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?
= 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
= 91
Possible answers: Student added = 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 = 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
= 514
Possible answer: The student put the 5 next to the 14, which is wrong. Answer should be 64.
Better answer: The student added 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
= 742
Possible answer: The student added = 12, but put the 1 on top of the 3 instead of the 6.
Better answer: The student added = 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. Number Line Model 7 - 9 = -2 Why is it important to start at 0?

8. Four related facts
If = 5, then = 4, = 9, and = 9
You try: If = 13, then …

9. Mental Subtraction Not as obvious as mental addition
Break apart the second number ( ) - 8
Adding up = 58, = 65,
Compensation (65 + 2) - (28 + 2)
Compatible Numbers ( , 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?

11. Regrouping
Show a diagram for

12.
Start with…

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

14. You try: draw pictures

15. Subtraction the way you learned it… 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 1 7

17. Why does this work? This way worked because
is the same as adding 10 to both numbers:
= =
= =

18. Use drawings or manipulatives to explain…
Here is another one. 1

19. Use drawings or manipulatives to explain…
Here is yet another one.
= 132

20. 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

21. A quick review of subtraction
Try this: Explain what the student is doing.
>
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…

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 Write in base 5. True or false: = 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
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 $ How many apples and how many oranges did I buy?
Answer: 10 apples and 17 oranges