Making Mathematical Connections Activity 1 6/3/2002
Activity 1 Isometries of the Plane
We will investigate transformations of the plane that preserve distance. These transformations will take the entire plane onto a "copy" of itself. For this activity you will need to draw a scalene triangle on a two transparencies so that you have two "identical" triangles. Here's a definition.
Definition: A one-to-one, onto function f from the plane to the plane is called an isometry of the plane if, for any two points P and Q, the segment PQ is congruent to the segment fPfQ()().
1. Give an explanation of the meaning of congruent as used above. Try to be precise.
2. Using your transparencies, experiment to find ways to move a triangle from one location to another. How many types of isometries of the plane can you identify? Is there one that is a combination of two others? What properties of your triangles are preserved by the isometries?
3. Consider an isometry of the plane as a function from the plane to itself. Craft a careful definition for each type of isometry of the plane.
4. Is there a type of transformation of the plane that is not an isometry but preserves some of the same properties preserved by isometries? Making Mathematical Connections Activity 1 6/3/2002
Teacher's Notes for Activity 1
• Allot a 50-minute class period for this activity.
• Students should work in groups of three or four.
• The idea of congruent triangles on transparency sheets is to give the students the image of a transformation taking the entire plane to itself.
• Item 2 is meant to allow students to "discover" the four basic isometries, rotation, reflection, translation, and glide. They might need some prompting on the last.
• Items 1 and 3 are to reinforce the place of definition in mathematics. Sharing definitions and the ensuing discourse is likely to bring out the importance of careful wording. There are other issues to be discussed regarding mathematical definitions. These include minimal conditions and elegance. These issues might begin to surface here.
• Item 4 tries to get students to identify similarity transformations. These will be addressed again in Activity 5.
• A minor goal of this activity is to help students begin to make connections between functions and geometric transformation. Making Mathematical Connections Activity 2 6/3/2002
Activity 2 Rotations, Reflections, Translations, and Glides
These are the four main types of isometries of the plane. They are called isometries ("same measure") because distance is preserved. Here are some exercises investigating isometries.
1. For a reflection F and a point P, if F(P)=P', what is F(P')?
2. Let F be a reflection in line m. What is F(m)?
3. For any transformation T, a line left invariant, k, is a line such that T(k)=k.
a) Find a line other than m that is invariant under the reflection F in line m.
b) What are the invariant lines for a reflection in line m? How many are there?
c) What are the invariant lines for a rotation of angle α radians with center O?
d) What are the invariant lines for a translation?
e) What are the invariant lines for a glide?
4. Given a geometric figure and it's image under a reflection, is there a way to find the line of reflection? Explain.
5. Given a geometric figure and it's image under a rotation, is there a way to find the angle and center of the rotation? Explain. Does the answer depend on the angle?
6. Given a geometric figure and it's image under a translation, is there a way to find the translation vector? Explain.
7. Given a geometric figure and it's image under a glide, is there a way to find the line of reflection and the translation vector? Explain. Making Mathematical Connections Activity 2 6/3/2002
Teacher's Notes for Activity 2
• Allot a 50-minute class period for this activity.
• Students should work in groups of three or four.
• Provide students with rulers, protractors, compasses, graph paper, and MIRAs (or some such device).
• The basic goal of this activity is to acquaint students with some basic properties of various isometries.
• Before Item 6, introduce the vocabulary "translation vector."
• Students might need guidance on Item 3a and Item 7, both pertaining to glides. Making Mathematical Connections Activity 3 6/3/2002
Activity 3.1 Compositions
Now we will experiment with composing isometries. 1. What kind of transformation is the composition of two translations? 2. What kind of transformation is the composition of two reflections? 3. What kind of transformation is the composition of two rotations? 4. Can you find a transformation of the plane that takes triangle ABC to triangle A'B'C'? Can you do it with a single transformation? What is the minimum number of translations, reflections, and/or rotations needed? Can you do it with only reflections?
A
B C C'
A' B'
Activity 3.2 • Complete the Connected Geometry Investigation 2.11, A Reflection Puzzle. • Analyze the investigation from a high school teacher's perspective (Activity A, item 2). • Try to prove your conjecture about reflections. You may need to use a lemma that states "the image of three non-collinear points determines an isometry." Making Mathematical Connections Activity 3 6/3/2002
Teacher's Notes for Activity 3
• Allot a 50-minute class period for Part 3.1. Students may need to finish as homework. • Students should work in groups of three or four. • Part 3.1, Item 3 might be hard. You may want to give a hint about using cases (same- center or different-center rotations). • The last question in Item 4 foreshadows the conjecture we'd like students to make in Part 3.2. That is, that any isometry can be expressed as the composition of three reflections. • Part 3.2 should be assigned for homework with class discussion of the results. Making Mathematical Connections Activity 4 6/3/2002
Activity 4 Proof with Isometries
Here are some theorems to prove using isometries.
1. An isometry preserves angle measure.
2. If ∆ABC and ∆DEF are two triangles such that AB = DE, AC = DF and ∠BAC = ∠EDF, then BC = EF, ∠ABC = ∠DEF, and ∠ACB = ∠DFE.
3. If ∆ABC and ∆DEF are two triangles such that AC = DF, ∠BAC = ∠EDF, and ∠ACB = ∠DFE, then BC = EF, AB = DE, and ∠ABC = ∠DEF.
4. Suppose that ∆ABC is a triangle in which ∠ABC = ∠ACB. Prove that AB = AC.
5. Suppose that A and B are two distinct points on a circle and that the tangents to the circle at A and B meet at point P. Prove that AP = BP. (Hint: Consider the line that passes through P and the center of the circle.)
6. Any isometry is a composition of reflections, translations, and rotations.
7. Any isometry is a composition of reflections. Making Mathematical Connections Activity 4 6/3/2002
Teacher's Notes for Activity 4
• This activity will take more than a 50-minute class period. For some parts students will probably need some hints or steering. Try to work with their ideas rather than give them too big a push. Make sure they work with isometries, not measure. • Students should work in groups of three or four. You might want to ask students to read this activity for homework before working on it in class. Ask them to bring in written ideas or sketches. • Finishing parts two and three requires the use of Euclid's postulates. For a hint, ask them which ones. • Part five has a hint. Part six will lay the foundation for Activity 6. Making Mathematical Connections Activity 5 6/3/2002
Activity 5 The Human Vertices
We will investigate transformations of a square of side one in the coordinate plane. Each of you will have a starting coordinate. You will connect the vertices with taught rope for straight lines. Person Starting Coordinates (x,y) A (1,0) B (0,0) C (1,1) D (0,1)
The square will be transformed under the functions described below.
Function (ti) ti (x,y) t1 ()xy, −
t2 ()2xyy+ ,
t3 ()xy−+21, t4 2 −+2 2 (),()xy2 xy
t5 ()−xy, t 1 1 6 ()2 yx, 2 t −1 +−+ 7 ()2 xyx, 2 y t8 ()yx, t9 1 + 3 − 3 + 1 2 xyxy2 , 2 2 t −+ −1 + 10 ()xy2 , 2 xy
1 Making Mathematical Connections Activity 5 6/3/2002
1) Describe the image and location of the square under each transformation. What happens in each transformation to characteristics such as distance and angle? What happens to pairs of parallel lines? Are there other characteristics that change or stay unchanged?
2) Classify the transformations into categories according to characteristics of the square that change or remain unchanged. How many different categories do you have? What are they?
3) Which transformations are isometries?
4) Write the transformations that are isometries as matrix equations. That is, think of a x vertex (x,y) as a vector, . Write an equation with a 2 by 2 matrix A and a vector y i e x x e so that t = A + . f ii y y f
5) What are the matrix equations for the non-isometric transformations?
6) Calculate the determinant of each matrix. What relationship do you notice between the determinant of the matrix and the type of transformation?
2 Making Mathematical Connections Activity 5 6/3/2002
Teacher's Notes for Activity 5
• The physical part of this activity should be completed in a 50-minute class period. The problems might need to be finished in another period or as homework. The purpose of this activity is to enable students to make connections (physically) between transformational geometry and linear algebra. Going through the motions induced by each transformation enables a powerful connection.
• You will need to create a coordinate plane for this activity. A tiled room works well. Use masking tape for the axes.
• Transformations t4 and t9 will be a bit difficult, but with some thought and review of right-angle trigonometry, the students should get through them.
• You might need to review how a matrix is obtained from the images of basis vectors.
• Stop the groups after problem 2 and have a whole-class discussion comparing the categories and characteristics.
3 Making Mathematical Connections Activity 6 6/3/2002
Activity 6 Isometries and Linear Algebra
x cosθθ− sin x e 1) Show that the function t : −> + (Eqn. 1) is an isometry y sinθθ cos y f
because it is a counter-clockwise rotation through angle θ about the origin followed e by a translation by the vector . f
x cosθθ sin x e 2) Show that the function t : −> + (Eqn. 2) is an isometry y sinθθ− cos y f
because it is a reflection in the line through the origin that makes an angle of θ/2 with e respect to the positive x-axis, followed by a translation by the vector . f
3) Find a formula for the inverse of a 2 x 2 matrix in terms of its determinant.
4) Here's a definition: A matrix U is called orthogonal if and only if U-1 = UT. Another way to say this is UT U = I.
a. Which matrices from Activity 5 are orthogonal? b. Show that U is orthogonal if and only if the columns of U are orthonormal. c. Show that if U is orthogonal, |det U| = 1.
5) Here's another definition: A Euclidean transformation of ℜ2 is a function t : ℜ→ℜ22 of the form tx()=+ Ux a where U is orthogonal. We have verified that every isometry of ℜ2 is a Euclidean transformation (since every isometry is a composition of reflections).
Verify the converse by showing that every Euclidean transformation can be written in the cosθ form of Eqn. 1 or Eqn. 2. Hint: Every orthonormal vector can be written as for sinθ some θ ∈ℜ.
6) Verify that the set of Euclidean transformation of ℜ2 (=E(2)) forms an algebraic group under the operation of composition of functions by checking that they satisfy the following axioms for a group.
∈⇒ ∈ 1) (Closure) tt12,() E22 t 1o t 2 E () ∈⇒ = 2) (Associativity) tt123,, t E ()2 () t 1oo t 2 t 3 t 1 oo() t 2 t 3
3) (Identity) ∃=eI()∈∋∀∈ E()22 tE (), tIIttoo == − 4) (Inverses) ∀∈tE()2 ∃ st ( ≡1 ) ∋ tsstI = = oo Making Mathematical Connections Activity 6 6/3/2002
Teacher's Notes for Activity 6
• This activity is meant to bring closure to the mathematical ideas connecting transformational geometry and linear algebra. • Assign parts 1 through 3 for homework before the class meeting devoted to this activity. Before this assignment make certain students have sufficient knowledge of matrix arithmetic. In particular, part 3 might be difficult as homework. • Work on parts 4, 5, and 6 in a 50-minute class. Students should develop the ideas for these proofs in class and finish them off for homework. One definition the students may not know is orthonormal. Recall, two vectors are orthonormal if they are orthogonal and of norm 1. • Give an example of how a particular group (say, integers) satisfies the axioms. Investigation for Unit 7 Name______The Height of Red Hill
Statement of Problem: You and a friend are preparing for a backpacking trip and are trying to shape up before your departure. The local mountain, Red Hill, has a great trail leading from the base of the mountain up to the very top. This is where you and your friend will do your training. Both of you would really like to know how much elevation you gain when you hike up this hill. In other words, you would like to find the height of Red Hill. Using a clinometer, you stand in the parking lot near the base of the hill. From here you see the top of the mountain at a 25° angle. You walk 740 feet towards the mountain and take another measurement. From here you see the top of the mountain at a 55° angle. Using only this information answer the following questions.
a.) Draw an accurate picture of what is going on in this situation. Fill in all values you know and all lengths that are unknown.
b.) Explain in words what it is you are trying to find and how you can use the given information to find that value. In other words, explain what your “game plan” is. Do not solve the problem yet; only explain how you are going to approach the problem.
Turn over c.) Use the space below to answer the question. Please show all necessary work. Provide pictures, equations and explanations for each step of the problem. Label your answers clearly.
d.) Make a list of all the previous math knowledge you had to have in order to be able to do this problem. (Example: You need to know that a triangle has 180°) Making Mathematical Connections Activity A 6/3/2002
Activity A Examination of Mathematical Tasks
1. First, complete the mathematical tasks from the high school curriculum. Then discuss with your group any variation in individual interpretations and solution strategies. Agree on a solution.
2. Here is a list of issues a teacher might consider when previewing a mathematical task. Consider the task you just finished and identify how each of these issues might come into play when a teacher is preparing to use the task with a class.
• Student preparation issues prior knowledge and ability scaffolding (providing appropriate incremental support) variation in student abilities • Type of instruction direct whole class discussion small groups or pairs individual seat work • Resources calculators manipulatives tools (straight edge, compass, etc.) materials (paper, colored pencils, etc.)
3. Consider the following questions. • How can we use this task to take advantage of opportunities to explore important mathematical ideas? For instance, different types of equivalence (e.g. equal values or congruent shapes), or differing meanings of the same vocabulary (e.g. "f of x" vs. "half of five").
• Are there opportunities to build on previous knowledge, even if it is not mathematical knowledge?
• Where are opportunities to develop problem solving skills?
1 Making Mathematical Connections Activity A 6/3/2002
• Consider the list of features of a rich mathematical activity. Which of these features does the task you are analyzing exhibit? Which features does the task lack?
• Consider the mathematical task you just completed. Here are some questions a teacher might ask about any mathematical task. Use the Unit 5 overview and discuss these questions. Compile a group response to each. • How do these tasks fit together?
• Is there a rearrangement that might be beneficial to students?
• Are there other tasks that might be useful to students before they attempt these?
2 Making Mathematical Connections Activity A 6/3/2002
Teacher's Notes on Activity A
• This activity is designed (in whole or in part) as a guide for students to examine high school curriculum materials. It will be beneficial to focus on specific examples from the curriculum materials to be studied (in items 3, 4 &5). • Item 2 is used in Activity 3.2. • The whole of Activity A is used to examine curriculum material that students will be seeing in use during their classroom visit. • The whole of Activity A is used to examine curriculum material focusing on transformational geometry. • Activity A is used to examine Height of Red Hill and accompanying student work.
3 Making Mathematical Connections Activity B 6/3/2002
Activity B Classroom Observations
You have spent considerable time analyzing the task in which the high school students will be engaged this afternoon. Each of you will observe one class, making note of various student and teacher behaviors. Each of you will have a separate observation assignment. We will debrief our observations with the teacher later this afternoon using the following framework.
Instruction Student Processes Resources Preparation direct skills practicing materials procedures whole class variation recalling facts calculators discussion small group or scaffolding reasoning, tools pairs communicating individual knowledge not involved manipulatives seatwork
A pair of you will visit each class. Each member of the pair will have a different assignment. One member will note the time involved in each type of instruction and will record all teacher questions and responses. The other member of each pair will note what resources are used and record all student questions and responses.
So, you should have a heading on your pad that looks like one or the other of these:
Instruction (D, W, G, or I) Teacher's questions and responses
Resources (M, C, T, or P) Students' questions and responses
1 Making Mathematical Connections Activity C 6/3/2002
Activity C Guidelines for Creating a One-Day Class Activity
You will prepare a mathematical activity for the high school geometry class. You will have one day to engage students in this activity. You will lead students through this activity in both afternoon classes. Here are the steps you should follow in creating an activity. Steps for creating the activity 1. Pre-assess students' prior knowledge 2. Choose a learning goal 3. Create learning sub-goals 4. Create tasks that will enable students to achieve the sub-goals 5. Plan for summary and closure for the entire activity
Possible choices for an activity • Tessellations – Students should learn to create a shape that can tessellate the plane and use the shape to tessellate a (large) page. Students should learn how to recognize a shape that cannot tessellate the plane. Which polygons tessellate the plane, which don't? • Frieze Patterns – Students should be able to create a Frieze pattern. Students should be able to classify their Frieze pattern in categories offered to them. • Geometric Exploration – Students should be able to use geometric transformations to explore and make conjectures. (An example is the Reflection Puzzle that you completed.) Questions to consider What makes a Rich Mathematical Activity? What is the range of students' prior knowledge? How difficult will it be to grade? How long will each task take? What to do if students are not engaged? What about students who finish early?
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Planning considerations: 1. Plan on having enough time to allow for closure each day. Each class period is 58 minutes long. 2. Have an activity that extends the lesson for those students who might finish early. 3. Give students a rubric so that they are aware of how they will be graded. 4. Have a starting place so that all students can begin. Also, have very challenging questions for all students. 5. Incorporate individual work followed by work in pairs or small groups. 6. Pre-assess the activity to identify possible student difficulties and have hints prepared to help students work through the difficulties. Be careful that the hints don't give away the answers. (Given enough time, many students can figure out problems.) 7. Have obvious goals (objectives) so that students know what they are learning. 8. Have a way to summarize the activity and achieve closure for the students
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