Transcript Day 1

Math 302A section 5
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Spring, 2009
Instructor: Laurie Varecka
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January 15, 2009
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Class policies
About the course
Sec 1.1: NCTM Principles and Standards
Sec 1.2: Problem solving
Sec 1.3: Patterns (Exploration 1.1)
Sec 1.4: Representations
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NCTM Principles & Standards for School Mathematics
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Process standards:
• Problem Solving
• Reasoning & Proof
• Communication
• Connections
• Representation
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Sec 1.2 – Problem Solving
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Polya’s Four Steps:
1. Understand the problem
2. Devise a plan
3. Monitor your plan
4. Look back at your work
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1.2 (cont’d)
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An example and discussion of
homework expectations.
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Ex: A school play charges $2 for
students and $5 for adults. For the
three days of the play, 20 tickets were
sold and $85 was raised. How many
student tickets were sold?
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1.2 - Homework expectations
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The following solutions to the “school play”
problem are unacceptable. Why?
• 15 tickets for $5, 5 tickets for $2
• The school sold fifteen $5 tickets and five
$2 tickets.
• I did the work in my head.
• I did the work on my calculator.
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1.2 - Homework expectations
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More unacceptable solutions:
• 2x + 5y = 85; x + y = 2 so x = 5
• I tried different numbers until I got
the answer.
• This problem is too basic to bother
with an explanation.
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1.2 - Homework expectations
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A Good solution might look like:
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There are 20 tickets total, so I need to find
two numbers that add to 20.
If I have 10 students and 10 adults, that gives
me 20 tickets, but $2 * 10 students and $5 *
10 adults gives me only $70.
Since the adult tickets are more expensive, …
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1.2 - Homework expectations
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… I need more than 10 adults. In fact, each
time I trade a student for an adult, the total
ticket sales go up by $3. So, I’ll try adding
5 adults (and subtracting 5 students):
5 students + 15 adults = 20 tickets sold
5 students * $2 + 15 adults * $5 = $85
Therefore, 5 student tickets were sold.
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1.2 - Homework expectations
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Another Good solution:
If we let S = the number of student tickets and
A = the number of adult tickets, then
S + A = 20
is the equation for the number of tickets and
2S + 5A = 85
is the equation for the amount of money
raised.…
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1.2 - Homework expectations
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… I can rewrite S + A = 20 as A = 20 – S and
then substitute into the other equation:
2S + 5(20 – S) = 85
2S + 100 – 5S = 85
100 – 3S = 85
15 = 3S
Therefore, S = 5, and there were 5 student
tickets sold.
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1.2 - Homework expectations
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In general,
• Explain what you did.
• Explain why you did it.
• Be sure to check that your answer
really does answer the question.
• Be sure to check for arithmetic errors.
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1.2 (cont’d.)
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Ex: A farmer looks out his window and
sees pigs and chickens. He says to his
daughter, “I count 14 heads and 48
feet. How many pigs and how many
chickens are out there?”
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(Introduce yourself to your table, then
work on this together.)
What approach did you take?
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Problem Solving Strategies
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Guess & check
Solve simpler problem
Solve similar problem
Draw diagram
Find pattern
Find counter-example
Induction
• Work backwards
• Make table
• Make graph
• Write equation(s)
• Estimate
• Act it out
• Organized list (try all
possibilities)
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1.3 - Patterns
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Ex: Find the next term in the sequence.
Describe the pattern you found (use a
sentence).
a) 1, 2, 4, 8, 16, …
b) 25, 5, 1, 0.2, 0.04, …
c) 1, 3, 6, 10, 15, …
d) 1, 0, -1, 0, 1, 0, -1, …
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Homework
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• Read sections 1.1 – 1.4
• Write down answers to the questions in
section 1.1 in your notebook (not to hand
in).
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