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S Grade 5 Mathematics Frameworks

Unit 4Geometry and Measurement – Plane Figures

Georgia Performance Standards FrameworkFifth Grade Mathematics Unit 4 1st Edition

Unit 4 Geometry and Measurement – Plane Figures

(6 weeks)

TABLE OF CONTENTS

Overview..............................................................................................................................3 Key Standards & Related Standards....................................................................................4 Enduring Understandings.....................................................................................................6 Essential Questions..............................................................................................................6

Concepts & Skills to Maintain.............................................................................................7

Selected Terms and Symbols...............................................................................................8

Classroom Routines.............................................................................................................9

Strategies for Teaching and Learning..................................................................................9 Evidence of Learning...........................................................................................................9 Tasks..................................................................................................................................10

Archimedes’ Box...................................................................................................11 Who Put the Tang in Tangram?.............................................................................23 What’s My Area?...................................................................................................36 King Arthur’s New Table......................................................................................41 It’s As Easy As Pi..................................................................................................48 The Circle’s Measure.............................................................................................52 Circle Cover-Up.....................................................................................................61

Culminating Task

Stained Glass Designs........................................................................................................74

Georgia Department of EducationKathy Cox, State Superintendent of Schools

MATHEMTATICS GRADE 5 UNIT 4: Geometry and Measurement – Plane Figures May 18, 2023 of 84

Copyright 2010 © All Rights Reserved


OVERVIEW

In this unit students will: derive the formula for the area of a parallelogram derive the formula for the area of a triangle find the area of regular and irregular polygons estimate the area of circles derive the formula for the circumference and area of a circle find the area of plane figures using formulae understand congruence of geometric figures and their corresponding parts understand the relationship of a circle’s circumference, diameter, and pi use variables for unknown quantities use formulae to represent the relationship between quantitiesAlthough the units in this instructional framework emphasize key standards and big ideas at

specific times of the year, routine topics such as addition and subtraction of decimals and fractions with like denominators, whole number computation, angle measurement, length/area/weight, number sense, data usage and representations, characteristics of 2-D and 3-D shapes and order of operations should be addressed on an ongoing basis.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.





STANDARDS ADDRESSED IN THIS UNIT

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

KEY STANDARDS

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures.b. Derive the formula for the area of a parallelogram.c. Derive the formula for the area of a triangle. d. Find the areas of triangles and parallelograms using formulae.e. Estimate the area of a circle through partitioning and tiling.f. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles,

and/or triangles and find the sum of the areas of those shapes.g. Derive the formula for the area of a circle.h. Find the area of a circle using the formula and pi ≈ 3.14.

M5M2. Students will extend their understanding of perimeter to include circumference.a. Derive the formula for the circumference of a circle.b. Find the circumference of a circle using the formula and pi ≈ 3.14.

M5G1. Students will understand congruence of geometric figures and the correspondence of their vertices, sides, and angles.

M5G2. Students will understand the relationship of the circumference of a circle to its diameter is pi (π ≈ 3.14).

RELATED STANDARDS

M5A1. Students will represent and interpret the relationships between quantities algebraically.

a. Use variables, such as n or x, for unknown quantities in algebraic expressions.b. Investigate simple algebraic expressions by substituting numbers for the unknown.c. Determine that a formula will be reliable regardless of the type of number (whole

numbers or decimals) substituted for the variable.

M5P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.





M5P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.

M5P3. Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and

others.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely.

M5P4. Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.

M5P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.





ENDURING UNDERSTANDINGS

Rectangles can be used to determine the area formula for parallelograms. Rectangles can be used to determine the area formula for triangles. The area of irregular and regular polygons can be found by decomposing the polygon into

triangles, squares, and rectangles. Congruent figures must have all corresponding vertices, sides, and angles congruent. “Pi” is the relationship between a circle’s circumference and diameter. Pi is consistent regardless of the size of the circle’s circumference and diameter. Parallelograms and rectangles can be used to derive the formula for the area of a circle.

ESSENTIAL QUESTIONS

How can we identify congruent figures? How can we use vertices, sides, and angles to name congruent figures? How can we find the area of figures? How can we use the area of a rectangle to find the area of a triangle? How can we cut and rearrange shapes to find the area? How can we use one figure to determine the area of another? Is there a common way to calculate area? How do you know? How can shapes be combined to create new shapes? How can a shape be broken down into smaller shapes? How do we figure the area of a shape without a formula for that shape? How are the perimeter and area of a shape related? How are the areas of geometric figures related to each other? How are the diameter and circumference of a circle related? How are the circumference and diameter of a circle related? What is pi? How does it relate to the circumference and diameter of a circle? How do we find the circumference of a circle? How do the areas of squares relate to the area of circles? Why is the area of a circle measured in “square units” when a circle isn’t square? How is the formula for the area of a circle related to the formula for

the area of a parallelogram? How can the formulae for the area of plane figures be used to solve problems? How can we find the area of regular and irregular polygons when you don’t have a

specific formula? How can we verify that two figures are congruent? How are circumference, diameter, and pi related?





CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

computation with whole numbers and decimals angle measurement measuring length and finding area or rectangles and squares add and subtract common fractions with like denominators characteristics of 2-D and 3-D shapes number sense data usage and representations order of operations





SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

Definitions for these and other terms can be found on the Intermath website, a great resource for teachers. Because Intermath is geared towards middle and high school, grade 3-5 students should be directed to specific information and activities.http://intermath.coe.uga.edu/dictnary/homepg.asp

Area: A measurement of the region enclosed by a closed plane figure. Area is always expressed in squared units.

Congruence (congruent): Having the same size and shape.

Circumference: The distance around a circle.

Diameter: A line segment passing through the center of the circle with both endpoints on the circle; the distance across a circle through its center.

Formula: A mathematical rule written using symbols, usually as an expression describing a certain relationship between quantities.

Irregular Polygon: A polygon with sides not equal and/or angles not equal.

Pi: The relationship of the circle’s circumference to its diameter, when used in calculations, pi is typically approximated as 3.14; the relationship between the

circumference (C) and diameter (d),Cd

≈ 3 17∨3.14

Polygon: A closed plane figure having three or more straight sides.

Radius: A line segment with one endpoint at the center of a circle and the other endpoint on the circle; the distance from the center of a circle to any point on the circle.

Regular Polygon: A polygon with all sides equal (equilateral) and all angles equal (equiangular).

Tiling: A repeating pattern of closed figures that covers a surface with no gaps and no overlaps.




http://phytosphere.com/treeord/measuringdbh.htm


Variable: A symbol representing a varying quantity that is often represented by a letter of the alphabet. Example: In the expression in z + 10, “z” is a variable.

CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include obvious activities such as estimating, analyzing data, describing patterns, and answering daily questions. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, and how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students’ number sense, flexibility, fluency, collaborative skills, and communication. These routines contribute to a rich, hands-on standards-based classroom and will support students’ performances on the tasks in this unit and throughout the school year.

STRATEGIES FOR TEACHING AND LEARNING

Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible by using graphs, tables,

pictures, symbols, and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback,

peer feedback, and metacognition which includes self-assessment and reflection. Students should write about the mathematical ideas and concepts they are learning.

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:

estimate and find the area of regular and irregular polygons understand the relationship of a circle’s circumference, diameter, and pi find the circumference of circles estimate and find the area of circles understand congruence of geometric figures and their corresponding parts substitute variables for unknown quantities use formulae to represent the relationship between quantities





TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all fifth grade students. These tasks or tasks of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

Task Name Task TypeGrouping Strategy Content Addressed

Archimedes’ Box Learning TaskPartner/Small Group Task

Using the pieces from Archimedes’ Box to work with congruence and area

Who Put the Tang in Tangram?

Learning Task Individual/Partner Task

Using tangrams to derive the formulae for the areas of triangles and

parallelograms

What’s My Area? Learning TaskIndividual/Partner Task

Determining area of a polygon through deconstruction to simple plane figures

King Arthur’s New Table Performance TaskIndividual/Partner Task Determining area using formulae

It’s As Easy As Pi Learning TaskPartner/Small Group Task

Discovering the relationship between circumference and diameter of a circle

Saving Sir Cumference Learning TaskSmall Group Task

Identifying the value of pi and deriving the formula for the circumference of a

circle

Circle Cover-up Learning Task Individual/Partner Task

Deriving the formula for the area of a circle

Culminating Task:Stained Glass Designs

Performance TaskIndividual/Partner Task Using area to create a mosaic design





LEARNING TASK: Archimedes’ BoxAdapted from an NCTM, Illuminations lesson, “Archimedes’ Puzzle” http://illuminations.nctm.org/LessonDetail.aspx?id=L720

STANDARDS ADDRESSED

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures.b. Derive the formula for the area of a parallelogram.c. Derive the formula for the area of a triangle. f. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles,

and/or triangles and find the sum of the areas of those shapes.









phenomena.




http://illuminations.nctm.org/LessonDetail.aspx?id=L720


ESSENTIAL QUESTIONS

How can we identify congruent figures? How can we use vertices, sides, and angles to name congruent figures? How can we find the area of figures? How can we use the area of a rectangle to find the area of a triangle? How can we cut and rearrange shapes to find the area?

MATERIALS

“Archimedes’ Box, Finding Congruent Figures” student recording sheet “Archimedes’ Box, Describing Congruent Figures” student recording sheet “Archimedes’ Box, Finding Areas” student recording sheet “Archimedes’ Box, Finding Areas Workspace” student sheet (each student will need 2-3

copies of this student sheet) Scissors Colored pencils or crayons

GROUPING

Partner/Small Group Task





TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION

Students explore Archimedes’ Box to develop an understanding of congruence. Then they work with the pieces of Archimedes’ Box to find the area of triangles and to further develop students’ understanding of area.

Part 1 – Congruent FiguresComments

This task may be introduced by giving a little history of Archimedes’ Box. A brief history can be found on the following web sites:

http://illuminations.nctm.org/Lessons/Stomachion/Stomachion-AS-ArchPuzzle.pdf http://illuminations.nctm.org/LessonDetail.aspx?id=L720 http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html Also, students may be given the opportunity to explore the pieces of Archimedes’ Box by

trying to create one of the many shapes possible using all 14 pieces. Shapes that can be made using the pieces can be found on http://illuminations.nctm.org/LessonDetail.aspx?id=L720.

Background KnowledgeStudents should know the definition of congruence before beginning this task. Two shapes

are congruent if they have the same size and shape. In Archimedes’ Box, there are two pairs of congruent triangles; A and N, B and I. Also, students should be aware that congruent parts of congruent figures can be identified using tick marks as shown below.

Task Directions Students will follow directions below from the “Archimedes’ Box, Finding Congruent

Figures” student recording sheet.

1. Cut out the shapes from Archimedes’ Box. 2. On the lines below:

a. List the shapes from Archimedes’ Box that are congruent.

b. Write to explain how you know these shapes are congruent.

Questions/Prompts for Formative Student Assessment What makes two shapes congruent? How do you know those shapes are congruent? Do both pieces have the same shape? Do both

pieces have the same size? How do you know?Part 2 – Describing Congruent Figures




http://www.nycswcd.net/files/Forestry%20Measurements.pdf

http://www.uen.org/

http://www.kathimitchell.com/pi.html

http://intermath.coe.uga.edu/dictnary/homepg.asp


Comments It is important not to give students the “Archimedes’ Box, Describing Congruent Figures”

student recording sheet for this part until they have completed Part 1 because the student sheet provides one of the pairs of congruent shapes in Archimedes’ Box.

Encourage students to label the vertices of each triangle, A and N that they cut out in part 1. Turn triangle A so that corresponding parts are oriented in the same way as triangle N. That way when they are writing congruency statements they can refer to the triangles, making it easier to identify corresponding parts. An example is shown below.

Line up the two triangles so corresponding parts are orientated in the same way. For example, corresponding vertices E and Y are both orientated in the bottom left of their triangle.

Background KnowledgeIt is important for students to be able to find congruent parts of two figures. To do this they

need to be able to read a congruency statement accurately. For example, in the congruency statement DEF XYZ, students need to know that D corresponds to X, E corresponds to Y, and F corresponds to Z. Therefore,∠D≅∠X ,∠Y ≅∠E, and ∠F≅∠Z. Also, EF≅YZ , FD≅ ZX, and DE≅ XY . When sides are congruent that also means they have the same length, or their lengths are equal in measure. When angles are congruent that also means they have the same measure, or their angle measures are equal.

Task DirectionsStudents will follow the directions below from the “Archimedes’ Box, Describing Congruent

Figures” student recording sheet.1. Look at triangle A and triangle N below from the Archimedes’ Box.

We know the two triangles are congruent because:

a. __________________________

b. __________________________

2. Follow the examples to write congruency statements for the angles and sides of DEF and XYZ.





3. Below or on the back of this paper, trace one of the shapes from Archimedes’ Box twice. Label the vertices and write a triangle congruency statement for the two triangles.

Questions/Prompts for Formative Student Assessment Which vertices correspond? Are the vertices of the triangles listed in corresponding order? How do you know? How do you know those angles are congruent? How do you know those sides are

congruent? How do you name a triangle?

Part 3 – Finding AreasComments

Students should be in groups of 4 to work this part of the task. There are 16 pieces of Archimedes’ Box, each pair of students of can find the area of 8 of those pieces and share their work and answers with their group-mates. Alternatively, pairs of students could be assigned to find the area of 2-3 of the pieces of Archimedes’ Box and then the areas can be shared with the class.

The “Archimedes’ Box, Finding Areas Workspace” student recording sheet can be used by students find the areas of the different pieces. Students may need multiple copies of this student sheet. Be sure the grid lines show clearly on the student copies. It may be helpful to suggest students use colored pencils to identify the triangle for which they need to find the area and to draw the rectangle that surrounds it in colored pencil, too. This will help to isolate the part of the box with which the students are working.

Background Knowledge




To find the area of the triangle shaded yellow, put a rectangle around the entire triangle and then considered the areas of two parts of the triangle.

The area of the small part of the yellow triangle is half of the area of the square that surrounds it. The square has an area of 4 square units. Half of 4 is 2, so the area of the small part of the yellow triangle is 2 square units.

The area of the large part of the yellow triangle is half of the area of the rectangle that surrounds it. The area of the rectangle (2 x 10) is 20. Half of 20 is 10. Therefore, the area of the large part of the yellow triangle is 10 square units.

Add the two areas together to find the area of the yellow triangle is 12 square units.

10 u2

2 x 2 = 4

2 x

10 =

20

2 u2

To find the area of the yellow triangle identified to the left, find the area of the rectangle that surrounds it and subtract from the area of the rectangle the area of the green part of the rectangle leaving the area of the yellow triangle.

To find the area of the green triangle identified to the left, find half of the area of the rectangle that surrounds it. The rectangle is 2 x 12, so it has an area of 24 square units. Therefore the area of the green triangle is 12 square units. 12 u2

16 u2

4 u2

The rectangle that surrounds the yellow triangle is 4 x 10 = 40 square units. The three green triangles have a total of 32 square units (12 + 4 + 16 = 32). Subtracting the area of the green triangles from the area of the rectangle leaves 8 square units (40 – 32 = 8). Therefore, the area of the yellow triangle is 8 u2. Students should notice that when dividing a triangle into two smaller triangles to find the area, the triangle is divided by a line segment from a vertex perpendicular to the opposite side. (This segment is the height of the triangle.) Using the strategies explained above and other strategies such as recognizing congruent triangles from part 1, students should find the areas of the pieces as shown.

To find the area of the green triangle identified to the left, find half of the area of the rectangle that surrounds it. The rectangle is 4 x 8, so it has an area of 32 square units. Therefore the area of the green triangle is 16 square units.

To find the area of the green triangle identified to the left, find half of the area of the rectangle that surrounds it. The square is 2 x 2, so it has an area of 4 square units. Therefore the area of the green triangle is 2 square units.





4 u2


Students should notice that the area of each piece of Archimedes’ Box is a whole number and a multiple of 3.

Task DirectionsStudents will follow the directions below from the “Archimedes’ Box, Finding Areas”

student recording sheet1. You will be working with your group to find the area

of each of the shapes in Archimedes’ Box. As your find the area of a shape, record the measure below and share the measure with your group. Be able to explain to your group how you found the area of the shape. Each square is one unit (1 u2 or 1 square unit).

2. What do you notice about the areas of all of the shapes?

3. Write to explain the strategies you used to find the area of each triangle.

Questions/Prompts for Formative Student Assessment Can you draw the rectangle that contains the triangle? Can you find a rectangle that is divided in half by one of the sides of the triangle? How can you break the triangle into two triangles? Can you find the area of each part of

the triangle? How many square units are in this triangle? How do you know? What is the unit of measure for area? How can it be written? Can you subtract the parts of the rectangle, leaving you with the area of the desired

triangle? Can you show me? How can you identify the triangle for which you are trying to find the area? How can you





Questions for Teacher Reflection Are students able to identify congruent figures? Do students recognize that corresponding parts need to be identified when describing

congruent figures? How many students have identified the relationship between the area of triangles and

rectangles? How can that relationship be made more explicit so that students are able to bring that understanding to future tasks?

DIFFERENTIATION

ExtensionChallenge students to arrange the 14 pieces of Archimedes’ Box in a different way to create a

square. The link below shows the 536 possible solutions to this puzzle.http://illuminations.nctm.org/LessonDetail.aspx?id=L720

Intervention Students can use the geoboard at the NLVM website below to find the area of most of the

pieces of Archimedes’ Box. Assign students those triangles that can be contained in rectangles with a length of no more than 10 units. (The geoboard on this website has dimensions of 10 x 10.) Then students can create the triangle on the website and use the rubber bands to create a rectangle around the triangle to find the area as described above. http://nlvm.usu.edu/en/nav/frames_asid_282_g_3_t_3.html?open=activities&from=category_g_3_t_3.html

TECHNOLOGY CONNECTION

http://illuminations.nctm.org/LessonDetail.aspx?id=L720 Provides a variety of shapes students can try to create using all 14 pieces of Archimedes’ Box.

http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html Provides the 536 solutions of Archimedes’ Box for the extension activity.

http://nlvm.usu.edu/en/nav/frames_asid_282_g_3_t_3.html? open=activities&from=category_g_3_t_3.html Provides a 10 x 10 geoboard.




http://www.worsleyschool.net/science/files/circle/area.html


http://www.uen.org/Lessonplan/preview.cgi

http://www.shodor.org/interactivate/activities/AreaExplorer/

http://www.fluorescentgallery.com/store/1842257/product/FG3201_Stained4.jpg


http://www.wiley.com/college/reys/0470403063/appendixc/masters/geoboard_recording.html


Name _________________________________________ Date __________________________

Archimedes’ BoxFinding Congruent Figures

1. Cut out the shapes from Archimedes’ Box.2. On the lines below:

a. List the shapes from Archimedes’ Box that are congruent.b. Write to explain how you know these shapes are congruent.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Archimedes’ Box




D E

F

Z

Y X


Name _________________________________________ Date __________________________

Archimedes’ BoxDescribing Congruent Figures

1. Look at triangle A and triangle N below from the Archimedes’ Box. We know the two triangles are congruent because:

a. _____________________________________________________________________

b. _____________________________________________________________________

DEF XYZ

2. Follow the examples to write congruency statements for the angles and sides of DEF and XYZ.

a. ∠D ≅∠X b. _____________________

c. _____________________

3. Below or on the back of this paper, trace one of the shapes from Archimedes’ Box twice. Label the vertices and write a triangle congruency statement for the two triangles.




a. DE≅ XY b. _____________________

c. _____________________


Name _________________________________________ Date __________________________

Archimedes’ BoxFinding Areas

1. You will be working with your group to find the area of each of the shapes in Archimedes’ Box. As your find the area of a shape, record the measure below and share the measure with your group. Be able to explain to your group how you found the area of the shape. Each square is one unit (1 u2 or 1 square unit).

Shape Area (in square units) Shape Area

(in square units)

A H

B I

C J

D K

E L

F M

G N

2. What do you notice about the areas of all of the shapes?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

3. Write to justify the strategies you used to find the area of each triangle.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________





Name _________________________________________ Date __________________________

Archimedes’ BoxFinding Areas Workspace





LEARNING TASK: Who Put the Tang in Tangram? Adapted from a lesson on the Utah Education Network www.uen.orgTangram clip art from http://www.who.int/world-health-day/previous/2005/infomaterials/en/

STANDARDS ADDRESSED

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures.b. Derive the formula for the area of a parallelogram.c. Derive the formula for the area of a triangle. g. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles,









phenomena.




http://nlvm.usu.edu/en/nav/frames_asid_163_g_3_t_3.html

http://firewall.evsc.k12.in.us/icats/projects/lessons/fifth/harp.ecs.hs/ecs5.2.htm


ESSENTIAL QUESTIONS

How can we use one figure to determine the area of another? Is there a common way to calculate area? How do you know?

MATERIALS

The Warlord’s Puzzle by Virginia Walton Pilegard, or similar book about tangrams “Who Put the Tang in Tangrams? Finding Areas” student recording sheet (2 pages) “Who Put the Tang in Tangrams? Deriving Formula I” student recording sheet (2 pages) “Who Put the Tang in Tangrams? Deriving Formula II” student recording sheet Tangram sets Geoboards, Rubber bands 9 x 12 art paper

GROUPING

Individual/Partner Task


This task will help students determine the area of tangram pieces without using formulae. Then students will use their knowledge to help them develop and use formulae to determine the area of squares, rectangles, triangles, and parallelograms.





Part 1 – Area of Tangram PiecesComments

As an introduction to this task, the book The Warlord’s Puzzle by Virginia Walton Pilegard or similar book about tangrams can be read to the students. After the story, guide students to create their own tangram pieces through paper folding. Directions with illustrations can be found below and at the following web site: http://mathforum.org/trscavo/tangrams/construct.html.

Fold a 9 x 12 piece of art paper to form a square. Cut off the extra piece at the bottom and discard.

Cut the square in half on the diagonal fold to form two triangles. Take one of the triangles and fold it in half to form two smaller congruent triangles. Cut

along the fold. Take the other large triangle and make a small pinch crease in the middle of the baseline

(longest side) to identify the center. Take the apex of the triangle (the vertex opposite the longest side) and fold it to touch the center of the baseline. This forms a trapezoid.

Cut along the fold line. This gives you a trapezoid and a small triangle. Fold the trapezoid in half (two congruent shapes) and cut along the fold line. Take one half of the trapezoid and fold the pointed end to form a small square. Cut along

the fold. This will give you a small square and a small triangle. Take the remaining half of the trapezoid. Fold one of the corners of the square end to

form a small triangle and a parallelogram. Cut along the fold. As you deconstruct the square, discuss the relationships between the pieces. Once complete with a full set (One small square, two small congruent triangles, two large congruent triangles, a medium size triangle and a parallelogram), ask students to experiment with the shapes to create new figures.

To start this task, give each student a set of plastic tangrams to use for this task. (They are easier to trace than the paper ones.) Ask students to find the two small congruent triangles and review the definition of congruent: same size, same shape. Put them together to make a square. Ask students what the area of this shape would be and ask them to explain how they know. (Because the square formed with the two small triangles is the congruent to the tangram square, its area must be the same 1u2 .¿ Next ask students to take just one of the small triangles. Ask students what its area would be and ask them to explain how they know. Remember to relate it to

the square. (The area of each small triangle is half of the area of the square, so its area is12

u2.)

Give students the time to try to determine how to make the shapes before students share their work. This process time for thinking and experimenting will help students develop their spatial reasoning.

Next, give each student the “Who Put the Tang in Tangram? Finding Areas” student recording sheet. Ask students to work with a partner to find the area of the given shapes. Once most students have completed finding the area of the figures, encourage partners to model how they found the areas and ask the class if they agree or disagree, requiring students to explain their thinking.

Background KnowledgeStudents will need to approach this task with the following prerequisite knowledge:

Experience with common plane figures and the identification of their sides and angles.




http://www.mathgoodies.com/lessons/vol2/circumference.html


Familiarity with how to use a geoboard and transfer shapes on the geoboard to geoboard paper.

Knowledge of area and congruence. Understanding of the area of a rectangle and its formula.

Students will be figuring the area of each of the tangram pieces by comparing them to the small square. The small square will be “one square unit”. Therefore, the length and width are both one unit because 1 x 1 = 1.

Sample solutions for the “Who Put the Tang in Tangram? Finding Areas” student recording sheet are shown below.





Task DirectionsStudents will follow directions below from the “Who Put the Tang in Tangram? Finding

Areas” student recording sheet.

Questions/Prompts for Formative Student Assessment What is the area of this shape? How do you know? What shapes have the same area as the area of this shape? What shapes did you use to create a figure congruent to this figure? What are the areas of

those shapes used? What could you add to find the area of this shape?

Questions for Teacher Reflection





Which students are able to find a relationship of each tangram piece to the area of the square?

Which students need to completely cover a figure with tangram pieces in order to find its area?

Which students are able to use the relationships between tangram pieces to find the area of figures?

Did students recognize that the area of a figure can be found by finding the area of pieces of the figure and then adding them together?

Part 2 – Area Formulae for Triangles and ParallelogramsComments

Students will derive the formula for the area of triangles and the area of parallelograms in this part of the task.

Once students are finding the area of triangles and parallelograms they need to transition from the dimensions of length and width to the dimensions of base and height. Therefore, it is important for students to know the characteristics of base and height. Allow students to experiment with the following web sites. Ask students to observe characteristics of the base and height of these figures.

First, students can explore the area of triangles using the GeoGebra web site, http://www.geogebra.org/en/upload/files/english/Victoria/TriangleArea.html. Students should observe that:

The height of a figure is a segment that is perpendicular to the base. The height of a triangle can be one of the sides of the right angle in a right triangle.

The height is an interior segment in an acute triangle, and it is an exterior segment (the base needs to be extended) in an obtuse triangle. See the screen shots below from the GeoGebra web site for examples of each of these types of triangles.

Next, students can explore the area of triangles using the following Illuminations web site: http://illuminations.nctm.org/ActivityDetail.aspx?ID=108. On this site, students are able to move all of the three vertices of the triangle. The program gives the length of the base and the height, as well as the area of the triangle. This information can be added to a table, allowing students to look for patterns. Students should recognize that no matter how the shape of the triangle changes, the height of the triangle is perpendicular to the base.

When exploring parallelograms, student can explore the areas of a rectangle and a parallelogram with the same base and height on the following




http://illuminations.nctm.org/ActivityDetail.aspx?ID=108

http://www.wiley.com/college/reys/0470403063/appendixc/masters/geoboard_recording.html


GeoGebra web site: http://www.geogebra.org/en/upload/files/english/Knote/Area/parallelograms.html.

Or students can explore the area of a parallelogram by “cutting” off a triangle and sliding it to the other side to create a square with the same area.

Background Knowledge

The area of a triangle can be written: b ×h

2 or12

×b × h. The area of a parallelogram and

rectangle can be written: b × h.From this point on, students will be expected to use the dimensions of base and height when

finding the area of rectangles, triangles, and parallelograms. In sixth grade students will explore surface area and volume of 3-D figures. To find the volume of any figure, students will be expected to understand they need to find the area of the base (B) and multiply it by the height of the figure. The powerful idea behind this is that it doesn’t matter what shape the base is. Once students find the area of the base, the volume can be found by multiplying the B×h.

Questions/Prompts for Formative Student Assessment How does folding help you recognize the relationship of area? Can you solve for area without a formula? How? Why would it be helpful to have a formula for area?

Questions for Teacher Reflection Which students are able to devise the formulae for area of a parallelogram and triangle? Which students are able to use correctly the formulae for area of a parallelogram and

triangle? Are students able to articulate the relationship between the base and height of a figure? Are students accurately using the appropriate vocabulary for area (i.e. base and height)? Are students able to label correctly the units of measure for area (i.e. square units, square

inches, square centimeters, etc.)?

DIFFERENTIATION

Extension There are different ways to create many of the shapes on the “Who Put the Tang in

Tangram? Finding Areas” student recording sheet. Allow students to explore these shapes to see if they can find different ways to create them using the tangrams.

Using a Geoboard and Geoboard recording paper (available at http://www.wiley.com/college/reys/0470403063/appendixc/masters/geoboard_recording.html). Ask students to create and record as many different parallelograms as they can that

have an area of12 , 1, or 2 square units. Ask student to identify and give the measure of a

base and height for each parallelogram.




http://www.geogebra.org/en/upload/files/english/Knote/Area/parallelograms.html


http://illuminations.nctm.org/LessonDetail.aspx


Using a Geoboard and Geoboard recording paper (available at http://www.wiley.com/college/reys/0470403063/appendixc/masters/geoboard_recording.html). Ask students to create and record as many different triangles as they can that have

an area of12 , 1, or 2 square units. Ask student to identify and give the measure of a base

and height for each triangle.

Intervention For students with fine motor control difficulties do not have them trace the shapes. Just

let them manipulate the tangrams. Also, students may be given a copy of the tangram puzzle so they just have to cut out the shapes, not fold to make the shapes.

It might be helpful to give some students two sets of tangrams in different colors so they can more easily see the relationships between the shapes.


http://illuminations.nctm.org/ActivityDetail.aspx?ID=108 Interactive triangle and parallelogram applets. Allows students to explore the relationship between the base and height in any type of triangle or parallelogram. (Do not use the trapezoid feature.)

http://www.geogebra.org/en/upload/files/english/Victoria/TriangleArea.html Interactive triangle, students can find the area given the base and height.

http://www.geogebra.org/en/upload/files/english/Knote/Area/Parallelogram2.html Interactive parallelogram, students can find the area given the base and height, as well as move a triangular piece to create a rectangle.

http://www.geogebra.org/en/upload/files/english/Knote/Area/parallelograms.html Interactive parallelogram, students can find the area given the base and height.




http://www.kathimitchell.com/pi.html



http://en.wikipedia.org/wiki/Pi

http://illuminations.nctm.org/Lessons/Stomachion/Stomachion-AS-ArchPuzzle.pdf

http://illuminations.nctm.org/Lessons/Stomachion/Stomachion-AS-ArchPuzzle.pdf


Name ________________________________________ Date ___________________________

Who Put the Tang in Tangram? Finding Areas

Find the area of the following figures.

Figure Show your work Area of Figure (in square units)

Small Triangle

Medium Triangle

Large Triangle

Parallelogram

Trapezoid

Two small and one medium triangles

Rectangle

Page 1Georgia Department of Education

Kathy Cox, State Superintendent of SchoolsMATHEMTATICS GRADE 5 UNIT 4: Geometry and Measurement – Plane Figures

May 18, 2023 of 84Copyright 2010 © All Rights Reserved


FigureSketch it below Show your work Area of Figure

(in square units)Triangle congruent to a large

triangle (Do not use the square)

Trapezoid (Different from the one page 1)

Parellelogram (Different from the one on page 1)

Pentagon

Square using all 7 pieces

Who Put the Tang in Tangram?Finding Areas (Continued)

Page 2





Name ________________________________________ Date ___________________________

Who Put the Tang in Tangram? Deriving Formula I

1. On your geoboard, make a square with an area of nine square units. Record it on the given geoboard. 1. Determine its length and its width.______________________

2. Write an equation for the area of the square.

___________________________________________________

3. Divide the square in half by drawing a diagonal in the square.

4. What two congruent shapes have you made?

_________________________________________________

5. What is the area of one triangle? ______________________

Explain how you found the area of one triangle. Show all work on the geoboard.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

2. Make a different rectangle on your geoboard. Record it on the given geoboard.

1. Determine its length and its width.______________________

2. Write an equation for the area of the rectangle.

___________________________________________________

3. Divide the square in half by drawing a diagonal in the square.

4. What two congruent shapes have you made?

_________________________________________________

5. What is the area of one triangle? ______________________

Explain how you found the area of one triangle. Show all work on the geoboard.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

Page 1





3. Make another different rectangle on your geoboard. How would you find the area of a triangle created in your rectangle by a diagonal? Explain how you found the area of the triangle. Record your work on the geoboard.______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________

4. What do patterns do you notice about finding the area of a triangle?

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

5. What is a formula we could use to find the area of a triangle?

___________________________________________________________________________

___________________________________________________________________________

6. Use the formula to find the area of the triangles below. Use another method to find the area of each triangle. Verify that the area is the same using both methods. Show all work.





Page 2





Name ________________________________________ Date ___________________________

Who Put the Tang in Tangram? Deriving Formula II

1. Use a straight edge to draw a parallelogram in one of the grids at the bottom of the page.

2. Carefully cut out your parallelogram.

3. Follow a line on the graph paper to cut off a triangle from one end of your parallelogram. See

the diagram below.

4. Slide the triangle to the opposite side of your parallelogram.

What shape is formed? ________________

5. What are the dimensions of the shape? ________________What is the area? _____________

6. Do you think this will always work? Explain your thinking.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

7. Use the grid paper below to draw a different parallelogram. Find the area of the area of the

parallelogram.

8. What is the formula for finding the area of a parallelogram? __________________________





LEARNING TASK: What’s My Area?

STANDARDS ADDRESSED

M5M1. Students will extend their understanding of area of geometric plane figures.

d. Find the areas of triangles and parallelograms using formulae.f. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles,










phenomena.





ESSENTIAL QUESTIONS

How can shapes be combined to create new shapes?

How can a shape be broken down into smaller shapes?

How do we figure the area of a shape without a formula for that shape?

MATERIALS

“What’s My Area?” student recording sheet Metric rulers

GROUPING



This task requires students to “break apart” an irregular geometric figure into smaller regular figures in order to find the area of the figure.

Comments This task can be introduced by asking students if there is a formula to find the area of the

figure. The students should recognize that there is no formula. Therefore, challenge students to identify a process they could use to find the area.

Background KnowledgeStudents should have had experience with common plane figures and the calculation of their

area. Also, students should also be familiar with how to use a ruler to measure lengths of segments. Finally, students should know how to measure to the nearest millimeter.

One way the design could be broken into rectangles, squares, and triangles is shown below. Students should measure each segment to the nearest millimeter and then use the appropriate formula to calculate the area of each shape. Once all the shape areas are found, the total area can be found by adding up the individual areas. The students should deduce that the formula for the area of a shape is the total of the areas for each individual shape that makes up the larger shape, or A = a1 + a2 + a3 + ... + an with A = area of the large shape and a = area of each small shape. Challenge students to record in their journals an explanation of the process used to calculate the area. In the example shown, the shape was separated into triangles, and rectangles. The total area is 13,262 mm2.




Area of Figure288 mm2

1050 mm2

(1320 x 3) 3960 mm2

(1100 x 2) 2200 mm2

4400 mm2

(682 x 2) + 1364 mm 2 13,262 mm2

13,262 mm2 = 132.62 cm2


Students can be asked to find the area of the figure using square centimeters. In this case, students could measure to the nearest tenth of a centimeter. For example the top triangle would have dimensions of 2.4cm x 2.4 cm and have an area of 2.88 cm2.

Ask students to include measurements of the dimensions of each shape, that way areas found within the range of the given solution can be verified and accepted.

Task DirectionsStudents will follow the directions below from the “What’s My Area?” student recording

sheet.Find the area of this figure in square millimeters. Measure each segment to the

nearest millimeter.

Questions/Prompts for Formative Student Assessment Do you see any rectangles or triangles that could be contained within this figure? If you don’t have a formula for area, how can you determine the area of a figure? How do you find the area of a triangle? Rectangle? Parallelogram? Square? What is the base of this shape? What is the height of this shape? How do you know this is

the height of the shape? (How is the height related to the base?)

Questions for Teacher Reflection Which students were able to divide the figure into shapes for which they could find the

area? Which students were able to use the formulae correctly to find the area of the shapes? Do students understand that the base and the height must be perpendicular?





DIFFERENTIATION

ExtensionHave students design their own shape made up of squares, rectangles, parallelograms, and

triangles. After they find the total area of their design, they can challenge a partner to find the area of their design.

InterventionProvide cutouts of the individual shapes for students to manipulate. Calculate the areas and

write it on the cutouts. Then allow students to combine the areas of each figure.


http://illuminations.nctm.org/LessonDetail.aspx?ID=L583 This web site provides a lesson for finding the area of irregular figures.

http://illuminations.nctm.org/LessonDetail.aspx?ID=U160 The lesson above is part of this unit on finding area. (Skip Lesson 2, Area of trapezoids, it is not part of the GPS for fifth grade.)

http://www.shodor.org/interactivate/lessons/Area/ Lesson to accompany an imbedded applet http://www.shodor.org/interactivate/activities/AreaExplorer/ This applet allows students to see the square units within the figure.





http://www.shodor.org/interactivate/lessons/Area/

http://www.geogebra.org/en/upload/files/english/Knote/Area/Parallelogram2.html

http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html


Name ________________________________________ Date ___________________________

What’s My Area?

Find the area of this figure in square millimeters. Measure each segment to the nearest millimeter.





PERFORMANCE TASK: King Arthur’s New Table

STANDARDS ADDRESSED

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures.d. Find the areas of triangles and parallelograms using formulae.f. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles,



a. Use variables, such as n or x, for unknown quantities in algebraic expressions.b. Investigate simple algebraic expressions by substituting numbers for the unknown.








phenomena.





ESSENTIAL QUESTIONS

How are the perimeter and area of a shape related?

How are the areas of geometric figures related to each other?

MATERIALS

Sir Cumference and the First Round Table by Cindy Neuschwander or similar book about plane figures

“King Arthur’s New Table” student recording sheet

“King Arthur’s New Table, 20 x 12 Grid Paper” student recording sheet (several copies per group)

One centimeter grid paper (optional)

GROUPING



This task requires students to solve a problem by finding the area of squares, rectangles, parallelograms, and triangles using formulae.

Comments One way to introduce this task is by reading Sir Cumference and the First Round Table by

Cindy Neuschwander or a similar book about plane figures. As the book is read, students may cut a piece of 1 cm grid paper as a “table” and follow Sir Cumference’s and Lady Di’s steps to modify their “tables.” Discuss with students whether or not the area changes as the table is transformed.

Background KnowledgeStudents should build on their understanding of area from the learning task, “Who Put the

Tang in Tangram?” to find the areas required in this task.




h = 4 cm

b = 4 cm

A = 16 cm2h = 3 cm

b = 4 cm

A = 12 cm2

A = b x h A = b x h


Reminder It is important to use base times height for the formula for the area of a rectangle or square

instead of length times width (l x w) or side times side (s2) so that students will be able to make the connection to the formulas for parallelograms and triangles.

If students know the formula for finding the area of squares and rectangles, they can use this knowledge to find the formula for parallelograms and triangles.

A rectangle can be turned into a parallelogram by cutting off a triangle and sliding it to the opposite side as shown below. Since a parallelogram can be created from a rectangle without changing the area, the formula for the area of a parallelogram is the same as the formula for the area of a rectangle, A = b x h.

A parallelogram can be turned into a rectangle by cutting off a triangle and sliding it to the opposite side as shown below. Since a parallelogram can be rearranged into a rectangle without changing the area, the formula for the area of a parallelogram is the same as the formula for the area of a rectangle, A = b x h.

A triangle can be formed by drawing a diagonal in a rectangle. Because two congruent

triangles are formed, a triangle is12 of a rectangle. Therefore, the formula for finding the area of

a triangle would be12 the formula for the area of a rectangle, A =

12 (b x h) or A =

b ×h2 .





Also, a triangle can be formed by drawing a diagonal in a parallelogram. If you draw a

diagonal in the parallelogram, you get two congruent triangles. So, a triangle is12 of a

parallelogram. And, the formula for finding the area of a triangle would be12 the formula for the

area of a parallelogram, A =12 (b x h).

Graphics above can be found at: http://firewall.evsc.k12.in.us/icats/projects/lessons/fifth/harp.ecs.hs/ecs5.2.htm

Task DirectionsStudents will follow directions below from the “King Arthur’s New Table” student recording

sheet. If King Arthur’s meeting room is 20 m x 12 m, what would be a perfect shape

and size for the table in his meeting room? The table must seat all twelve knights and leave least 3 m of space between the table

and the wall for the knights to walk and each knight will need approximately 1.5 meters of space at the table. Use the grid paper to sketch each table and the charts below to record the information for each table your group considers creating for the knights.

Questions/Prompts for Formative Student Assessment How are parallelograms and rectangles alike? Is there anything I can do to make a parallelogram look like a rectangle without changing

the area? Is there anything I can do to make a rectangle look like a parallelogram without changing

the area? How are triangles and rectangles alike? How is the area of a triangle related to the area of a rectangle? How are triangles and parallelograms alike? How is the area of a triangle related to the area of a parallelogram? How much space does each table require? (What’s its area?) How can you use a formula to find the area of each shape? Which shape is the best one to use to make a table for the room? Why do you think so? Did you meet the requirement for space to walk around the table? How do you know? Does each knight have at least 1.5 m of space at the table? How do you know? Why did you choose the table you did for the knights?

Questions for Teacher Reflection Where students able to find the area of all table shapes?






Did students choose a table for the room and defend their choice using mathematical reasoning?

Which students are able to explain how to derive the formula for the area of a rectangle, square, parallelogram, and triangle?





DIFFERENTIATION

ExtensionChange the size of the room and/or the number of knights.

InterventionAllow students to focus on just the square and rectangle first. Students may choose to work

with 1” square tiles. Then have students move onto the parallelogram and triangle. Students may find it easier to work with dot paper instead of the grid paper provided.


http://www.geogebra.org/en/wiki/index.php/Area_Formulas Several interactive applets for exploring the formulae for rectangles, triangles, and parallelograms.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=108 Area tool allows students to explore the area of triangles and parallelograms.




http://illuminations.nctm.org/ActivityDetail.aspx?ID=108

http://www.who.int/world-health-day/previous/2005/infomaterials/en/


Name ________________________________________ Date ___________________________

King Arthur’s New TableIf King Arthur’s meeting room is 20 m x 12 m, what would be a

perfect shape and size for the table in his meeting room? The table must seat all twelve knights and leave least 3 m of space

between the table and the wall for the knights to walk and each knight will need approximately 1.5 meters of space at the table. Use the grid paper to sketch each table and the charts below to record the information for each table your group considers creating for the knights.

Table Shape: Rectangle Table Shape: Square

Formula: _______________________ Formula: _______________________

Table Shape: Parallelogram Table Shape: Triangle

Formula: _______________________ Formula: _______________________





Name ________________________________________ Date ___________________________

King Arthur’s New Table20 x 12 Grid Paper





LEARING TASK: It’s as Easy as Pi

STANDARDS ADDRESSED









phenomena.





ESSENTIAL QUESTIONS

How are the diameter and circumference of a circle related?

MATERIALS

Different size circular objects (e.g. soda cans, records, CDs, wastebaskets, paper plates, etc.)

Floral ribbon or string (floral ribbon is inexpensive and will not stretch.)

Scissors Tape

GROUPING

Partner/Small Group Task


This task is the first opportunity students have to explore the relationship between the circumference and diameter of a circle.

Comments This task is an introduction to pi. The important aspect of

this task is for students to see that it takes a little more than 3 diameters to make the circumference of a circle. “The Circle’s Measure” (the next task) fully develops the relationship between the circumference and the diameter of a circle.

To view Jennifer Jackson and her 5th grade students at Chattanooga Valley Elementary School in Walker County work this task, click on the link below. http://gadoe.georgiastandards.org/mathframework.aspx?PageReq=MathQuad

Allow groups to share their findings, then as a class come to a consensus regarding the relationship between a circle’s circumference and its diameter. Remember, this task is an introduction to pi. The important aspect of this task is for students to see that it takes a little more than 3 diameters to make the circumference of a circle. “The Circle’s Measure” (the next task) fully develops the relationship between the circumference and the diameter of a circle.

Background KnowledgeBefore beginning this task, students should be able to identify the circumference and

diameter of a circle.






Task DirectionsStudents will follow the directions below from the “It’s as Easy as Pi” student recording sheet.

1. Carefully wrap string around the circumference of your circular object.

2. Cut the string when it is exactly the same length as the circumference.3. Now take your “string circumference” and pull it across the diameter of your

circular object. Cut as many “string diameters” from your “string circumference” as you can.

4. Repeat steps 1-3 with at least 3 different circular objects.Tape your string sets to the bottom of this paper and explain what you noticed. Compare your data with others. What do you notice?

5. How are diameter and circumference related? How do you know?

Questions/Prompts for Formative Student Assessment How many diameters are you able to cut from the string representing the length of the

circumference? Was there any string leftover? How much? Do you think the diameter and circumference of a circle are related? How are they related?

Why do you think so? Does the size of the circle make a difference in the number of diameters you are able to cut?

Questions for Teacher Reflection Which students described the relationship as consistent regardless of the circle’s size? Which students described the relationship of circumference divided by diameter as a little

more than 3?

DIFFERENTIATION

Extension Students can be asked to estimate what fraction of the diameter the small piece represents

(about 1/7).

Intervention Students can revisit the relationship being studied in this task by taking the three

diameters and the small leftover piece and re-create the circle with the pieces. Allow pairs of students to turn in one student recording sheet for this task.


http://www.kathimitchell.com/pi.html A comprehensive list of sites about pi.






Name________________________________________ Date ____________________________

It’s as Easy as Pi

1. Carefully wrap string around the circumference of your circular object.

2. Cut the string when it is exactly the same length as the circumference.3. Now take your “string circumference” and pull it across the diameter of your circular

object. Cut as many “string diameters” from your “string circumference” as you can.4. Repeat steps 1-3 with at least 3 different circular objects.5. Tape your string sets to the bottom of this paper and explain what you noticed.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

6. Compare your data with others. What do you notice?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

7. How are diameter and circumference related? How do you know?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________





LEARNING TASK: Saving Sir CumferenceAdapted from http://www.uen.org/Lessonplan/preview.cgi?LPid=15436

STANDARDS ADDRESSED












M5P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.




http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html


b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

How are the circumference and diameter of a circle related?

What is pi? How does it relate to the circumference and diameter of a circle?

How do we find the circumference of a circle?

MATERIALS

For the class Sir Cumference and the Dragon of Pi

by Cindy Neuschwander or similar book about the relationship between circumference and diameter

Class graph to record the circumference and diameter data groups collect (See the Comments section)

For each group Several circular objects (e.g., soda cans, CD’s, wastebaskets, paper plates, coins, etc.).

For each student “Saving Sir Cumference” student recording sheet Measuring tape (metric) Narrow ribbon or string that doesn’t stretch Centimeter grid paper

CommentsA narrow ribbon is recommended for this task because it doesn’t stretch, whereas yarn (and

some types of string) will stretch making measurements inaccurate. Also, floral ribbon is a good choice because it is less expensive.

GROUPING

Small Group Task




or

C ÷ d = C ÷ = d d = Cd = C

= C ÷ dd = C ÷ C = d C = d



This task allows students to further define the relationship between the circumference and diameter of a circle as (approximately 3.14) and to derive the formula for the circumference of a circle.

Comments One way to introduce this task is by reading the first part of the book, Sir Cumference and

the Dragon of Pi, by Cindy Neuschwander. STOP after reading “The Circle’s Measure” on page 13. Tell the students that it is their job to solve the riddle and save Sir Cumference before the knights go to slay the dragon.

After giving each student the “Saving Sir Cumference” student recording sheet, students may work in small groups to complete Part A. It may be helpful to ask groups to use a “Think-Pair-Share” strategy. Individually think about the answers to part A (Think). Next, students share their ideas and thoughts with their group (Pair). Finally, allow 2-3 students to discuss their ideas with the class (Share). Time dedicated to class discussion is critical. If no student brings up the connection between “Measure the middle...” with the diameter of a circle and “...circle around” with the circumference of a circle, help students make this connection.

If using the book with this task, continue reading pages 14-18 after students finish part A of the student recording sheet. STOP after reading page 18. At this point the students should understand that there is a relationship between diameter and circumference and that the measure of the circumference is very close to 3 times the length of the diameter. If this is not generally understood, hold a brief discussion before continuing with the task.

Students may work in groups to make the measurements, but should record the data individually. As students work, circulate around the room taking notes on the various strategies students are using.

To prepare for the class graph, create a coordinate grid on a large piece of grid paper or on a computer prior to the start of the task. Make sure the intervals used will allow the circumference and diameter of all of circular objects to be recorded on the graph. Have each group plot their data points on the class graph in addition to on their own student recording sheet.

Once students have identified a more exact number (3.1 or 3.2) for pi, the symbol π (pi) may be introduced to represent the ratio of the circumference to the diameter of a circle. Note: For our purposes, 3.14 is a close enough approximation of pi, however, for the curious student, the value of π to nine decimal places is 3.141592654. This is still an approximation of the number whose decimal expansion has no end.

Once students have finished the task and the findings have been discussed and clarified, the rest of the book may be read aloud to the students.

Allow students to build on their understanding of fact families to think about how they could find the circumference of a circle, given the diameter. Students have determined that the circumference (C) divided by the diameter (d) equals pi (approximately 3.14). This equation and its “fact family” are shown.

Provide opportunities for students to find the circumference of a circle given the diameter or radius, using the approximate value for pi as 3.14.





Background KnowledgePi is defined as the relationship between the circumference (C) and diameter (d);

Cd

≈ 3 17∨3.14 .

Review how to measure the circumference and diameter of an object in centimeters and how to record the data in a table. (Ribbon or string may be used to wrap around circular objects and then held against a measuring tape.)

When graphing the data collected during the task, students create a collection of points that, if connected, should be very close to a straight line. This indicates that there is a direct/linear relationship between diameter and circumference. Students will study direct/linear relationships in sixth grade, but this is a good preview of that topic. For teacher information only, the equation of the line would be y = 3.14x. Because it is a direct relationship, the graph of the equation would pass through the origin (0, 0) making the y-intercept 0 (y = 3.14x + 0).




In this task, students make a valuable connection between the circumference and diameter of circles. For any given diameter, the circumference of the object is the product of its diameter and pi (π). This relationship is often expressed in the formula,

Circumference (C) = Pi () x Diameter (d)or

C = π x dThus, if an object has a diameter of 2 in., the circumference of that object is approximately 6.28 in.

Pi (π) was probably discovered sometime after people started using the wheel. The people of Mesopotamia (now Iran and Iraq) certainly knew about the ratio of diameter to circumference. The Egyptians knew it as well. They gave it a value of 3.16. Later, the Babylonians figured it to 3.125. But it was the Greek mathematician Archimedes who figured that the ratio was less than 22/7, but greater than 221/77. But pi wasn’t called “pi” until William Jones, an English mathematician, started referring to the ratio with the Greek letter or “p” in 1706. Even so, pi really didn’t catch on until the more famous Swiss mathematician, Leonhard Euler, used it in 1737. Thus, pi evolved through the contribution of several individuals and cultures.

Reference: http://www.uen.org/Lessonplan/preview.cgi?LPid=15436



Task DirectionsStudents will follow the directions below from the “Saving Sir Cumference” student

recording sheet. Sir Cumference has been turned into

a dragon! Help Radius and Lady Di of Ameter break the spell and save Sir Cumference. The answer to this problem lies in this poem. Can you solve the riddle?A. Read “The Circle’s Measure”

What do you think is meant by “measure the middle and circle around”?

What two numbers should you divide? How do you know?

B. Use a measuring tape to find the diameter and circumference of at least 5 different sized circular objects. (You may want to measure the circumference with narrow ribbon and then measure the ribbon to find the measure of the circumference.) Discuss with your group how to record your data in a table, and then create the table below.

C. Use the grid paper below to make a coordinate graph. Use the horizontal axis for diameter and the vertical axis for circumference. Plot your data for each object your group measured on the graph.

D. Enter your diameter and circumference data on the class graph.E. What do you notice about the points you plotted on your graph? How is your graph

similar to/different from the class graph that is being created?

Questions/Prompts for Formative Student Assessment How are the diameter and circumference of a circle related? How much bigger is the circumference than the diameter? How can you be more precise? How can you organize the information you will collect? What increments will you use on your graph’s scale to allow all of the data to fit on your

graph? How do you know where to plot the points? What do you notice about the points you plotted? If you know that C ÷ D = , how could you use this information to find the circumference

of a circle given its diameter? What is the formula for finding the circumference of a circle? How do you know?

Questions for Teacher Reflection Do students understand the relationship between circumference and diameter? Do students recognize that pi () is a constant and not a variable? Do students understand

that ≈ 3.14? Are students able to explain how to derive the formula for the circumference of a circle? Are students able to use the formula for circumference to find the circumference of a

circle given the diameter?




The Circle’s MeasureMeasure the middle and circle around,Divide so a number can be found.Every circle, great and small – The number is the same for all.It’s also the dose, so be clever.Or a dragon he will stay…

forever…

2 3 4 5DiameterInches 1

This length of the circumference represents a diameter measure of one inch. It is found by multiplying 1” by (1” x 3.14 = 3.14”). Therefore, every 3.14” on the diameter measuring tape represents 1” of diameter. Students will need to approximate a measure of 3.14 which is approximately. Using a traditional ruler, that would be a tiny bit more than.


DIFFERENTIATION

Extension Encourage students to explore a tape measure used in forestry management. Trees can be

cut when they reach a certain diameter. However, it is impossible to measure the diameter of a tree using a traditional method with a tape measure without cutting it first. Therefore, a diameter ruler was created to measure the diameter using the relationship of diameter and circumference. More information on measuring trees can be found at the following web sites:

http://www.nycswcd.net/files/Forestry%20Measurements.pdf http://phytosphere.com/treeord/measuringdbh.htm

Ask students to create a diameter tape measure. How would a tape that measures diameter when wrapped around the circumference be created? Once students have created the ruler, allow them to try it out on circular objects in the classroom. Alternatively, students could use their rulers outside to find the diameter of trees.

Background InformationTo create a diameter tape measure, students would need to identify the length of the

circumference when the diameter is 1”, 2”, 3”, etc. To do so, students would need to multiply the diameter by 3.14, giving them the length of the circumference. Then they would need to mark the length on a ribbon or a roll of paper (such as adding machine tape). Instead of listing the actual measure, each interval would be labeled as the number of inches representing the diameter. See example below.

Another way to make this type of tape measure is to use a nonstandard unit of measure such as “soup can diameter”. In this way, students can cut off the label of a can and lay it flat to mark off segments along a ribbon or roll of paper equal to the circumference of the can. However, the scale can be labeled as “can-diameter” units. The diameter of other objects can be measured using the tape measure and labeled in terms of the nonstandard unit “can-diameters.”

An exploration of this type is an effective way for students to think more deeply about the relationship between, diameter, and circumference. Also, it allows to students to measure units in terms of diameter (or radii) which is the basis for radians in trigonometry.Intervention






Have students cut a strip of paper equal to the circumference of a circular object (i.e. a can). Then have the students place the can on the paper and trace it. Ask students to determine the number of times they can trace the can. Repeat with different sized cans. Ask students to write about what they notice and explain what their results mean regarding the relationship of diameter and circumference. As in the examples below, students should notice that a little more than three diameters fit on the circumference of a circle.

Hand out five index cards to each student or group of students. Write the words circumference, radius, pi, and diameter on the board. Ask the students to write one word on the top of each card. Encourage students to use a thesaurus or other reference material to write synonyms, definitions, and examples of each word on the back of the card. Students then arrange the cards in a manner that makes sense to them. (The students may arrange alphabetically, from least to greatest, or cluster the cards in groups.) Have several groups present and justify their arrangements.


http://www.uen.org/Lessonplan/preview.cgi?LPid=15436 A link to the activity on which this task was based.

http://en.wikipedia.org/wiki/Pi The Pi entry provides a graphic for pi, a circle’s circumference is measured on a ruler created using increments equal to the diameter of the circle.

http://www.mathgoodies.com/lessons/vol2/circumference.html This site provides background information on circles and allows students to practice finding the circumference of circles.

http://www.kathimitchell.com/pi.html A comprehensive list of sites about pi. http://www.joyofpi.com/pi.html The First 10,000 digits of pi. http://www.eveandersson.com/pi/digits/ Digits of pi up to 1 million.




http://www.eveandersson.com/pi/digits/

http://www.joyofpi.com/pi.html

http://mathforum.org/trscavo/tangrams/construct.html

http://www.geogebra.org/en/upload/files/english/Victoria/TriangleArea.html


http://gadoe.georgiastandards.org/mathframework.aspx


Name _________________________________________ Date _________________________

Saving Sir CumferenceSir Cumference has been turned into a dragon! Help Radius and Lady Di of Ameter break the spell and save Sir Cumference. The answer to this problem lies in this poem. Can you solve the riddle?

A. Read “The Circle’s Measure.” What do you think is meant by “measure the

middle and circle around”?

___________________________________________________________________________

______________________________________________________

What two numbers should you divide? How do you know?

___________________________________________

___________________________________________

___________________________________________

B. Use a measuring tape to find the diameter and circumference of at least 5 different sized

circular objects. (You may want to measure the circumference with18” ribbon and then

measure the ribbon to find the measure of the circumference.) Discuss with your group how to record your data in a table, and then create the table below.




The Circle’s MeasureMeasure the middle and circle around,Divide so a number can be found.Every circle, great and small – The number is the same for all.It’s also the dose, so be clever.Or a dragon he will stay…

forever…


Page 1





C. Use the grid paper below to make a coordinate graph. Use the horizontal axis for diameter and the vertical axis for circumference. Plot your data for each object your group measured on the graph.

Cir

cum

fere

nce

Diameter

D. Enter your diameter and circumference data on the class graph.

E. What do you notice about the points you plotted on your graph? How is your graph similar to/different from the class graph that is being created?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Page 2





LEARNING TASK: Circle Cover-Up

STANDARDS ADDRESSED

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures.e. Estimate the area of a circle through partitioning and tiling.g. Derive the formula for the area of a circle.h. Find the area of a circle using the formula and pi ≈ 3.14.









phenomena.





ESSENTIAL QUESTIONS

How do the areas of squares relate to the area of circles?

How is the formula for the area of a circle related to the formula for the area of a parallelogram?

Why is the area of a circle measured in “square units” when a circle isn’t square?

MATERIALS

“Circle Cover-Up” student recording sheet “Circle Cover-Up, Cut and Cover” student

recording sheet “Circle Cover-Up, Circles and

Parallelograms” student recording sheet “Circle Cover-Up, Circles to Cut” student

sheet (One per group of four) Scissors Crayons or colored pencils Tape or glue stick

GROUPING



Students will extend their understanding of area and derive the formula for the area of a circle by rearranging the area of a square and by adapting the formula for the area of a rectangle.

Part 1Adapted from a task created by Michelle Parker, Gordon County Schools, Georgia

CommentsOne way to introduce this task is to review the area formulae for squares, rectangles,

triangles, and parallelograms.





Background KnowledgeStudents need to bring to this task their understanding of area of rectangles, including the

ability to determine area by tiling. Also, students need to recognize an approximate value of (3.14) and understand that it represents the relationship between the diameter and area of a circle.

Students will need to recognize that if the length of the radius of the circle is represented by r, then the length of the large square would be2×r and the area of the large square would be 2×r ×2×r ,or4 ×r ×r , (or4 r2). Therefore, the area of the three smaller squares in terms of the radius (r) of the circle can be written as shown.

34

× 4 ×r × rwhich is equal to 3×r × r

When students cut the shaded area and paste it inside the empty quadrant of the circle, they should notice that the area of three squares is not enough to fill the circle. There needs to be a little bit more shaded area to fill the blank quadrant, as shown in the example below.

The area of one square would be s x s. But the length of each side is the same as the length of the radius of the circle, so it could be written as r x r. There needs to be three areas of the square plus a little bit more. Ask students what relationship in a circle is equal to three and a little bit more.

Students should remember that pi () is a little more than three. Therefore the area of a circle could be found by multiplying π × r ×r=¿Area of a circle. This is also written as π × r2 or simplyπ r2.

Task DirectionsStudents will follow the directions below from the “Circle Cover-Up” student recording sheet.

1. Compare the area of the square and circle below. Which one has the larger area? Write to explain how you know.

2. Compare the areas of the two figures below. Do you think one of the areas is larger than the other? Write to explain your thinking.

3. If the radius of the circle is r, what is the length of the large square in terms of r?





4. What is the approximate area of the three

smaller squares ( 34

of the large square)in

terms of r? 5. Follow the directions on the “Circle Cover-

Up Cut and Cover” student recording sheet. Which figure has the larger area? Write below to explain your findings.

6. How do you think finding the area of a circle is related to finding the area of a square?

7. What role do you think pi plays in finding the area of a circle?

Also, students will follow the directions below from the “Circle Cover-Up, Cut and Cover” student recording sheet.

1. Color the area of the squares that are outside the circle. 2. Cut out the circle, but save the colored area that was not

inside the circle.3. Paste the area outside the circle into the blank quadrant in a

mosaic design. Try not to overlap pieces. You may need to cut your colored pieces into smaller pieces so they will fit.

Questions/Prompts for Formative Student Assessment What do the square and circle have in common? Why do you think the square (or the circle) has a larger area? How much of the blank quadrant did you fill with the shaded area? Did you have any left

over? Did you have enough? Why do you think pi plays a role in the area of a circle?

Questions for Teacher Reflection Did students recognize that the area of the circle is a little more than the area of three

squares with the square’s side length equal to the circle’s radius? Did the students recognize that pi is required to find the area of a circle?

Part 2Comments

Part two of this task may be introduced by reading Sir Cumference and the Isle of Immeter by Cindy Neuschwander or a similar story about finding the perimeter and area of plane figures.

Background KnowledgeStudents need to know how to find the area of a parallelograms using the formula b x h

before completing this task. By cutting a circle into 8 (or more) equal sectors, the individual pieces can be arranged so

that they begin to resemble a parallelogram. The smaller the pieces, the more it looks like a parallelogram.




A = b x h

Formula for area of rectangle


The images below are screen shots from http://curvebank.calstatela.edu/circle/circle.htm.

Below is our “parallelogram” made of the circle sectors. Notice the radius drawn in one of the middle sectors.

We know the formula for the area of a parallelogram is A = b x h. So, now we need to use what we know about the characteristics of a circle (radius, circumference, and pi) to find the formula for the area of a circle.

The circumference (C =2 × π × r) is the distance around the circle. However, when the sections of the circle are arranged into a parallelogram, the circumference becomes the TWO bases of the parallelogram. Each base is 12 the circumference orπ × r.

Since we know the formula for the area of a parallelogram, and the circle sectors now resemble a parallelogram, we can begin with the area formula for a parallelogram.




http://curvebank.calstatela.edu/circle/circle.htm


The formula for the area of rectangle can also be used to derive the formula for the area of a circle. As the sectors of the circle get smaller and smaller, the parallelogram gets closer and closer to the shape of a rectangle. For more information on using the area of a rectangle, go to the following link. http://www.worsleyschool.net/science/files/circle/area.html

Task DirectionsStudents will follow the directions below from the “Circle Cover-Up, Circles and

Parallelograms” student recording sheet.1. Cut out one circle from the “Circle Cover-Up, Circles to Cut” student sheet.2. Cut the sectors of the circle apart and arrange them on

the grid paper as shown to form a parallelogram.3. Use the grid paper to help you approximate the area of

the “parallelogram” formed.4. Write the formula for the area of a parallelogram.

5. Rewrite the formula for the parallelogram replacing the base (b) with12the length of

the circumference of the circle (π× r).6. Rewrite the formula above, replacing the height (h) with the radius of the circle (r). 7. If the radius of the circle is 5 units, find the measure of its diameter. 8. Compare the area of the parallelogram you approximated above with the area of the

circle you found using the formula. What do you notice?

Questions/Prompts for Formative Student Assessment How can you arrange the sectors of the circle to create a shape that looks like a

parallelogram? What is the measure of the radius in units? What is the measure of one of the bases of the “parallelogram”? What is the formula for the area of a parallelogram? What parts of the circle can be replaced in the parallelogram formula? How do you know? How do you find the length of the base of the “parallelogram” you created? How do you find the height of the “parallelogram” you created? How did you approximate the area of the “parallelogram” you created?

Questions for Teacher Reflection Which students were able to connect the base and height from the parallelogram formula

with12 the length of the circumference of the circle (π × r) and the radius (r) respectively?

Which students were able to use the formula for a parallelogram and the grid paper to approximate the area of the “parallelogram” created with the circle sectors?

DIFFERENTIATION

ExtensionAsk students to explore and prepare an explanation of other ways to derive the formula for

the area of a circle. Two web sites that show alternative methods are given below.






http://curvebank.calstatela.edu/circle2/circle2.htm Uses animation to derive the formula for the area of a circle based on the area of a triangle.

http://www.worsleyschool.net/science/files/circle/area.html Uses graphics to derive the formula for the area of a circle.

InterventionProvide students with some scaffolding for the “Circle Cover-Up, Circles and

Parallelograms” student recording sheet as shown in the examples below.4. Write the formula for the area of a parallelogram.

b h = Area of parallelogram5. Rewrite the formula for the parallelogram replacing the base (b) with

12the length

of the circumference of the circle (π × r). ( __ __ ) h = Area of parallelogram

6. Rewrite the formula above, replacing the height (h) with the radius of the circle (r).

( r) ___ = Area of parallelogram7. If the radius of the circle is 5 units, find the measure of its diameter.

( r) r = Area of parallelogram

Please find the intervention version of the students recording sheet titled “Circle Cover-Up, Circles and Parallelograms, Version 2” at the end of this task.


http://curvebank.calstatela.edu/circle/circle.htm Uses animation to derive the formula for the area of a circle based on the area of a parallelogram.

http://www.rkm.com.au/ANIMATIONS/animation-Circle-Area-Derivation.html Uses animation to derive the formula for the area of a circle based on the area of a parallelogram.

http://curvebank.calstatela.edu/circle2/circle2.htm Uses animation to derive the formula for the area of a circle based on the area of a triangle.

http://www.worsleyschool.net/science/files/circle/area.html Uses graphics to derive the formula for the area of a circle.




http://beststainedglassart.com/wp-content/uploads/image/Geometric.jpg

http://curvebank.calstatela.edu/circle2/circle2.htm

http://www.rkm.com.au/ANIMATIONS/animation-Circle-Area-Derivation.html


http://www.wallpaperforwindows.com/pc/catalog/ea5021-thumb.jpg



Name _________________________________________ Date _________________________

Circle Cover-Up1. Compare the areas of the square and circle below.

Which one has the larger area? Write to explain how you know.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

2. Compare the areas of the two figures below.

Do you think one of the areas is larger than the other? Write to explain your thinking.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 1





3. If the radius of the circle is r, what is the length of the large square in terms of r?

________________________________________________________________________

4. What is the approximate area of the three smaller squares( 34

of the largesquare)in terms

of r?

________________________________________________________________________

5. Follow the directions on the “Circle Cover-Up Cut and Cover” student recording sheet.Which figure has the larger area? Write below to explain your findings.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

6. How do you think finding the area of a circle is related to finding the area of a square?

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

7. What role do you think pi plays in finding the area of a circle?

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 2





Name _________________________________________ Date _________________________

Circle Cover-UpCut and Cover

1. Color the area of the squares that are outside the circle. 2. Cut out the circle, but save the colored area that was not inside the circle.3. Paste the area outside the circle into the blank quadrant in a mosaic design. Try not to overlap

pieces. You may need to cut your colored pieces into smaller pieces so they will fit.





Name _________________________________________ Date _________________________

Circle Cover-UpCircles and Parallelograms

1. Cut out one circle from the “Circle Cover-Up, Circles to Cut” student sheet.

2. Cut the sectors of the circle apart and arrange them on the grid paper as shown to form a parallelogram.

3. U

s

e

the grid paper to help you approximate the area of the “parallelogram” formed.

4. Write the formula for the area of a parallelogram. _________________________________

5. Rewrite the formula for the parallelogram replacing the base (b) with12the length of the

circumference of the circle (π × r). _________________________________

6. Rewrite the formula above, replacing the height (h) with the radius of the circle (r).

_________________________________

7. If the radius of the circle is 5 units, find the measure of its diameter.

_________________________________

8. Compare the area of the parallelogram you approximated above with the area of the circle you found using the formula. What do you notice?

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________





Name _________________________________________ Date _________________________

Circle Cover-UpCircles to Cut





Name _________________________________________ Date _________________________

Circle Cover-UpCircles and Parallelograms

Version 2

1. Cut out one circle from the “Circle Cover-Up, Circles to Cut” student sheet.

2. Cut the sectors of the circle apart and arrange them on the grid paper as shown to form a parallelogram.

3. U

s

e

the grid paper to help you approximate the area of the “parallelogram” formed.

4. Write the formula for the area of a parallelogram. b h = Area of parallelogram ________________________

5. Rewrite the formula for the parallelogram replacing the base (b) with12the length of the

circumference of the circle (π × r). (__ __) h = Area of parallelogram

________________________6. Rewrite the formula above, replacing the height (h) with the radius of the circle (r).

( r) ___ = Area of parallelogram________________________

7. If the radius of the circle is 5 units, find the measure of its diameter. ( r) r = Area of parallelogram ________________________

8. Compare the area of the parallelogram you approximated above with the area of the circle you found using the formula. What do you notice?

___________________________________________________________________________

___________________________________________________________________________





___________________________________________________________________________

___________________________________________________________________________

Unit 4 Culminating Task

PERFORMANCE TASK: Stained Glass Designs

STANDARDS ADDRESSED

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures.b. Derive the formula for the area of a parallelogram.c. Derive the formula for the area of a triangle. d. Find the areas of triangles and parallelograms using formulae.e. Estimate the area of a circle through partitioning and tiling.f. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles,

and/or triangles and find the sum of the areas of those shapes.g. Derive the formula for the area of a circle.h. Find the area of a circle using the formula and pi ≈ 3.14.








M5P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.





c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.






phenomena.

ESSENTIAL QUESTIONS

How can the formulae for the area of plane figures be used to solve problems? How can we find the area of regular and irregular polygons when you don’t have a

specific formula? How can we verify that two figures are congruent? How are circumference, diameter, and pi related?

MATERIALS

“Stained Glass Designs” student sheet “Stained Glass Designs, Invoice” student

recording sheet 1 cm grid paper (can be found at

http://www.etacuisenaire.com/pdf/gridpaper.pdf Colored construction paper or transparency

sheets Rulers Scissors Glue sticks

Comments





While this task may serve as a summative assessment, it also may be used for teaching and learning. If this task is used as a summative assessment, it is important that all elements of the task be addressed throughout the unit so that students understand what is expected of them.

GROUPING



This task is a summative assessment and includes multiple standards addressed in the previous tasks.

Background KnowledgeStudents should have had many opportunities to determine area of geometric figures with

and without formulae. Students should also be familiar with how to derive a formula and use formulae in an appropriate context.

This task requires work with decimals, so students should be able to operate with one- and

two-digit decimals accurately. Also, it may be helpful for students to recognize that34 is

equivalent to 0.75. This may make the computation of the commission easier for students.

Comments Students can create a finished product of their design using colored construction paper or

transparency sheets that they color. Student work will vary but should include a detailed description of how they determined the amount of glass in each color to purchase. Students should also include a “map” of how the shapes were cut from each piece of colored glass, including measurements. One-centimeter grid paper can be provided for this.

Students need to be aware of the costs for the sheets of glass. Even if they only have one small piece in a specific color, they MUST purchase an entire square foot of glass in that color. The total price of the materials should reflect the number of whole sheets of glass purchased, not fractions of sheets. If necessary, model how to fill in the invoice. Show students what to put in each of the columns. To complete the invoice, calculators may be made available.

Task DirectionsStudents will follow the directions from the “Stained Glass Designs” student sheet and

complete a “Stained Glass Designs, Invoice” student recording sheet.Windows to the World is a locally owned company specializing in stained glass

windows. You have just been hired to create stained glass designs for circular windows. As part of your agreement, you need to submit a design. You may include both regular and irregular shapes in your original stained glass window design.

Windows to the World has certain requirements for their circular windows: All designs must fit inside windows with an area ranging from 200 in2 to 400 in2. All designs must include at least three different sets of congruent figures. Each design must include a supplies/materials list and cost of materials.





Windows to the World increases the selling price for their windows for resale. The selling cost for each window is the cost of the materials multiplied by 1.5. Create a formula to calculate the selling price, and then use the formula to find the selling price for your design.

Windows to the World pays for the supplies you used to create the window design. For

each design, your commission is34 the cost of the materials. Create a formula for finding the

commission for each design. You need to submit an invoice with the formula you used to calculate your commission as well the total cost including the cost of the materials and the commission.

IMPORTANT: Sheets of glass are only sold in square foot sections (12” x 12”). Prices range from $6.12 to $11.46, depending on the color. A price list is included for your use.

Questions/Prompts for Formative Student Assessment Can you find the area of each of the figures you used in your design? How many of pieces of glass will you need for your design? How do you know? What is the total area of your stained glass design? How did you find the area?

What decimal is equivalent to34 ? How do you know?

What cost is different for each design? Can the cost be represented by a variable? How do you know?

Questions for Teacher Reflection Which students were able to complete all aspects of the task? Are students able to work with decimal numbers with fluency and efficiency? Which students used the formulae for the figures with fluency and accuracy? Were students able to determine the area of irregular figures and figures for which they

did not have a formula? Which students were able to represent the variable costs using formulae?

DIFFERENTIATION

Extension Ask students to make a second design for a rectangular window with the same area

requirements.

Intervention Provide students with a “Stained Glass Designs, Invoice” student recording sheet with the

formulae provided. Formula for the selling price of each design

Cost for materials = mSelling price = 1.5 m

Formula for the total cost of each designCost for materials = mCost for commission = 0.75 mTotal cost = m + (0.75 m)





Allow students to compose their design on the following web site http://nlvm.usu.edu/en/nav/frames_asid_163_g_3_t_3.html?open=activities&from=category_g_3_t_3.html. Students can then print and determine the area of the figures and the design.




http://www.bing.com/images/search

http://www.bing.com/images/search



Some sample stained glass designs are shown below. Quilt patterns may also provide ideas for students. http://www.quilterbydesign.com/Classes/StainedGlassQuilt.jpg

http://www.wallpaperforwindows.com/pc/catalog/ea5021-thumb.jpg

http://www.charlestonmuseum.org/N5content/ProdImages/Green-blue-glass-panel.jpg http://beststainedglassart.com/wp-content/uploads/image/Geometric.jpg


http://www.bing.com/images/search?q=stained+glass+designs

%2c+geometric&FORM=IGRE#focal=a4202d664007de0c945b869eeaa4e137&furl=http%3A%2F%2Fwww.connectworld.net%2Froyal%2Fcrystal_damonds.jpg




http://www.geogebra.org/en/upload/files/english/Victoria/TriangleArea.html#focal=a4202d664007de0c945b869eeaa4e137&furl=http%3A%2F%2Fwww.connectworld.net%2Froyal%2Fcrystal_damonds.jpg





http://illuminations.nctm.org/ActivityDetail.aspx

http://illuminations.nctm.org/ActivityDetail.aspx

http://www.charlestonmuseum.org/N5content/ProdImages/Green-blue-glass-panel.jpg

http://www.geogebra.org/en/wiki/index.php/Area_Formulas

http://www.quilterbydesign.com/Classes/StainedGlassQuilt.jpg


Name _________________________________________ Date _________________________

Stained Glass Designs

Windows to the World is a locally owned company specializing in stained glass windows. You have just been hired to create stained glass designs for circular windows. As part of your agreement, you need to submit a design. You may include both regular and irregular shapes in your original stained glass window design.

Windows to the World has certain requirements for their circular windows: All designs must fit inside windows with an area ranging from 200 in2 to 400 in2. All designs must include at least three different sets of congruent figures. For each figure used, a sample of the figure needs to be submitted with the area of the

figures listed. Each design must include a supplies/materials list and cost of materials.

Windows to the World increases the selling price for their windows for resale. The selling cost for each window is the cost of the materials multiplied by 1.5. Create a formula to calculate the selling price, and then use the formula to find the selling price for your design.

Windows to the World pays for the supplies you used to create the window design. For each

design, your commission is34 the cost of the materials. Create a formula for finding the

commission for each design. You need to submit an invoice with the formula you used to calculate your commission as well the total cost including the cost of the materials and the commission.

IMPORTANT: Sheets of glass are only sold in square foot sections (12” x 12”). Prices range from $6.12 to $11.46, depending on the color. A price list is included for your use.

Stained Glass SuppliesSheets of Glass are sold by the square foot.

Color Cost per square foot

Clear/White $6.12Blue $6.93Pink $10.26Black $7.14Gray $6.12Green $6.93Orange $11.01Purple $6.12Red $10.74

Yellow $11.46White $6.12




INVOICE

Designer’s Name Date

Item Description Quantity Price (each) Total Price

Total Price of Materials: ___________________

Formula for the selling price of each design: _____

Selling Price: ______

Formula for the total cost of each design: ______ _____

Total cost for the design (Cost of materials + Commission) ______

Attach your complete design to the invoice. On the back of the invoice, attach a sample of each figure used in your design and list the

area of each figure. Include a description of how you found the area of each figure using words and numbers.


Name _________________________________________ Date _________________________

Stained Glass DesignsInvoice




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