Content Area: Math
Index: 4.3A Grade 12 CPI 2
Standard: 4.3 - Patterns and Algebra
Strand: A - Patterns
Cumulative Progress Indicator: 2 - The student will analyze and explain the
general properties and behavior of functions of one variable, using appropriate
graphing technologies
·
Slope of a line or
curve
·
Domain and range
·
Intercepts
·
Continuity
·
Maximum/minimum
·
Estimating
roots of equations
·
Intersecting
points as solutions of systems of equations
·
Rates of change
Grade: 12
Sample Activities:
·
As regular parts of their assessments, students make up
graphs to represent specific problem situations, such as the cost of pencils
that sell at two for a dime, the temperature of an oven as a function of the
length of time since it was turned on, their height from the ground as they ride
a ferris wheel as a function of the amount of time since they got on, the time
it takes to travel 100 miles as a function of average speed, or the cost of
mailing a first-class letter based on its weight in ounces.
·
Students use a string of constant length, say 30
inches, and list all possible lengths and widths of rectangles with integral
sides which have this perimeter. They determine the perimeter and area for each
rectangle. Then they make three graphs from their data: length vs. width,
length vs. perimeter, and length vs. area. They look for equations to describe
each graph, determine an appropriate range of values for each variable, and
then graph the functions using graphing calculators or computers. The rectangle
of maximum area, a square, does not have integral values, but can be found
using the trace function or algebraic procedures. Students also investigate the
area of a circle made with the same string and compare it to the areas of the
rectangles.
·
Students take on the role of "forensic
mathematicians," trying to determine the height of a person whose femur
was 17 inches long. They measure their own femurs and their heights, entering
the class data into a graphing calculator or computer and creating a scatterplot.
They note that the data are approximately linear, so they use the built-in
linear regression procedures to find the line of best fit and then make their
prediction.
·
Students collect data about the height of a ball that
is thrown in the air and make a scatterplot of their data. They note that the
points lie on a quadratic function and use their graphing calculators to find
the curve of best fit. Then they make some conjectures about the speed at which
the ball is traveling. They think that the ball is slowing down as it rises,
stopping at the maximum point, and speeding up again as it falls.
·
Students take on the role of "forensic
mathematicians," trying to determine how tall a person would be whose
femur is 17 inches long. They measure their own femurs and their heights,
entering this data into a graphing calculator or computer and creating a
scatterplot. They note that the data are approximately linear, so they find the
y-intercept and slope from the graph and generate an equation that they think
will fit the data. They graph their equation and check its fit. They also use
the built-in linear regression procedure to find the line of best fit and
compare that equation to the one they generated.
·
Students plot the data from a table that gives the
amount of alcohol in the bloodstream at various intervals of time after a
person drinks two glasses of beer. Different groups use different techniques to
generate an equation for the graph; after some discussion, the class decides
which equation they think is best. The students consider the following
questions: What information does the slope give for this situation? Would that
be important to know? Why or why not?
·
Students investigate the effect of changing the radius
of a circle upon its circumference by measuring the radius and the
circumference of circular objects. They graph the values they have generated,
notice that it is close to a straight line, and use the slope to develop an
equation that describes that relationship. Then they discuss the meaning of the
slope in this situation.
·
Students work through the On the Boardwalk lesson found
in the Introduction to this Framework. A quarter is thrown onto a grid made up
of squares, and you win if the quarter does not touch a line. A grid is drawn
on the floor using masking tape, and a circular paper plate is thrown onto the
grid several hundred times to simulate the game. The activity is repeated
several times, varying each time the size of the squares in the grid. The
students collect data and make a graph of their results (size of squares vs.
number of wins out of 100 tosses). The graph looks like a straight line,
suggesting that as the size of the squares increases without bound, so does the
percentage of "hits". But, of course, the percentage of hits cannot
exceed 100%, so the line is actually curved, with an asymptote at y=100.
·
The school store sells pencils for 15 cents each, but
it has some bulk pricing available if you need more pencils. Ten pencils sell
for $1, and twenty-five pencils sell for $2. The students make a table showing
the cost of different numbers of pencils and then generate a graph of number of
pencils vs. cost. The students note that the graph has discontinuities at ten
and twenty-five, since these are the jump points for pricing. They also note
that if you need at least seven pencils, it is better to buy the package of ten
and if you need 17 or more, you should get the package of 25.
·
Students make a table, plot a graph (number of people
vs. cost), and look for a function to describe a situation in which the Student
Council is sponsoring a Valentine's Day dance and must pay $300 to the band, no
matter how many people come. They also must pay $4 per person for refreshments,
with a minimum of 50 people. The students note that the cost will be $500 for
anywhere from 0-50 people and then increase at a rate of $4 per person. They
decide that this is a function with a corner and needs to be defined in pieces
