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Drawing vs. Constructing
Mathematical Goals: Teachers will be able to
 Use and understand common geometric objects such as points, lines, and segments.
 Use and understand properties of triangles and quadrilaterals.
Pedagogical Goals: Teachers will be able to
 Classify the cognitive demand of tasks as low or high.
 Explain how different tools can shape the ways in which students think about particular
geometric objects.
 Begin to consider tasks appropriate for use with dynamic geometry software.
Technological Goals: Teachers will be able to use a technological tool to
 Describe and use common features of dynamic geometry environments (DGEs) including
dragging, measuring, and calculating.
 Understand the difference between drawing and constructing.
 Generate constructions of objects using a DGE including right triangles, squares,
rectangles, rhombi, trapezoids, etc.
Mathematical Practices:
 Make sense of problems and persevere in solving them.
 Construct viable arguments and critique the reasoning of others.
 Use appropriate tools strategically.
 Attend to precision.
 Look for and make use of structure.
Length of session: 90 minutes
Materials needed: Computer with Geometer’s Sketchpad, Drawing vs. Constructing Participant
Handout, DrawingVsConstructingSolution.gsp file
Overview:
In this session participants will be introduced to Geometer’s Sketchpad (GSP) and consider ways it might be used in
the classroom to discover properties and theorems by exploring invariances. Participants will also learn about
drawing versus constructing and practice constructing various objects in GSP.
Estimated #
Activity
of Minutes
40 minutes
Introduction to Geometer’s Sketchpad
 Briefly introduce participants to Geometer’s Sketchpad (GSP). Show them
the menu bar at the top and the tool bar (typically found on one side).
Allow participants time to explore what types of things they can do in
Geometer’s Sketchpad (GSP). Encourage them to try creating points, lines,
segments, and circles. See if they can determine how to measure, drag, or
transform objects. The goal here is to let them get comfortable with where
things are located in GSP. They should be actively engaged and talking
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Introduction to dynamic geometry environments. In Preparing to teach mathematics
with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.
2
50 minutes
about what they are doing. They should spend at least 15 minutes playing
in GSP before considering the questions below.
 Questions to consider:
1. What types of things were you able to do in GSP? Answers will vary.
2. Discuss the dependent and independent relationships of objects in
GSP. How can you determine if an object is dependent or
independent? In GSP, if an object depends on another, that means it
was created after the object it is dependent upon. Thus, the behavior
of the dependent object is influenced by the manipulation of the
independent object. For example, if you have line segment AB and
create point C to be on this segment, point C is dependent on line
segment AB. Point C can only move along segment AB. Deleting
point C will not affect segment AB, but deleting A, B or the segment
between them will automatically delete point C. GSP refers to these
independent and dependent relationships as parent and child,
respectively. Information regarding the independent or dependent
status of an object can be found by right clicking on the object and
examining its properties.
3. Use GSP to determine the angle sum of a triangle. First, create a
triangle and measure each of its angles. Then, using the calculate
tool, click on each of the angle measurements to generate the sum.
You will see that the angle sum of a triangle is 180 degrees. Then, by
dragging any of the vertices of the triangle, we can see that while
each individual angle within the triangle changes, the angle sum
remains the same. This is an example of using GSP to explore
invariances. See sample solution on Angle Sum of a Triangle tab of
DrawingVsConstructingSolution.gsp file.
4. How would you classify the cognitive demand of the task in question
3? As stated, the task in question 3 would be of high cognitive
demand. More specifically, it can be classified as doing mathematics
since there are multiple ways students might approach creating the
triangle and then determining if the sum holds constant for all cases.
5. What other geometric properties of triangles could you have students
explore using your sketch from question 3 or with a slight
modification of the sketch? Special types of triangles according to
angles, lengths or both angles and lengths could be explored. For
example, students may be asked to turn a triangle into another, by
dragging, (for example, maybe creating an isosceles right triangle)
Also, the length-angle relationships among a triangle’s sides could
be explored, which could give students some ideas before exploring
triangle inequality.
Drawing versus constructing
 Have participants create a right triangle. Ask them to measure the angles
and then drag a vertex to determine if the triangle always remains right. In
most cases, the triangle will probably not keep this property, thus we
would say the figure “broke.”
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Introduction to dynamic geometry environments. In Preparing to teach mathematics
with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.
3

Questions to consider:
1. How can we create figures so that they pass a drag test (i.e. keep the
properties we want it to have) and don’t break? This is really a
question that will be investigated throughout the questions below.
Ultimately, we want participants to realize that we must construct
figures based on the particular properties we want them to keep.
2. Can you create a triangle that is a right triangle and will always be a
right triangle when dragged? To create a right triangle that remains a
right triangle, first create a segment, say AB. From here, we need to
construct a perpendicular segment to generate our right angle. To do
this, select the segment and one endpoint, say A, and select
“perpendicular line” from the construct menu. Now, we can
construct a point (C) on the perpendicular line and complete the
triangle by constructing segments AC and BC. Note that point C can
only be dragged along the perpendicular line but A and B can be
dragged anywhere within the sketch. Finally, select the perpendicular
line and then “hide perpendicular line” from the display menu (or
use control/command h as a shortcut). By performing a drag test
dragging any of the vertices, we can see that triangle ABC will
always remain a right triangle with a right angle at vertex A. See
sample solution on the Right Triangle Construction tab of the
DrawingVsConstructingSolution.gsp file.

After participants have successfully constructed a right triangle, discuss
drawing versus constructing. Ask them, what is the difference? A
construction keeps the properties of the figure that we want it to have, thus
we create our figure based on these properties. A drawing may be able to
represent the particular properties we desire but will not maintain these
properties under a drag test.

Questions to consider:
1. Construct the following objects:
a. Isosceles triangle
b. Isosceles right triangle
c. Equilateral triangle
d. Rectangle
e. Square
f. Rhombus
g. Kite
h. Trapezoid
i. Isosceles trapezoid
j. Cyclic quadrilateral
If time doesn’t permit participants to construct each of these, have
them select at least 3, one triangle, one quadrilateral, and one of
their choosing. Samples of each object can be found in the
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Introduction to dynamic geometry environments. In Preparing to teach mathematics
with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.
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2.
3.
4.
5.
DrawingVsConstructingSolution.gsp file on their respective tabs. If
you are unsure how to construct a particular object, there are videos
and directions all over the internet! Be sure to stop after participants
have constructed the above objects and have some come up and show
how they constructed various ones. Be sure to highlight that there are
different ways to construct the same object and try to choose two
different people who constructed the same object using two different
approaches and have them demonstrate/explain their approach.
In what ways is constructing in GSP similar to and different from
using a compass and straightedge?
Answers will vary. Some possible responses are that GSP allows you
to create a dynamic version of an object where a compass and
straightedge create just one object at a time. In both cases, in order
to construct an object, the user needs to have an understanding of the
properties of the object to generate appropriate constructions.
Why is it important for students to understand the difference between
drawing and constructing?
While drawing geometric shapes can be fun and useful at times, the
properties of the intended shape are frequently not withheld under a
drag test. Constructing shapes, instead, insures that the properties of
the geometric shape are maintained. If a shape is drawn, then any
relationships and findings the students discover may not hold for that
particular shape.
How would you characterize the cognitive demand of the
construction task? Why?
This is a high-cognitive demand task if the students use the properties
of the shape in constructing it. In order to accomplish the task the
students must be able to draw upon their previous understanding of
the properties of the shape and determine how to appropriately
utilize GSP to construct it.
How can you use GSP to have students explore properties of
particular objects? Provide an example.
By providing a sketch with the pre-constructed objects, you can have
students explore the properties of an object by having them measure
sides, angles, diagonals, etc. and perform a drag test to explore for
invariances. This will allow students to determine if their conjecture
regarding a specific property holds true. One example would be to
have all of the quadrilaterals in a sketch, maybe on different tabs,
and have a chart where students can record the properties as they
discover them. Then, they could generate a hierarchical model of
relating the quadrilaterals based on their properties.
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Introduction to dynamic geometry environments. In Preparing to teach mathematics
with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.
5
Drawing vs. Constructing
Participant Handout
Introduction to Geometer’s Sketchpad (GSP)
Dynamic geometry environments (DGEs) are particular technology tools that have been used in
the learning and teaching of geometry to assist students in moving beyond the specifics of a
single drawing to generalizations across figures. The Geometer’s Sketchpad is one of many
DGEs. Although they all are different, they do share some common features which include
among others: a set of primitive elements of Euclidean geometry; the ability to construct other
geometric objects using these primitive objects; the ability to act on constructed objects via
transforming tools, measurement tools, and calculation tools; and the ability to explore relations
among constructed objects via dragging, animation, and hiding and showing objects and
measurements.
Questions to consider:
1. What types of things were you able to do in GSP?
2. Discuss the dependent and independent relationships of objects in GSP. How can you
determine if an object is dependent or independent?
3. Use GSP to determine the angle sum of a triangle.
4. How would you classify the cognitive demand of the task in question 3?
5. What other geometric properties of triangles could you have students explore using
your sketch from question 3 or with a slight modification of the sketch?
Drawing versus constructing
Create a right triangle. Measure the angles and then drag a vertex to determine if the triangle
always remains right.
Questions to consider:
1. How can we create figures so that they pass a drag test (i.e. keep the properties we
want it to have) and don’t break?
2. Can you create a triangle that is a right triangle and will always be a right triangle
when dragged?
Discussion of drawing versus constructing
Questions to consider:
1. Construct the following objects:
a. Isosceles triangle
b. Isosceles right triangle
c. Equilateral triangle
d. Rectangle
e. Square
f. Rhombus
g. Kite
h. Trapezoid
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Introduction to dynamic geometry environments. In Preparing to teach mathematics
with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.
6
2.
3.
4.
5.
i. Isosceles trapezoid
j. Cyclic quadrilateral
In what ways is constructing in GSP similar to and different from using a compass
and straightedge?
Why is it important for students to understand the difference between drawing and
constructing?
How would you characterize the cognitive demand of the construction task? Why?
How can you use GSP to have students explore properties of particular objects?
Provide an example.
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Introduction to dynamic geometry environments. In Preparing to teach mathematics
with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.