Extension Lesson Plan

 

Name:  Leah Coutorie

 

Title of lesson:  Using Trig with Catapults

 

Date of lesson:  Unknown (written 3-2-05)

 

Length of lesson:  2 one-hour periods

 

Description of the class:

                     Name of course:  Algebra II

                     Grade level:  10th – 12th

                     Honors or regular:  regular

 

Source of the lesson:

            My own Research Methods inquiry

 

TEKS addressed:

            ¤111.33. Algebra II (One-Half to One Credit).

(a) Basic understandings.

(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra.

(3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.

(4) Relationship between algebra and geometry. Equations and functions are algebraic tools that can be used to represent geometric curves and figures; similarly, geometric figures can illustrate algebraic relationships. Students perceive the connections between algebra and geometry and use the tools of one to help solve problems in the other.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof.

(b) Foundations for functions: knowledge and skills and performance descriptions.

(1) The student uses properties and attributes of functions and applies functions to problem situations.

 

(2) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. 

c) Algebra and geometry: knowledge and skills and performance descriptions.

¤111.34. Geometry (One Credit).

(1) The student connects algebraic and geometric representations of functions.

(2) The student analyzes geometric relationships in order to make and verify conjectures. Following are performance descriptions.

 (A) The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships.

(B) The student makes and verifies conjectures about angles, lines, polygons, circles, and three-dimensional figures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

(3) The student understands the importance of logical reasoning, justification, and proof in mathematics. Following are performance descriptions.

(B) The student constructs and justifies statements about geometric figures and their properties.

(C) The student demonstrates what it means to prove mathematically that statements are true.

c) Geometric patterns: knowledge and skills and performance descriptions.

The student identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures. Following are performance descriptions.

(1) The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.

(2) The student uses properties of transformations and their compositions to make connections between mathematics and the real world in applications such as tessellations or fractals.

(3) The student identifies and applies patterns from right triangles to solve problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

e) Congruence and the geometry of size: knowledge and skills and performance descriptions.

 

(2) The student analyzes properties and describes relationships in geometric figures. Following are performance descriptions.

(B) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of polygons and their component parts.

 

 

I.      Overview

With this lesson I want the students to review some trigonometric properties that they have learned prior to this project by utilizing them in the project setting.  This extension lesson will hopefully show the students how trig is used outside of fictional scenarios, but in a scenario that they actually created.  This will be a good mathematical representation of the project and its usefulness in learning math because it is very important for students to see a purpose in what they do.

 

II.  Performance or learner outcomes

            Students will be able to:

                        Recall trigonometric properties

                        Use these properties to solve problems

                        Apply this knowledge to further problems

                        Discuss their findings

   

III. Resources, materials and supplies needed

                Marble Launcher (Cambridge Physics outlet) found in UTeach lab

                Black, plastic CPO marble – found in UTeach lab

                Measuring tape

                Carbon paper

 

IV. Supplementary materials, handouts.

                Evaluation handout

 

Five-E Organization

Teacher Does                    Probing Questions                                Student Does      

Engage:

I will do a short demonstration with the marble launcher that mimics the catapult launches they have been doing.  The marble launcher is set up in the middle of the classroom (desks scattered on either side of launch area so all can see) and the measuring tape will lay along the trajectory of the marble with thick strips of carbon paper aligning it.  The launcher is set at 60 degrees from the floor.  The carbon paper is so the marble will make a mark so that we can tell the distance it traveled.  I will have volunteers help in this demonstration.

 

      

 

1.  [after the marble has been launched] How high up did the marble go?

 2.  How can we find out how high it went?

 3.  Can someone draw on the board the trajectory of the marble?

4.  Where is the point that would show us the height it traveled?

 

1.  About as high as my desk, we donÕt know, this high, [may tell me the horizontal length it traveled]

 2.  Stand here with a meter stick, try to guess, watch to see where it starts coming down

3.  [a volunteer draws a symmetrical arc]

4.  the peak of the arc, the height of travel is the distance from the floor to the peak of the arc

 

                                                   

Explore:

I will put them in groups of 3 to have them explore this question.  They will use knowledge to answer how to find the height of this arc.

 

 

 

1.  How can we use math to find the height of travel?

 2.  Get into groups of 3 to figure this out.  [go around the room and offer hints and guidance, after a while ask them what some trig properties are]  What mathematical properties are you using to answer this question?

[IÕll tell them what they should be using so to steer any of them away from the wrong line of thinking and let them continue]     

1.  [not put in to groups yet]  something with right triangles, use trig properties

2.  a^2+b^2=c^2 , SOH CAH TOA , special triangle: 30-60-90

 

 

    

Explain:

After allowing the students to explore and come to an answer or way of finding the answer I will choose the group who, from my observation, has the greatest understanding to present their solution to the class.  I will then have another group re-tell the first groupÕs answer to ensure understanding.

 

 

 

1.  Will so-and-so group please come to the board to explain their solution?

2. Will another so-and-so group take it form there?

3.  Will a third so-and-so group explain to me what your classmates just told us?

1.  The horizontal length of travel is L and the angle of launch is theta.  The height is here (draw a point at the peak of the arc on their diagram) and the so it makes a right triangle with side L/2.  Since we know L because we measured it and we know theta because we set it, we can use these to find h, the height.

2.  We use TOA, or tan(theta)=h/(L/2)Éput it in the calculator like soÉ

3.  [using the diagram thatÕs now on the board, quickly point out the main points and summarize what has already been explained]

                                               

Extend / Elaborate:

At this point I will have the class do a similar problem with me giving them different data to work with.  I will do it at the overhead only by their commentary.  This way I can see who looks confused and who is quick to offer answers.  Then I can try to quiet the eager students and help along the weaker ones.

 

      

 

1.  What if [label a diagram] the horizontal length traveled was 8 feet and the angle was set at 45 degrees?  What would we do first?

2. Now what?

3.  What if I changed the angle to 30 degrees, do you think the marble would travel further or shorter, higher or lower?

4.  What if I changed the angle to 70 degrees, do you think the marble would travel further or shorter, higher or lower?

5.  At what angle do you think the marble will travel the furthest?

 

 

1.  Label the diagram and mark that L/2 is 4 and draw this triangle with sides 4, h, and unknown hypotenuse and label the angle 45 degrees.

2.  Now we can see that since tan theta equals opposite over adjacent, we can plug in those numbers to the calculator and get the answer.

3.  shorter and lower, because it will reach the ground sooner and because it is aimed lower

4.  shorter and higher, because since the marble travels symmetrically, it will come right down and so not go as far out, and higher because it is aimed higher

5.  at 45 degrees because itÕs right in the middleÉ

   

  Evaluate:

After a detailed discussion of this concept and after any questions are discussed, I will hand out an assessment to be done individually.  It will consist of about 3 or 4 questions that are set up similarly to the scenarios we worked with throughout class.  It will be taken up and discussed afterward.

 

 

 

      

 

1.  What is the height when the horizontal distance is 100 feet and the angle it is launched at is 60 degrees?

2.  What is the height when the horizontal distance is 150 feet and the angle it is launched at is 50 degrees?

3.  What is the height when the horizontal distance is 50 feet and the angle it is launched at is 35 degrees?

4.  Bonus:  At what angle is an object launched where it will travel its furthest?

 

 

1.  173.2 feet

2.  178.8 feet

3.  35.01 feet

4.  45 degrees, explanations will vary along the lines of ÒitÕs in the middleÓ

 

Percent effort each team member contributed to this lesson plan:

100%___       ____Name of group member:  Leah Coutorie