Grade Level Geometry (High School) 1 40-minute class period

Topic/Title Perimeter and Area Relationships on a Line Segment

Materials Computers with Geometer’s Sketchpad, student handouts, chalk or white board, calculator

Standards Content Standards

High School Geometry: Introduction

An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts—interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material. Although there are many types of geometry, school mathematics is devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates). During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent.

In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades. These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent.

Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof. Geometric transformations of the graphs of equations correspond to algebraic changes in their equations.

Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.

Congruence G.CO

Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and

straightedge, 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.

Similarity, Right Triangles, & Trigonometry G.SRT

Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor:

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

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.

Prove theorems involving similarity 5. Use congruence and similarity criteria for triangles to solve problems and to prove

relationships in geometric figures.

Circles G.C Understand and apply theorems about circles

1. Prove that all circles are similar. Standards (cont.) Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

Anticipatory Set DiscussionStudents will be asked to find the largest-area rectangle with a perimeter of 80 units. The students should easily discover the answer can be

determined by 80 units4 sides

= 20 units per side. Using this, the students can

conclude that the area would be 20 units * 20 units = 400 units squared (or 400 square units).

Students will then be conversely asked to find the largest-perimeter rectangle with an area of 400 square units.

Students will discover that there is no such answer. A rectangle with an area of 400 square units can have a perimeter greater than any desired value. Students will provide examples of different perimeters that equate to the area of 400 square units.

Learning Activity The teacher will first demonstrate how to do the construction of equilateral triangle on a line segment using Geometer’s Sketchpad.

The teacher will first construct a line segment and name it AB. Next, an arbitrary point C will be placed on the line segment. An equilateral triangle will be created using the custom tool for Equilateral Triangles. (If the tool does not exist, proceed as follows:Select point A and C. Construct circle by point and radius.

Select point C and A. Construct circle by point and radius.Select both circles. Construct intersection points.Create line segments between A and the point above the line, and another segment from C to the point above the line segment AB.Hide circles and intersection point below the line segment AB.Select point A, C, and D (the top point), and the line segments AD and DC. Select Custom Tool; Create; name Equilateral Triangle.)Using the custom tool, an equilateral triangle can be easily created for the segment CB.

Now, select segment AC and measure the distance. Also measure segment CB.Select points A, D, C and construct the interior of the triangle. Measure the perimeter. Similarly, select points C, E, B and construct the interior of the triangle. Measure the perimeter. Then, sum the perimeters together.Next, select the interior for triangle ADC and measure the area. Similarly, select the interior for triangle CEB and measure the area. Then, sum the areas together.

At this point, we will now drag point C along the line segment AB. The students will be told to watch the calculation values just found as the point is moved. Students should notice that all of the values are changing except for the sum of the perimeters.

Next, construct the midpoint of the line segment AB, and name the point F. Drag C toward F. Stop when C falls directly on top of F. Students should realize that the areas and perimeters are the same. Thus, the triangles must not just be similar any more, but they are actually congruent.(Animate point C to a speed of .25, and have students examine the values of the calculations. The students should notice that the area is minimized when C and F are overlapping (meaning that when the triangles are congruent, the areas are the smallest that they can be), while the sum of the perimeters never change.

Students will now work on the sheet that is provided in pairs. Students will recreate the construction shown, but this time with squares and circles.

For the squares, students should construct the line segment AB with an arbitrary point C. Students should then use the Custom Example tool called Square given two vertices. Students can then use the same process to conclude the conjecture holds.

For the circles, students should construct the line segment AB with an arbitrary point C. Students should construct the midpoint of AC and CB, and label them D and E respectively. Students should then construct the circles by selecting D and A, and then selecting Construct by center and point. Students will similarly do the same for circle E, by selecting E and B. They will then follow the same procedures as before. Students should conclude that the conjecture holds.

Finally, the teacher will guide the students in a formal write-up of the conjecture.Conjecture: Given a line segment AB and a point C on the segment. If a N-sided regular polygon has the segment AC as one of its sides, and a similar N-sided regular polygon has the segment CB as one of its sides, then the sum of the perimeters of the two polygons will remain constant as point C is moved along the original segment AB; and the sum of the areas of the two polygons will be minimized when C is positioned at the midpoint of AB.

The students will be shown the general algebraic formula of the sum of the perimeters (N*AC + N*CB = N*AB, where N is the number of sides). Students will provide and work on some examples to justify that their answers are correct. Students should be able to explain that the result is independent of the position of C on the original line segment AB. (This can also be thought of as a composition.)

The lesson will be visual, auditory, and kinesthetic learning. The students will be given the opportunity to explore a conjecture that is presented through technology. Auditory and a part of visual will be experienced through lecture, while kinesthetic and a part of visual will be experienced through working with Sketchpad. Previous concepts and ideas of similarity among polygons has been studied and mastered.

The lesson will be taught as a combination of investigation (student recollection of area and perimeter), instruction (modeling an example), and exploration (students developing their own conjecture). The primary

theory of teaching illustrated in this lesson is constructivist. The students will be assessed on their work on-the-spot while being actively engaged in the lesson. The students will be encompassed in a class discussion.

Throughout the lesson, the NCTM Standards for Teaching Mathematics will be paramount, i.e.,

The students will be engaging in worthwhile mathematical tasks both in class work and assigned work. All assigned work is relevant to the NYS standards and will involve realistic applications.

The teacher’s role in discourse will be positive, engaging, challenging, and inspiring.

The students’ role in discourse will be encouraged, important, and mathematical and will fully satisfy the communication process strand.

Multiple methods will be used for enhancing discourse including class work, technology and directed activities.

The learning environment will constantly be one that will foster each student’s mathematical power. Students will be encouraged and assisted in active problem solving, making connections, and in understanding and creating representations while employing strong reasoning and proof skills.

The teacher will engage in a constant analysis of teaching and learning pre- and post-lessons to ensure that all objectives are met. Strengths of the lesson will be identified as well as areas needing adjustment.

Provision for Gearing DownDiversity There is some provision for gearing down. Students may need a more in

depth review of definitions (such as perimeter, area, minimize). They may even need a review of the Similarity, Right Triangles, & Trigonometry strand. For students who may have difficulty with note-taking, a copy of the lesson will be provided. Also, there will be an option for a video lesson if the written one is not sufficient enough (since Sketchpad will be used, it may be more helpful for visual learners).

Gearing UpFor students who may have a better understanding of the material being addressed, they will have the opportunity to explore the lesson using different polygons. Also, the students could work on the general formula for the area of a square.(Let the length of segment AB be represented by L, and the length of segment AC by x. Then the length of segment CB can be represented by L-x, and the sum of the two areas can be given by x2 + (L - x)2. I would have the students use algebraic manipulation to solve for the sum of the areas.

Sum of Areas = x2 + (L - x)2 = 2x2 -2Lx + L2

= 2[x2 – Lx] + L2

=2 [x2 – Lx + L2

4] + L2 - L2

2

= 2[x - L2

]2 + L2

2.

The expression inside the brackets is non-negative, so the smallest value it

can have is 0. That occurs when x = L2

which is exactly when point C is at

the midpoint of segment AB.)

Questions for KnowledgeUnderstanding What is the definition of similarity?

What is the definition of congruence?

What is the formula for perimeter and area: of a square; of a triangle; of a circle?

ComprehensionClearly express the procedure for doing the constructions.

Identify the dependent and independent sum.

ApplicationUsing Sketchpad, illustrate your constructions and solutions.

AnalysisWhen are the polygons congruent?

Why does this occur?

SynthesisCompose a generalization that may be used for all polygons with N sides.

EvaluationIs it possible to translate the conjecture into an algebraic formula for the perimeter and area?

Does this conjecture always hold?

Was the construction using Sketchpad more efficient than doing hand calculations? Could they easily be done?

Practice GuidedStudents will work on the problems provided in class.

IndependentStudents will be asked to construct each polygon and explore the conjecture using the worksheet provided.

Technology Geometer’s Sketchpad will be used during class for exploration and Integration discussion. For homework, if Sketchpad is unavailable, students will be

encouraged to use GeoGebra, which is an open source application that is similar to Sketchpad. It can be found at http://www.geogebra.org.

Students can use tutorials found at http://wiki.geogebra.org/en/Tutorial:Main_Page and searching for help for constructions. Also, Guillermo Bautista has video tutorials at http://mathandmultimedia.com/geogebra/ which may be more helpful.

Students can refresh their memory of polygon names at http://www.mathsisfun.com/geometry/polygons.html and see the number of sides, interior angles, and shape so that they can reconstruct them using the dynamic software.

Closure In summary, the area of a polygon on a line segment is minimized when the point in common of the two polygons is at the midpoint of the line segment. However, the sum of the perimeters is independent of the sum of the areas. Students will be encouraged to determine if this works with other polygons besides those presented during the lesson. This will be useful when students are asked to find the maximum area of a fenced yard with a perimeter of x.

Assessment Immediate (Formative)Students will be expected to take notes on how the original construction is done. The students will be assessed on their ability and understanding of the exploration of the topic. Students will be given the opportunity to share their findings with the class and get feedback from classmates. The step-by-step constructions on the worksheet will provide their reasoning and understanding of the topic.

The following holistic rubric will be used during class and on assigned work for my information only. Students will not receive a grade based upon the rubric.

4 - Engaged in the lesson and actively asking questions tofurther knowledge of material. Works with partner and challenges their ideas.

3 - Engaged in the lesson and asks some questions to further knowledge of material. Works with partner actively.

2 - Engaged less in the lesson and asks minimal questions to further knowledge of material. Works minimally with partner.

1 - Not engaged in the lesson and asks no questions to furtherknowledge of material. Does not do any work with partner.

Long Range (Formative)Students will practice the knowledge gained by finding the perimeter and areas of polygons algebraically, as well as using technology to check their work. Students will provide a step-by-step construction of one different polygon than those produced in class, and print out a result of the conjecture. This homework assignment will be collected, graded, and returned with comments the following day. Students will be expected to use mathematical language and express their ideas and understandings

completely and clearly. Students are aware of the point value of homework as significant to their overall grade.

Perimeter and Area Relationshipson a Line Segment

1) Construct a line segment. Place a point on the segment that divides the segment into two pieces. Construct a square on each part of the segment, as shown below, and measure the perimeter and area of each square. Explore the sum of the perimeters and the sum of the areas. (Note the construction process in steps.)

GF

ED

A BC

2) Construct a line segment. Place a point on the segment that divides the segment into two pieces. Construct a circle on each part of the segment, as shown below, and measure the circumference and area of each circle. Explore the sum of the circumferences and the sum of the areas. (Note the construction process in steps.)

EDA BC

3) Provide a conjecture to share with the class.

Homework Set

Directions: Using Sketchpad, answer the following questions.

1) Construct a line segment AB with an arbitrary point C that divides the segment in two parts. Construct two Heptadecagon (17 sided polygon) using the 17-gon example custom tool in Sketchpad, one with side AC and one with side CB. Find the perimeters, areas, sum of the perimeters, sum of the areas, and when the area is minimized. Write a step-by-step explanation. Provide a print out of your construction with values. (Hint: It should be similar to the circle construction.)

2) Given two equilateral triangles, one with a side of 4cm and the other with a side of 6cm, construct the triangles and calculator the sum of the perimeters and areas. For what value is the area minimized?

3) Given the sum of the perimeters = N*AC + N*CB = N*AB, what is the sum:a) of a pentagon with segment AC length 7cm and segment CB length 13cm?

b) of an octagon with segment AC length 2cm and segment CB length 12cm?

c) of a hexagon with segment AC length 12

cm and segment CB length 2.5cm?

d) of a decagon with segment AC length 135

cm and segment CB length 423

cm?

Homework Set Answer Key

1) Sketchpad printout.2) Sketchpad printout

3) Given the sum of the perimeters = N*AC + N*CB = N*AB, what is the sum:a) of a pentagon with segment AC length 7cm and segment CB length 13cm?

Sum of Perimeters = 5(7cm) + 5(13cm) = 35cm + 65cm= 100cm

ORSum of Perimeters = 5(7cm + 13cm)

= 5(20cm)= 100cm

b) of an octagon with segment AC length 2cm and segment CB length 12cm?

Sum of perimeters = 8(2cm) + 8(12cm)= 16cm + 96cm= 112cm

ORSum of perimeters = 8(2cm + 12cm)

= 8(14cm)=112cm

c) of a hexagon with segment AC length 12

cm and segment CB length 2.5cm?

Sum of perimeters = 6(12

cm) + 6(2.5cm)

= 3cm + 15cm= 18cm

OR

Sum of perimeters = 6(12

cm + 2.5cm)

= 6(3cm)= 18cm

d) of a decagon with segment AC length 135

cm and segment CB length 423

cm?

Sum of perimeters = 10(135

cm) + 10(423

cm)

= 10(85

cm) + 10(143

cm)

= 805

cm + 140

3cm

= 240+700

15cm

= 94015

cm

= 6223

cm or 62.6666667cm

OR

Sum of Perimeters = 10(135

cm + 423

cm)

= 10(85

cm + 143

cm)

= 10(24+70

15cm)

= 10(9415

cm)

= 94015

cm

= 6223

cm or 62.6666667cm