From Patterns to Relations to Functions
Since elementary school, your students have worked with patterns. The lessons
and activities featured here move students mathematically forward to
consideration of the rules underlying the patterns and then to formulation of
those rules in algebraic terms.
Looking for a "real-world" example of a linear function? In this lesson,
students model a bungee jump using a Barbie doll and rubber bands. They measure
the distance the doll falls and find that it is directly proportional to the
number of rubber bands. Since the mathematical scenario describes a direct
proportion, it can be used to examine linear functions.
Operating under a secret rule, the function machine uses numbers input by the
students to generate output. Students compare the input (domain) to the output
(range) to find the function rule. The analogy of a function machine is a
basic, strong visual that holds up even in advanced study of functions. (MSP full record
One end of a wooden board is placed on a bathroom scale and the other end is
suspended on a textbook; students can literally "walk the plank" and record the
weight shown on the scale as their distance from the scale changes. It turns
out that the relationship between the weight and distance is linear, and this
investigation leads to a real-world occurrence of negative slope. An activity
sheet, its solutions, and questions for class discussion are included in this
one-period lesson.
This two-lesson unit allows students to discover patterns in a fictional but
real-world scenario: How many handshakes occur when the nine Supreme Court
justices shake hands with each other? Students explore—through a table, a
graph, and finally an algebraic formula—the number of handshakes in any size
group. A second pattern is explored, that of triangular numbers; again,
students generalize the pattern with variables. The lessons are well
illustrated and include background information for the teacher.
Designed expressly for middle school classes, this lesson is built on the
premise that "teachers need help in building a bridge between their current
instructional goals and new goals that emphasize an earlier introduction to
algebraic thinking." As students work through tasks, they organize values into
tables and graphs as they move toward symbolic representations of the functions
involved. The problem situations, carefully explained, employ linear,
quadratic, and exponential models.
This lesson connects statistics and linear functions. Students construct
scatterplots, examine trends, and consider a line of best fit as they graph
real-world data. They also investigate the concept of slope as they model
linear data in a variety of settings that range from car repair costs to sports
to medicine. Handouts for four activities, spread out over three class periods,
are provided.
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Copyright
June 2006 — The Ohio State University. This material is based upon work
supported by the National Science Foundation under Grant No. 0424671. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of
the National Science Foundation.
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This work is licensed under a
Creative Commons License.
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