Subject Verb Agreement

Mini Lesson: Subject-Verb Agreement

Subject: English Language Arts

Grade Level: 6th-8th

Objective: Students explore subject–verb agreement using real-life examples and then talk about the difference between formal and informal language and how to use this important grammatical rule.

Common Core Standards:
•CCSS.ELA-Literacy W.6.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.
•CCSS ELA-Literacy W.6.5: With some guidance and support from peers and adults, develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach.

Time: 15 Minutes

Materials: Internet Access to watch video clips
Student Handouts Mini Lesson (Attached)

Rationale: After reviewing subject-verb agreement, middle school students will explore newspaper and song lyrics to identify both correct and incorrect subject-verb agreement. The emphasis on the lesson is on asking students to discover how this important grammatical rule is used (or deliberately ignored) in a variety of settings.

Assessment: Students comprehension of Subject-Verb agreement will be conduct during class/small group collaborative discussion and written work to be handed in.

Launch:
Students will watch the following clip: http://www.schooltube.com/video/b7de4895bc086b3b83f7/

Explore: After the brief review is given regarding subject/verb agreement, students will he given a worksheet with song lyrics to discuss in small working groups.
I will play the song for the class to hear the music as it sounds:

Upon completion, volunteers will be asked to share their corrections with classmates.

Summarize:
Hand out the “Making Subjects and Verbs Agree” to students for their Daybook. Class discussion will include the following questions:
• What is a subject? What is a singular subject? What is a plural subject?
• What is a verb? What is a singular verb? What is a plural verb?
• What is a pronoun?
• What does it mean to have subject-verb agreement?
• Can you think of any examples of songs, headlines, or quotes that lack subject-verb agreement?
• What sounds better – a sentence with or without subject-verb agreement?
• Does anyone know another language? If so, how does subject-verb agreement work in Spanish, Italian, etc?
• When you are grammatically correct in your writing and speech, what kind of impression do you give? Likewise, when you are grammatically incorrect, what kind of impression do you give?

Extension Assignments: Challenge students to find examples in media (music, newspaper articles, ads, etc) where subject verb agreement is not present. Students can complete the assignment found here for more practice.
http://www.englishwsheets.com/subject-verb-agreement-1.html

Resources:
Celce-Murcia, M., & Larsen-Freeman, D. (1999). The copula and subject–verb agreement. In The grammar book: An ESL/EFL teacher’s course, (2nd ed., pp. 53-78). Boston: Heinle & Heinle.

Paiz, Joshua M. and Chris Berry. (2013) Making Subject and Verbs Agree. http://owl.english.purdue.edu

Mini Lesson

EXTRA – FORM ANY SHAPE ON A GEOBOARD; FIND ITS AREA BY MAGIC

Bibliographic Information:
Lobsco, Michael L. “Form Any Shape On This GeoBoard, and Find Its Area By Magic.” Mental Math Workbook, Scholastic Inc, 1998. Pages 75-77. (ISBN # 0-439-14863-4)

Mathematical Concept: Using a Geoboard, Calculating Area

Grade Levels: 6th – 8th

NCTM Standards and Principles of School Mathematics:• Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
• Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
• Use geometric models to represent and explain numerical and algebraic relationships.
• Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Materials:
Geoboard Pattern
Piece of Wood 10 in x 10 in (25 cm x 25 cm) x ¼ inch (1.9 cm) thick
Small Nails
Hammer
Rubber Bands
Linseed oil, Shellac, or paint (optional)

Detailed Description:
With this common grid of nails, you can do more than form many geometric shapes and symmetrical designs. You can perform the spectacular feat of finding the area of regular or jagged intricate shapes using the magical formula learned in this activity. With this secret formula you can actually do what appears to be impossible. Have a friend make the most complicated shape on the board with a rubber band. You can relay how large it is in less than a minute!

Construction Procedure:
1. Photocopy the pattern attached and place it on top of the square board. Be sure you enlarge the photocopy to 300%.
2. Tape the corners of the sheet onto the board to keep it in place.
3. Nail nails on the dots into the board so that each protrudes ½ in (1.3 cm) above the surface. Remove the paper.
4. Sand the board and finish with shellac, oil, or paint. Dry completely.

Activity Procedure:
1. Work through Activities One and Two in class.
2. Once activities are completed, divide class into working partners.
3. Have each partner form an intricate figure on the geoboard.
4. The other partner should us Pick’s Formula to Solve the Area.

Attachment: Worksheet/Activity Student Page
• Geoboard Pattern to be Enlarged by 300%
• Activity One
• Activity Two
• Answer Key for Activity One and Two

TRANSLATION: ACTING OUT TRANSLATED FIGURES

Bibliographic Information:
Killian, Elaine. Acting Out Translated Figures.

Mathematical Concept: Translation

Grade Levels: 7th – 8th

NCTM Standards and Principles of School Mathematics:
• Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
• Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
• Use geometric models to represent and explain numerical and algebraic relationships.
• Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Materials:
Masking Taped Grid on the floor of the class
Yarn
Different pre-cut figures written down in (x, y) notation
Copies of Translation Worksheet

Detailed Description:
The following activity will describe translation of figures. By completion of this lesson, students should understand that a translated image is an image that “slides” in a given direction. The original object and its translation have the same shape, size, and face in the same direction. Students will “Act Out” a figure in the exercise given below. Students will participate by being the original figure and a translated figure.

Procedure:
1. Before class, the teacher will create an x and y axis grid on the floor using masking tape.
2. Divide the class into small working groups.
3. Students will come up to the table groups and choose a figure to represent.
4. The figure will be written down in (x, y) ordered notation form.
5. Each student will represent a point on the polygon. The polygon will be completed using yarn to connect the points (students)
6. The next group will come up and perform the given translation.
7. Each group should get a chance to be the original figure and the translated figure.
8. To follow up student comprehension, students will be given the attached “Translation” worksheet for homework.

Attachment: Worksheet/Activity Student Page
Translation – Handout

ROTATION

Bibliographic Information:
Harrington, Carla. Alexander County Public Schools, Teacher.

Mathematical Concept: Rotation

Grade Levels: 7th – 8th

NCTM Standards and Principles of School Mathematics:
• Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
• Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
• Use geometric models to represent and explain numerical and algebraic relationships.
• Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Materials:
Pencils
Computer with access to Smartboard
Copies of “Rotational Symmetry in the Alphabet” and “Rotational Symmetry in Road Signs” Handouts
Rotation of Shapes Handout

Detailed Description:
These activities are designed to introduce students to Rotational symmetry. Upon completion of these activities, students will know that an image is said to have rotational symmetry if it looks exactly the same when rotated. To illustrate examples of Rotational Symmetry, I will be using the SmartBoards application for Rotation sent to me by Mrs. Carla Harrington. Using this interactive tool, an image may be selected. After covering with virtual “tracing paper” the image is outlined. Then, you may click on Rotate to see how the “traced image” would appear if the original was rotated at 90°, 180°, 270° and 360°. It can be found at the following website:
http://www.teacherled.com/2008/01/28/rotational-symmetry/

Activity Procedure:
1. Give out the sheets titled “Rotational Symmetry in the Alphabet” and “Rotational Symmetry in Road Signs”
2. Have students tell below each letter/sign if it has rotational symmetry.
3. If a letter/sign does have rotational symmetry, have the students to mark the centre of rotation
4. Once students have completed the sheets, discuss the results with the class.
5. Assign Rotation of Shapes Handout for Homework.

Attachment: Worksheet/Activity Student Page
Rotation Handout

REFLECTION: PRACTICE USING MIRA’S

Bibliographic Information:
Woodward, Ernest and Hamel, Thomas. Geometric Constructions and
Investigations with a Mira. Maine: J. Weston Walch, 1992.

Mathematical Concept: Reflection and using Mira’s

Grade Levels: 6th – 8th

NCTM Standards and Principles of School Mathematics• Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
• Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
• Use geometric models to represent and explain numerical and algebraic relationships.
• Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Materials:
Graphing Paper
Colored Pencils
Sharp Writing Pencils
Mira’s
Copies of pages 3 and 5 from Geometric Constructions and Investigations

Detailed Description:
Completion of the following activities will allow students to learn the language associated with reflections and draw a reflected image of a figure over a line of reflection. Students will also be able to perform two-dimensional reflections while using a Mira, as well as recognize when a figure is not symmetrical. By the end of this lesson, students should realize that if the reflective figure looks reversed from its original, it is not symmetrical.

Procedure:
1. Activity One:
a. Hand out graphing paper to each student.
b. Have the students divide a piece of graph paper into four quadrants by drawing the axis lines.
c. In any one quadrant, create a pattern of polygons from the lines on the paper and diagonals.
d. Have students to copy the patterns into the other three quadrants by reflecting the pattern over the axis lines.

2. Activity Two:
a. Divide the class into working partners.
b. Hand each student a piece of graphing paper.
c. Divide the graph paper into four quadrants by drawing the axis lines.
d. Have each student draw a pattern in any one of the quadrants.
e. Each student should pass their graph paper to their partner.
f. The partner is to reflect the pattern over the axis lines.

3. Activity Three:
a. Pass out copies of page 3 from Geometric Constructions and Investigations
b. Pass out a mira to each student. Direct students to find the beveled edge (the edge that seems to cut inward) and that this beleved edge should be placed down.
c. Allow students to practice using the mira as they complete page 3 from Geometric Constructions and Investigations.
d. Students should use the mira to find the line of reflection which puts the boy on the swing.
e. Have students to trace the line of reflection.
f. Finally, students will trace the boy on the other side of the Mira while looking through it as the teacher monitors the activity and gives assistance as necessary.

4. Activity Four:
a. Pass out copies of page 5 from Geometric Constructions and Investigations
b. Instruct students to complete this activity the same way as Activity 3 was completed. They are to follow the handouts instructions to place the hats on the woman’s head.
c. Remind students to trace the line of reflection onto the handout

Attachment: Worksheet/Activity Student Page
Geometric Constructions and Investigations page 3

Geometric Constructions and Investigations page 5

SYMMETRY: POLYGONS AND LINES OF SYMMETRY

Bibliographic Information:
Muschla, Judith A. and Gary R. Muschla. “Polygons and Lines of Symmetry.” Geometry Teachers’s Activities Kit: Ready-to-use Lessons & Worksheets for Grades 6-12. West Nyack, New York. © 2000, pages 130. (ISBN 0-13-016777-0)

Mathematical Concept: Symmetry

Grade Levels: 6th – 7th

NC Standard Course of Study:
Competency 3: The learner will understand and use properties and relationships of geometric figures in geometry.
3.01: Using three-dimensional figures:
– Identify, describe, and draw from various views (top, side, front, corner).
– Build from various views.
– Describe cross-sectional views.

3.02: Identify, define, and describe similar and congruent polygons with respect to angle measures, length of sides, and proportionality of sides.

3.03: Use scaling and proportional reasoning to solve problems related to similar and congruent polygons.

Materials:
Pencils
Rulers
Protractors
Unlined Paper (optional)
Handout

Detailed Description:
Students will draw polygons that have a specific number of lines of symmetry. Students should work individually to complete this activity. Because the sides and angles of many kinds of polygons are congruent, they can be used to illustrate symmetry. In this activity, as students draw polygons, they will also consider their lines of symmetry.

Procedure:
1. Introduce this activity by explaining that symmetry is a relationship in which opposite sides of an object or figure are mirror images of each other.
2. Ask students if they can name examples of symmetry in the world/nature? (Human beings have 2 eyes, 2 ears, 2 arms, 2 legs; most animals are symmetrical; many flowers; buildings; butterflies)
3. Handout copies of the worksheet titled “Polygons and Lines of Symmetry.”
4. Review the instructions with the students.
5. Students are to complete the activity independently.
6. Once all students are finished with the assignment, the class will go over the answers.

Extension Assignment:
As an extension to this assignment, the class may be assigned to form a poster of symmetrical images found in magazines. Students should cut out the figure, glue it onto poster board, and draw its line of symmetry.

Attachment: Worksheet/Activity Student Page
Symmetry – Polygons and Lines of Symmetry – Handout

NETWORKS: FROM POINT TO POLYHEDRON

Bibliographic Information:
Thomas, Christine D. and Carmelita Santiago. (2002). “Building Mathematically Powerful Students Through Connections.” Mathematics Teaching in the Middle School, May 2002. Vol. 7, Iss. 9, p. 484-488.

Mathematical Concept: Exploring Networks with Polyhedron

Grade Levels: 6th – 8th

NCTM Principles and Standards for School Mathematics:
• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects;
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Materials:
75 colored non-bendable straws
Yarn, twine, dental floss, or string
Scissors
Rulers
Protractors
Bamboo Skewers
Sharp Knife
Instruction Sheet

Student Materials:
Notebook/Portfolio for Research
Calculator
Colored Pencils/Markers

Detailed Description:
This activity is based on parts from the “Building Mathematically Powerful Students through Connections” article. Prior to completing this unit, the instructor is encouraged to design each of the straw polyhedron and display them around the room. The models will be used to investigate concepts such as: point line, line segment, rays, angles, and networks. While engaged in this project, students will view themselves as mathematicians and researchers. As models are developed by the students, they will explore two-dimensional figures through investigations of polygons and their properties. Finally, the teams will delve into three-dimensional polyhedrons. Students will examine their string of triangles in terms of network, which is a figure made up of points (vertices) connected by non-intersecting edges (arcs).

Procedure:
1. Organize students into research teams of 2-4 students
a. Each student’s mission will be to contribute to the team’s development of the model, acquire an understanding of geometry concepts and relationships among the concepts, extend the team’s thinking about mathematics through research, and share and discuss insights with the team.
i. Through investigations, students should be able to find congruent angles: alternate interior angles, alternate exterior angles, corresponding angles, and supplementary angles.
b. The students will maintain a research portfolio in which they will keep:
i. A log of daily investigations, facts, and findings
ii. Charts, diagrams, and classroom notes to support investigations, facts, and findings
iii. Make inferences or speculations based on conclusions from the investigations
iv. A vocabulary list of new terms
v. Written reflections for each day’s activities

2. Students will be assigned to find potential connections to real-life situations through research, exploration, and mathematical discoveries.
a. Each team will have a person to work on the following tasks, or the teacher may divide each group of 4 as one specific team (i.e. Group 1 will be the Research Team, etc.)
i. The research team will investigate how the structures changes from adding new straws and/or vertices, analyze the properties of triangles, parallelograms, and trapezoids, and include the study of parallel lines .
ii. The measurement team will find the following measurements: the length of the straw, angle measures, and the area formulas for the triangle, trapezoid, and parallelogram.
iii. The algebraic team will create a chart and look for patterns to determine the number of straws required for a string of equilateral triangles of any length, as well as determine a formula for finding the number of straws to create a string of n triangles. (2n + 1)
iv. The discrete mathematics team will use the horizontal string of triangles to define and investigate networks. This group will also conduct internet research on mathematician Leonhard Euler, specifically looking for information on his work in graph theory.

3. Network: After gaining some background knowledge of the study of networks, the students will examine their string of triangles in terms of networks.
i. Students will examine the number of edges leading into each vertex and number each vertex accordingly.
ii. Discussion should ensue about finding an Euler path (traversing all edges without using any edge twice)
iii. Students should discover that they must begin at a certain vertex to completely travel the network and pass through every edge once and only once.
iv. Euler’s Theorem: If a network has 2 or fewer odd vertices, then it has at least 1 Euler path.
v. Students should conclude that because the string of connected triangles was a network with 2 odd vertices, they must begin and end at an odd vertex to trace an Euler path.

4. Pass out the instruction sheet titled Straw Polyhedra & Other Nets

5. Pass out copies of Polyhedra outlines as accessed on “Let’s Face It” (from PBS.org Mathline) for students to use as visual guides while constructing the Polyhedron networks. Students can use these handouts as guides for what the polyhedra should look like when it is lying flat.

Attachment: Worksheet/Activity Student Page
• Instructions as taken from the following website:
http://www.math.nmsu.edu/~breakingaway/Lessons/straw/straw.html

• Instructions

GEOMETRY VOCABULARY PROJECT

Bibliographic Information:
Harrington, Carla. Teacher for Alexander County Public Schools.

Mathematical Concept:
All geometric vocabulary terms and concepts

Grade Levels: 6th – 8th

NCTM Principles and Standards for School Mathematics:
• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects;
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Materials: Handout listing required terms and concepts

Detailed Description:
This activity idea was given to me by a seasoned teacher for Alexander County Public Schools. Students will be asked to complete a Geometry Vocabulary Project. As students encounter new vocabulary and geometric concepts, they should note them in their notebooks. Students will be given a required list of terminology and concepts to include. (As the school year progresses, teacher will want to revise the list to suit the material covered.)

Procedure:
1. Students will be given a list of required vocabulary terms and concepts to include in their notebook.
2. Students will be required to list the word and its definition, as well as include a picture to illustrate.
3. Terms/Concepts marked with an * require additional instructions. Students will need to refer to the handout for complete directions.
4. Since this is an independent assignment, students will need to be periodically reminded to be working on it. The teacher may want to allot the last few minutes of class for students to add the present day’s terminology information and/or include it as part of the day’s homework assignment.

Attachment: Worksheet/Activity Student Page
• Geometry Vocabulary Project Handout

What’s Regular About Tesselations?

Bibliographic Information:
Miles, Victoria. “What’s regular about Tessellations?” Mathematics Teaching in the Middle School, NCTM, Feb 2000.

Mathematical Concept: Regular Tiling

Grade Levels: 6th – 8th

NC Standard Course of Study:
Competency Goal 3: The learner will understand and use properties and relationships in geometry.
3.01: Represent problem situations with geometric models.
3.03: Identify, predict, and describe dilations in the coordinate plans.

Materials:
Scissors (optional)
Glue (optional)
Colorful Poster Paper (optional)
Snack Size Zip-Lock Bags (1 per student)
Computer with Internet Access
Overhead: What’s regular about this Polygon?
Handout: What’s regular about Tessellations (activity sheet)
Handout: Multiple copies of “Regular Polygons Activity Sheet” per student on (optional – colorful paper)
Answer Key: What’s Regular about Tessellations?

Detailed Description:
In this lesson, students explore regular and semi-regular tessellations. Students use manipulatives to discover which regular polygons will tessellate and which will not. Students will use geometry and measurement to investigate the three regular and eight semi-regular tessellations. This lesson works best when used with the Tessellation Creator. However, if computers with Internet connections are not available, you may use the Regular Polygons Activity sheet in its place. Have several copies available for each student so they are not limited in their explorations by having too few shapes. Each student will also need scissors, glue, and poster paper if using the cut-outs.

Prior to presenting this lesson, the teacher should become familiar with regular and semi-regular tessellations prior to teaching the lesson. There are 3 regular and 8 semi-regular tessellations (as shown on the “What’s Regular About Tessellations?” answer key. The teacher will want to explore how to create the various tessellations using the Tessellation Creator manipulative website.

Procedure:
1. If a computer with Internet access is not available, hand out the “Regular Polygons” activity sheet, scissors, and poster paper; have students multiple copies of this handout so that their exploration isn’t limited by having too few shapes.

2. Pass out copies of the “What’s Regular About This Polygon?” handout and/or use an overhead projector, to introduce the lesson.
a. Allow students time to answer the questions in class.
b. Review the definition of a regular polygon.
c. Introduce/Review methods for calculating the measure of one interior angle of a regular polygon.
d. There are 2 ways to show students the answer to Question 3:
i. Select a vertex and draw the two diagonals from that vertex. The pentagon is now divided into 3 triangles, each has 180°. Therefore, the sum of the interior angle measure of the pentagon is 180 x 3 = 540°.
ii. Draw a point inside the pentagon, and draw 5 line segments connecting that point to the 5 vertices. Notice the pentagon has been divided into 5 triangles. This time 180° x 5 = 900°, but 360° must be subtracted away from the 900° since the angles inside the pentagon don’t pertain to the pentagon’s interior angle.

3. Distribute one “What’s Regular About Tessellations?” activity sheet to each student. (Optional: at this time consider allowing students to work with partners so they can assist each other when using the technology tools and thinking about angle measures.)
a. Question 1 on this worksheet should be a review for students. They should already be familiar with the sum of the interior angles of polygons before beginning this lesson.
b. Students will discover that there are 3 regular tessellations: equilateral triangles, squares, and regular hexagons.
c. The reason these polygons tessellate on their own is the measure of a single interior angle in each polygon is a factor of 360°.
d. Stress connection between interior angle measures and tessellations.
e. If the teacher feels as if the students need more review, consider having them explore the Angle Sums manipulative website before beginning the formal lesson.

4. If students complete the table for regular tessellations, let them know there are 8 semi-regular tessellations. Have students find as many semi-regular tessellations as time permits.
a. Assign groups of 4-6 students a different semi-regular tessellation to share with the class.
b. Have students tape or glue the paper polygons onto poster paper.
i. Each poster should include:
1. a tiling of their semi-regular tessellation using colorful paper polygons
2. The vertex configuration
3. The sum of the interior angles surrounding any vertex.
ii. During the brief presentations, ask students to state:
1. What regular polygons are used?
2. If there are any other ways to classify the vertex configuration
3. Why it’s a semi-regular tessellation
iii. Collect the posters to be used a class bulletin board to review tessellation. (This is not included in the original lesson plan from www.NCTM.org )

Attachment: Worksheet/Activity Student Page
Original “What’s Regular About Tessellations?’ lesson from NCTM.org (for accommodation suggestions,
questions for students, and extension assignments)
Handout/Overhead: What’s Regular About This Polygon?
Activity Sheet: What’s Regular About Tessellations?
Activity Sheet (optional): Regular Polygons
Answer Key: What’s Regular about Tessellations?

Similarity and Congruence

Bibliographic Information:
“Similarity and Congruence.” School Improvement in Maryland. Maryland Mathematics Educators
http://mdk12.org/instruction/clg/lesson_plans/geometry/SimilarityCongru_211.html

Mathematical Concept: Similarity and Congruence

Grade Levels: 7th – 8th

NC Standard Course of Study:
Competency Goal 2: The learner will understand and use measurement concepts.
2.01: Determine the effect on perimeter, area, or volume when one or more dimensions of two- and three-dimensional figures are changed.
2.02: Apply and use concepts of indirect measurement.

Competency Goal 3: The learner will understand and use properties and relationships in geometry.
3.01: Represent problem situations with geometric models.
3.02: Identify, define, and describe similar and congruent polygons with respect to angle measures, length of sides, and proportionality of sides.
3.03: Use scaling and proportional reasoning to solve problems related to similar and congruent polygons.

Materials:
Calculator
Paper
Ruler
Formula sheet
Scissors for each student
Protractor
Patty Paper
Worksheet: Prediction Guide
Worksheet: Comparing Sizes of Figures
Worksheets: Congruent and Similar Triangle Investigation Activity 1-4
Worksheet: Practice with Congruent and Similar Triangles
Worksheet: Applications

Detailed Description:
Students will know how to determine that two figures are similar or congruent by investigating figures that are similar and figures that are congruent. Upon completion of the activities, students will know how to prove that two figures are similar or congruent by using definitions, postulates, and theorems.

Procedure:
1. Use the prediction guide to determine what students already know.
a. Feel free to add/delete/change statements as they fit your class.
b. Accept students’ answers and justifications and ask if the class thinks they are reasonable.

2. Divide the students into small working groups of 2.
a. Have students pull out the handout titled “Comparing Sizes and Figures”
b. Ask students to measure all the segments listed using a ruler. Students should use centimeters to measure.)
c. It may be helpful to have students change their ratios to a decimal (to show that the ratios are equal)

3. An introduction for the SSS and SAS postulates of congruency and the SSS, SAS, and AA postulates of similarity are presented.

a. Activity 1:
i. Pass out the handouts for Activity 1 and scissors.
ii. Each student needs to cut out the three strips of paper.
iii. Place the strips together corner-to-corner to create a triangle
iv. Each student should compare their triangles to their partner’s and to the other triangles in the class.
v. Are they congruent? Encourage students to prove the triangles congruent using the definition of congruent triangles.
1. Measure each side with a ruler and each angle with a protractor.
2. Measuring each side may not be necessary if students reason that everyone started with the same 3 strips of paper.)
3. Have students listed the minimum amount of information used to create these 2 congruent triangles.

b. Activity 2:
i. Pass out the handouts for Activity 1 and scissors.
ii. Using the three strips of paper from Activity 1, have ONE member from each group fold and cut each strip of paper in half.
iii. Students should create a triangle using the one-half pieces of each of the original strips.
iv. Have groups to compare this new triangle to the first one and describe any similarities and differences. [The two triangles are not congruent, but are the same shape.]
v. Since the new half-pieces create triangles which are similar to the original, see if students realize that all NEW triangles are congruent to each other.
1. Have students prove this by using SSS.
2. Help students trace the new triangles and label the sides the same as Activity 1.
vi. Students will justify the similarity of the old and new triangles.
vii. Have students measure each side of the triangle and place the measure in the space requested on the worksheet. [Note: the ratios of the lengths of each pair of corresponding sides are proportional.]
viii. Have students measure each pair of corresponding angles. [Note: the measures of each pair of corresponding angles are congruent. The answers in #4 and #5 show that the triangles are similar.]

c. Activity 3:
i. Pass out Activity 3 handout and patty paper to each student.
ii. Have each student copy the two sides and included angle using patty paper.
iii. Students will draw a segment AC to create a triangle.
iv. A group discussion will take place about what parts of the triangle were given and how their measures compare with everyone else’s [SAS]
v. Students will compare their triangle to the other triangles in the group.
vi. Have students justify that the triangles are congruent, similar, or both. [By definition, they are both. Encourage them to show that the corresponding three sides of the triangles are congruent and therefore the triangles are congruent.]
vii. Ask students to locate the midpoint of AB and of BC and label these points D and E respectively.
viii. Draw segment DE. Compare this triangle to triangle ABC [the two are similar]. Use the definition of similar triangles to justify that the triangles are similar.
ix. Lead a group discussion about what parts of the triangle were given and how this is the minimal amount of information needed in order to prove two triangles are similar [SAS].

d. Activity 4:
i. Pass out Activity 4 handout.
ii. Students will now trace two angles and use these to create triangles. Since these directions may be confusing, demonstrate using the overhead and transparencies.
iii. Have the students create 2 different triangles.
iv. Lead a class discussion about why the 2 triangles are not congruent and that AA [or AAA] is not a way to prove 2 triangles are congruent.
v. The 2 triangles the students create will be similar. Give each student time to measure the sides and show that the ratios of the corresponding sides are proportional. [This will be a great revelation because the ratios of the corresponding sides will be different than the others they have seen in Activities 1-
3. Activities 1-3 ratios will be 2:1.]
vi. Have the students who that the 3 angles are congruent to each other.
vii. Lead a discussion about how the minimum amount of information needed to prove 2 triangles similar is that 2 corresponding angles must be congruent. [AA]

e. Upon completion of Activities 1-4, students will be ready to believe that the ASA theorem of congruence works. Describe that ASA works for similarity as well and can really be the same as the AA theorem.

f. As an extension assignment, hand out copies of the “Practice with Congruent and Similar Triangles” and the “Applications” worksheets.

g. Concluding this study, refer to the Prediction Guide. Allow students to make corrections to their Prediction Guides since they have worked through the entire lesson.
i. Discuss any corrections and the correct answers.

Attachment: Worksheet/Activity Student Page
• Prediction Guide
• Comparing Sizes and Figures
• Congruent and Similar Triangle Investigation: Activity 1
• Congruent and Similar Triangle Investigation: Activity 2
• Congruent and Similar Triangle Investigation: Activity 3
• Congruent and Similar Triangle Investigation: Activity 4
• Practice with Congruent and Similar Triangles
• Applications
• Formula Sheet