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

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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?

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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

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AREA: FERTILIZING A LAWN

Bibliographic Information:
Muschla, Judith A. and Gary R. Muschla. “Finding Area: Fertilizing a Lawn.” Geometry Teacher’s Activities Kit: Ready-to-use Lessons & Worksheets for Grades 6-12. West Nyack, New York. © 2000, pages 289-291. (ISBN 0-13-016777-0)

Mathematical Concept: Calculating Surface Area

Grade Levels: 6th – 7h

NC Standard Course of Study:
Competency Goal 1: The learner will understand and compute with rational numbers.
1.03: Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil.

Competency Goal 2: The learner will select and use appropriate tools to measure two- and three-dimensional figures.
2.01: Estimate and measure length, perimeter, area, angles, weight, and mass of two- and three-dimensional figures, using appropriate tools. Draw objects to scale and use scale drawings to solve problems.

2.02: Solve problems involving perimeter/circumference and area of plane figures.

Competency Goal 3: The learner will understand and use properties and relationships of geometric figures in the coordinate plane.
3.02: Identify the radius, diameter, chord, center, and circumference of a circle; determine the relationships among them.

Competency Goal 5: The learner will demonstrate an understanding of simple algebraic expressions.
5.03: Solve simple (one- and two-step) equations or inequalities.

Materials:
Calculators (optional)
Finding Area: Fertilizing a Lawn Handout

Detailed Description:
Students will find the areas of rectangles and circles to determine how much fertilizer is required for a lawn; they are also to calculate cost and decide which bag of fertilizer is the better buy. Students should work individually or in pairs to complete this activity.

Procedure:
1. Introduce this activity by asking students how many of them help their parents or friends with yard work, particularly working on the lawn.
2. Explain that in this activity students are to assume they are buying fertilizer for a yard.
• They must determine how much to purchase.
• They must determine what size bags of fertilizer represent the better bargain.
3. Students will be given the worksheet titled “Finding Area: Fertilizing a Lawn.”
4. Direct students to study the diagram carefully and note that areas of the yard covered by the house, garage, driveway, deck, shed, hedges, and flower gardens are to be deducted from the total areas of a rectangle.
5. Have students to note the following 2 formulas:
a. Area = L x W
b. Area of a Circle = (pi) x r^2
c. Diameter of a circle = 2 x r
6. Students should be cautioned that although all of the dimensions they will need are not labeled, the missing measures can be determined from the dimensions that are given.

Attachment: Worksheet/Activity Student Page
Area – Fertilizing a Lawn – Handout

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ANGLES: TRIANGLE ANGLES

Bibliographic Information:
Jenkins, Robert H. “Triangle Angles.” 61 Cooperative Learning Activities for Geometry Classes. J. Westin Walch, © 1998, pages 15-16.

Mathematical Concept: Sizes of angles and lengths of sides of triangles

Grade Levels: 6th

NC Standard Course of Study:
Competency Goal 2: The learner will select and use appropriate tools to measure two- and three-dimensional figures.
2.01: Estimate and measure length, perimeter, area, angles, weight, and mass of two- and three dimensional figures using appropriate tools.

Materials: Linear Measuring Tools
Protractors
Triangle Angles Handout

Detailed Description:
This activity enables students to develop knowledge of triangle relationships through a process of exploration. Students will work independently, as well as collaboratively as they are assigned to work in small groups. After teams are formed, each member of the team will generate data. From the data pool, the team will be challenged to discover the relationships between the lengths of the sides of the triangle and the sizes of the angles opposites.

Procedure:
1. Form Teams
2. Students/Teacher will review and clarify the directions on the handout.
3. Students will work independently to generate data.
4. Students will collaborate within their small working group to describe the relationships found in their data.
5. Teacher will collect completed student work for formative assessment.

Attachment: Worksheet/Activity Student Page
Angles – Triangle Angles – Worksheet

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Investigating Volume

Bibliographic Information:
Coy, Erin. “Investigating Volume” Lesson Plan Dacusville Elementary School, Easley, SC. Copyright © 2006 Education World.

Mathematical Concept: Volume

Grade Levels: 7th

NC Standard Course of Study:
Competency Goal 2: The learner will select and use appropriate tools to measure two- and three-dimensional figures.
2.02 Solve problems involving volume and surface area of cylinders, prisms, and composite shapes.

Materials:
A large number of 1” cubes made of wood or plastic
A collection of different size household rectangular boxes (shoe, cereal, crackers, etc)
Rulers
Pencil & Paper
Student Worksheet (as attached)

Detailed Description:
For this lesson, students will work in small working groups. Each group will be given 2-3 different-sized rectangular boxes. Since the students will fill each box with the 1” cubes, the boxes should be manageable in size. (It may also be a good idea for the teacher to number each box and create a master answer key.) Have each working group to measure each box by filling with the 1” cubes. Provide each student with the worksheet and instructions on the handout titled “Investigating Volume.”

Introduce students to the way of writing volume. For example a box that is 4 inches wide, 6 inches long, and 2 inches high has a volume of…
4 x 6 x 2 = 48 cubic inches or 48 in.3
At the end of the lesson, review the vocabulary volume and cubic inch to be certain students understand them.

Extension Activity
Now that students know how to figure the volume of a container without manually placing 1-inch cubes in it, provide much bigger boxes for students to use as they calculate volume. They can use 1-inch cubes or rulers to do that. Observe that they are able to correctly calculate the volume of those boxes

Attachment: Worksheet/Activity Student Page
Investigating Volume – Handout

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POLYGONS: USING POLYGONS TO CREATE A WINTER BULLETIN BOARD

Bibliographic Information:
Helton, Sonia M. “December: Winter Holidays.” Math Activities for Every Month of the School Year. Prentice Hall, 1991. (ISBN: 13-9780876285671

Mathematical Concept: Constructing Images with Polygons

Grade Levels: 6th or 7th

NC Standard Course of Study:
COMPETENCY GOAL 2: The learner will select and use appropriate tools to measure two- and three-dimensional figures.
Objectives
2.01 Estimate and measure length, perimeter, area, angles, weight, and mass of two- and three-dimensional figures, using appropriate tools.

2.02 Solve problems involving perimeter/circumference and area of plane figures.

COMPETENCY GOAL 3: The learner will understand and use properties and relationships of geometric figures in the coordinate plane.
Objectives
3.01 Identify and describe the intersection of figures in a plane.

3.02 Identify the radius, diameter, chord, center, and circumference of a circle; determine the relationships among them.

3.03 Transform figures in the coordinate plane and describe the transformation.

3.04 Solve problems involving geometric figures in the coordinate plane.

Teacher Materials:
Brown Paper to Enlarge Reindeer
Image
Light Blue Paper for Sky
White Paper or Cotton for Snow
Red/Green Construction Paper for
Letters for “Winter Shapes”
1 Circle (various colors) per student
1 String (various lengths) per student
Christmas Tree

Student Materials:
Colorful Construction Paper
Scissors for each Student
Measurement Supplies: Ruler
Compass
Protractors
Glue or Tape for connecting circle/string
White Construction Paper
Markers or Colored Pencils

Detailed Description:
This activity allows students to express their creative abilities. Ask students to cut out various geometric shapes which can be used to construct winter images (i.e. circles for snowmen, rectangles for sleds, triangles for pine trees). Each student will be asked to construct at least one image for the bulletin board. Small working groups can work together to use a variety of geometric shapes to construct more complex images (buildings, houses, etc). As students begin to complete winter images, the class will use spatial visualization to design a class bulletin board.

Each student will then be given one of the precut circles and asked to select a geometric term from an envelope. Students will be required to look up the vocabulary term and write the definition on the circle. The circles will be hung as “holiday balls” from the Christmas Tree and/or Reindeer antlers. The Holiday Balls will be used to reinforce geometry concepts.
Geometry Bulletin Board” was excerpted from Math Activities for Every Month of the School Year written by Sonia M. Helton. This book has over 400 ready-to-use activities to bring math concepts and skills to life, month by month. Each activity features a theme for bulletin boards, as well as word and calculator problems, a motivational game, and “mixed practice” activity sheets.

Attachment: Worksheet/Activity Student Page
Vocabulary & Geometric Concepts
Precut Circles for Terms/Concepts

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