Constructions
by
Ana Kuzle
Geometric constructions
Day 1-Introduction
Objectives:
Prerequisite knowledge:
Construction as a word has a very specific meaning. Any drawing is restricted to use of only compass and a straightedge, whilst using a ruler to measure lengths and a protractor to measure angles is not allowed. Why is this so? In antiquity, geometric constructions were restricted to the use of only a straightedge and a compass. Notice, that here we used the term straightedge, and not a ruler. Why not measure? Greeks knowledge of mathematics was basic, including little arithmetic. So, faced with the problem of finding the midpoint of a line, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge. It is also why the straightedge has no markings. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.
Instead of the term geometric structures we more often use the term Euclidean constructions because the majority of those are held in Euclid's (300 BC) Elements. Geometric constructions are highly connected to problems of antiquity that include squaring a circle, duplication of a cube and angle trisection that have been proved to be geometrically constructed impossible hundreds of years later.
In this technological era it is no wonder students have never seen a compass let alone hold it in their hands. Since majority of the work is going to be done using construction tools, it is important for the students to get familiar with those.
Compass is a drawing instrument often used to draw circles and arcs. It has two legs, one with a point, and the second one with a lead or pencil. The openness of the compass is easily regulated by opening or closing both legs, and will remain in a particular setting until changed.
Straightedge is a tool used for drawing straight lines, and has no markings. If ruler is being used it must be cleared to students that using markings on the ruler is not allowed during constructions.
Patty paper (vax paper) will be used in this lesson as a representation for students to grasp more easily through geometrical concepts.
During this first day it is important for students to learn and develop appreciation for the context of geometric constructions. Also, it is important for students to have time to play with the tools, especially with the compass.
Discuss with the students the meaning of the words to sketch, to draw and to construct. Let them carry the meaning on their own and correct when needed. By the end of the day student should know the difference between those terms.
Explanation of the terms:
Neither ruler nor protractor is ever used to perform geometric constructions. They are measuring tools, and not construction tools.
Note: The lesson plan can be developed either using compass and ruler or using technology or both together. It is on the teacher to make that decision. However, I strongly encourage using the old fashion compass and ruler =).
Day 2-Basic constructions
1. Start off the discussion with two simple problems.
1. a. copy a given segment
1. b. copy a given angle
If necessary discuss with your students what it means to copy something. It should be clear to them that by copying we understand constructing an object congruent to a given one. At the beginning it is important to set the rules. Develop a discussion what would a proper construction include. Let them derive that there is more to construction than mare performing of steps and memorizing of the same ones. It should be clear that prior to starting the constructions they have to understand the problem and a good way for that is to make a sketch and start thinking how they would solve the problem. Afterwards the students can perform their steps using the geometry tool. However, the process does not end here. Students should justify their steps using proper geometry language using concepts learnt prior. Depending on the type of problem discussion will be needed on the uniqueness of the solution. Also, precision here is not the center of the universe, but make sure that they should manage compass and the ruler to be as precise as possible. Agreement on notation is also important. Make sure they mark the points with a little circle, and not dot, segment, or cross.
2. Ask your students what is a segment bisector. Ask the to draw it. Make a discussion depending on the solutions. It is possible they have drawn just one and possibly a perpendicular one. This is a good place for a discussion on the difference between a segment bisector and a perpendicular bisector. Student should be able to deduce that perpendicular bisector of a segment is unique for that segment, whilst there is infinitely many bisectors of a given segment.
The following problem is going to be constructed using first patty paper and then the geometry tools. Patty paper is going to be used for students to come up with the construction using the geometry tools. As the students use the patty paper develop a discussion on this activity. Discuss with them how will they use the patty paper? What is the line they get when folding the paper? What is the property of any point on that line in relation to the endpoints of the segment. If they were able to answer all of these questions they should be able to do the construction using the geometry tools. However challenge the students with the questions of explaining why the constructions is correct as well as whether the openness of the compass is important, can we change both or just one leg and still get a perpendicular bisector. The purpose is as for all activities regarding constructions for students not to take them for granted.
2. a. construct a perpendicular bisector using a patty paper
2. b. construct a perpendicular bisector using geometry tools
3. Since the students learnt the concept of a segment bisector it is advisable to use these concepts in more challenging problems that will help us introduce some geometrical concepts. For instance we could give the student the following problems:
3. a. Given triangle ABC construct three medians. What do you notice about all three medians?
3. b. Given triangle ABC construct the perpendicular bisectors of each side. What do you notice about all three bisectors?
These problems could be a small group work where students work on one of these problems. Both problems are rich with discoveries, especially when using a software. Students should be able to discover that all of the three medians or perpendicular bisectors coincide. Introduce terms centroid and circumvent. Let your student grasp through the meaning of those words to help them understand the newly discovered points and concepts.
Homework problems:
Day 3
1. Start the discussion by discussing how the perpendicular bisector to a given line could be constructed. Ask them how they can use patty paper to do that if getting there by connecting previous lesson is not going that well. Make sure they are aware that they can construct the perpendicular line to a given line from a point outside and both on the given line. Discuss with them how are these constructions the same or different. They should mathematically argument their construction decisions using geometric concepts and properties.
1. a. construct a line perpendicular to a given live through a given point outside the line
1. b. construct a line perpendicular to a given live through a given point on the line
2. The next activity connects the concepts of constructing a perpendicular and the shortest distance. Also, it closely connects construction of the angle bisector. Let the students draw an angle ABC. Let them place point P inside the angle, and construct perpendiculars from point P to both legs of the angle. Discuss with them which side is closer to point P? Make sure they justify any statement they make. Ask them where point P must be so that the distances to the legs are the same. If they started with an obtuse triangle would the conclusions be the same? If yes, why if not, why not? Discussion should lead by discovering that P should lie on the bisector of the angle ABC. For this activity one can either use technology and give student time to discover any patterns or by mare drawing and trying to deduce backwards the actual angle bisector construction. After justifying that point P lying on the angle bisector, ask them to construct angle bisector of ABC.
3. As mentioned before the previous activity is closely connected to constructing an angle bisector. If they were not able to construct it after the first activity, give them a patty paper and let them discuss how they would by using a patty paper get an angle bisector. They should be able to explain why by folding the patty paper in such way that the legs of the angle overlap (coincide) by connecting vertex B to the crease they got the angle bisector. Ask them how does this help them construct the angle bisector? Discuss with them whether the openness of the compass influences the construction. Does the openness has to have the same radius as the radius of the arc? What if the compass collapse. It is important for all of these questions to be discussed in details for the students to develop understanding of the nature of the constructions and the construction of the angle bisector in particular.
4. a. Given triangle ABC construct the angle bisectors of each angle. What do you notice about all three angle bisectors?
4. b. Given triangle ABC construct the altitudes of each angle. What do you notice about all three altitudes?
These problems could be a small group work where students work on one of these problems. Both problems are rich with discoveries, especially when using a software. Students should be able to discover that all of the three angle bisectors and lines containing the altitudes coincide. Introduce terms incenter and orthocenter. Let your student grasp through the meaning of those words to help them understand the newly discovered points and concepts. Especially try to develop understanding for why the orhocenter is defined as an intersection of lines containing the altitudes and not just intersection of the three altitudes.
Homework problems:
a. Trace this bisector as you drag point C along side AB. Describe the shape formed by this locus of lines.
b. Trace the midpoint od CF as you drag C. Describe the locus of points.
Day 4
1. Give your students a patty paper and let them derive to strategy to construct a line parallel to given line through a given point. They should be able to deduce that they should fold the paper to construct a perpendicular so that the crease runs through the point and then make another fold that is perpendicular though the first crease thought the given point and match the pairs of the corresponding angles created by folding. Discuss whether angles obtained are congruent. If so why? What conclusions can be made about the lines? Based on this activity they should be able to construct a line parallel to given line though a given point. Also, discuss with them whether the second line is unique and why.
2. Challenge them to think of another way to do the same construction. Since they should know that when two parallel lines are intersected by a transversal that we get congruent corresponding angles. Therefore, they should be able to derive that by copying the angle between the given line and point at that point, they will get a line parallel to the given line.
3. At this point it can be beneficial to give them problems that will connect several constructions.Try to give problem that employ different strategies. For instance, pose them the problem of constructing the angle bisector of two line segment in finite plane. Hence, two line segment that do not intersects on the paper, and by not extending those lines. This problem can be solved in different ways. One of the ways would be by constructing lines parallel to given line segments closer for them to intersect and then by doing the known angle bisector construction. Second, they could employ concepts about incenter and incicrcle. Discuss with them the strength and weakness of each method they derive.
Day 5 & 6-Construction problems
After knowing the basic constructions, we can construct more complex geometric figures.Here we will list several problems that could be a two day plan. Students could work individually, or in pairs depending on classroom environment. However, the teacher should be here only as a manager of the classroom, and let students struggle on their own. Before jumping and solving the construction problems, discuss the properties of triangles and quadrilaterals. Also, after each problem make a classroom discussion where students justify their construction processes. Be sure to discuss whether the solution in each of the following constructions is unique or not, and why.
Problems:
Construct the triangle ABC using all three segments.
Construct the triangle ABC using two segments and one angle. How many solution there are?
Construct the triangle ABC using one segment and two angles.
Construct an isosceles triangle with perimeter o and length of the base equal to a.
Construct a square given the length of the diagonal.
Construct a rhombus given lengths of both diagonals.
Construct a rhombus given the altitude and one diagonal.
Construct a parallelogram given one side, one angle and a diagonal.
Construct an isosceles trapezoid given the bases and ne angle.
Construct an isosceles trapezoid, given the median, altitude, and base angle of 30º.
There are numerous more challenging problems. The above one are just a proposal but not necessarily the only one appropriate for middle and high school students. For instance, since we explored to some extent the center of gravity for a triangle, a natural question can arise on the center of gravity for a quadrilateral. How do we construct it etc. It is on the teacher to decide how deep they want to go and pursue certain ideas and concepts.
Listed below are some web pages with interesting construction problems:
http://mathforum.org/library/problems/sets/geo_constr_locus.html
http://www.cut-the-knot.org/geometry.shtml
http://whistleralley.com/construction/reference.htm
http://jwilson.coe.uga.edu/EMT668/Asmt6/EMT668.Assign6.html (Problem #3)
http://www.nvcc.edu/home/tstreilein/constructions/
Algebraic constructions
Day 7 & 8-Introduction and construction of basic algebraic expressions
Objectives:
Prerequisite knowledge:
This topic naturally follows after teaching basics of geometry constructions connecting it to algebraic concepts. However, since high school textbooks rarely include algebraic method for most of the problems explanations will be given in .gsp files. Let students discuss the strategies employed and reasoning in each of the following problems.
1. The best way to introduce this method is by simply posing a problem where knowledge of geometric methods do not help us, but instead have to depend on different method.For the discussion we can choose among variety of problems. Here, as a motivation task will suffice the following problem:
Construct the triangle ABC if the relation among its sides a, b and c is given by
Give the students time to explore this problem. They should realize that trying to employ geometric methods learnt so far cannot help them is solving this problem. However, discuss with them the expression given that relates its sides. They should be able to realize that on the right side of the expression they have geometric mean of certain sides, G (a, c) and G(b, c). Connecting the geometric mean with the arithmetic mean and knowing that for any non-negative x and y, G(x, y)<=A(x, y), where we have equality of and only if x=y, they should conclude that all of the sides must be equal to satisfy the given expression. From this point the problem is trivial since in the previous lesson they learnt how to do basic constructions .
Thus, through this problem students should become aware that algebraic method can help us with construction problems when geometric ones fails. This problem also gives a great picture what does the algebraic method includes. Students should on their own understand how and why does this method work, and not be given to them per se.
2. Before looking at complex algebraic expressions and their constructions,we will observe constructions of basic expressions. Based on these application of the algebraic method is grounded. In the following list of basic constructions a, b and c are lengths of given segments, m and n are natural numbers, whilst x is the segment we want to construct.
Constructions for problem 4-11 with explanations can be found here. All of the above problems should develop connections between numbers and their geometric representation, or in other words geometric representation of algebraic expressions. Discuss each of the problems with the students.
Day 9-Applications of algebraic method in construction problems
Through the following problems, students will gain insight into efficacy of the algebraic method in construction problems, its positive and negative sides. The problems vary in difficulty and each of the problem explanation are given in .gsp file here. It is on the teacher to decide how to organize these activities. Thus, individual work, work in pairs or groups. However, make sure your students understand the problem, and construction itself and discuss as a class each of the problems given.
Problem selection:
Homework problems:
Day 10-Conclusions
1. Start off the today's discussion on regular polygons and their contractibility. Ask them if and how they can construct n-gon when n=3, 6, 12, 4, 8, etc. They should say yes. However ask them if any n-gon is constructible. Probably you will here yes, no or nothing. At this point you can talk about Gauss, child prodigy who as the age of 15 proved that a regular 17-gon can be constructed. He was so fascinated with regular 17-gons that he wished for it to be engraved on his grave. However, his wish never came true, but in it was on a statue of him in his hometown Braunschweig. Not all regular polygons are constructible like 7-, 9-, 11-gon, etc. However, they are with so called neusis constructions that allow using a marked ruler. Let your students explore neusis constructions and give them a group project that involves approximation construction of regular 7-, 9, and 11-gon.
2. Summarize the results of the lesson. At this point discuss with the students the difference between the conventional construction, and Euclidean consecution. Also discuss the algebraic method and its relevance to constructions.
Teacher reflection
Teacher should reflect on their teaching asking question such as:
Did I challenge the students? How?
Were students actively engaged in the learning process? How do I know?
Did the students exceed my expectations in some areas, and not meet them in others?
Did I have to adjust teaching? If so, what adjustments did I make, and were they effective?
Reference list
Serra, M. (2003). Discovering geometry: An investigative approach. Emeryville, CA: Key Curriculum Press.