NCTM 2015 Regional Meeting, Session 245
Geometric Transformations and Linear Functions: Two Sides of a Coin
Summary: In this presentation, Scott Steketee and Daniel
Scher show a sequence of geometric functions activities. In these activities
students transfer their understanding of geometric
functions that take a point in the plane as input and produce
another point as output—reflection, rotation, dilation, and
translation in “Flatland”—to linear functions that take a real
number as input and produce another real number as output.
Students restrict the domains of the geometric functions to number
lines (“Lineland”), measure the values of the variables, and
discover that composing and restricting just two geometric
functions enables them to create any linear function
geometrically, and to represent it in either Dynagraph or
Cartesian graph form.
Link to the presentation web sketch.
Download the handout.
The presentation involved six activities. When used with students,
each activity is expected to take one or two class periods. Each activity
is accompanied by a student worksheet.
Dilate Family: This activity comes from the introductory unit
on Functions and Function Families, and is included here because it
provides students with important prerequisite understandings about geometric functions
and gives them practice working with Web Sketchpad tools.
the Dimension: Students begin with the already-familiar
geometric functions in Flatland and figure out how to move each
function into Lineland by restricting the domain of the
independent variable to a line and adjusting the function so
that the range of the dependent variable lies on the same line.
the Domain: Once the functions are in LineLand, students
turn the restricted domain into a number line, measure the
values of the independent and dependent variables, and figure
out the numeric operation that corresponds to each geometric
on a Line: Students compose two Flatland functions (the
two that correspond to multiplication and addition), again
restricting the domain to a number line, and examine the
behavior of the resulting function both geometrically and
- Create a Dynagraph:
Students shift the output
variable vertically to create a Dynagraph representation of
their linear function. They drag the independent variable to
study the behavior of various linear functions and to solve
puzzles in which they adjust the scale factor (for dilation) and
the vector (for translation) to match given mystery functions.
- Connect to Cartesian:
Students rotate the
output variable of their composed dilation/translation function
by 90° and use the resulting Cartesian axes to create and
analyze their function’s Cartesian graph, and to solve puzzles
in which they adjust the original dilation and translation
functions to match given mystery graphs.
These activities are designed to accomplish a number of
objectives. In doing them, students will:
- Drag independent variables of geometric functions while
observing the motion of dependent variables
- Attend to function behavior, and particularly to the relative
rate of change of the independent and dependent variables
- Use function notation with geometric transformations
- Restrict the domains of functions and observe the effect on
- Figure out how to move 2D geometric functions onto a 1D number
- Analyze the numeric behavior of the resulting 1D number-line
- Relate the 2D behavior of the original geometric functions to
the corresponding numeric behavior of the number-line functions
- Create dynagraph representations of number-line functions
- Use dynagraphs to analyze and explain the behavior of linear
- Solve mystery-function dynagraph puzzles by matching the
behavior of a constructed function to the observed behavior of a
- Create perpendicular-axis representations of number-line
- Use perpendicular-axis representations to construct Cartesian
- Solve mystery-function Cartesian puzzles by matching the shape
of a constructed graph to the observed shape of a mystery
These activities require web access using a browser that
30 October 2015: Revised based on the presentation at the Atlantic City Regional.
15 April 2015: Derived this page from the original Cartesian Connection
page, to support the presentation at the 2015 NCTM Annual Meeting.
* Dynagraphs were invented by Paul
Goldenberg, Philip Lewis, and James O’Keefe. See their classic
1992 article “Dynamic representation and the development of a
process understanding of function.” In G. Harel and E. Dubinsky
(Eds.), The Concept of Function: Aspects of Epistemology and
Pedagogy. MAA Notes V. 25 (1992), Mathematical Association