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Week 5

Week 5. Warm up Problem: What did the student do wrong? 457 - 29 = 338 Explain the error in a sentence. What makes a good answer?. 457 - 29 = 338. Incomplete answers: was supposed to cross out the 5 and make it a 4.

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Week 5

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  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. Agenda--finish subtraction and get ready for exam • Exam (50 minutes) next class. Bring calculator, pen or pencil, and colored pens, pencils, markers. Short survey after exam. • 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 and HW on D2L. • Buy the class notes!!!

  7. -2 -1 0 1 2 3 4 5 6 7 8 9 10 Number Line Model • 7 - 9 = -2 • Why is it important to start at 0?

  8. Four related facts • If 9 - 4 = 5, then9 - 5 = 4, 4 + 5 = 9, and 5 + 4 = 9 • You try: If 74 - 61 = 13, then …

  9. Mental Subtraction • Not as obvious as mental addition • 65 - 28Break apart the second number (65 - 20) - 8 • Adding up 28 + 30 = 58, 58 + 7 = 65, 30 + 7 • Compensation (65 + 2) - (28 + 2) • Compatible Numbers (65 - 25, add back 3)

  10. The name of the strategy is not important… • You try… and be ready to explain how you did it. Are there other ways? • 91 - 82 • 97 - 39 • 301 - 293

  11. Regrouping • Show a diagram for • 302 - 84

  12. 302 - 84 • Start with…

  13. Now, indicate what you are subtracting • 302 - 84: Let red be the part you take away

  14. You try: draw pictures • 58 - 37 • 47 - 29

  15. Subtraction the way you learned it… 6 7 8- 3 9 2 2 8 6 5 1 Why did you cross out the 6? Why did you put a little “1” next to the 7? Can you show this with pictures?

  16. Here are three other ways to think about subtraction • Explain why this works--use pictures or manipulatives • 9 8 4- 3 6 8 • 6 1 6 1 7

  17. Why does this work? • This way worked because • 984 - 368 is the same as adding 10 to both numbers: • 984 + 10 = 980 + 4 + 10 = 980 + 14 • 368 + 10 = 360 + 8 + 10 = 370 + 8

  18. 1 1 Use drawings or manipulatives to explain… • Here is another one. • 7 8 2 7 8 2 7 8 2- 3 4 7- 3 4 7- 3 4 7 • 4 4 4 4 5 • 4 3 5

  19. Use drawings or manipulatives to explain… • Here is yet another one. • 3 2 6- 2 9 4 2 -7 01 0 0 • 100 - 70 + 2 = 132

  20. A real problem • Place the digits 1, 2, 3, 6, 7, 8 in the boxes to obtain: • Greatest sumLeast sumGreatest differenceLeast difference

  21. A quick review of subtraction • Try this: Explain what the student is doing. 8 7 8 8- 3 9 --> - 4 0 4 8

  22. 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

  23. Use manipulatives or diagrams to show or use words to explain why… • 9 8 4 9 8 14- 3 6 8- 3 7 8 6 1 6

  24. 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.

  25. Use pictures or diagrams to explain… • Write 1e716 in base 10.Write 39010 in base 5.True or false: 425 = 346

  26. Why is this a pattern? • Find the one’s digit for 329. • Can you find the one’s digit for the first 10 powers of 3? • 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, …

  27. 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.

  28. 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

  29. 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: 10 apples and 17 oranges

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