# American Institute of Mathematical Sciences

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:
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[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.
Compass constructions made easy [11]
Circle arc template
A given line segment $\overline {AB}$
Circle arc template with centre at $A$
Circle arc template with centre at $A$
A first arc
Circle arc template with centre at $B$
Second arc with intersection points labelled
Intersection points, $D$ and $E$
Straightedge aligned with points $D$ and $E$
Line segment $\overline {DE}$ constructed
Finished construction. $\overline {DE}$ is the perpendicular bisector of $\overline {AB}$
Line segment $\overline {AB}$
Circle arc template with centre at $A$
Marking point of intersection made with $\overline {AB}$
Point of intersection labelled $P$
Circle arc template with centre at $B$
Marking point of intersection made with $\overline {AB}$
Point of intersection labelled $S$
Circle arc template with centre at $P$
Marking point of intersection made with $\overline {AB}$
Point of intersection labelled $Q$
Circle arc template with centre at $S$
Marking point of intersection made with $\overline {AB}$
New point of intersection labelled $R$
Circle arc template with centre at $Q$
Arc with centre at point $Q$
Circle arc template with centre at $R$
Arc with centre at point $R$ and intersection points labelled
Intersection points $D$ and $E$
Straightedge aligned with points $D$ and $E$
Straightedge aligned with points $D$ and $E$
Finished construction. $\overline {DE}$ is the perpendicular bisector of $\overline {AB}$
$\angle AOB$
Circle arc template with centre at $O$
Marking point of intersection, $P$
Marking point of intersection, $Q$
$\angle AOB$ with points $P$ and $Q$
Circle arc template with centre at $P$
'Small' section of arc traced, with centre at $P$
Circle arc template with centre at $Q$
Arc with centre at $Q$ intersecting previous arc to form point $C$
Line segment $\overline {OC}$ is constructed using straightedge
Finished construction. $\overline {OC}$ is the bisector of $\angle AOB$
Line Segment $\overline {AB}$ and point $P$
Circle arc template with centre at $P$
Points $Q$ and $R$ constructed
Drawing arc with centre at point $Q$
Drawing arc with centre at point $R$
New point of intersection labelled $S$
Line segment $\overline {PS}$ is constructed using straightedge
Finished construction. $\overline {PS}$ is perpendicular to $\overline {AB}$
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