Week 5Week 5Warm up Problem: What did the Warm up Problem: What did the

student do wrong?student do wrong?

457 - 29 = 338457 - 29 = 338

Explain the error in a sentence.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.• 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.

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.

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.

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.

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!!!

Number Line Model• 7 - 9 = -2

• Why is it important to start at 0?

-2 -1 0 1 2 3 4 5 6 7 8 9 10

Four related facts• If 9 - 4 = 5, then

9 - 5 = 4, 4 + 5 = 9, and 5 + 4 = 9

• You try: If 74 - 61 = 13, then …

Mental Subtraction• Not as obvious as mental addition• 65 - 28

Break 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)

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

Regrouping• Show a diagram for

• 302 - 84

302 - 84• Start with…

Now, indicate what you are subtracting

• 302 - 84: Let red be the part you take away

You try: draw pictures• 58 - 37

• 47 - 29

Subtraction the way you learned it…

6 7 8- 3 9 2 2 8 6

51

Why did you cross out the 6? Why did you put a little “1” next to the 7? Can you show this with pictures?

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

7

1

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

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

1

1

Use drawings or manipulatives to explain…• Here is yet another one.

• 3 2 6- 2 9 4 2 -7 0 1 0 0

• 100 - 70 + 2 = 132

A real problem• Place the digits 1, 2, 3, 6, 7, 8 in the

boxes to obtain:

• Greatest sumLeast sumGreatest differenceLeast difference

A quick review of subtraction

• Try this: Explain what the student is doing.

8 7 8 8- 3 9 --> - 4 0

4 8

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

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

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.

Use pictures or diagrams to explain…

• Write 1e716 in base 10.

Write 39010 in base 5.

True or false: 425 = 346

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, …

• 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.

• 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

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