February 5, 2009

1. February 5, 2009 “Trust yourself. You know more than you think you do.” -Benjamin Spock

2. February 5, 2009 • Buy Class Notes in bookstore • Exam 1: Thursday, 2/12 *50 minutes* Covers 1.2 – 1.4, 1.7, 2.3, 3.1, 3.2 in text, explorations 1.1, 1.4, 1.7, 2.7, 2.9, 3.1, 3.6 and notes/problems from class. • Sample questions online (answers will be posted) • Drop deadline: Tuesday, 2/10 • Section 3.2 covered today

3. 3.2 – Subtraction Ways to represent subtraction (“pictorial models”): • Take away • Comparison

4. -2 -1 0 1 2 3 4 5 6 7 8 9 10 3.2 (cont’d) • Number line model Ex: 7 – 9 = -2 Why is it important to start at 0?

5. 3.2 (cont’d) Vocabulary: for A – B = C • A is called the Minuend • B is called the Subtrahend • C is called the Difference Do the commutative and associative properties from addition apply here, too?

6. 3.2 (cont’d) Examples: Is A – B = B – A? Is 6 – 4 = 4 – 6? What is true of 6 – 4 and 4 – 6? One of the worst things we can do is lie to kids: “You can’t take a bigger number away from a smaller number.” Or “You have to put the bigger number first (in the minuend).”

7. 3.2 (cont’d) (NO, subtraction is neither commutative nor associative.)

8. 3.2 (cont’d) It’s important that students see connections between computations like these: Ex: If 9 – 4 = 5, then 9 – 5 = 4 and 4 + 5 = 9 and 5 + 4 = 9 You try: If 74 – 61 = 13, then…

9. 3.2 (cont’d) Mental Subtraction (Less obvious than mental addition) Ex: 65 – 28 • Break subtrahend apart: (65 – 20) – 8 • Adding up: 28 + 30 = 58 and 58 + 7 = 6 then 30 + 7 = 37is the total difference • Compensation: (65 + 2) – (28 + 2) • Compatible numbers: (65 – 25) – 3

10. 3.2 (cont’d) As with mental addition, the name of the strategy is not important. Ex: (you try, and write down an explanation of how you did these): • 91 – 82 • 97 – 39 • 201 – 293

11. 3.2 (cont’d) Another strategy for subtraction: Regrouping Ex: Use base 10 blocks to demonstrate the subtraction problem 302 – 84

12. 3.2 (cont’d) 302 – 84: Start with:

13. 3.2 (cont’d) 302 – 84: Now, look at the part you are taking away (shown here in black). We need to regroup one flat into 10 longs, and one of those longs into 10 units to remove 84 from 302. The result? 218

14. 3.2 (cont’d) Ex: What did the student do wrong? 457 – 29 = 338 (Explain the error in a sentence.)

15. 3.2 (cont’d) Ex: What did the student do wrong? 457 – 29 = 338 (Explain the error in a sentence.) An incomplete answer: “The student was supposed to cross out the 5 and make it a 4.” (This does not explain mathematically why it is wrong. It only says which rule was broken in the standard algorithm.)

16. 3.2 (cont’d) 457 – 29 = 338 An better answer: “In the minuend, we need to exchange a ten for some ones. So, 5 tens and 7 ones is the same as 4 tens and 17 ones. This student did not trade the ten away, and so thinks that 5 tens and 7 ones is the same as 5 tens and 17 ones.”

17. 3.2 (cont’d) As you consider student errors like this, ask yourself… • Was the student applying correct “rules” incorrectly? • Would the student have made this mistake if he/she was using base 10 blocks? • If the student thought of each number as more than a collection of digits (e.g. 56 is 5 tens and 6 ones), would this mistake be made? • In my answer, does my explanation focus on rules or on the mathematical reason why this error does not produce the correct answer?

18. Homework Due Tuesday, 2/10: Link to Online Homework List