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Introductory Geometric Activities
An introductory unit in the Algebra Project high school curriculum
David W. Henderson, lead writer
Intro1. What is Geometry?
Intro2. Can we describe this?
Intro3. Introductory Experiences with paper-folding and ‘Parallel’ and ‘Perpendicular’.
This material is based upon work supported by the National Science Foundation
under Grants #IMD0137855 and #IMD0628132. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the authors
and do not necessarily reflect the views of the National Science Foundation.
Please send all comments, suggestions, questions, and other feedback to the lead writer of this
module: David Henderson, <dwh2@cornell.edu>.
A teacher who is using this material in a classroom may modify this material to suit their
classroom. Copies of such modifications should still bear in the footer on every page: “
Algebra Project, Inc, <current year>.”
 Copyright, 2011, by Algebra Project, Inc. Do not
copy or duplicate without written permission from the
Algebra Project, Inc. <info@algebra.org> .
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INTRO – INTRODUCTORY GEOMETRY ACTIVITIES
Objectives:
In each of the following sections, students should be able to…
INTRO1
o Identify and discuss the various human activities that involve geometry.
o Classify their own experiences and activities into the strands of human
activity and justify those classifications.
INTRO2
o Use language to describe shapes, patterns, and sizes.
o Interpret descriptions and draw what is being described.
INTRO3
o Problem solve how to fold arbitrary perpendicular lines, parallel lines,
squares and rectangles on a sheet of paper
o Justify and communicate, in ways that can be followed by others, why
these constructions work.
Prerequisites: None
Timing of Unit
1-4 Days (2-4 hours per section). More if extensive writing is assigned.
Comments from Material Developer on Whole Unit:
These short units are designed to prime the students to think creatively about geometry and
to provide the teachers with some feedback as to what geometric knowledge the students are
entering with. In particular:
Intro1 introduces five strands of human experiences that lead to geometric ideas.
Intro2 provides activities to challenge student abilities to describe geometric shapes and to
recognize shapes from descriptions. This Section can be used as both a Pre-test and as a Posttest to access some of the effects of the geometry units that cannot be accessed by Standards
Exams.
Intro3 presents a paper-folding challenge that will bring out in the students previous
knowledge about perpendicular and parallel, start the students communicating their geometric
thinking.
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INTRO1 – What is Geometry?
Objectives of Intro1: students should be able to…
o Identify and discuss the various human activities that involve geometry.
o Classify their own experiences and activities into the strands of human activity
and justify those classification
Timing of unit: 1-2 days
Materials needed:
chart paper, access to dictionaries and Wikipedia and geometry texts
Comments from Material Developer on Intro1:
These short units are designed to prime the students to think creatively about geometry and
to provide the teachers with some feedback as to what geometric knowledge the students are
entering with. In particular, this section introduces five strands of human experiences that lead
to geometric ideas.
Teaching Tips from teachers:
o In the beginning of the materials, students are given a definition for geometry. Have students
(as an extended assignment) find three different sources such as wikipedia, dictionary and
geometry text to find other definitions of geometry. Then have students define what they
think geometry is.
o After discussing the strands, student stories need to be guided. Have students to write about
what they did over Christmas break or what they did last summer. This will help to bring
more of the strands out for later use.
o After stories are complete, have 5 to 6 sheets of chart paper up with strands labeled on each
chart. Students will share stories with group and place experiences on the chart paper where
they think it fits. After students have shared with partners, they must explain to class why
they placed the experiences in certain strands. Also have students explain why (if happened)
they could not fit their experience in a particular strand.
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Intro1. What Is Geometry?
Mathematician:
Geometry is the visual study of shapes, sizes, patterns, and positions that have apparently
occurred in all cultures through these five strands of human activities:
1.
2.
3.
4.
5.
building/structures (houses, laying out a garden, …)
machines/motion (crow bar, bicycle, saw, swing, …)
navigating/star-gazing (How do I get from here to there? …)
art/patterns (design, representations, perspective…).
counting/measurement (How many? How large is it? ...)
Many mathematicians think that all of geometry developed from these activities in various
cultures around the world. We are now going to look at our experiences with ideas from
geometry. And we will try to classify each of our experiences into these five strands.
Worksheet 1 (Individual Work). Write a story about your various experiences with ideas from
geometry. You can talk about experiences in the classroom, but more important write about
experiences outside of the classroom. Think broadly – if you are not sure whether something
“counts” as geometry then write about it and tell why you are in doubt.
Worksheet 2 (Group Work). Share the individual experiences with the group and the group
produces a combined list of geometric experiences and explain in what ways they relate to
geometry.
Mathematician: Now let us try to classify each of our experiences into these five strands.
Student 1: But some of our experiences seem to fit under more than one strand.
Mathematician: Can you give us an example?
Student 2: I want to go quickly to the park on my bicycle but first I have to repair the basket,
which has fallen partially off.
Student 1: Yes. Repairing the basket is in the building strand.
Student 3: Deciding the quickest route to the park is in the navigating strand.
Student 2: And pedaling, steering, and balancing my bicycle is in the machines and motion
strand.
Student 4: And I like to have my bike painted with cool designs and that would be in the
art/pattern strand.
Student 5: We would use measurement to know how far we traveled to the park and how long it
took to get there.
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Worksheet 3 (Group Work). Take the list of experiences that the group has developed and
classify them into one of the five strands and include an “other” strand for experiences that don’t
seem to fit. Also, some experiences and activities may fit into more that one strand – in this case
list it under each of the strands that apply but identify what aspect of the experience puts it into a
particular strand. As you think about each strand, are there other activities that humans do that
would fit into that strand?
Worksheet 4 (Class Work). Each group reports back to the whole class and the lists are
combined into a class list. Discuss as a class the things put under “other”, if any. If there is
something that does not fit into any of the five strands, then can you agree on a description of a
sixth strand that it would fit in?
Note: The class may come up with other strands, if so accept them as long as the students can
agree on a rough description of the new strand.
Mathematician: If you find what you think is a sixth strand within geometry then write up your
description of it and describe why you think it does fit with the other strands. Share this with
other students in other classes and share it with me. Maybe you will show me something I
haven’t thought of.
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Intro2. Can we describe this?
Objectives: In INTRO2, students should be able to…
o Use language to describe shapes, patterns, and sizes.
o Interpret descriptions and draw what is being described.
Comments from Material Developer:
These short units are designed to prime the students to think creatively about geometry and
to provide the teachers with some feedback as to what geometric knowledge the students are
entering with. In particular:
Intro2 provides activities to challenge student abilities to describe geometric shapes and to
recognize shapes from descriptions.
This Section can be used as both a Pre-test and as a Post-test to access some of the effects
of the geometry units that cannot be accessed by Standards Exams. It would be useful to use
this as a pre-test at the beginning of the geometry and then again at the end. The effectiveness of
the descriptions should be better at the end and there should be more use of geometric terms.
Basic Idea of Intro2. (There are 2 variations to this.)
First variation:
1. The students divide into pairs or small groups.
2. Each group is supplied with an opaque bag containing a collection of different 2-d shapes.
3. Each student takes turns being ‘it’. The person who is ‘it’ sticks his/her hand into the bag
and handles an shapes (keeping it in the bag) and describes (using only words – not gestures)
what the shape is that s/he is handling and what its size is.
4. The other students attempt to draw a picture of what ‘it’ has described. The other students
may ask questions of ‘it’ as needed.
5. The handled shape is removed from the bag and compared with the drawings.
6. A list is made of the words and phrases that were useful to the members of group in ‘its’
description.
Second variation:
1. The students divide into pairs or small groups.
2. Each group is supplied a collection of cards blank on one side and with a picture/photo on
the other side of a pattern or design.
3. Each student takes turns being ‘it’. The person who is ‘it’ takes one of the cards and holds
it so others in the group can not see the picture and describes to other members of the group
what the depicted design or pattern is.
4. The other students attempt to draw a picture of what ‘it’ has described. The other students
may ask questions of ‘it’ as needed.
5. The picture on the card is then shown to the group and compared with the drawings.
6. A list is made of the words and phrases that were useful members of the group in ‘its’
description.
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INTRO3 – Introductory Experiences with Parallel and Perpendicular
Objectives: In each INTRO3, students should be able to…
o Problem solve how to fold arbitrary perpendicular lines, parallel lines,
squares and rectangles on a sheet of paper
o Justify and communicate why these constructions work in ways that can be
followed by others.
Comments from Material Developer on Intro3:
These short units are designed to prime the students to think creatively about geometry and
to provide the teachers with some feedback as to what geometric knowledge the students are
entering with. In particular: Intro3 presents a paper-folding challenge that will bring out in
the students previous knowledge about perpendicular and parallel, start the students
communicating their geometric thinking.
Lines via paper folding (origami). The idea is to get the students starting to think geometrically
and learn the powerful technique of constructions by paper folding and to get them to start
thinking about parallel and perpendicular. Give them as much hints as needed to get them
started and encourage the students to share with each other as they go along.
This is a group of basic activities concerning lines and their properties. We suggest that
they be given to students verbally by the teacher in a form of series of puzzles. Every student
should have a few sheets of paper to play with. In the figure below, numbers indicate the
corresponding puzzles.
?
1
2
3
4
5
6
7
As an introductory warming up question, the teacher could ask (depending your students’
background): Which concept do you find easier: “parallel” or “perpendicular”.
Puzzle 1. How can you make a straight line on the sheet of paper? Can you make an
“arbitrary” straight? – one that is not parallel to any of the edges.
Remarks for teacher: Students will typically respond with the right answer: by folding. But
they will typically fold the paper along one of the edges. The teacher should then ask: “What if
the sheet of paper has form-less, ragged edges? Let us try to ignore the edges and make an
arbitrary line”. (See Figure 1 above.)
Puzzle 2. Now, try to make a new line that is parallel to the arbitrary line just produced.
Remarks. Students will – most probably – find this task undoable. If a student requests a
ruler, remind that the puzzle is restricted to the sheet of paper without other instruments. When
students produce approximate parallels, the teacher should point to the imprecision of the results
(with a ruler?) The intention is to end this puzzle with a fiasco. Suggest the next puzzle as
possibly easier:
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Puzzle 3. How do you make a line perpendicular to a given line?
Remarks: Start with a new sheet of paper. Make sure that the first line is “arbitrary” with
respect to the edges of the sheet. Students should quickly discover the method by folding so that
the semi-lines of original line coincide. (If it is not obvious for them, ask them to compete who
gets it first).
After success, you may also discuss/remind the students about the concept of symmetry, if it
has already been introduced.
Puzzle 4. Now, back to puzzle 2: try to make a parallel line.
Remarks. This time students should find the task easy. The teacher could tease students with
recalling their answer to the initial question (“which concepts do they find easier”). Remark
how a prejudgment can be misleading and that the initial intuition about difficulty of a problem
may occasionally be wrong.
Puzzle 5. Make a rectangle with sides not parallel/perpendicular to any sides of the sheet of
paper. (Such a rectangle is said to be in general position .)
Remarks: This is now easy. Students may simply continue with the same sheet of paper they
ended the last task.
Announce to the students that they have learned how to make a rectangle by paper folding
that is located in a general position (That is, with sides not parallel/perpendicular to the edges
of the sheet).
Puzzle 6. Make a square piece of paper from a rectangular piece of paper.
Starting with a new sheet of paper, point out that one edge is longer than the other. The task
is to tear off a rectangle from one side so that a perfect square is left. Students should come out
easily with solution: start with diagonal bend.
Puzzle 7. Make a square in general position
Announce that this is a “megapuzzle” that summarizes all we learned in the previous steps.
Each student should be able to perform the necessary steps.
Compare the results of different students.
Extra question: How would you double-check your result? (By bending along the other
diagonal).
The students will likely come up with other notions such as ‘mid-point’ and ‘perpendicular
bisector’. This should be encouraged but allow the students to use their own ways of saying it.
Teaching Tips from teachers:
INTRO 3.1

This section asks students to fold paper and make an arbitrary line. Teachers
should make sure that students understand what “arbitrary” means in this
context (a straight line anywhere on the page). It might be helpful to discuss
other contexts of “arbitrary” or “random,” perhaps outside of mathematical
Intro-8
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contexts.
The definition of parallel and perpendicular may differ in the minds of
students (and some students may not have definitions of parallel and
perpendicular). This section should help students formulate or redefine an
understanding of these terms in a geometric context.

Not every student is an artist and may feel discouraged if they can not get
the folds correct. Have a student that is mastering the task help others in the
room or stand in front of the class to demonstrate to students.

INTRO 3.3
Optional Activity for “Final Task”
Divide the class in half. One half works on Puzzle #5 and one half works on
Puzzle #6. Break the Puzzle #5 half into small groups and break the Puzzle #6
half into the same number of small groups (so that each Puzzle #5 group has a
Puzzle #6 group to swap papers with later). Each small group writes the
instructions for their construction. Groups will then exchange instructions and
see if they can construct the same figure using the instructions given. At the
end, groups will compare figures to see whose instructions were the most precise.
This is a time to praise precision of language, and point to the fact that this is
something that will be useful throughout the course, remembering that this unit
is a time to lay groundwork for setting habits, routines, and norms for the year.
Then groups can come back together and individuals have time to solve
Puzzle #7. Techniques from Puzzle #5 and Puzzle #6 come together to allow you
to solve Puzzle #7.
Technology Usage
Technology, where available, can be useful to construct lines, line segments,
rays, angles, etc and also learn how to label. There are many different
possibilities, such as: Geometer’s Sketchpad (program for teachers can be
bought on www.amazon.com for $50-$60), Geogebra (free, web-based, and
powerful found at www.geogebra.org), software with SmartBoards and other
interactive white boards, Cinderella (find at cinderella.de). However, it is
important for students to have the experience of making the constructions by
hand in Intro3 before the use of technology.
Remarks/Suggestions from Kelly Gaddis (mathematics educator who works with teachers in
the South Bronx):
Zenon said that he found folding to be a tool kids are comfortable with, and so he tries to
bring it in as much as he can in later activities…this led me to think about how I would use IGA
at the beginning of the school year…
I would end this short unit explaining that during the past three classes students have
experienced what this class will be about, what we’ll be doing throughout the year, and write
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those up as a poster or in some form like this:
* Thinking Like a Mathematician
* Tackling and solving puzzles
* Looking or visualizing, and then describing what you see
* Asking and communicating “Why?”
* Using physical action to create and investigate ideas
* Using ideas you create as the basis for new ones: Ideas built upon other ideas
* Revisiting ideas across contexts: symmetries, assumptions, arbitrariness, point of view
* Capturing ideas using detailed diagrams and specific terms
* Laying out findings and justifications in ways that can be followed, and even replicated, by
others
I want to put a list visually in the room so that I can point to the elements of it regularly.
For example, when students are floundering or stuck, or when they ask questions such as: “Is
this right?” “What should I do next?” “How do I start?” I will point to the poster and suggest,
“Think like a mathematician” or ask, “Are you thinking like a mathematician?” I’ll refer
specifically to one of the items on the list as a suggestion, such as “Look on your sphere and
describe what you see;” “You have a lot of ideas here already that you can use to build your
definition” or “Is the process you used laid out in a way that can be replicated by a friend?”
Later we as a class can add to the list each time we bring in a new general habit or way of
doing things. In fact, perhaps we’ll start with a four-element list and add four or five more
elements to it over the course of the year. Most important: Each element in the list captures an
experience we have had in class after we’ve had it: Start from an experience and name it; don’t
give a name and then try to provide meaning for it.
(Suggested wordings for items in the list always welcome)
TRM Resources
See TRM Website (www.algebra-trm.org) to access a supplemental lesson
plan/activity that expands on the puzzles that are introduced in Intro 3.2 titled
“Learning to think like Mathematicians: Experiencing Geometry on a Unitless
Plane”, along with the additional files for Unit 2
Dictionary terms in this file:
general position 
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Intro3. Introductory experiences with paper folding
and ‘parallel’ and ‘perpendicular’
Intro3-1. Opening Activity: Use a sheet of paper to fold a paper airplane, or bird, or ship, or
whatever you most enjoy to make.
Unfold the sheet and look at the fold lines on the paper. Do you see straight lines? Do you see
angles? Do you see perpendicular lines? Do you see parallel lines? Do you see any shapes?
Share with you group and then with the whole class.
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Intro3-2. Puzzles: There follows a series of puzzles about paper folding. These will introduce
us to techniques that will be useful as we learn more geometry. After each puzzle draw a sketch
of what you did and number the lines in the order that you make them.
Puzzle 1. How can you make a straight line on the sheet of paper? Can you make an “arbitrary”
straight? – one that is not parallel to any of the edges.
Puzzle 2. Now, try to make a new line that is parallel to the arbitrary line just produced.
Puzzle 3. How do you make a line perpendicular to a given line?
Puzzle 4. Now, back to puzzle 2: try to make a parallel line.
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Puzzle 5. Make a rectangle with sides not parallel/perpendicular to any sides of the sheet of
paper. (Such a figure is said to be in general position .)
Puzzle 6. Make a square piece of paper from a rectangular piece of paper.
Puzzle 7. Make a square in general position.
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Intro3.3. Final task: Write a story that describes how to use folding to make a square in
general position. Note: All of the concepts of Intro3.2-(Puzzles 1 – 7) will be used
Intro-14