Tantalizing Triangles

Grade Level:

/ 10 /

Subject:

/

Geometry

/

Prepared By:

/ Carol Clinton

Taught at

/ Hughes High School /

Teacher/Bell

/

Richmond/Bells 4 & 5

/

Source

/ Original lesson, incorporating information from references listed at bottom of lesson plan

Analyze Learners

Overview & Purpose

In this lesson students are exposed to engineering design fundamentals using Cincinnati structures as examples. They gain a physical understanding of triangle, side, angle concepts; lines in coordinate system; congruence, and similarity. They practice constructing triangles using Geoboards, dot paper, and paper and pencil; calculating and measuring angles; and classifying triangles.
An extra credit extension with tantagrams was offered. /

Education Standards Addressed

Number, Number Sense and Operations Standard –
G.  Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions.
Measurement Standard –
D. Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements.
E. Estimate and compute various attributes, including length, angle measure, to a specified level of precision.
Geometry and Spatial Sense Standard –
A. Formally define geometric figures.
B. Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence.
C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines.
D. Use coordinate geometry to represent and examine the properties of geometric figures.
E. Draw and construct representations of two- dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.
F. Represent and model transformations in a coordinate plane and describe the results.
I. Use right triangle trigonometric relationships to determine lengths and angle measures.
Patterns, Functions and Algebra Standard -
C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.
Select Goals and Objectives /

Teacher Guide

/

Student Guide

Objectives

(Specify skills/information that will be learned.) / Students will
·  define triangles by Cartesian coordinates of their vertices;
·  Construct triangles on geoboard
·  Classify triangles;
·  Solve missing angle problems; and
·  Create congruent and similar triangles. /

Materials Needed

·  Paper
·  Pencil
·  Geoboards and bands
·  Dot Paper
·  Protractor
·  Ruler
·  Handouts
o  Pretest
o  Bridge sections
o  Worksheet
o  Post-test
o  Tantagrams

Select Instructional Strategies –

Information

(Give and/or demonstrate necessary information) / Direct Instruction – demonstration of desired behaviors; lecture on key information
Questioning – during lecture and during observations of group work / Group Work – collaborative learning
Utilize Technology / Demonstrate use of geoboards and dot paper / Construct triangles with geoboards. Draw triangles on dot paper. Use protractor and ruler to measure triangle angles and sides. Use calculator to solve missing angle. /

Other Resources

(e.g. Web, books, etc.)
Preparing for the OGT in Mathematics, pages 219 - 309

Require Learner Participation

Activity

(Describe the independent activity to reinforce this lesson) / Introduction to objectives and activities for the day. Show photos of local bridges, airports and other landmarks that are based on triangle geometry.
Catch: Teams of two, one describes a section of the Brent Spence Bridge truss, the other draws it based on the description. Switch. Triangles are important features of many engineering structural designs. We need good ways to describe and construct them.
Activity:
Distribute pre-test (if not done on prior day)
Distribute geoboards (give warning about no rubberband horseplay)
Distribute worksheet and dot paper
Demonstrate constructing triangle based on vertices, measuring sides with ruler and angles with protractor.
Demonstrate drawing the triangle on dot paper.
Review types of triangles: acute, obtuse, isosceles, right; 30-60, 45-45
Review sum of angle rule
Review rules of congruent and similar
Discuss bisectors
Discuss worksheet
Discuss tantagram handout.
if lesson goes short, discuss altitudes, medians, demonstrate tantagrams, discuss tessellations – if lots of time, calculate slopes of sides, show that slopes of sides adjacent to right angles are perpendicular; if more time, do a problem regarding how much shore line will be lost for a given slope if sea level rises
if lesson goes long they will do worksheet as homework rather than classwork / Complete worksheet exercises.
1. Construct triangles on geoboard by plotting specified points by Cartesian coordinates.
2.  Sketch the triangles on dot paper.
3.  Classify triangles,
4.  Solve missing angles, check on geoboard with protractor and ruler.
5.  Construct similar and congruent triangles. Measure side lengths
6.  Define angle bisector.

Evaluate (Assessment)

(Steps to check for student understanding) / Observe students’ classwork, question them about the work and give immediate feedback.
Graded classwork
Pre-test/post-test / Graded classwork
Pre-test/Post-test /

Additional Notes

Extra credit – define additional angle bisectors, calculate triangle altitudes and medians, construct tantagram figures.

Reflection after teaching this lesson twice:

For the most part, I was pleased with the lesson. The amount of content was pretty good – a few students finished and were able to do extra credit work. Some didn’t finish. But at least they were all engaged in it and seemed to enjoy the work, The students were mostly enthusiastic. A few seemed overwhelmed with the concepts and/or terminology, but through working with them one-on-one they got on track. (And best of all, no problems with flipping rubber bands at each other or other horseplay that my own classmates would have done in the 10th grade!)

The second time I taught the lesson, I spent more time explaining the worksheet – reading nearly every instruction word for word rather than giving it a general description and referring to the printed instructions on the worksheet.

Data:

Only partial data is available. Student scores rose an average of 14% from the pre-test to the post test. Roughly half of the students showed no change. Roughly one third saw increases up to 200%. However, four students actually showed decreased performance on the post-test (the classroom teacher attributed this to illness and/or distraction of those individual students). A slight positive correlation was observed between classwork grades and test performance.

Information on student attitudes is not currently available.

References:

•  Bass, Laurie E. and Johnson, Art; Prentice Hall Mathematics Geometry, Pearson Education, Inc.; Saddle River, New Jersey; 2004

•  Andres, Richard J. and Bernstein, Joyce; Preparing for the OGT in Mathematics; Amsco School Publications, Inc.; 2004

•  Lund, Charles; Dot Paper Geometry With or Without a Geoboard; Cuisenaire Co. of America, Inc.; New Rochelle, NY; 1990

•  Cech, Joseph P. and Tate, Joseph B.; Geo-board Activity Sheets; Ideal School Supply Company; Oak Lawn, IL; 1971

•  Dimmerling, Amy; Vice Bowling, Bethany; and Massie, Emma; Brent Spence Bridge STEP lesson; University of Cincinnati, http://www.eng.uc.edu/STEP/activities/descriptions/brent_spence_bridge.htm; 2004