Transcript 302appw5

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.
• 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…
5
1
678
- 392
286
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
•
•
1
87 4
9
3 6 8
6 1 6
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.
•
•
782
- 347
782
7 8 12
- 3 4 7- 3 4 7
4 4
4 4 5
1
•
4 3 5
Use drawings or
manipulatives to explain…
• Here is yet another one.
•
-
•
3 2 6
2 94
2
-7 0
1 00
100 - 70 + 2 = 132
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
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
3 6 8
9 8 14
- 3 7 8
6 1 6
Alphabitia…
CBA
+ DCA
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