the language of the problem

  24. Subtraction- Errors • Confusion between subtraction and addition • Pre-operational behavior • Strategies • Give students a variety of subtraction problems • Tasks that include open number sentences • 17 + ___ = 35 • 35 - ___ = 17

  25. Subtracting Errors • Regrouping strategies • Recognizing you need to regroup • Conceptually-based procedures • Traditional algorithm: Slash and dash • Compensation method

  26. Subtraction- Errors • Renaming errors • Explicit trade: 1 hundred=10 tens, 1 ten=10 ones • Regrouping in the wrong place • 1 hundred = 10 ones?? • Modifications? • Base-10 blocks and place value charts • Bundled straws (calendar manipulative)

  27. About Basic Facts • Fluency through Playing Math Games • 20 • Closer to 100 • Closest to 20 (or another number)

  28. Number lines • Open number line (Dutch) • 19 + 27 • 27 gets split into 20+1+6 • Move along the number line • Why does +1 get done first? +1 +20 +6 19 20 30 40 46

  29. Number lines • Open number line (Dutch) • 32-15 • 15 gets split into 2+3+10 • Move backwards along the number line • Why does -2 get done first? 3 10 2 17 20 30 32

  30. Alternative Algorithms (Addition) • Partial addition • Expanded form 33+29 30+20 = 50 3+9= 12 50+12=62 + +

  31. Alternative Algorithms (Addition) • Making 10 33+ (7+20+2) = 40 + 20 + 2 = 62 • Compensation 33-1 = 32 29+1 = 30 32+30= 62 +

  32. Alternative Algorithms (Subtraction) • Compensation - - 36

  33. Alternative Algorithms (Subtraction) +1 • Compensation 29+1= 30 53+1 = 54 54-30= 24 - - +1 29 30 53 54

  34. Alternative Algorithms (Subtraction) • Reverse subtraction • You must borrow 1 from the 5 • Instead of making the 5 a 4 and the adding 10 to the 3 • Students add 10 to both numbers. Make the 2 into a 3. • The first number gets 10 ones. The second number gets 1 ten. -

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