Grade Level:  6

Can You Make a Soccer Ball with Recycled Materials?

This inquiry unit takes students through a sequence of lessons that focus on polygon properties addressed in CCSS.Math.Content.6.G.A.1.

Lessons include:  empathy for using recycled materials; investigating areas of Dream Catcher circle hoops based upon the areas of string polygons contained within the circles; analyzing  traits and identifying regular polygons that tessellate into floor patterns (including symmetries, angular measurements);  examining how to make a curved bowl out of regular polygons; to eventually ideate, prototype, and test a new type of soccer ball design that includes different regular polygons and is made out of recycled materials.

Content Standard addressed in this inquiry unit:

CCSS.Math.Content.6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Core Concepts

Useful Items can be made from recycled materials.

Analyze polygons:  traits, areas, composition, decomposition, appearing on flat & curved surfaces.

Create a durable soccer ball with polygons made of recycled materials.

 

EmpathizeUnderstanding a real-world situation and feeling what another person is feeling.

How can we make useful items out of recycled materials?

Lesson Overview:

This lesson focuses on teaching students about the need to recycle and how items made out of recycled materials can help solve problems. The Empathize Phase of this Inquiry Unit helps students understand real-world reasons for recycling. Students watch videos that feature models of items made from recycled materials, then are asked to ponder and explain why recycling materials into re-purposed items is a worthwhile cause.

Professional Preparation:

Arrange all recycled materials in a common area for students to view.

Plan to facilitate whole group discussions first with circle or u-shaped configurations. In discussions consider some of the following suggestions:

  • Engage students in sharing of ideas, reasoning, and approaches, using a variety of representations.
  • Encourage students to present and explain ideas and reasoning to one another in pair, small group, and whole class discourse.
  • Motivate students to listen carefully to and critique the reasoning of peers, using examples to support or counterexamples to refute arguments.
  • Ask clarifying questions; try out others’ strategies and approaches.
  • Help identify how different approaches to solving a task are the same and how they are different.

Download the following two videos and preview prior to showing the students:

Choose questions to ask students after they watch the first video. Examples:

  • What does it mean to recycle? To repurpose?
  • What did you notice about the use of recycled materials in this video?
  • Which repurposed items were the most interesting to you and why?
  • In what ways do you think recycling is important/useful?
  • Did this video help you think differently about recycling? How? Why?

Choose questions to ask students after they watch the second video. Examples:

  • What did you think about this boy’s use of recycled materials?
  • Why do you think he decided to make a soccer ball out of plastic bags and string?
  • What properties of the ball do you think he was looking for? (e.g., strength, ability to kick it around and not break, large enough to use, easy to make, inexpensive, available to anyone, etc.)
  • What did he do to make the soccer ball? (e.g.,  he kept adding bags within bags, then tightening them up with string so the “ball” was strong enough to kick)
  • What other recycled materials could he have made a soccer ball out of?
  • How was his soccer ball different from a real soccer ball? How was is similar?
  • In what ways do you think the way this boy recycled is important/useful?
  • Did this video help you think differently about recycling? How?  Why?

Provide enough space in the room to form student teams at tables or in small desk clusters (3-4 students per team). Room arrangement should allow for observation and safety in getting around to students.

Make copies of the Our Recycle Sheet, 1 per team.

our-recycle-sheetProcedure:

Ask students if they recycle and, if so, what they recycle.  Gather a list and organize the information of commonly recycled items from the class input.

Observe Student Reactions to the two videos:

  • Start with a whole group discussion.
  • Have students watch the first video. Ask students questions about the video.
  • Have students watch the second video. Ask students questions about the video.

Recycled Materials:

  • Form teams of 3 to 4 students. Pass out one Our Recycle Sheet to each team.  Have teams of students investigate the materials in the common area.
  • Ask each student to choose and bring back one recycled item from the materials in the common area to their team.
  • Ask teams to investigate and discuss how they might combine their recycled items to make something new and useful for someone else, and then to list and record ideas on the Our Recycle Sheet (20 minutes).  Then share findings with the class.

 

Materials

  • RAFT Makerspace-in-a-box

-or-

Various adhesives and fasteners (staplers, tape, glue)

Scissors, pens or pencils

Assorted polygon shapes made from recycled materials

Handout:

Our Recycle Sheet, 1 per team

A variety of recycled items (e.g., plastic bags, rubber bands, fabric, paper, cups, straws, paper clips, cup lids, plates, string, file folders, etc.)

Prior Knowledge

Ask students if they recycle and, if so, what they recycle. Gather a list and organize the information of commonly recycled items from the class input.

Core Concepts

Determine why it might be useful to make a items out of  recycled materials

Learning Targets

Identify assorted recycled materials that could be made into other items.

Useful items can be made from recycled materials.

Re-purposed recycled materials can help solve problems.

 

Assessment and identifying relationships:

Ask student teams to share out their ideas about recycling.

Have students explain (via written, oral, or other report) to the class how their ideas could be useful and what they learned about recycling materials.

Based on their demonstration, assess students on how well they understand the reasons for recycling with the following rubric (or one of your own):

assessment-rubric

Define –In order to create solutions, students need time to investigate various approaches and to hear differing viewpoints from one another. In this lesson phase, students will investigate attributes of polygons, polygons that can be composed of or decomposed from other polygons, and determine their areas: useful information in defining the final Design Challenge project.

Unit Inquiry Question:

How many different polygons can you find in the string patterns in a Native American Indian Dream Catcher?  How can you use the areas of each polygon to approximate the area of the whole circular hoop?

Lesson Overview:

This activity “weaves” Native American Indian tradition with mathematics!

The Focus in this Define phase of the Inquiry Unit is on teaching students about varieties and areas of special quadrilaterals, triangles, and other polygons, and how these shapes can be reconfigured into and be made from other polygons. In making a Native American Indian Dream Catcher, students will explore polygon configurations within the string patterns inside the hoop. Students will estimate the inner area of the Dream Catcher hoop by adding up individual areas of each polygon within the hoop. They will notice as more polygons fill the space within the hoop, the more accurately they can estimate of the area of the circular hoop.

Professional Preparation: 

Arrange all materials in a common area for students to access. Organize same items together (e.g., a bag of all feathers, a box of beads, hoops, etc.).

Prepare 1 copy per student of the Dream Catcher Recording Sheet.

drcatcher-recording-sheeet

Prepare to share information about the history of the Native American Indian Dream Catcher with the class — if possible, show actual Dream Catcher examples.

Suggestion: Begin the lesson with the entire class.

Engage students in sharing ideas, reasoning, and approaches, as they explain a representations.

Encourage students to present and explain ideas and reasoning to one another in pair, small group, and whole class discourse.

Motivate students to listen carefully to and critique the reasoning of peers, using examples to support or counterexamples to refute arguments.

Ask clarifying questions; try out others’ strategies and approaches.

Help students identify how different approaches to solving a task are the same and how they are different.

Prepare methods of assessment that include:

  • Monitoring students’ progress to promote student learning
  • Making instructional modifications to instruction for student learning
  • Evaluating students’ achievement to summarize and report students’ demonstrated understanding
  • Making the students effective self-assessors, teaching them to give feedback that they can act on to advance their own learning and help them attain their goals

Provide enough space in the room to form teams of 3 to 4 students.

Room arrangement should allow for observation and safety in getting around to students.

Procedure :

After students enter the room, assign student teams.  Then allow all students a 5 minute gallery walk to look at the all materials in a common area and then ask them to prepare for a whole group discussion.

Distribute one Dream Catcher Recording Sheet to each student.

Group Discussion —

Begin with a whole group discussion 

Review and show examples of special quadrilaterals, triangles, and other polygons.  Hand out polygon shapes and scissors. Ask each group to observe and to explain the following: 

    • What is meant by area?  
    • What types of triangles are there besides right triangles? How can you find their areas?
    • What are quadrilaterals?
    • What are the names and properties of special quadrilaterals?  Of other polygons? How can you find their areas?
    • How could you find the area of a pentagon?  Of a hexagon? (e.g. break them into smaller triangles or quadrilaterals,  then add up those areas to find the area of the whole)
    • Say you know the area of a whole polygon (e.g., hold up a large paper square for students to see.  Tell them the area of the square is 36 units, meaning each side is 6 units in length).  Now fold that shape into smaller shapes (e.g., fold along one diagonal, then unfold and show students two triangles within the square).  How could you find the area of each partial shape (e.g., how could you find the area of one of the triangles?  Answer: Since the whole square has area 36 units, and one triangle is half the size of the whole square, then the area of one triangle is ½ of 36 units, which is 18 units).
    • Say you have smaller polygonal shapes that piece together into one whole shape. How could you find the area of the whole shape by knowing the area of each piece? (Answer:  by adding up the areas of each smaller polygon you will get the area of the whole shape).

Share the tradition of the Native American Indian Dream Catcher with the class. Mention recycled materials will be used to make our Dream Catchers today. Solicit ideas from the students on types of materials they could recycle to create a Dream Catcher with, and what they remember about the importance of recycling.

Student Teamwork:  

Break students into teams of 3 to 4 students.  Ask teams to choose one member to gather all needed materials for their team, and then return to their team to create Dream Catchers.   Hold up a hoop and instruct as follows while making a model of a Dream Catcher with the students to see:

  • Mark 3 points anywhere along rim of the hoop and attach a clip to each point.
  • Fold down the inner binder clip loop so it points towards the center of the circular hoop (Suggestion:  place hoop on a table or other flat surface while adding clips along the rim).
  • Slip the end of one string through the loop of the first binder clip and tie a knot.
  • Pass the other end of the string through a loop on each binder clip.

drcatcher-with-one-string

Say: Discuss and confirm within your team:

  • How many and what type(s) of polygons can be created with the string inside the hoop?  (answer:  1 triangle).

drcatcher-with-triangle

  • Untie the string.  Add one more binder clip along the rim of the hoop. Now you have 4 binder clips on the outer edge.
  • Slip the end of one string through the loop of one binder clip and tie a knot.
  • Pass the string through each of the other binder clips without crossing strings inside the center of the hoop.  What shapes do you make?  (answer:  quadrilaterals.  Have students show and name the types they make).

drcatchers-with-quadrs

 

  • Untie the string.  Keep 4 binder clips on the outer edge.  This time, cross strings when you string from any binder clip to another binder clip. What polygons do you see?  (For example, a trapezoid and 8 triangles below):

drcatcher-with-crossed-stringsAsk students to untie their strings and remove binder clips.

Have students create their own dream catchers using up to 10 binder clips and the string lengths. Mention: if you run out of string, tie and cut it off at the nearest binder clip.  Start a new string through another loop, tie a knot, and begin again. Also at any time add beads, feathers,  or other embellishments to the strings.  

Skip points around the hoop (see examples below).  Start new strings at different locations to create a variety of patterns.

various-drcatcher-arrangemtsAsk students to look for polygons within the string patterns and to record what information about the shapes on the Dream Catcher Recording Sheet.drcatcher-recording-sheeet

Have students add as many polygon shapes within their hoop as possible. Pass out extra binder clips and string if needed.  Ask students what they notice as more clips are added around the hoop and more polygons are made with the string (the string polygon shapes fill up the inside of the hoop).

 

Materials

  • RAFT Makerspace-in-a-box

-or –

(for each student):

Hoop, 15 to 25 cm (6” to 10”) in diameter (e.g., an embroidery hoop)

Small or Medium size Binder clips, 10

String, 8 lengths ~1 m (3 ft) each

Feathers,  beads, or other embellishments, 3 to 5

Rulers, scissors, pens, pencils

Handout:

Dream Catcher Recording Sheet

 

 

 

Prior Knowledge

Students recognize properties of 2 dimensional shapes, including understanding the definitions of vertices, edges, angles, faces, area, congruence, similarity, and symmetry.

Core Concepts

Identify polygons within the string patterns of a Dream Catcher.

Find the total inside area of a Dream Catcher hoop by adding the areas of string polygons within it.

Learning Targets

Analyze and identify traits, including areas, of triangles and of special quadrilaterals and polygons

Analyze and identify polygons that can be composed into rectangles and decomposed into triangles and other shapes.

Background Information about Dream Catchers:

Prepare to share information about the history of the Native American Indian Dream Catcher with the class — if possible, show actual Dream Catcher examples.  The basic story is as follows:

The first dream catchers were crafted by the Ojibwe (Chippewa) tribe.  Legend tells of a “spider woman” named Asibikaashi whose magical web had the power to trap the Sun.  Dream catchers were traditionally hung above sleeping babies to catch bad dreams and let only good dreams pass through to the child. Later, the bad dreams would disappear when the first rays of sunlight struck the web.

When the native Ojibwe nation dispersed to the four corners of North America, Asibikaashi found it hard to share her webs with everyone who wanted one. So, mothers, sisters, and grandmothers started creating their own webs using flexible hoops made from willow branches.

Traditional native dream catchers have 8 holes along the outer rim to represent a spider’s 8 legs.  Today, many different Native American Indian tribes make dream catchers in a wide variety of styles.

For more information refer to resources such as:  http://www.firstpeople.us/FP-HtmlLegends/TheLegendOfTheDreamcatcher-Chippewa.html

dream-catcher-illustration

types-of-dream-catchers

 

 

 

 

drcatcher-recording-sheeet

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Assessment and identifying relationships:

Ask student groups  to analyze and interpret their designs with the following questions:

  • How many shapes did you discover within your Dream Catcher weaving?
  • What polygons in your design fit together to create a new polygon shape?
  • As you add more binder clips to the hoop, and you attach string to all of them, what do you notice about the number of polygons inside the hoop?  (Answer:  the inner area of the hoop fills up with more polygons)
  • Explain how knowing the area of polygons within the hoop would help you to find the total inside area of the circular hoop?
  • What do you estimate to be the inner area of your hoop?
  • What changes could you have made to your design and why?
  • How does your design incorporate recycled materials and what did you learn about Dream Catchers?

Proficiency:  Choose form(s) of assessment that illustrate student understanding and proficiency in the following:

  • Students demonstrate (via written, oral, or other report) their method for finding the area of the circular hoop, based on the collective areas of the polygons inside it.  Students describe each polygon and their areas, and whether those polygons were constructed and/or deconstructed from other shapes.
  • Based on what they learned, student groups discuss, estimate, and then share out ideas on how they might find the surface area of a curved shape, such as a soccer ball made of polygons.
  • Student groups present their findings to their peers, and/or a parent audience.

Sample Rubrics:

assessment-rubric

 

snip-assessment-rubric

Define –In order to create design solutions, students need time to investigate various approaches and to hear differing viewpoints from one another with a series of investigations. In this lesson phase, students investigate attributes of regular polygons that tessellate together to tile a floor, focusing on symmetries and angular measurements, and take this information to help further define concepts useful for the final Design Challenge project.

Unit Inquiry Question:

How could you create a floor out of polygon tiles?

Lesson Overview:  Students investigate which regular polygon “tiles” could be placed together to create a floor design. Tiles on a tessellated floor cannot overlap and cannot have spaces between them. The Define phase in this lesson consists of exploring polygonal placement with the following investigations; symmetry and angular measurement of regular polygons; determining equivalent regular polygons that tile a floor; ascertaining combinations of different regular polygons that tile a floor; and finding areas of floors made of tessellating polygon “tiles”.

Professional Preparation: 

Arrange all materials in a common area for easy student access. Arrange polygon shapes into sets (each set consists of the following:  10 equilateral triangles, 10 squares, 10 pentagons, 10 hexagons, and 10 octagons; Each team will have one set).  Obtain one rectangular mirror for each team to use (note: if not readily available, then before the lesson make mirrors by taking the mylar from half of one hinged mirror to use as a rectangular mirror for each team. After examining reflective symmetry with the class, pass out additional mylar and tape and ask each team to assemble the hinged mirrors).

If needed, prepare to review how to read and use a protractor and compass with the students.

Suggestion: Begin the lesson with the entire class, but have students sit with team partners; plan to facilitate whole group discussions:

Make enough copies of each of the following handouts: Drawings #1,  Drawings #2, Hinged Mirror Kaleidoscope, and Interior Angles of Regular Polygons Chart.

Prepare or have student teams make hinged mirrors, 1 per team of students:

hinged-mylar-mirrorPlace the two 7.5 cm x 5 cm (3” x 2”) Mylar pieces side-by-side lengthwise, highly reflective sides down.  Leave a small gap between the strips -the gap should be about twice the thickness of the Mylar.  Attach the pieces together with tape lengthwise – the tape will act as a hinge.

Prepare methods of assessment.

Provide enough space in the room to form student teams at tables, or in small desk clusters (2 students per team).  Room arrangement should allow observation and safety in getting around to students.

After students enter the room, assign student teams or allow students to pick team members. Have student teams sit together, pass out mylar mirrors and handouts Drawings #1 and #2.

Procedure :

Define phase— Explore Symmetry — group investigation

Reflective Symmetry: Look at the Drawings #1 handout

  • What do these two pictures have in common?  (Answer: they can be cut in half by a vertical line, making two identical halves).
  • How could you illustrate half of each picture?
    • Idea: Draw a vertical line on the butterfly to divides it into two matching halves.
    • Place the mylar mirrors upright and lengthwise along the line. Ask students to look in the mylar mirrors to notice the reflection.
    • Record this type of reflection as “vertical symmetry” on the top of your Drawings #1 handout.
    • Repeat this activity with the face on the Drawings #1 handout.
    • Notice: the features on one side of each drawing have matching features on the other side. Tell them the lines they have drawn are called lines of reflection, or mirror lines.

Remember: it is important for students to understand there can be more than one line of reflection and the orientation is not always vertical!

Look at the Drawings #2 handout  Discuss horizontal and Vertical symmetry. Allow them time to discover horizontal (reflectional) symmetry on their handout, using rulers and mylar mirrors.

  • Ask: How many lines of reflection can you find?  Add all the lines of reflection onto the pictures and then verify your findings with the mirror.(answers:  A, B, C, E , F, and H all have vertical (reflectional) symmetry.  D, E, F, and H all have horizontal (reflectional) symmetry. This shows that some pictures have both vertical and horizontal (reflectional) symmetry. G does not have any reflectional symmetry).

Rotational Symmetry:

A figure has rotational symmetry if it appears the same after being turned less than 360 degrees (a full turn — the turn must be less than a full one otherwise all figures would have rotational symmetry!)

  • Ask students to look again at the Drawings #2 handout. Ask them to rotate the handout 180 degrees.  What do they notice?  (E and F look the same as they started).
  • The pictures that look exactly the same have rotational symmetry. Which ones did not have rotational symmetry?   (Answer:  In order for G to rotate and look the same, you have to rotate it 360 degrees, meaning it does not have rotational symmetry).

Define phase— Explore  Angular Measurement of Regular Polygons

Pass out (or have student teams assemble if not made yet) the hinged mirror Kaleidoscopes, protractors, and copies of the Hinged Mirror Kaleidoscope handout.

If needed, review how to read and use a protractor and compass with the students.

Discuss:  how many degrees are in a circle?  (answer: 360°).

Refer to the Hinged Mirror Kaleidoscope handout. Place the hinged mirror so that the edges rest against the black lines. Explore different shapes and images while varying the angle of the mirrors.

The mirror angle (also called the apex angle or central angle of a polygon) is the angle between the mirrors when the regular polygon is formed.

Use protractors and notice the angle between the mirrors to establish a relationship between the angle and the regular polygon that appears in the reflection (e.g., the mirror angle of an equilateral (or regular) triangle has a measure of 120°.   The mirror angle of a square is 90°. The mirror angle of a regular hexagon is 60°, and the mirror angle of a regular octagon is 45°.  Notice the smaller the mirror angle, the greater the number of equal sides in the polygon).

Ask: how do these measures relate to 360°? (Answer:  the number of degrees surrounding a point is always 360.  That means, for example 120° = 360°/3, 90° = 360°/4, 60° = 360°/6, 45° = 360°/8, and so on.

What is the mirror angle for any regular polygon having n sides? (Answer: the mirror angle of a regular polygon having n sides has a measure of 360°/n).

Ask: how could knowing this help you find the mirror angle of a regular pentagon? (Answer:  there are 5 equal sides in a regular pentagon, so the mirror angle = 360°/5 = 72°)

Pass out the Interior Angles of Regular Polygons Chart. Ask students to draw line segments for each polygon from the center of the regular polygon to each of its vertices. What do you notice? (Answer:  each polygon is divided into identical (congruent) isosceles triangles.  The central angle is the mirror angle of the polygon.

How can you use this information to find the base angles of each inner equilateral triangle? (Answer: the sum of the angles in a triangle is 180°.  Subtracting the mirror angle measure from 180 results in the measure of the total of both base angles. But both base angles are equal in measure, so half of that amount is the measure of each base angle in the inner triangles.  In the equilateral triangle below, the mirror angle is 120°.  180° – 120° = 60°.  So the base angles of each smaller triangle within the equilateral triangle are each 30°.equilateral-triangle-anglesHow can you find the angle measure for the equilateral triangle? (Answer: each angle measures 60°). 6o-degree-triangle

Find and record the interior angle measure for each regular polygon on the Interior Angles of Regular Polygons Chart. (e.g., the interior angle measure for a regular pentagon is:  180° – 72° = 108°; so, that means   ½ (108) = 54).  Hint: for n sided polygons, open a hinged mirror to the degree measurement for that polygon (e.g., 360°/n).  Lay the mirror on top of the polygon so its angle measurement is at the center of the polygon and the bottom edges of the mirror point to two consecutive vertices (showing the polygon’s reflection in the mirror). Mark the center on the polygon. mark-a-central-angleinterior-angles-of-pentagons

Define phase — Explore tiling a floor with equivalent regular polygons

Student Teamwork — Brainstorm, sketch, rapid prototype

One team member gathers needed materials and then returns to the team:

  • Adhesives
  • Regular Polygon shapes, 10 each: equilateral triangle, square, pentagon, hexagon, and octagon.
  • Scissors, pens or pencils

Student teams look for polygons that tile together by themselves (e.g., tile a floor with  all squares, or all equilateral triangles, or all hexagons).

What do you notice about the intersections at the corners of each tiling arrangement? (at each corner, the angular measurements add up to 360°)

Define phase— Explore tiling a floor with a combination of different regular polygons

Optional: Mention teams may tape together regular polygon shapes to create a tiling design, or choose to trace the outlines of each polygon onto paper to show the tiling pattern.

Using what they have learned, student teams create a sample “tile” floor out of a combination of regular polygons.

Students look for patterns and use repeated reasoning in designing a tiled floor with a combination of regular polygons.

Students determine how to estimate the area of the entire pattern based upon the areas of each polygon or parts of polygonal shapes.

Advanced research:  What are irregular polygons, and which irregular polygons can be used to tile the floor?  Link to:

http://www.raftbayarea.org/ideas/What%20are%20Penrose%20Tiles.pdf

Materials

  • RAFT Makerspace-in-a-box

-or –

Various adhesives and fasteners (staplers, tape, glue)

Precut regular polygon shapes, 10 each per team (all lengths the same size)

Scissors, pens or pencils

Rulers and Protractors

Rectangular mirrors, approx 4 in by 6 in, 1 per student team

Mylar 2-hinged Kaleidoscopes, 1 per student team

Handouts:

Drawings #1, 1 per student team

Drawings #2, 1 per student team

Hinged Mirror Kaleidoscope, 1 per student

Interior Angles of Regular Polygons Chart, 1 per student

Optional:  Paper, 11 in by 17 in., and/or calculators

Prior Knowledge

Properties of polygons especially angular measures of regular triangles and regular quadrilaterals & types of symmetries/angular measurements.

How to use protractors and compasses.

Core Concepts

Define and recognize regular polygons.

Symmetries and angular measurements of regular polygons.

Floors(planes) consisting of regular polygon tiles have no gaps & no overlaps.

Find combinations of different regular polygons that tessellate together.

Find the area of a floor that consists of polygon tiles.

Learning Targets

Analyze and identify traits of regular polygons, including lines of symmetry and angular measurements.

Apply these techniques to create a tessellated floor pattern made of regular polygons, so there are no overlaps and no spaces between them.

Explain how to find the area of a floor covered by a combination of regular polygons.

Assessment : Choose form(s) of assessment that illustrate student understanding and proficiency of the following:

  • Students demonstrate (via written, oral, or other report) their method for finding central angles in the regular polygons that comprise a tiled floor.
  • Students clearly identify traits of the regular polygons, including lines of symmetry and angular measurements, in the their tile flooring
  • Students created a floor pattern made of various regular polygons, so that no polygons overlapped and there are no spaces between them (e.g., together they formed a tessellation)
  • Students demonstrate how they would estimate the area of an entire floor based upon the areas of each polygon or parts of polygonal shapes in their floor design.
  • Students reflect on any changes they would have made to their design and why.
  • Students explain what inspired them to create the design.
  • Students explain why some regular polygon shapes were not used in their design.

Sample Rubrics:

assessment-rubric

 

snip-assessment-rubric

Define –In order to create solutions, students need time to investigate various approaches and to hear differing viewpoints from one another. In this lesson phase, students investigate attributes of polygons that fit together into making a curved surface that forms into a bowl to help further define the area of a curved surface that will be useful in the final Design Challenge project.

 

Unit Inquiry Question:

How can you create a bowl made out of polygons?

Lesson Overview:  This lesson focuses on students formulating and sharing ideas on how to create a curved shape by attaching different regular polygons together. Students will then create a bowl-shaped object having regular polygons for surfaces. To determine the surface area of their bowls, students will add the individual areas of each polygon within its design. The polygons will be made from recycled materials and will include a cost per polygon used. Given a budget, students will balance cost restrictions, and examine reasons why they chose certain polygons over others in their bowl design. They will also share other uses for bowl shapes and how it might be beneficial to make objects out of recycled materials.

Professional Preparation: 

Provide enough space in the room to form students into teams (3-4 students per team). Room arrangement should allow for visual supervision and safety in moving around to the student teams.

Plan to set up teams at tables, or in small desk clusters.

Make copies of The Polygon Bowl Data Sheet, 1 per student.the-super-polygon-bowl-data-sheet

Bring in sample bowls for each team to examine.

Arrange all materials in one area of the room for easy accessibility (suggestion: call this area the “End Zone” — making this a fun connection to “The Super Polygon Bowl”).

Organize polygon shapes in sets (each set consists of the following:  50 equilateral triangles, 50 squares, 50 pentagons, 50 hexagons, and 50 octagons). Put each set of shapes into its own container.

Prepare a container of tokens:  15 tokens per team.

After students enter the room, assign students to teams or allow students to pick own teams. Ask teams to choose names and record this information on The Super Polygon Bowl Contest Tally Sheet.

the-super-polygon-bowl-contest-tally-sheet

Allow teams 5 minutes to do a gallery walk to see all materials in the “End Zone”, and then ask them to prepare for a whole group discussion.

Suggestion: Have students sit with team partners.

Procedure: 

Define phase — Reflect on what you already know — Group Discussion:

  • Ask students to recall the lesson on Dream Catchers. Have them give examples of special quadrilaterals, triangles, and other polygons. How did you find the area of a shape made from other shapes?
  • Ask students to recall tiling a floor: Ask students to define and give examples of regular polygons.
  • Ask what would happen if you try to tile a floor using only regular pentagons?  (Answer: you get gaps between them) Why? (Answer:  the central point does not add to 360°):

pentagons-will-not-tessellate-by-selves

  • Ask: What would happen if you bring the edges of the regular pentagons on either side of the gap together? (Answer:  The pentagons will raise creating a bowl-like shape).

Hold up a bowl. Present the following scenario:

  • Your team is taking part in the great “Super Polygon Bowl” competition!
  • Your team is commissioned to create a new bowl made entirely out of a combination of different polygons!
  • Your team can use precut regular polygons in your design or decompose regular polygons into smaller polygons that result in fractional parts of the precut regular polygons .
  • Your team’s  job is to create the best bowl costing closest to but no more than $1 to produce.
  • Criteria:
    • Each team has 15 tokens. Each token represents 10 cents.
    • You purchase polygons only from me (the teacher) anytime until the contest is over.
    • Each unit polygon* has a value of 10 cents.  (*a unit polygon is the name for the original regular polygon shape before it is cut into parts).
    • Each team must keep track of the original number of regular polygons used if any are cut into parts.
    • Each student must fill out all information on The Polygon Bowl Data Sheet.
    • You may cut, fold, and/or create smaller polygons from given precut regular polygons, but if you do, you must figure out the cost of each smaller part based on its fraction of the original unit polygon cost.
    • The teacher will announce when it is time to stop (“touchdown”).
    • One member from each team will turn in a team The Super Polygon Bowl Data Sheet to the teacher.
    • To win:
      • the bowl must be made of different types of regular polygons (it is ok to have duplicate different types of regular polygons)
      • All the following must appear on The Super Polygon Data Sheet:
        • the names of each polygon on the final bowl surface
        • the areas of each polygon on the final bowl surface
        • the price of each polygon on the final bowl surface
        • the final total price of the polygon bowl.
    • The winner of the Super Polygon Bowl will be announced!
    • Each team will explain to the class how they determined which polygons to use and the surface area of their bowl.
    • Each team will also describe other uses for bowl shapes, and explain benefits for making objects out of recycled materials
  • Questions? (Remind students they can use all materials from the “End Zone”, not to hoard materials, etc.).

Student Teamwork — 

  • One student member from each team gathers needed materials and returns them to the team.
  • Student teams decide which polygons to use in making a bowl and rapidly prototype designs.
  • Student teams create bowl designs and test for criteria. Each student records information on the The Super Polygon Bowl Data Sheet.
  • Students show teacher results of their data sheet.
  • Teacher records each team’s required information on The Super Polygon Bowl Contest Tally Sheet form.
  • Teacher gathers class to announce winner(s) of the “Super Polygon Bowl”
  • Winning team members present how they determined the surface area of their bowl and what they learned about making a polygon bowl.
  • Class discusses what was learned, including other uses for a bowl shaped out of recycled materials (e.g., larger bowls could be made into inexpensive domed housing, which collapse into flat, easy to store/ship compartments when not in use)
  • What to notice: student engagement, collaborating, explaining, critiquing and sharing ideas, etc.

Further investigations: 

Have student teams make a polygon bowl out of a mixture of different polygons, costing the least with the largest overall area.

Have student teams make a polygon bowl out of a mixture of different polygons, costing the most with the smallest overall area.

 

Materials

  • RAFT Makerspace-in-a-box

-or –

Precut regular polygons made from recycled materials; 50 each of the following:

equilateral triangles, squares, pentagons, hexagons, & octagons

Same color bead tokens : 15 per team

Containers for precut polygon sets and for tokens

Various adhesives and fasteners (staplers, tape, glue)

Handouts:

The Super Polygon Bowl Data Sheet, 1 per student

The Super Polygon Bowl Contest Tally Sheet, 1 for teacher

Scissors, pens or pencils

Sample bowls

Optional: calculators

Prior Knowledge

Properties of polygons especially angular measures of regular triangles and regular quadrilaterals & types of symmetries/angular measurements.

Which polygons fit together without gaps or spaces to create a flat surface.

How to properly use a protractor and a compass.

 

 

Core Concepts

Symmetries and angular measurements of regular polygons.

Surfaces made of regular polygon tiles having no gaps & no overlaps.

Combinations of different regular polygons that tessellate.

Finding the surface area of a curved bowl that consists of polygons.

Learning Targets

Analyze polygons that can be composed into rectangles and other quadrilaterals, and those that can be decomposed into triangles and other shapes.

Determine the surface area of a curved bowl made from different polygons

Find the surface area of a bowl with a constraint on the cost of each polygon used in its design

Determine why it might be useful to make a cost-effective bowl out of recycled materials

Investigate other uses for domed shaped products made of recycled materials.

the-super-polygon-bowl-contest-tally-sheet

 

the-super-polygon-bowl-data-sheet

 

 

 

 

 

 

Assessment : You may use the results of The Super Polygon Bowl Contest Tally Sheet or choose other form(s) of assessment that illustrate student understanding and proficiency of the following:

  • Their polygon bowl, including shapes used, their areas, how they determined the surface area of the bowl, and which shapes were constructed and/or deconstructed from other shapes.
  • Further ideas that extend from what they learned into other re-purposed, useful items (e.g., create inexpensive precut shapes out of recycled materials for a “do-it-yourself” geodescent dome kit that could be made into temporary housing, a roof, or a playground climbing structure; sold in flat containers and shipped anywhere).
  • Student groups present their findings to their peers, and/or a parent audience.
  • Students reflect on any changes they would have made to their design and why.
  • Students explain what inspired them to create the design.
  • Students explain why some regular polygon shapes were not used in their design.

Some general rubrics that you might include:

snip-assessment-rubric

assessment-rubric

 

IdeateIn order to create a final design, students need to work together to brainstorm ideas that suits the needs of the problem. In this phase they decide on a model of a soccer ball to create, create a list of attribute requirements, and sketch ideas for a model made from recycled materials.

Prototype & Test – In order to choose the best solution for the Design Challenge project, students iterate building, testing, redesigning, and retesting the model until all requirements are met; resulting in a final model of a soccer ball made from recycled materials.

Design Challenge Question:

Can you make a soccer ball out of recycled materials?

Lesson Overview:

This lesson combines all the information that students have learned from each in the previous lessons into a new challenge of designing a nearly spherical-formed innovative type of soccer ball made up of different polygons.  They will estimate the area of the soccer balls, based on finding individual component polygonal areas as before, decide which attributes make for durable soccer ball, and investigate how using recycled materials is helpful in this final project. Students show proficiency at the end of this lesson demonstrating their findings to peers, other classes, and/or to a parent audience.

Professional Preparation: 

  • Provide enough space in the room to form student teams (3-4 students per team). Room arrangement should allow constant eye contact and safety in getting around to students.
  • Prepare to set up teams at tables, or in small desk clusters.
  • Suggestion: facilitate whole group discussions with circle or u-shaped configurations
  • Make copies of Design a Soccer Ball Data Sheet.

design-a-soccer-ball-data-sheet

  • Bring in sample soccer balls for each team to examine.
  • Arrange all materials in one area of the room for easy accessibility (suggestion: call this area the “Goal Line”)
  • After students enter the room, assign student teams or allow students to choose own teams. Allow students a 5 minute gallery walk to look at the all materials at the “Goal Line”, and then ask them to prepare for a whole group discussion.

Procedure: 

Define phase — Reflect on what you already know — Group Discussion:

  • Ask students to recall previous lessons: Dream Catchers, tiling the floor, and designing bowls.
  • Optional: Ask for ideas on how their bowl concepts might help in designing a ball?
  • Ask the following:
    • How did you find the area of the Dream Catcher hoop? (answer: by adding up the areas of the string polygons inside the hoop)
    • How did you find the area of a tiled floor pattern? (answer: by adding up the areas of each polygon within the floor design)
    • How did you find the surface area of a bowl made out of polygons? (answer: by adding up areas of each polygon on its surface)
  • Hold up a soccer ball. Ask:
    • What can you tell me about the shapes on the surface of this ball? (Answer: they are pentagons and hexagons)
    • How could you figure out the surface area of this soccer ball based on what you know about pentagons and hexagons? (Answer: you could find the surface area of the ball by adding up the areas of each pentagon and hexagon).
    • What else do you notice about this ball? (Answers vary.  For example:  each polygon’s surface is slightly curved to make the ball spherical;  each shape’s edge is same length;  it has a firm surface for kicking, each polygon edge fits next to another edge without gaps or overlaps; it is black and white; etc.)
  • Present the challenge  (NOTE: times for activities are suggestions):
    • Your team is commissioned to create a new soccer ball having a surface having different types of polygons (not just pentagons and hexagons). You can use any of the materials from the “End Zone” to design your soccer ball.
    • You choose how to create your polygons: e.g., by connecting straw lengths,  or by taping together precut polygon shapes.
    • List on your Design a Soccer Ball Data Sheet the following information: the name of each polygon you use; the number of sides in each polygon; the area of each polygon; the number of times each polygon appears on your soccer ball; your soccer ball’s estimated surface area; and any changes you made/would have made to your design and why.
    • You may cut, fold, and/or create smaller polygons from other premade polygons.
    • You will have 10 minutes to brainstorm ideas, including gathering ideas for a prototype.
    • After that time, you will have 50 minutes to build, design, apply concepts, and test your design.  (Remind not to hoard materials, look first, ideate, prototype, test, and then make a “super” bowl).

Student Teamwork — Ideate Phase

  • Student teams brainstorm ideas for ball designs. Teacher: walk among the teams and encourage evidence of students using the 4 C’s, engaged conversations, involvement, and cooperation.  Call time after 10 minutes are up.

Student Teamwork — Prototype/Test Phase

  • Student teams build a prototype soccer ball. They test and refine their design, retest, and so on until desired results are obtained.
  • Each student team records all necessary information on the Design a Soccer Ball Data Sheet. Each student must be ready to explain to the class the following:
    • How they composed and decomposed polygon shapes to generate new polygons, and how they found the areas of these shapes.
    • How they found the surface area of their soccer ball based on the areas of the polygons it is made from.
    • How their soccer ball holds up after kicking and tossing it around
    • Any changes they could have made to their design.
    • How their design incorporates recycled materials and how that might be beneficial.
  • What to notice: student engagement, collaborating, explaining, critiquing and sharing ideas, etc.

Proficiency

  • Students write a paragraph describing their soccer ball, including shapes used, their areas, how they determined the surface area of the ball, and which shapes were constructed and/or deconstructed from other shapes.
  • Students explore and report on further ideas that extend from what they learned into other re-purposed, useful items.
  • Student groups present their findings to their peers, and/or a parent audience.

 

 

 

Materials

  • RAFT Makerspace-in-a-box

-or –

Precut regular polygons made from recycled materials; 50 each of the following:

equilateral triangles, squares, pentagons, hexagons, & octagons

Handout:

Design a Soccer Ball Data Sheet, 1 per student

Various adhesives (staplers, tape, glue)

Scissors, pens or pencils

Examples of actual soccer balls

Straws, connectors, etc.

Plastic bags

Shredded paper clippings

Fabric swatches

Optional: calculators

Prior Knowledge

Students demonstrate understanding of composed and decomposed polygon shapes to generate new polygons, and how to find the areas of these shapes in flat and curved surfaces

 

 

Core Concepts

Analyze the surface of a soccer ball made of pentagons and hexagons

Prototype, test, and design a durable soccer ball containing different types of polygons and/or items that can be assembled into polygon shapes, and is made out of recycled materials.

Determine why it might be useful to make a items out of  recycled materials

Learning Targets

Discover polygons that can be composed into rectangles and other quadrilaterals

Analyze polygons that can be decomposed into triangles and other shapes.

Analyze and find the surface area of a real soccer ball made of pentagons and hexagons.

Analyze and find the surface area of a soccer ball made of different types of polygons

Determine why it might be useful to make a soccer ball out of recycled materials

Assessment : …choose other form(s) of assessment that illustrate student understanding and proficiency of the following:

Ask each team to share their soccer ball with the class and to answer the following:

  • What inspired them to design it that way?
  • Did they compose and decompose polygon shapes to generate new polygons?  If so, to explain what they did and how they found the areas of these shapes.
  • How they determined the surface area of their soccer ball.
  • How durable is their soccer ball after kicking and tossing it around
  • Explain any limitations to their design and what they would do to correct that if given other choices.
  • How their design incorporated recycled materials and how that might be beneficial.
  • Compare and contrast their designs with an actual soccer ball. How are the two alike and how are they different?

 

A Review of Polygons — properties and areas:

Define polygons: The word “polygon” derives from the Greek poly, meaning “many,” and gonia, meaning “angle.”  Review, find examples or draw pictures of polygons, and then discuss with students (and record explanations)  the following:

  • What is a polygon? (answer: it is a flat, 2- dimensional closed geometric shape bounded by 3 or more straight line segments (called sides), and an equal number of points (called vertices) — note: a circle is not a polygon because it does not have straight sides).
  • A vertex (plural “vertices”) is a corner. A side joins one vertex with another.
  • An angle is the amount of turn between two straight lines that have a common end point (the vertex).
  • convex-vs-concaveA convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.
  • If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a “cave” in it)
  • For two-dimensional figures, any side can be a base. Typically, however, the bottom side, on which the polygon ‘sits,’ is called the base.
  • Equilateral means all sides are equal in length. Equiangular means all angles are equal in measurement.

A regular polygon has all angles equal and all sides equal, otherwise it is irregular.Polygons are named for their number of sides and angles:

  • 3: Triangle
  • 4: Quadrilateral
  • 5: Pentagon
  • 6: Hexagon
  • 7: Heptagon
  • 8: Octagon
  • 9: Nonagon
  • 10: Decagon
  • 11: Undecagon
  • 12: Dodecagon
  • n: n-gon

Types of triangles: (a polygon with 3 sides)…. See examples below (remember some triangles belong to more than one category of triangle):

types-of-triangles

 

 

 

 

 

 

 

 

 

 

 

 

The sum of the interior angles in a triangle always equals 180 degrees.

How can you show the sum of the interior angles in any triangle is 180°?  See example below:

sum-of-triangle-anglesTypes of quadrilaterals (a polygon with four sides) …. See examples below (remember some quadrilaterals belong to more than one category of quadrilaterals):

types-of-quads

 

 

 

 

 

 

 

 

 

 

 

 

Interior angles of quadrilaterals equal 360 degrees:

How can you show the sum of the interior angles in any quadrilateral is 360°?  See folding example below:

 Color or label the vertices of a quadrilateral.

starting-colored-quads

Cut off the corners, and then fit the vertices together showing they all fit around a point.  The sum of the interior angles in any quadrilateral is 360°.

colors-of-quad-fold-in

Interior Angles of Regular Pentagons:

To find the central angle use a hinged mirror:

mark-a-central-angle

The central angle of a regular pentagon is found also by noticing it can be split into five regular triangles that meet up in the center whose angles add to 360 degrees.

That means one central angle is:    360°/5 = 72°

interior-angles-of-pentagons

Polygons from other polygons:

  • How can you make one type of triangle out of another type? (pass out various paper triangles, fold them, check) — e.g., fold an isosceles triangle in half to make two smaller right triangles.
  • How can you make one type of quadrilateral out of another type? (Pass out paper quadrilaterals, fold them, check) — e.g., see illustrations below:

two-polgons-from-one

  • How can you make a triangle from a quadrilateral? Vice versa? (e.g., experiment with folding different polygons to make them into other polygons).  Here is an illustration:

Quadrilateral into a triangle(s):

  • Cut a rectangle along diagonal line to form 2 triangles.

quad-to-triangles

 

 

 

Triangles into a quadrilateral:

  • Fold an equilateral triangle as shown below to create a trapezoid (& another triangle):

triangle-into-2-polygons

  • If you have one shape, such as a pentagon, could you fold it to make other polygons? (Pass out a variety of different paper shapes, fold them, check).
  • Can you make one larger polygon from smaller polygon shapes?

(Pass out paper shapes, fold them, check).

Define Area:

  • The area of a polygon is the number of square units inside that polygon.
  • The area of a triangle is ½ (base)(height). This is determined from the fact that a parallelogram can be divided into 2 triangles.  The area of a parallelogram is (base)(height).  For instance, in the parallelogram below, the area of each triangle equals ½ the area of the parallelogram. The base and height of a triangle must be perpendicular to each other.

 

area-of-a-parallelogram

  • How could you find the area of a regular pentagon?

 (answer: one way is to deconstruct it into triangles, find the areas of those, and then add their areas to reconstruct the original larger shape).  For example let’s find the area of a regular pentagon, having each side 6 units in length and an apothem of approximately 4 units (the apothem is the line from the center of the pentagon to a side, intersecting the side at a 90º right angle) —see illustration below where the pentagon is split into 5 identical equilateral triangles, and then into 10 equivalent right triangles:

area-of-reg-pentagon

  • How could you find the area of a regular hexagon?

(answer: again, one way is to deconstruct it into triangles, find the areas of those, and then add their areas to reconstruct the original larger shape).  For example:

area-of-reg-hexagon

Use trigonometry to find the value of the apothem in a regular pentagon and regular hexagon, for instance, when the side value is known (not required to teach at this grade level, but just noted as a way to find the approximate apothem value).

  • What is meant by surface area? (answer: The surface area of a solid object is a measure of the total area that the surface of an object occupies). The surface area for polyhedra (i.e., objects with flat polygonal faces), is the sum of the areas of its faces.  The surface area of curved surfaces involves more than just adding up areas of flat surfaces.
  • Ask students to contemplate: How do soccer balls roll when their surface area is not smooth: it is made up of hexagons and pentagons?  How might you estimate the surface area of a soccer ball? What strategies could you use?

 

 

 

Materials

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