Algebra for Precalculus and Calculus Students Mindful Manipulation. Philip Cheifetz, Ellen Schmierer Nassau CC. “Recent” Events. 1960 - 1970s New Math Viet Nam War Deferments Open Enrollment: less talented student pool 1980s Graphing Calculators
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Algebra for Precalculus and Calculus StudentsMindful Manipulation
Philip Cheifetz,
Ellen Schmierer
Nassau CC
1960-1970s
1980s
1990s
2000s
44% = passing
34% = passing
Mistakes with Fractions My Students Routinely Make (A Joke)
The chef instructs his apprentice:
"You take two thirds of water, one third of cream, one third of broth..."
The apprentice:
"But that makes four thirds already!“
"Well -just take a larger pot!"
Mistakes with Commutativity Young Students Make (Not a Joke)
Teacher: "Who can tell me what 7 times 6 is?"
Student: "It's 42!"
Teacher: "Very good! - And who can tell me what 6 times 7 is?"
Same student: "It's 24!"
UNFORTUNATELY,
Students do not have a number sense.
Therefore, they do not have an algebra sense,
Nor do they have the sense to ask “Does the answer seem reasonable?”
I’m Angry!
A non-trivial number of high school mathematics teachers and/or supervisors:
~cheifp/Itseemstome1.htm
A car trip costs $1.50 per fifteen miles for gas and 30¢ per mile for other expenses, plus $20 for car rental. The total cost for the trip is given by theequation
A company uses two different sized trucks to deliver sand. The first truck can transport x cubic yards and the second y cubic yards. The first truck makes S trips to the job site and the second makes T trips. What do the following expressions represent in practical terms?
You plan to drive 300 miles at 55 miles per hour, stopping for a two-hour rest. You want to know t, the number of hours the journey will take. Which of the following equations would you use?
Match each graph with a description of its zeros
a. 0 b. 1 c. 2 d. 3 e. none of these
g(x)
h(x)
a. 0 b. 1 c. 2 d. 3 e. none of these
g(x)
h(x)
In the equation 3x + 2y = 9, if x increases by 2 then y
a. Increases by 2
b. Decreases by 6
c. Increases by 3
d. Increases by 6
e. Decreases by 3
Match the equation to the description
a. Astronomical growth
b. A 6% increasing growth rate
c. A 6% decay rate
d. Quickly approaches zero
Which of the following tables shows data from an invertible function?
Which of the following are possible equations for the graph below?
d
a b c