Unit 2 - Solid Figures (10 sessions)

Unit Objectives Students will be able to: ● Associate two-dimensional drawings with three-dimensional shapes ● Visualize pyramids, prisms, and cylinders from two-dimensional drawings ● Identify nets of cubes, cuboids, prisms, and pyramids. ● Identify the solid represented by a net. ● Determine whether a figure can be the net of a given solid. ● Determine whether a solid can be formed from a given net.

Unit Standards M6.C.1: Analyze characteristics and properties of two- and three- dimensional geometric shapes and demonstrate understanding of geometric relationships. M6.C.1.1: Define and/or use basic properties of triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons and circles. Session 1: Drawing Solid Figures Lesson Objectives Students will be able to: ● Associate two-dimensional drawings with three-dimensional solids. ● Determine the number and shapes of the faces of a two-dimensional drawing of a solid.

Materials ● Textbook p. 15-16, tasks 1-3 ● Workbook Ex. 4 ● Different 3D Shaped sponges and wooden solid figures

Technology: ● BrainPop: Polyhedrons.

Anticipatory Set: Brainpop video. ● Watch the video twice.

○ The first time students do not take notes. ○ In the second viewing, students will take notes and answer fill out the vocabulary sheet that goes

along with the video. ● Complete the review quiz (as a class) that follows the video ● Go over the vocabulary sheet (informal assessment).

○ Make sure everyone has the correct definitions down.

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Lesson: 1. Examine three-dimensional models of the following solid figures:

1.1. cubes 1.2. cuboids 1.3. prism 1.4. pyramids 1.5. cylinders 1.6. sphere ● Provide students with three-dimensional models of cubes, cuboids, prisms, pyramids, spheres,

and cylinders. Students take out notes and label their notes Solid Figures

● Define a three-dimensional figure as a polyhedron. ● Define a polygon

Hands-on visual for students

● Have students examine the faces, edges, and curved surfaces of the models. ● pass around solid figures and have students count the edges and faces and vertices ● Using the cylinder, demonstrate what a face is.

○ A face is a flat surface. ○ Try balancing the cube on the side of the cylinder. ○ Allow the cube to fall off. ○ This will give the visual effect that the cylinder’s side is not a face because it is not flat.

● Demonstrate using a sphere as well to show that the sphere does not have a flat surface, and therefore does not have a face.

In student notes: Draw a Venn Diagram on the board.

● As a class, compare and contrast polygons and polyhedrons by creating a Venn Diagram ● Use real life examples as references

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Session 2: Real-life Examples of Solid Figures Lesson Objectives Students will be able to: ● Associate two-dimensional drawings with three-dimensional solids. ● Determine the number and shapes of the faces of a two-dimensional drawing of a solid.

Anticipatory Set

● Show real life examples of solid figures through the overhead projector ● Have students practice drawing different solid figures in their notes ● Ask students about the difference between pyramid and prism by looking at real-life examples. ● Students will count the number of faces and name the shapes of the faces. ● Make sure the students know the following: (Add to vocab sheet)

○ A face is a flat surface ○ An edge is a line formed by two faces ○ A vertex is the point where two or more edges meet. ○ Similarly, discuss faces and edges or curved surfaces of objects (e.g. tin cans, boxes,

prisms for breaking up light waves) or pictures of objects. ○ Discuss what it means to be regular and irregular for a polygon ○ Discuss the difference between a polygon and a polyhedron and show real-life examles ○ Discuss how a cube and a cuboid are alike and how they are different.

● Is a cube always a cuboid? yes ● Is a cuboid always a cube? no

2. Relate 3-dimensional models to 2-dimensional drawings. ● Discuss page 15 in the textbook.

○ Discuss the number of faces, vertices, edges, and shapes of the faces. ○ Get students to tell you which of these figures are prisms and which figures are pyramids.

● Provide students with the 3-dimensional models of prisms and pyramids and have students try drawing a 2-dimensional representation, using the figures on p.15 to guide them. ○ For task 3, guide students to see that A, B, and C are prisms, but D is not. ○ Point out that prisms have two bases and pyramids have only one base. ○ Define a base in their notes

● Is a base always a face? yes! ● Is a face always a base? No!

Exit Slip: In your own words, describe what is a polyhedron. Workbook Exercise 4, Task #1-2 pp 12-13 Students with match the figures with the solids. Students will have to count the number of faces for 4 different solid figures.

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Session 3: Drawing Solid Figures Continued… Objectives: Students will be able to... ● Visualize prisms, pyramids, and cylinders from two-dimensional drawings. ● Identify prisms, pyramids, and cylinders from three-dimensional drawings. ● Identify vertices, edges, and faces of solid figures. ● Compare and contrast Irregular and Regular Polyhedrons ● Understand the difference between polygons and polyhedrons ● Understand the difference between cuboid and cube

Anticipatory Set: Read the answers from the exit slip “What is a polyhedron?” Ask the students to point out real life examples of polyhedrons in the classroom and name the solid figure shape if they can. Procedure: Take notes on vocabulary while reviewing terms from previous day. ● Prisms have two faces opposite each other (bases) that are the same shape and size.

○ We name prisms by the shape of its base ● A pyramid has only one base, which may or may not be a triangle.

○ We name pyramids by the base (rectangular or triangular) ● The faces of prisms and pyramids are polygons (2D figure) ● A polygon is a plane figure with at least three straight sides and angles.

○ They typically have five or more straight sides. ● The Apex is the highest point in a 2D or 3D figure.

○ Highest point relative to the base line or bottom polygon base. ● Relate 3-dimensional models to 2-dimensional drawing

Guided and Independent Work: Fill in a chart on solid figures:

1. Model 1-2 shapes. 2. Model how to count each f, v, e, answer the yes/no questions ("classifying" figures - prisms, regular

polyhedrons, etc) and 3d drawing strategies. 3. For the remaining solid figures, students will work with hands on solid figures.

a. Students will work in groups of 2-3 students b. Students will move around to different stations to work with different solid figure shapes

4. Tell students to study their notes and vocabulary for a quiz tomorrow.

5. Exit Slip:

In terms of a polyhedron, what is a base? A face? An edge?

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Session 4: Quiz on Solid Figures Anticipatory Set Index cards : review base, face, and edge Note-Taking: Create a Venn Diagram using the terms base and face. Review Activity: Minds in Motion to Review for quiz: Apex, Vertice, Face, Edge, Base, Polyhedron.

Apex yoga tree Highest point

Vertice turkey vulture created by edges can be an apex

Face peace can be a base a polygon

Edge time out formed by two faces or bases

Base owl eyes is always a face gives the name of a prism or pyramid

Polyhedron antlers a 3D figure made up of polygons

Ask any questions about Solid Figures or vocabulary before taking the quiz. Take Quiz (15 minutes) Accommodations: Shae was able to read out loud the questions and re-word the questions. Shae was allowed to work with Miss Paula and I on the side after class using 3D solid figures. On the back if you have extra time draw some solid figures. Show off your 3D skills. We will share some of your drawings tomorrow morning. Group work Class finishes their chart on shapes. If you finish early you can identify solid figures in the room and draw the figures on the back sheet of your chart. Go over the chart as a class. Use 3D solid figures to help with any solid figures that cause confusion.

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Session 5: Quiz corrections and Riddles Objectives: Students will be able to identify solid figures based off of clues in riddles Students will be able to create their own riddle about solid figures Anticipatory Set Part 1: Students play I-Spy using the shapes they drew on the back of their quiz. Students in the class guess their object. Anticipatory Set Part 2: Give two riddles and have students guess a solid figure Students may use their chart to solve the riddle Lesson Give directions on how to create a riddle:

1. choose your shape 2. Write 3 or 4 clues 3. Each clue should eliminate shape(s). 4. 1st clue should be general and broad

Independent work: Hand out index cards. Students write their final draft of their riddle on the lined side of the index card. Students write their shape on the blank side of the index card. If students finish early they may write a second riddle. Once everyone is finished with their riddles, have students share with their partners and have partners guess their shape. Students are encouraged to utilize their charts to help come up with their riddle and guess their partner’s shape. Technology: Khan Academy: Assign math students to a coach(teacher) on Khan Academy. Review expectations with students about Khan Academy. Introduce the concept of watching a video and taking notes on a video. Informal assessment: Collect riddles from the students

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Session 6: Introduction to Nets of Solid Figures Objectives: ● Students will be able to form solids from nets. ● Students will be able to identify shapes based off of hints in math riddles

Anticipatory Set: Read some of the riddles students created from the day before and have students solve the riddles Materials: Cut outs of nets for hexahedrons (A-N) Blank pieces of paper Procedures: ● Hold up a paper net and attach it to the whiteboard for all to see.

○ Ask students how many faces are on the net. 6… ○ Ask students what are the shapes or polygons of the faces. squares… ○ Ask students if all the faces are the same? Yes.. ○ Ask students if I folded up the figure, what solid figure would I have? Hexahedron, cube, or

cuboid. ● Have students refer to their charts. ● Question 1: What is a hexahedron? A solid figure with six faces. ● Question 2: What other name can we give a solid figure with six faces? A cube and cuboid. ● Question 3: How do you know this 2D figure produces a hexahedron?

○ Hexahedrons have six faces which this figure does. There are 6 faces or bases. Hands on Demonstration: Ask two volunteers. One student receives a net that can be made into a cube while another student receives a net that will not work for a cube. Have the students attempt to fold up the net and see if the net works for a hexahedron. Display both of the nets on the board. Do not mention the vocabulary word “net.” Have students work with nets and see which ones do and do not work. The next lesson introduce the vocabulary term and strategies for nets. Point out on the two figures the different types of panels. There can be three panels across, or four, or even as low as two. Tell students that they will be working with different figures and folding them figures to find out which ones work and which ones do not. Hand out pieces of computer paper and have students divide up the paper into eight equal sections by folding the paper. I will model how to fold the paper step by step.

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Label each section in alphabetical order on the front and back of the paper. Model on the board labeling A-N. There will be two extra spaces on the back of the sheet. Tell students what you expect them to do at each station. Model drawing the figures and checking yes, the figure (net), works or “no” for no the figure does not work. Show how to draw the models on the board by first drawing the basic shape of the figure and then filling in the lines for all the squares (faces). Students will clear off their desks for the stations. Place different nets around the room. Draw names from straws to determine partners. Allow students to work at about seven stations giving about two minutes per station. Show a diagram of the how you want the class to rotate around the room. Front row moves to the left, middle row moves the the right, and back row moves to the left in a snake-like shape. Pairs move about every two minutes. Make sure both partners are trying on their own to manipulate the faces into cubes and checking yes or no if they work. Homework Assessment: For the last 10-15 minutes of class, students will go onto their chromebooks (laptops). We will guide them through how to log onto Khan Academy using their email address. Watch part of a video on Nets of polyhedra (video). Have students set up their notes by giving a title of Nets of polyhedra and the date. Remind students to watch a video once without taking notes. Independent work online: Nets of 3D figures (exercise) They will watch a video all about Nets for homework. They must answer at least 5 answers in a row correctly. The website will allow the teacher to know if the students watched the video and if they answered at least 5 answers in a row correctly.

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Session 7: Net Strategies for a Hexahedron Objectives: Students will be able to identify nets of cubes. Students will identify strategies for nets of cubes. Students will be able to define the term net in their own words. Anticipatory Set Hold up a paper net and ask what we call this 2D figure. A net THINK - PAIR- SHARE - Students talk to their classmates about the video they watched and come up with their own definition for a net. Regroup: Come up with a formal definition for a net and put in their notes. Lesson: Net - a two-dimensional figure that will create a 3D solid when folded on the solid lines. Hand out cut out nets of a rectangular pyramid and a rectangular prism. Model steps for labeling and glueing into their interactive notebook. Students will label the base by the name of the polygon, label the faces and the number of faces. Students will only glue the base of the net to their notebook. Once they glue the nets into their notebook, students will label the nets by the solid figures they create. Net Strategies: 2D model of a solid with its faces and bases arranged in a panel like manner. Some nets do not work according to the number of faces there are or how the faces are arranged. I like to think of it has a 3D puzzle. Some of the pieces are missing and sometimes the faces overlap when a net does not work. When all the pieces fit, it is a true net. Consider the following when finding a net that works: Edges: the number of free edges are key to look at when figuring out if a net works. Panels: The faces and bases lined up together.

Questions to ask yourself when considering panels: What can the longest panel be? What can the shortest panel be?

Explore nets of cubes and strategies for identifying nets that work for Hexahedrons Go through all the different nets that students worked with the day before. Use the overhead projector (ELMO) to show the different nets and then identify strategies and rules for different panels

***Point out that we are going to be systematic on how we go about this - meaning we are going to sort through all the nets by their similar characteristics and check for certain elements in all the nets (free edges, number of squares in panels, number of faces, shapes of faces)

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Guide and Model students in listing some general rules for determining if a net of 6 squares could form a cube.

1. Ask the students who had all the nets with four panels. (start with four panel strategies) 2. Project the 4-paneled nets on the ELMO. Have students copy them down. Ask the groups if the panels

worked or did not work. Then model the first strategy for the 4-paneled nets. See what the ones that worked had it common and the ones that did not work have in common. Use the words “edges” and “panels.”

3. Move onto the groups who had the nets with panels as long as three faces across. Ask them which nets worked and have students copy the panels. 3.1. Have Students work in pairs trying to figure out patterns for strategies with three panel nets.

3.1.1. Go over the strategies and add to the list 3. Then work with panels that have only two across. Students work in pairs to figure out strategies and then report back to the teacher to add onto the list. Assessment Workbook exercise 5 problem 1 - circle correct nets for cubes. Workbook exercise 5 problem 2 - circle correct nets for cuboids Teacher Task - Sort out the good and the bad strategies that students came up with and compose a final list of strategies for students to have. Students will be either given the list on Friday or Monday when they can finish up the workbook exercises. Strategies: ● The longest panel cannot have more than 4 squares. ● If the longest panel has 4 squares, the remaining 2 have to be placed on opposite sides. ● If the longest panel has 3 squares, the remaining 3 can be placed on either side but there must always be

at least 2 free edges on either side of the longest panel. ● If the longest panel has 2 squares, the remaining 4 must be evenly distributed on either side such that

there will be one free edge on either side of the panel. Exit Slip: What is a net? What is the longest panel for a net of a hexahedron? What is the shortest panel for a net of a hexahedron?

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Session 8: Nets Continued and Euler’s Law Objectives: Students will be able to identify nets of cuboids, pyramids, and prisms. Students will be able to identify the solid represented by a net Students will test out the rules and strategies for nets of solid figures Students will use different solid figures to prove Euler’s Law Anticipatory Set: Display a solid figure and ask the name of the figure, the number of bases, the number of faces, and the number of vertices. Ask the students to decide on which net of the solid works for the solid figure on page 19 of the textbook. Have students explain why the two nets work and why the other two nets do not work. Page 20: Display a net and ask which solid matches the net. Have students explain why or why not they the solids match the net. Handout the strategies worksheet that they students came up with from the day before. Classwork Workbook Exercises #’s 5-7 Exercise 5. Students test out the strategies they came up with to select nets that work and do not work. Exercise 6. Students are given a solid figure and must figure out which nets work for the solid figure. Exercise 7. Students are given a net and must figure out which solid figure matches the net. Closure: Euler’s Law for Proving True Polyhedrons Use solid figure examples to test the Euler’s Law to see if it actually works! Have students write down Euler’s Law in their notes. Students use their charts to plug in the numbers for a solid figure to prove the Law.

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Session 9: Review Day for Test Anticipatory Set: Minds in Motion Review the workbook answers using the ELMO. Have students come up and display their answers. Work with students who need extra help. Students who do not need help can actively study. Actively Study includes the following: -create note cards -go around their house finding real-life solid figures -Go over vocabulary -create nets -study net strategies -actively study their chart Extra time: Students may work on making Buckyballs, Tetrahedrons, Hexahedrons, and Icosahedrons for Extra Credit. Minds in Motion for Solid Figures

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Session 10: Test on Solid Figures Provide hands-on solid figures. Students may take only one solid figure at a time without the label. -Be able to apply Euler’s Law (Formula is given).

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Unit 2 - Solid Figures (10 sessions)

Unit Objectives Students will be able to: ● Associate two-dimensional drawings with three-dimensional shapes ● Visualize pyramids, prisms, and cylinders from two-dimensional drawings ● Identify nets of cubes, cuboids, prisms, and pyramids. ● Identify the solid represented by a net. ● Determine whether a figure can be the net of a given solid. ● Determine whether a solid can be formed from a given net.

Unit Standards M6.C.1: Analyze characteristics and properties of two- and three- dimensional geometric shapes and demonstrate understanding of geometric relationships. M6.C.1.1: Define and/or use basic properties of triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons and circles. Session 1: Drawing Solid Figures Lesson Objectives Students will be able to: ● Associate two-dimensional drawings with three-dimensional solids. ● Determine the number and shapes of the faces of a two-dimensional drawing of a solid.

Materials ● Textbook p. 15-16, tasks 1-3 ● Workbook Ex. 4 ● Different 3D Shaped sponges and wooden solid figures

Technology: ● BrainPop: Polyhedrons.

Anticipatory Set: Brainpop video. ● Watch the video twice.

○ The first time students do not take notes. ○ In the second viewing, students will take notes and answer fill out the vocabulary sheet that goes

along with the video. ● Complete the review quiz (as a class) that follows the video ● Go over the vocabulary sheet (informal assessment).

○ Make sure everyone has the correct definitions down.

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