1. Week 5 Warm up Problem: What did the student do wrong? 457 - 29 = 338 Explain the error in a sentence.

2. What makes a good answer? 457 - 29 = 338. Incomplete answers: 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. 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.

3. Types of errors 36 + 28 = 91 Possible answers: Student added 6 + 8 = 14, and put the 1 down and the 4 up when it should be the 4 down and the 1 up. You need to explain why this is wrong--otherwise, addition is just a bunch of rules! Better answer: Student added 6 + 8 = 14, but thought that 14 is 4 tens and and 1 one. But 14 is really 1 ten and 4 ones. Correct answer is 1 ten and 4 ones and 5 tens which is 64.

4. Types of errors 36 + 28 = 514 Possible answer: The student put the 5 next to the 14, which is wrong. Answer should be 64. Better answer: The student added 6 + 8 which is 14, or 1 ten and 4 ones. But then the student added 3 tens and 2 ones, and recorded this as 5 hundreds instead of 5 tens. Answer should be 6 tens and 4 ones, or 64.

5. Types of errors 365 + 287 = 742 Possible answer: The student added 5 + 7 = 12, but put the 1 on top of the 3 instead of the 6. Better answer: The student added 5 + 7 = 12, which is 1 ten and 2 ones. But the student recorded this as 1 hundred and 2 ones. The correct answer is 652.

6. Exam (50 minutes) next class. Bring calculator, pen or pencil, and colored pens, pencils, markers. Don’t memorize the names of strategies for mental calculation or estimation. Do memorize the number systems (Roman, Egyptian, Mayan, Babylonian, and Alphabitian) through hundreds. Answers to sample questions Buy the class notes!!!

7. A real problem Place the digits 1, 2, 3, 6, 7, 8 in the boxes to obtain: Greatest sum Least sum Greatest difference Least difference

8. A quick review of subtraction Try this: Explain what the student is doing. 8 78 8 - 3 9 --> - 4 0 4 8

9. A quick review of subtraction Find a, b, c, and d that will make this subtraction problem work. (a, b, c, d all different numbers.) 6 a b - c 8 b 1 d a

10. Use manipulatives or diagrams to show or use words to explain why… 9 8 49 8 1 4 - 3 6 8 - 3 7 8 6 1 6

11. Alphabitia… C B A + D C A NO NEW SYMBOLS! You may only use A, B, C, D, and 0. In any base, base 7, you may use 0, 1, 2, 3, 4, 5, 6. In base 9: 0, 1, 2, 3, 4, 5, 6, 7, 8. In base N: 0, 1, …, N-1.

12. Use pictures or diagrams to explain… Write 1e7 16 in base 10. Write 390 10 in base 5. True or false: 42 5 = 34 6

13. Why is this a pattern? Find the one’s digit for 3 29. Can you find the one’s digit for the first 10 powers of 3? 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, …

14. So, there are 4 terms that repeat in this sequence. To find the 29th term, do 29 ÷ 4 = 7 R 1. So, since the remainder is 1, we look for the first term, which is 3. Now, how do we know this patterns continues? Be able to write a sentence for this.

15. It continues because when we multiply the ones digits of each factor by 3, we are always going to have one of these: ….3 3 = …..9 ….9 3 = …..7(from 27) ….7 3 = …..1(from 21) ….1 3 = …..3

16. Problem Solving I bought 27 apples and oranges. Apples cost $0.59 each. Oranges cost $0.69 each. Before tax, I spent $17.63. How many apples and how many oranges did I buy? Answer: You bought 10 apples and 17 oranges.