February  2022, 2(1): 1-36. doi: 10.3934/steme.2022001

Beyond the compass: Exploring geometric constructions via a circle arc template and a straightedge

1. 

School of Mathematics and Statistics, The University of New South Wales, UNSW, Sydney, 2052, Australia; cct@unsw.edu.au

2. 

School of Mathematics and Statistics, The University of New South Wales, UNSW, Sydney, 2052, Australia; d.beeolmedo@student.unsw.edu.au

* Correspondence: Email: cct@unsw.edu.au; Tel: +61-02-9385-7111

Academic Editor: Suzieleez Syrene Abdul Rahim

Received  January 2022 Revised  February 2022 Published  March 2022

For thousands of years, the compass-and-straightedge tools have dominated the learning and teaching of geometry. As such, these inherited, long-standing instruments have gained a lustre of naturalized pedagogical value. However, mounting evidence suggests that many learners and teachers struggle to efficiently, effectively and safely use compasses when constructing geometric figures. Compasses are difficult for learners to use, can lead to inaccurate drawings, and can be dangerous. Thus, there is value in reconsidering the role of the compass in the learning and teaching of geometric constructions and to offer better tools as alternatives. The purpose of this work is to address the aforementioned need by proposing an alternative tool to the compass that is safer, more efficient and more effective. We will argue that a circle arc template forms such an alternative tool, and we will illustrate how learners and teachers can add value to their classrooms by using it, in conjunction with a straightedge, to establish the well-known constructions seen in geometry curricula around the world.

Citation: Christopher C. Tisdell, David Bee Olmedo. Beyond the compass: Exploring geometric constructions via a circle arc template and a straightedge. STEM Education, 2022, 2 (1) : 1-36. doi: 10.3934/steme.2022001
References:
[1]

Department of Education. Mathematics programmes of study: Key stage 3. National curriculum in England. Document DFE-00179-2013. 2013; Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/SECONDARY_national_curriculum_-_Mathematics.pdf.

[2]

Common Core State Standards Initiative. High School: Geometry - Congruence- Make Geometric Constructions.; Available from: http://www.corestandards.org/Math/Content/HSG/CO/#CCSS.Math.Content.HSG.CO.D.12.

[3]

Assessment and Reporting Authority [ACARA] Australian Curriculum. F-10 Australian Mathematics Curriculum-Mathematics (Version 8.4). 2021; Available from: https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?year=11758&strand=Measurement+and+Geometry&elaborations=true&elaborations=false&scotterms=false&isFirstPageLoad=false.

[4]

Ministry of Education Singapore. MATHEMATICS SYLLABUSES. Secondary One to Four. Express Course. Normal (Academic) Course. 2019; Available from: https://www.moe.gov.sg/-/media/files/secondary/syllabuses/maths/2020-express_na-maths_syllabuses.pdf?la=en&hash=95B771908EE3D777F87C5D6560EBE6DDAF31D7EF.

[5]

NZMaths. Ruler and Compass Constructions. 2021; Available from: https://nzmaths.co.nz/resource/ruler-and-compass-constructions.

[6]

Martin, G.E., Geometric constructions. 1998, New York: New York : Springer.

[7]

Brown, P., Evans, M., Hunt, D., McIntosh, J., Pender, B., and Ramagge, J., Construction - Measurement and Geometry : Module 13, 2011, The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics, Melbourne. Available from: http://amsi.org.au/teacher_modules/Construction.html.

[8]

Schaaf, W.L.E., Reprint Series: Mascheroni Constructions. RS-4. 1967, Stanford, California, USA: Stanford University.

[9]

Albrecht, W.A.J., A Critical And Historical Study Of The Role Of Ruler And Compass Constructions In The Teaching Of High School Geometry In The United States, PhD Thesis, 1952, The Ohio State University, Ann Arbor.

[10]

Hendroanto, A. and Fitriyani, H., Analyzing the need of math geometry drawing tools in mathematics classroom. Journal of Physics: Conference Series, 2019, 1188: 012051. https://doi.org/10.1088/1742-6596/1188/1/012051. doi: 10.1088/1742-6596/1188/1/012051.

[11]

Orlin. B., Math with Bad Drawings- Compass Constructions Made Easy. 2018; Available from: https://mathwithbaddrawings.com/2018/01/10/compass-constructions-made-easy/.

[12]

Richeson, D.S., Tales of Impossibility - The 2000-Year Quest to Solve the Mathematical Problems of Antiquity. 2019: Princeton University Press.

[13]

Vernon, J., Student stabbed with compass, 2013, Hutt News, New Zealand, Available from: http://www.stuff.co.nz/dominion-post/news/local-papers/hutt-news/8865591/Student-stabbed-with-compass.

[14]

Jiang, E., Chinese teacher is detained after 'using a drawing compass to stab and injure dozens of students because they failed to finish their homework', 2020, Daily Mail Australia, Australia. Available from: https://www.dailymail.co.uk/news/article-8369935/Chinese-teacher-detained-using-compass-stab-dozens-students.html.

[15]

Bhattacharya, S., 14-year-old stabbed to death with compass 'over girl', 2017, The Times of India, India. Available from: https://timesofindia.indiatimes.com/city/delhi/14-yr-old-stabbed-to-death-with-compass-over-girl/articleshow/58466802.cms.

[16]

Welkowitz, S., Tenth Year Geometry for all American Youth. The Mathematics Teacher, 1946, 39(3): 99‒112. https://www.jstor.org/stable/27953069.

[17]

Day Ashley, L., Case Study Research, in Research Methods & Methodologies in Education, R. Coe, et al., Editors. 2017, pp. 114‒121. Los Angeles, CA: SAGE.

[18]

Dounskaia, N. and Goble, J.A., The role of vision, speed, and attention in overcoming directional biases during arm movements. Experimental Brain Research, 2011,209(2): 299‒309. https://doi.org/10.1007/s00221-011-2547-9. doi: 10.1007/s00221-011-2547-9.

[19]

Dounskaia, N.V., Ketcham, C.J., and Stelmach, G.E., Influence of Biomechanical Constraints on Horizontal Arm Movements. Motor Control, 2002, 6(4): 366‒387. https://doi.org/10.1123/mcj.6.4.366. doi: 10.1123/mcj.6.4.366.

[20]

Severi, F., Sui problemi determinati risolubili colla riga e col compasso (Estratto da una Lettera al Prof. F. Enriques). Rendiconti del Circolo matematico di Palermo, 1904, 18(1): 256‒259. https://doi.org/10.1007/BF03014102. doi: 10.1007/BF03014102.

[21]

Steiner, J. and Archibald, R.C., Geometrical constructions with a ruler, given a fixed circle with its center. Scripta mathematica studies: no. 4. 1950: Scripta Mathematica, Yeshiva University.

[22]

Mackay, J.S., Solutions of Euclid's Problems, with a rule and one fixed aperture of the compasses, by the Italian geometers of the sixteenth century. Proceedings of the Edinburgh Mathematical Society, 1886, 5: 2‒22. https://doi.org/10.1017/S0013091500001334. doi: 10.1017/S0013091500001334.

[23]

Gibb, A.A., There are more ways than one to bisect a segment. The Mathematics Teacher, 1977, 70(5): 390‒393. http://www.jstor.org/stable/27960865.

[24]

Hlavaty, J.H., Mascheroni constructions. The Mathematics Teacher, 1957, 50(7): 482‒487. http://www.jstor.org/stable/27955488.

show all references

References:
[1]

Department of Education. Mathematics programmes of study: Key stage 3. National curriculum in England. Document DFE-00179-2013. 2013; Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/SECONDARY_national_curriculum_-_Mathematics.pdf.

[2]

Common Core State Standards Initiative. High School: Geometry - Congruence- Make Geometric Constructions.; Available from: http://www.corestandards.org/Math/Content/HSG/CO/#CCSS.Math.Content.HSG.CO.D.12.

[3]

Assessment and Reporting Authority [ACARA] Australian Curriculum. F-10 Australian Mathematics Curriculum-Mathematics (Version 8.4). 2021; Available from: https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?year=11758&strand=Measurement+and+Geometry&elaborations=true&elaborations=false&scotterms=false&isFirstPageLoad=false.

[4]

Ministry of Education Singapore. MATHEMATICS SYLLABUSES. Secondary One to Four. Express Course. Normal (Academic) Course. 2019; Available from: https://www.moe.gov.sg/-/media/files/secondary/syllabuses/maths/2020-express_na-maths_syllabuses.pdf?la=en&hash=95B771908EE3D777F87C5D6560EBE6DDAF31D7EF.

[5]

NZMaths. Ruler and Compass Constructions. 2021; Available from: https://nzmaths.co.nz/resource/ruler-and-compass-constructions.

[6]

Martin, G.E., Geometric constructions. 1998, New York: New York : Springer.

[7]

Brown, P., Evans, M., Hunt, D., McIntosh, J., Pender, B., and Ramagge, J., Construction - Measurement and Geometry : Module 13, 2011, The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics, Melbourne. Available from: http://amsi.org.au/teacher_modules/Construction.html.

[8]

Schaaf, W.L.E., Reprint Series: Mascheroni Constructions. RS-4. 1967, Stanford, California, USA: Stanford University.

[9]

Albrecht, W.A.J., A Critical And Historical Study Of The Role Of Ruler And Compass Constructions In The Teaching Of High School Geometry In The United States, PhD Thesis, 1952, The Ohio State University, Ann Arbor.

[10]

Hendroanto, A. and Fitriyani, H., Analyzing the need of math geometry drawing tools in mathematics classroom. Journal of Physics: Conference Series, 2019, 1188: 012051. https://doi.org/10.1088/1742-6596/1188/1/012051. doi: 10.1088/1742-6596/1188/1/012051.

[11]

Orlin. B., Math with Bad Drawings- Compass Constructions Made Easy. 2018; Available from: https://mathwithbaddrawings.com/2018/01/10/compass-constructions-made-easy/.

[12]

Richeson, D.S., Tales of Impossibility - The 2000-Year Quest to Solve the Mathematical Problems of Antiquity. 2019: Princeton University Press.

[13]

Vernon, J., Student stabbed with compass, 2013, Hutt News, New Zealand, Available from: http://www.stuff.co.nz/dominion-post/news/local-papers/hutt-news/8865591/Student-stabbed-with-compass.

[14]

Jiang, E., Chinese teacher is detained after 'using a drawing compass to stab and injure dozens of students because they failed to finish their homework', 2020, Daily Mail Australia, Australia. Available from: https://www.dailymail.co.uk/news/article-8369935/Chinese-teacher-detained-using-compass-stab-dozens-students.html.

[15]

Bhattacharya, S., 14-year-old stabbed to death with compass 'over girl', 2017, The Times of India, India. Available from: https://timesofindia.indiatimes.com/city/delhi/14-yr-old-stabbed-to-death-with-compass-over-girl/articleshow/58466802.cms.

[16]

Welkowitz, S., Tenth Year Geometry for all American Youth. The Mathematics Teacher, 1946, 39(3): 99‒112. https://www.jstor.org/stable/27953069.

[17]

Day Ashley, L., Case Study Research, in Research Methods & Methodologies in Education, R. Coe, et al., Editors. 2017, pp. 114‒121. Los Angeles, CA: SAGE.

[18]

Dounskaia, N. and Goble, J.A., The role of vision, speed, and attention in overcoming directional biases during arm movements. Experimental Brain Research, 2011,209(2): 299‒309. https://doi.org/10.1007/s00221-011-2547-9. doi: 10.1007/s00221-011-2547-9.

[19]

Dounskaia, N.V., Ketcham, C.J., and Stelmach, G.E., Influence of Biomechanical Constraints on Horizontal Arm Movements. Motor Control, 2002, 6(4): 366‒387. https://doi.org/10.1123/mcj.6.4.366. doi: 10.1123/mcj.6.4.366.

[20]

Severi, F., Sui problemi determinati risolubili colla riga e col compasso (Estratto da una Lettera al Prof. F. Enriques). Rendiconti del Circolo matematico di Palermo, 1904, 18(1): 256‒259. https://doi.org/10.1007/BF03014102. doi: 10.1007/BF03014102.

[21]

Steiner, J. and Archibald, R.C., Geometrical constructions with a ruler, given a fixed circle with its center. Scripta mathematica studies: no. 4. 1950: Scripta Mathematica, Yeshiva University.

[22]

Mackay, J.S., Solutions of Euclid's Problems, with a rule and one fixed aperture of the compasses, by the Italian geometers of the sixteenth century. Proceedings of the Edinburgh Mathematical Society, 1886, 5: 2‒22. https://doi.org/10.1017/S0013091500001334. doi: 10.1017/S0013091500001334.

[23]

Gibb, A.A., There are more ways than one to bisect a segment. The Mathematics Teacher, 1977, 70(5): 390‒393. http://www.jstor.org/stable/27960865.

[24]

Hlavaty, J.H., Mascheroni constructions. The Mathematics Teacher, 1957, 50(7): 482‒487. http://www.jstor.org/stable/27955488.

Figure 1.  Compass constructions made easy [11]
Figure 2.  Circle arc template
Figure 3.  A given line segment $ \overline {AB} $
Figure 4.  Circle arc template with centre at $ A $
Figure 5.  Circle arc template with centre at $ A $
Figure 6.  A first arc
Figure 7.  Circle arc template with centre at $ B $
Figure 8.  Second arc with intersection points labelled
Figure 9.  Intersection points, $ D $ and $ E $
Figure 10.  Straightedge aligned with points $ D $ and $ E $
Figure 11.  Line segment $ \overline {DE} $ constructed
Figure 12.  Finished construction. $ \overline {DE} $ is the perpendicular bisector of $ \overline {AB} $
Figure 13.  Line segment $ \overline {AB} $
Figure 14.  Circle arc template with centre at $ A $
Figure 15.  Marking point of intersection made with $ \overline {AB} $
Figure 16.  Point of intersection labelled $ P $
Figure 17.  Circle arc template with centre at $ B $
Figure 18.  Marking point of intersection made with $ \overline {AB} $
Figure 19.  Point of intersection labelled $ S $
Figure 20.  Circle arc template with centre at $ P $
Figure 21.  Marking point of intersection made with $ \overline {AB} $
Figure 22.  Point of intersection labelled $ Q $
Figure 23.  Circle arc template with centre at $ S $
Figure 24.  Marking point of intersection made with $ \overline {AB} $
Figure 25.  New point of intersection labelled $ R $
Figure 26.  Circle arc template with centre at $ Q $
Figure 27.  Arc with centre at point $ Q $
Figure 28.  Circle arc template with centre at $ R $
Figure 29.  Arc with centre at point $ R $ and intersection points labelled
Figure 30.  Intersection points $ D $ and $ E $
Figure 31.  Straightedge aligned with points $ D $ and $ E $
Figure 32.  Straightedge aligned with points $ D $ and $ E $
Figure 33.  Finished construction. $ \overline {DE} $ is the perpendicular bisector of $ \overline {AB} $
Figure 34.  $ \angle AOB $
Figure 35.  Circle arc template with centre at $ O $
Figure 36.  Marking point of intersection, $ P $
Figure 37.  Marking point of intersection, $ Q $
Figure 38.  $ \angle AOB $ with points $ P $ and $ Q $
Figure 39.  Circle arc template with centre at $ P $
Figure 40.  'Small' section of arc traced, with centre at $ P $
Figure 41.  Circle arc template with centre at $ Q $
Figure 42.  Arc with centre at $ Q $ intersecting previous arc to form point $ C $
Figure 43.  Line segment $ \overline {OC} $ is constructed using straightedge
Figure 44.  Finished construction. $ \overline {OC} $ is the bisector of $ \angle AOB $
Figure 45.  Line Segment $ \overline {AB} $ and point $ P $
Figure 46.  Circle arc template with centre at $ P $
Figure 47.  Points $ Q $ and $ R $ constructed
Figure 48.  Drawing arc with centre at point $ Q $
Figure 49.  Drawing arc with centre at point $ R $
Figure 50.  New point of intersection labelled $ S $
Figure 51.  Line segment $ \overline {PS} $ is constructed using straightedge
Figure 52.  Finished construction. $ \overline {PS} $ is perpendicular to $ \overline {AB} $
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