campinile lesson plans

7/23/2019 Campinile Lesson Plans

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SIMILARITY USING GEOGEBRAIntroduction & Prior Classwork

This project is a culminating project on similarity for a high school level geometry

course. Before undertaking this final project, students will have proficiency withGeoGebra and will have completed the following activities:

1) Measured the height of the school hallway ceiling using a mirror. They will have

modeled this method of indirect measurement on GeoGebra and be able to explain theequivalent ratios used in this method. In the diagram below, point B is the location of the

mirror and point E represents a spot on the ceiling that students are using for sighting

their measurements.

2)Measured the height of the school hallway ceiling by sighting its measurement on ameter stick and modeling the similar triangles on GeoGebra. In the diagram below, point

E once again represents a spot on the ceiling that students are using for sighting theirmeasurements.

INSTITUTE FOR ADVANCED STUDY/PARK CITY MATHEMATICS INSTITUTE SUMMER 2011


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3) Students will have discussed each of these methods of indirect measurement tohighlight the strengths and weaknesses of each method and where each is subject to error.

These discussions will also focus on discovering which measurements were known (or

able to be measured) for each method and the realities of getting those measurements inother contexts. Important points which should be emphasized in discussions with

students:

Mirror Method of Indirect Measurement:

Known Measurements:1) Horizontal distance between student and mirror

2) Horizontal distance between mirror and sighting spot on ceiling3) Height of student

Major Accuracy Challenges:1) Getting sighting spot on ceiling exactly in center of mirror.2) Finding spot on floor directly under sighting spot on ceiling.

Meter Stick Method of Indirect Measurement:

Known Measurements:1) Horizontal distance between eye and outstretched arm.

2) Distance between meter stick and sighing spot on ceiling.

3) Sighted height of ceiling on the meter stick.

Major Accuracy Challenges:1) Measuring horizontal distance between eye and meter stick.2) Marking sighted height measurement on meter stick (its quite awkward to lay on floor

and see exactly floor and ceiling measurements on meter stick.

3) Finding spot on floor directly under sighting spot on ceiling.

At this point in the school-year, students will not have studied the trigonometric ratios.



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PROJECT OVERVIEWTeacher Notes

GUIDING QUESTION:What is the best method ofindirect measurement to most accurately determinethe height of the UC Berkeley Campanile?

Knowing the distance between you and the object you are measuring is a key element in

the traditional methods of indirect measurement using similar triangles. At this point,

students will have used several methods to measure the height of the ceiling. Eachmethod used in class required that the students measure the horizontal distance between

them and the target point on the ceiling. These methods of measurement are only useful

if students know the distance between themselves and the object whose height is being

measured.

What if the distance between the student and the tall object is unknown or is far enoughaway that measuring it would be cumbersome and therefore challenging to complete with

a good degree of accuracy?

DAY 1:

This project begins with a dynamic GeoGebra sketch that models a way to find the height

of an unknown tall object without knowing the distance between you and the object.Students are given a sketch (on paper) and will need to construct it on GeoGebra. Using

both relationships from similar triangles and algebra, they will discover some non-intuitive equivalent relationships between segments in various triangles.

DAY 2:Class begins with a discussion of what equivalent relationships they found yesterday and

proving why they are true. Then, students will discuss if the model on GeoGebra

provides another method for indirect measurement. The benefit of this new model is thatit provides a way to find the height of an unknown object without knowing your distance

from the object.

Students will then be provided with their project: Use a method ofindirect measurement to determine the height of theUC Berkeley Campanile.

At this point, students will have discussed three methods of indirect measurement and can

choose whichever method they believe will give them the most accurate results.



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Students will form groups of no more than 4 students. They will decide on which method

they want to use for determining the height of the Campanile and will use GeoGebra to

import a photo of the Campanile and create a dynamic sketch, labeling which featuresthey will measure. Students will have to turn in the document CampanileBlueprints before they can head out to conduct their measurements (student

worksheet #3 in the file).

DAYS 3-5:No class time will be spent on the project. During these three days, students will have thefollowing tasks to complete:

1) Go to UC Berkeley and take their measurements.

2) Create a dynamic sketch on GeoGebra to model their work. This sketch should be toscale and use their actual measurements.



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LESSON PLANS:

COMMON CORE HIGH SCHOOL GEOMETRY STANDRDS COVERED INTHIS PROJECT:

G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass andstraightedge, 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.

G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to

decide if they are similar; explain using similarity transformations the meaning of similarity for

triangles as the equality of all corresponding pairs of angles and the proportionality of all

corresponding pairs of sides.

G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two

triangles to be similar.

G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove

relationships in geometric figures.

G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or

structure to satisfy physical constraints or minimize cost; working with typographic grid systems

based on ratios).

**NOTE**My students have block periods which are 98 minutes inlength. Teachers using this project with a traditional schedule will needto modify accordingly the amount of work given in each day.

DAY 1: Students will receive a worksheet GeoGebra Modeling (student

worksheet #1 in the GeoGebra files) which will ask them to construct the followingfigure:

They will then be asked to use the spreadsheet tool in GeoGebra to find GH*FD and

BE*AD. After constructing this figure, they will discover that these two relationships are

equal, no matter where you slide points A or F on their dynamic sketch. An example ofthis dynamic sketch is fileGGsimilarityfinalproject_GGmodelingdynamicsketch.



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Students will have had previous experience with GeoGebra and will know how to use the

features needed for this construction.

Once students dynamic sketches are completed, students will discuss in pairs whyGH*FD=BE*AD.

Class will end with each student recording his/her ideas on why this relationship is

equivalent. Any students who didnt finish their sketches will be required to do so forhomework so that by the following day, all students have discovered this equivalent

relationship through an accurately constructed sketch.

At this point, there will be no mention of how (or even if) this model explains a method

of indirect measurement. The initial goal is for students to discover why this equivalent

relationship exists. The second day of this project will be focused on how this model can

be used in the field to indirectly measure the height of a tall building.

Below is the algebra to explain why GH*FD=BE*AD:

by AA.

So, because corresponding parts of similar figures are similar.

Additionally,

by AA.

So, because corresponding parts of similar figures are similar.

When cross multiplying the each equation, the following results are obtained:AE*B1D=AD*BE

FH* B1D=FD*GH

And its given that



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So, by substitution,

AD*BE= FD*GH

DAY 2:

Class will begin with a class discussion on proving why AD*BE= FD*GH.

One key mathematical point to be made is that even though AD*BE= FD*GH

is NOT similar to

and is NOT similar to .

Following this discussion, students will be given the following prompt to think about

individually. They will receive the prompt on a worksheet so that they can have thesketch to write their thoughts on. This worksheet is entitled, A Third Method ofIndirect Measurement (student worksheet #2 in the GeoGebra files).

During this unit on similarity, we found the height of the hallway ceilingusing two distinct methods of indirect measurement. We modeled

each method on GeoGebra and discussed the accuracy of eachmethod. Using the sketch you created yesterday on GeoGebra, canyou figure out how this could be used as a model for a new method ofindirect measurement?

Students will have 5-10 minutes to work on this prompt individually. Then, students willpair up and compare their ideas with a colleague. Finally, the whole class will have a

discussion on how this model can be used to measure the height of a tall building.

Below is an explanation of how this model can be used. Just like in the earlier model, a

student will lie flat on the ground, holding a meter stick in an outstretched hand. He/she

will sight the height measurement of the tall building and record it. But this time, thestudent will make a second measurement by walking backwards several feet and usingthe same method again.



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All lengths in red type are distances which can be measured.All lengths in blue type are distances which can be calculated using the now proven

relationship that AD*BE= FD*GH.

Using this equation, and the known measurements, students can solve for ED. ED is the

distance from the meter stick to the tall building. Although this could be measured, it is

quite challenging to measure it accurately since it is several feet (at a minimum) from thelocation of the student sighting the height measurement.

Once ED is calculated, one set of the similar triangle ratios can be used to calculate B1D,

the height of the building.



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Class on Day 2 will end with a projection of this image and a presentation of the project:

Use a method of indirect measurement to determine

the height of the UC Berkeley Campanile.

Students will form groups of no more than 4 students. They will decide on which method

they want to use for determining the height of the Campanile and will use GeoGebra toimport a photo of the Campanile and create a dynamic sketch, labeling which features

they will measure. Students will have to turn in the document CampanileBlueprints before they can head out to conduct their measurements (studentworksheet #3 in the GeoGebra file).

DAYS 3-5

Students will be conducting their measurements and making their final sketch onGeoGebra to illustrate the model they used. No class time will be spent on the project

other than answering questions which arise as they do their field work.

On the day that the final project is due, students will turn in both an electronic and printed

copy of their GeoGebra model. They will have a write up explaining exactly how theydid their measurements and will show all calculations. On that day, the actual height of

the Campanile will be revealed and well have a discussion around what factors led to



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some groups having a more accurate measurement of the Campaniles height than other

groups.

The student rubric for the final project write-up is entitled, Campanile Rubric andcan be found in theGeoGebra files as student worksheet #4.


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