9.3.2C Technology Tools
Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets.
Overview
Standard 9.3.2 Essential Understandings
Examples suggest, whispering, "It might be true." One counterexample, however, thunders, "It is false." (attributed to Polya)
Using logic to construct and refute arguments is the essence of proof and distinguishes formal high school geometry from previous geometric learning. Students should continue to explore proposed conjectures using dynamic geometry software, but then use formal arguments to move from proposed conjectures to proved theorems. The structure of geometric proof helps the students to organize their thinking and make connections between ideas. "Students should see the power of deductive proof in establishing the validity of general results from given conditions. The focus should be on producing logical arguments and presenting them effectively with careful explanation of the reasoning, rather than on the form of proof used (e.g., paragraph proof or two-column proof)." (NCTM, PSSM, p. 310) Students need to center on creating, evaluating and communicating accurate mathematical arguments in a valid logical progression.
All Standard Benchmarks
9.3.2.1 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.
9.3.2.2 Accurately interpret and use words and phrases such as "if...then," "if and only if," "all," and "not." Recognize the logical relationships between an "if...then" statement and its inverse, converse and contrapositive.
9.3.2.3 Assess the validity of a logical argument and give counterexamples to disprove a statement.
9.3.2.4 Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations.
9.3.2.5 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets.
9.3.2.5 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Copy any segment using construction tools.
- Copy any angle using construction tools.
- Construct the angle bisector of any angle.
- Construct the midpoint of any segment.
- Construct the perpendicular bisector of any segment.
- Use dynamic geometry software (Geometer's Sketchpad, Geogebra, TI-nSpire, Casio fx-CG 10, etc) to generate variable examples to formulate and test conjectures.
- Use a compass and straightedge to construct a general example to formulate and test conjectures.
Work from previous grades that supports this new learning:
- Students have made conjectures and developed informal arguments related to shape and space.
- Students have used dynamic geometry software, diagrams and measurements to explore two-dimensional geometric concepts.
NCTM Standards: Geometry
Instructional programs from prekindergarten through grade 12 should enable all students to:
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
In grades 9-12 all students should-
- analyze properties and determine attributes of two- and three-dimensional objects;
- explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
- establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
Use visualization, spatial reasoning, and geometric modeling to solve problems
In grades 9-12 all students should-
- draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;
- visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections;
- use geometric models to gain insights into, and answer questions in, other areas of mathematics;
- use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSS)
HS.G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
HS.G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
HS.G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
HS.G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
HS.G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
HS.G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
HS.G-SRT.6 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
HS.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
Misconceptions
Student Misconceptions and Common Errors
- Students might argue that a conjecture must be valid simply because it worked in all the examples they tried.
- Students may use "drawing" techniques instead of actual construction.
Vignette
In the Classroom
The goal of this lesson is to have students generate and explore a variety of examples for a conjecture using dynamic geometry software. In this example, students are working individually at computer stations. A similar lesson could be accomplished using an interactive whiteboard or a set of TI-Nspires or Casio fx-CG 10's within the classroom.
The students are asked to draw a triangle, construct a new triangle by joining the midpoints of its three sides, and calculate the ratio of » the area of the midpoint triangle to the area of the original triangle (see fig. 7.13a). As they drag one vertex to create many different triangles, the students notice that the ratio of the two areas appears to remain constant at 0.25.
Fig. 7.13. Exploring and extending the results of connecting the midpoints of the adjacent sides of polygons
Jake says he thinks that this relationship will always hold. He says that since the base of each of the four small triangles is a midline, each side of the midpoint triangle should be half as long as the parallel side of the large triangle. Each midline cuts the altitude in half, so the height of each small triangle is half that of the large triangle. Dividing each of these lengths by 2 divides the area by 4, so the area of the small triangle is one-fourth of the area of the large one.
Berta agrees with Jake's answer and thinks she can show that it must be true. She explains how she has extended the midlines and the sides of the triangles to form three pairs of parallel lines and is now able to determine many pairs of congruent angles. She reasons, using parallelism, that the corresponding sides are congruent and determines that the three small triangles formed at the vertices of the original triangle are congruent by angle-side-angle. She is confident that the midpoint triangle should be congruent to the other three, but when the teacher asks her how she can be sure, she is unable to give an explanation.
The teacher asks one question: "Do you know anything about the sides of that triangle?" Her friend Dawn quickly notes that all its sides are shared with the sides of the other three, which indicates that it would have to be the same size.
Hope has a somewhat different way of looking at the situation. She notices that the lengths of the corresponding sides of the midpoint triangle and the original triangle are in a ratio of 1:2, so they must be similar. Thus, the area of the midpoint triangle must be one-fourth the area of the original on the basis of the class's earlier observation that the areas of similar triangles are related by the square of their scale factor. The teacher asks the class to think about the relationship between Hope's method and Jake's method.
The students decide to test whether a constant ratio exists for the area of a "midpoint" quadrilateral to the area of a convex quadrilateral (see figure 7.13b). It appears that the area ratio in this case is 0.5. They are able to prove this relationship by dividing a quadrilateral into two triangles and employing the methods they used in the previous investigation. The students begin to wonder whether they have discovered a big idea. Does a constant ratio hold for other polygons?
For the first few convex pentagons they try (see fig. 7.13c), the area ratio appears to be constant at 0.7 (see Zbiek [1996] for a discussion about how this problem can be solved using technology). When they are unable to generate a proof, they decide to check more examples. When they do so, they begin to see some variation in the ratios. This development is disappointing to the students, who were hoping that the result they proved for triangles and quadrilaterals would be general. Their teacher points out that they really should be encouraged by their results. They have made a series of conjectures, produced proofs of some conjectures, and produced counterexamples to show when other conjectures do not hold. That kind of careful thinking, he says, is truly mathematical.
(Vignette is from the National Council of Teachers of Mathematics: NCTM, PSSM, p. 311-312)
Resources
Teacher Notes
- Teachers need to stress the difference between conjecture supported by example and proof supported by formal logic.
- Teachers need to remind students that producing a diagram by drawing and measuring creates only a single example drawn to scale and cannot be used for dynamic variations.
Additional Instructional Resources
- This is a resource bank of activities to use with a variety of TI products. Students could link and download them individually to their handheld devices or they could be used with SmartView interactive presentations.
- Geogebra is a free software program useful for both algebra and geometry.
- Dan Meyer's blogsite includes his version of a complete Geometry course. Many of the activities can be incorporated for use with any textbook.
Reflection - Critical Questions regarding teaching and learning of these benchmarks:
- What other instructional strategies and activities (technology related) can I use to engage my students to reach the goals of this benchmark?
- How do I scaffold my instruction for my students?
- Do the tasks I've designed connect to underlying concepts or focus on memorization?
- How can I differentiate the lesson?
- How can I tell if students have reached this learning goal?
Kimberling, C. (2003). Geometry in action. Emeryville, Ca: Key College Publishing.
Marzano, R. J., & Pickering, D. J. (2005). Building academic vocabulary. Alexandria, Va: ASCD.
National Council of Teachers of Mathematics. (2000.) Principles and standards for school mathematics. Reston, Va: NCTM.
Assessment
Teacher's Note: The first set of assessment problems are written for compass-straightedge work. The second assessment set is written for dynamic geometry software. This assessment is usually completed before beginning the section of the textbook on points of concurrency. Student constructions are then used to create illustrative examples for the proofs related to concurrency and their associated circles.
The "Cool Line" is the Euler Line and the "Cool Circle" is Feuerbach's Nine Point circle.
Triangle Construction Project
Goal: Use Geometry Tools to accurately determine cool points related to triangles.
Use: Sharp pencils (multiple colors would be great), compass, protractor, ruler, unlined paper
Part 1 (Cool Line)
- Draw an acute scalene triangle with sides approximately 11 cm, 14 cm, and 16 cm in the center of your paper. Make 5 copies of this page.
- On page 1, construct the perpendicular bisectors of each side of the triangle. Find the point where they intersect and label it point Q.
- On page 2, construct the angle bisectors of each angle of the triangle. Find the point where they intersect and label it point R.
- On page 3, construct the line segments connecting each vertex with the midpoint of the opposite side. Find the point where they intersect and label it point S.
- On page 4, construct the line segments from each vertex perpendicular to the opposite side. Find the point where they intersect and label it point T.
- On page 5, trace your four points onto one copy of the triangle. Three of the points should be collinear. Draw the line connecting them and label it L.
Part 2 (Cool Circle)
- Draw an acute scalene triangle PQR with sides approximately 11 cm, 15 cm, and 16 cm in the center of your paper.
- Carefully measure and find the midpoints of the sides of triangle PQR. Draw them in with one color and label them A, B, and C.
- Construct the altitudes of triangle PQR using a second color. Label the vertices of the right angles D, E, and F.
- Label the point where the altitudes of triangle PQR meet as point T.
- Find the midpoints of PT, QT, and RT. Draw them in with color three and label them G, H, and J.
- Connect any pair of points (not P, Q, R or T) with color four. Construct the perpendicular bisector of this segment.
- Connect another pair of points (not P, Q, R or T) with color five. Construct the perpendicular bisector of this segment. The point where the lines from steps 6 and 7 meet is the center of the circle.
- Draw the circle that contains points A, B, C, D, E, F, G, H, and J with color six.
- Repeat steps 1-8 with an equilateral triangle. How does this change your results?
SOLUTIONS:
Euler's Line:
Nine-Point Circle:
- Nine-Point Circle starting from an equilateral triangle: The altitudes coincide with the midpoints of the sides, so it becomes a six-point circle with T at the center.
Geometry lab - Points of concurrency.
In this lab, you will be working with triangles, and trying to determine an Euler line (this is pronounced "oiler"). The Euler line is named after a Swiss mathematician, Leonhard Euler. You will also work with something called the "nine-point circle." It is important to read the directions, and follow them as written.
- Draw any triangle. Construct the midpoints of each of the sides, and a median from any one vertex. Construct a median from a second vertex. Click on the point of intersection to construct it. Will the third median also pass through this point of intersection? Try it and see. Move the triangle around to see if your conjecture is always true. Sketch the diagram below.
You will now hide all three medians, but keep the point of intersection on your screen. Select all three medians, and go to "Hide Segments" under the "Display" menu. You should now have your original triangle, along with the midpoints of the three sides, and one additional point somewhere.
You will now name this additional point. Select the "text" tool (the "A" tool) from the tool box, and point it at the new point (the hand will turn black). Click on the point, and a letter will appear. Now, take the tool, and point it directly at the label of the point (not at the point itself). Double-click on the label, and you will get a window on your screen. You will label this point "ce," and it will be the centroid, which is the point of intersection of the medians of a triangle.- As you drag the vertices and sides of the triangle, ask: Will the centroid ever be outside the triangle? On the triangle? If so, when? If not, why not? Explain below.
- Select a midpoint and that side of the triangle. Construct the perpendicular bisector of a side of your triangle. Construct the perpendicular bisector of another side. Construct the point of intersection. Will the third perpendicular bisector pass through that point? Construct it, and check. Move the triangle around to try and verify your conjecture. Sketch the diagram below.
- Hide all three perpendicular bisectors, but leave the point of intersection on the screen. Label this point "ci." This is the circumcenter, which is the point of intersection of the perpendicular bisectors of a triangle.
- Again, move the vertices and sides of the triangle, and ask: Will the circumcenter ever be outside the triangle? On the triangle? If so, when? If not, why not?
- Select the three vertices of the triangle in any order. Construct the bisector of an angle of the triangle. Construct the bisector of another angle of the triangle. Construct the point of intersection. Will the third angle bisector pass through that point of intersection? Check it out. Move the triangle around to try and verify your conjecture. Sketch your entire diagram below.
- Hide all three angle bisectors, but leave the point of intersection on the screen. Label this point "in." This is the incenter, the point of intersection of the angle bisectors of a triangle. Will the incenter ever be outside the triangle? On the triangle? If so, when? If not, why not?
- Select a vertex and the side opposite that vertex. Construct the altitude to a side of the triangle (or, to the line that contains that side of the triangle). Construct the altitude to another side. Construct the point of intersection. Will the third altitude pass through that point? Check it out. You might have to move the triangle to see the point of intersection. Sketch the diagram below.
- Hide all three altitudes, but keep the point of intersection on the screen. Label this point "or." This point is the orthocenter, which is the point of intersection of the altitudes of a triangle. Will the orthocenter ever be outside the triangle? On the triangle? If so, when? If not, why not?
- You should now have a triangle on your screen, and four other points, the centroid, the circumcenter, the incenter and the orthocenter. Move the vertices of the triangle. Move the sides of the triangle. Describe what is happening to the four points.
- Will any three of these four points always be collinear? If so, which ones? Select the circumcenter and orthocenter, and construct a segment between them. How many of the four points are collinear? __________
- Is it ever possible to have all four points collinear? If so, when?
- Can you move the diagram so that the four points become "one point?" If so, in what type(s) of triangle(s) will this occur? If not, why not?
Differentiation
Use partners in computer labs or while doing compass-straightedge constructions in the classroom. Make sure that each student is held accountable by alternating which student performs each step.
Create a glossary of terms used for your dynamic geometry software and during compass-straightedge constructions so that all students will know how to perform each step.
Add the following to their assignment, creating the nine-point circle. Here's where the sketch will get a bit complex, and possibly tricky.
- Keep your original triangle on your screen, with midpoints still marked, and with the four "special" points still on the sketch. Drag the vertices and/or sides of the triangle, to make it as big as possible on your screen.
- Select the three midpoints, and select "Segments" from the "Construct" menu. This will form a triangle within your original triangle. While the three sides of your new triangle are still selected, choose "Midpoints" from the "Construct" menu. This will give you midpoints of the three sides of your new triangle.
- Construct the perpendicular bisectors of two of the sides of your new triangle. Select the two new lines, and construct their point of intersection. Hide the two perpendicular bisectors, and name their point of intersection "n," which is the center of your nine-point circle (we'll talk about why it's called a "nine-point circle" later on). Draw as accurate a sketch as possible of your screen.
- Select in order your new point "n," then one of the midpoints of your original triangle. Under the "Construct" menu, go to "Circle By Center+Point." Describe what happened.
- Next, reconstruct an altitude of your original triangle, and construct the point of intersection of the altitude with the side it hits. Do the same with the other two altitudes of your original triangle. Hide the three altitudes. What do you notice?
- As an added bonus, construct the three segments which connect the three vertices of your original triangle to the orthocenter (labeled "or"). Construct the midpoints of these three new segments. Hide the three segments. Describe what you notice.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
creating and manipulating geometric models with their software. | assisting students with steps in the program's construction process. |
writing down conjectures and testing them by changing their geometric models. | asking leading questions to help students focus on potential conjectures. |
using compass and straightedge techniques to draw figures. | checking students for correct compass techniques. |
Parent Resources
- Sophia and Khan Academy are websites with uploaded lessons teaching multiple topics. Most of these lessons were developed by teachers and reviewed.
- Teacher Tube and You Tube include multiple uploaded lessons on most school topics.
- Geogebra is a free software program for both algebra and geometry that you and your child can download and use at home.
- Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.