Similarity - Using Mirrors to Find Heights

Transcription

1 Similarity - Using Mirrors to Find Heights AUTHOR(S): DANA SHAMIR TEACH # 2 MENTOR: NANNETTE STRICKLAND DATE TO BE TAUGHT: 11/29/2007 LENGTH OF LESSON: 45 MINUTES GRADE LEVEL: 8 SOURCE OF THE LESSON: Connecting Mathematics Stretching and Shrinking Understanding Similarity TEKS ADDRESSED: (8.3)Patterns, relationships, and algebraic thinking. The student identifies proportional relationships in problem situations and solves problems. The student is expected to: (A) compare and contrast proportional and nonproportional relationships; and (B) estimate and find solutions to application problems involving percents and proportional relationships such as similarity and rates. (8.9)Measurement. The student uses indirect measurement to solve problems. The student is expected to: CONCEPT(S): (B) use proportional relationships in similar shapes to find missing measurements. In Grade 8 mathematics, students focus on using basic principles of algebra to analyze and represent proportional and non-proportional relationships. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other; and they connect verbal, numeric, graphic, and symbolic representations of relationships. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Knowledge of similarity is important to the development of children s understanding of the geometry in their environment. It is generally understood that understanding proportional reasoning is an important stage in cognitive development. OBJECTIVES: Students will apply their knowledge about similarity of triangles to a real-world problem. Students will use the mirror method to find the height of a tall object.

2 Students will develop a technique for indirect measurement. RESOURCES: Small mirrors for each group of students. Measuring tapes for each group. A transparency displaying a colorful Sierpinski Small ziplocs with candy bars hanging on the walls at the same height, one for each group. Pre-assessment paper for each student. Post-assessment paper for each student. Instructions for the measurement procedure for each group. ENGAGEMENT Time: 15 minutes Before we begin I need you all to write your names on the paper in front of you (the pre-assessment paper) and solve these two questions. Collects the papers after 5 minutes. Please take a piece of paper and draw a triangle on it. The teacher does the same on the board. It can be any triangle: draws a rightangle triangle, an acute triangle and an obtuse Please divide the triangle into four identical triangles. How can we make sure the triangles are all the same? [Mark the midpoints of each side. Connect the midpoints to form four identical triangles]

3 Please shade the center (teacher does the same on the board) For each unshaded triangle, divide it again to form four identical triangles. Shade the centers. Now, we can repeat these steps over and over many times. Shows a transparency with a colorful Sierpinski This is called a Sierpinski How are these four smaller triangles related to the large triangle that we started with? Can we go on doing this forever? What did we do? Did we draw anything that is different than the original? Does anyone know how we call these figures that are similar to the original but can be subdivided an infinite number of times? [self-similar fractals] EXPLORATION Time: 25 minutes We need to figure out the height of the Ziploc bags. The teacher asks for different ways to do it. Presents a mirror and asks if they can think of a way to measure the height using the mirror. Does any one have any idea how to measure the height of the bags? What s another way to do it?

4 I need one of you to come here and help me demonstrate this. We will measure the height of the board as an example. We will place a mirror on the floor of the room and the student will stand straight and look into the mirror, then either move forward or backwards until the top of the board is reflected in the center of the mirror. Have the class observe and sketch this setup: student, mirror, top of the board, floor. Can you tell me when two triangles are similar? Sketch a picture of the setup with the two triangles on the board. Mark the angle of reflection and the angle of incidence. Use: d distance from the person to the mirror. D - distance from the board to the mirror. h height of the person s eyes. H the height of the board. Do you see any triangles being formed in what you see here? Is it enough that the angles are congruent? Do the sides have to be congruent? What can you say about the sides if the triangles are similar? Are these two triangles similar? Which angles are congruent? If the two angles are congruent what can we tell about the last pair? Are the sides of the triangles the same size? What measurements do you need to make to use similar triangles to estimate the height of the board? [The top of the board to the floor forms a triangle with the ground distance to the mirror and the line drawn from the top of the board to the mirror. Sides and angles are the same. [Corresponding angles are the same size (congruent). The side lengths of one triangle are multiplied by the same number (scale factor), to get the corresponding side lengths in the second triangle] [They are similar because their corresponding angles are equal] [angle of incidence and angle of reflection] [two right angles] [Distance from the person to the mirror. Distance from the board to the mirror. Height of the person s eyes.]

5 O.k., let s get some measurements done. Makes the students count 1,2,3,4,5,6. All the 1 s are a group, all the 2 s are a group, etc. Each group needs a mirror, a measuring tape and to stand under one of the Ziploc bags. You need to decide who the observer is, who the recorder is and who does the measurement. Please record your measurements on a paper and be ready to explain how you calculated the height of the bags. Each group gets a detailed description of the required steps. The units that we will use will hopefully be cm, but that can change according to the units on the measuring tapes. The students form the groups and start the activity. The teacher circulates around the groups and offers help. If there is enough time, ask them to measure the ceiling too. Does it make it easier if we stand within a nice (whole number) distance from the mirror? Say, 2 meters, or 3? Does it simplify the calculation? EXPLANATION Time: 10 minutes

6 Ask students once again to explain what triangles were formed, why the triangles are similar, and how the fact that they are similar allows us to estimate the height. Repeat the calculation on the board: EMBED Equation.3 Multiply each side of the equation by h: EMBED Equation.3 Collect the class data from each group including the different triangle sizes and write it on the board. What is the proportion factor of the similar triangles? Why do we have different triangles? Why do we have different results? What can be a source of error? What is the best estimate for the height of the bags? [ EMBED Equation.3 ] [Measurements are not accurate, people have different heights, the mirrors were positioned in different distances, etc ] [average of the students calculations] EVALUATION Time: 5 minutes Hands out the postassessment papers. Please write your names on the papers and solve the two questions. Thanks you!

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