Welcome to Math 302B
Fall 2009
Agenda
• Go over syllabus, class procedures,
etc.
• How to answer a homework problem
• Begin algebraic thinking
• Assign homework
A good answer…
• Shows how you got the answer (all the
work).
• Explains why you did what you did.
• Where appropriate, explains what a
child did correctly/incorrectly, and what
mathematics the child did/did not
understand.
Here’s a problem.
• The first five terms of a sequence are
2, 5, 10, 17, 26, …
• What is the 10th term?
• What is the nth term?
– Introduce yourselves at your table.
– In your groups, try to figure out the tenth term.
Then, write a sentence or two that you think is
perfect for a homework answer.
Unacceptable Answers
• 10th term is 101
• 2, 5, 10, 17, 26, … 101, … ?
• 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, …
• This is a trick question. There is no answer.
More unacceptable
answers
• The 10th term is 101. I did the math in my
head.
• The 10th term is 101. I did the math on my
calculator.
• The 10th term is 101. I just did trial and
error--lucky me! I got it on the first try.
Acceptable answer
• I see the pattern 2, 5, 10, 17, 26, …. So I
made a table:
Term: 1 2 3 4 5 6 7 8 9 10
Pattern: 2 5 10 17 26 37 50 65 82 101
• Then, I saw that 5 - 2 = 3; 10 - 5 = 5;
17 - 10 = 7; 26 - 17 = 9. The differences
increase by 2, so to continue, I would add 11,
then 13, then 15, and so forth. So the 10th
term is 101.
Acceptable answer
• I see the pattern 2, 5, 10, 17, 26, ….
By itself, there is nothing that leaps out
at me. But, if I subtract 1 from each
number, I get 1, 4, 9, 16, 25, … which
are the perfect squares. So, each
number is n2 + 1. So the 10th term is
102 + 1 = 100 + 1 = 101.
Algebraic Reasoning
• Organizing data and generalizing data:
– Mathematical modeling through graphs
– Mathematical modeling through tables.
• Generalizing patterns
• Relationships involving variables--things that
vary.
• We use the rules of algebra to express
algebraic reasoning.
Does a variable “vary”?
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•
•
•
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•
5x = 60
y = 3x - 4
C=π•d
n/n = 1 if n ≠ 0.
y = kx
sin2(x) + cos2(x) = 1
The Equal Sign =
• Signifies that the expressions on each side
are equivalent.
• If 3x + 4 = 12, then 3x + 4 - 4 = 12 - 4.
• Look back at the properties: commutative,
associative, distributive, identity, inverses
• Now look at algebraic “formulas”
• (x + 3)(x - 4) = 0 means either
x + 3 = 0 or x - 4 = 0. Why?
Function
• A special relationship found in algebra is the
function. A function is a relation where every
value of input has just one value of output.
• Your age today: your height today.
• Students name: Students ID number
• Number of hours worked: Paycheck
• Diameter of a circle: area of the circle
These are not functions
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•
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•
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Shoe size: number of shoes you own.
Your height: Your age
Date: price of gasoline in the area
Length of a football field: Year of Superbowl
Number of courses you are registered for:
number of students in your largest class
In any function
• The input is called the independent variable.
We usually put this on the x-axis.
• The output is called the dependent variable.
That is, it depends on the independent
variable. We usually put this on the y-axis.
• Example: age vs. height; typically, your
height depends on your age.