Introductory Geometric Activities



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, .

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, .”

( Copyright, 2011, by Algebra Project, Inc. Do not copy or duplicate without written permission from the Algebra Project, Inc. .

INTRO – INTRODUCTORY GEOMETRY ACTIVITIES

|Objectives: |

|In each of the following sections, students should be able to… |

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|INTRO1 |

|Identify and discuss the various human activities that involve geometry. |

|Classify their own experiences and activities into the strands of human activity and justify those classifications. |

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|INTRO2 |

|Use language to describe shapes, patterns, and sizes. |

|Interpret descriptions and draw what is being described. |

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|INTRO3 |

|Problem solve how to fold arbitrary perpendicular lines, parallel lines, squares and rectangles on a sheet of paper |

|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 Post-test 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. |

INTRO1 – What is Geometry?

|Objectives of Intro1: students should be able to… |

|Identify and discuss the various human activities that involve geometry. |

|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: |

|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. |

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|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. |

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|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. |

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. building/structures (houses, laying out a garden, …)

2. machines/motion (crow bar, bicycle, saw, swing, …)

3. navigating/star-gazing (How do I get from here to there? …)

4. art/patterns (design, representations, perspective…).

5. 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.

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.

Intro2. Can we describe this?

|Objectives: In INTRO2, students should be able to… |

|Use language to describe shapes, patterns, and sizes. |

|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.

INTRO3 – Introductory Experiences with Parallel and Perpendicular

|Objectives: In each INTRO3, students should be able to… |

| |

|Problem solve how to fold arbitrary perpendicular lines, parallel lines, squares and rectangles on a sheet of paper |

|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. |

| |

|[pic] |

|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. |

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|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. |

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|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). |

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|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. |

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|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. |

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|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: |

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|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 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. |

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|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  for $50-$60), Geogebra (free, |

|web-based, and powerful found at ), 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 those up as a poster or in some form like this: |

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|* 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 |

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|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 (algebra-) 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 ( |

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.

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. |

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|Puzzle 2. Now, try to make a new line that is parallel to the arbitrary line just produced. |

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|Puzzle 3. How do you make a line perpendicular to a given line? |

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|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 (.) |

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|Puzzle 6. Make a square piece of paper from a rectangular piece of paper. |

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|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

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