and Transitional Comprehensive Curriculum. Grade 6 Mathematics Unit 6: Geometry, Perimeter, Area, and Measurement

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1 Grade 6 Mathematics Unit 6: Geometry, Perimeter, Area, and Measurement Time Frame: Approximately five weeks Unit Description This unit focuses on measurement and the application of perimeter and area concepts to irregular and regular polygons. Students are introduced to volume as a measure of filling and to surface area as a measure of wrapping. After exploring nets for measuring the surface area of rectangular prisms, students use their new knowledge to develop strategies to measure the surface area of triangular prisms and pyramids. Student Understandings Students work with applying the relationship between dimensions of figures and perimeters and areas for regular and irregular polygons. Students develop, understand, and apply the surface area and volume formulas for prisms and pyramids. Guiding Questions 1. Can students make appropriate estimates of area and perimeter? 2. Can students use various strategies to find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. 3. Can students apply what they know about finding the area of one geometric shape to help them find the area of other shapes? 4. Can students model and identify perfect squares up to 144? 5. Can students recognize that 3-D figures can be represented by nets? 6. Can students represent three-dimensional figures using nets made up of rectangles and triangles? 7. Can students combine the areas for rectangles and triangles in the net to find the surface area of a 3-dimensional figure? MATH: Grade 6 Grade-Level Expectations and Common Core State Standards Grade-Level Expectations GLE # GLE Text and Benchmarks Algebra 14. Model and identify perfect squares up to 144 (A-1-M) 16. Evaluate simple algebraic expressions using substitution (A-2-M) Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-1

2 Measurement 22. Estimate perimeter and area of any 2-dimensional figure (regular and irregular) using standard units (M-2-M) CCSS for Mathematical Content CCSS # Geometry 6.G.1 6.G.2 6.G.4 CCSS Text 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. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. ELA CCSS Reading Standards for Literacy in Science and Technical Subjects 6 12 RST Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 6 8 texts and topics. College and Career Readiness Anchor Standards for Writing 6.W.4 Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. Sample Activities Activity 1: Vocabulary Cards (CCSS: RST.6-8.4) Materials List: index cards, pencil To develop students knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to geometric properties, formulas, and measurement. Distribute 3 5 or 5 7 inch index cards to each student. Model the directions for creating the sample card. On the board, place a targeted word in the middle of the card, as in the example that follows. Ask the students to provide a definition. It is best if a word can be defined in the students own words, but make sure the definitions are correct and complete. Write the definition in the appropriate space. Next, have students list the characteristics or a description, give one or two examples, and illustrate the term. Have students create cards for the following Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-2

3 terms: perpendicular, angle, right angle, obtuse angle, acute angle, area, perimeter, edge, vertices, face, and base, surface area, and nets. Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards and in preparation for other class activities and quizzes. Definition An angle of less than 90 Characteristics less than 90 more than 0 Acute Angle Examples Illustration Activity 2: That s Perfect! (GLE: 14) Materials List: Square Tile BLM or square tiles, Grid Paper BLM, Perfect Squares BLM, pencil, scissors Distribute square tiles or the Square Tile BLM. If using the Square Tile BLM, have students cut out the tiles. Each group of 2 students needs about 150 tiles. Ask students what makes a polygon a square. It is important to emphasize that all sides must be equal, and all angles must be right angles to form a square. Have students work with partners to create as many different squares as possible and then draw them on the grid paper BLM. Have students label the dimensions of each square and the number of tiles it took to create the square and record the information on the Perfect Squares BLM. dimensions of the square number squared area 1 1 1² 1 After the students have created as many squares as possible, have each group share one of the squares they created and display it for the class to see. Discuss that when the dimensions are multiplied together the product equals the number of tiles needed to create the perfect square. This product is the area of the square. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-3

4 Is 44 a perfect square? To be a perfect square, there must be a whole number that times itself is = = 49 There is no perfect square between 36 and 49. So no, 44 is not a perfect square. What are some squares that would not be perfect squares? A square with side lengths of 2.5 units would be an example of a square that is not a perfect square. The side lengths must be whole numbers to be a perfect square. Activity 3: Room Measurement (GLEs: 22) Materials List: Grid Paper BLM, meter stick or calibrating wheel, various measuring tools, list of objects to measure, paper, pencil If a door is two meters high, ask students to estimate how many doors (if laid down end-to-end) would go across the room s width and how many would go across the length. Check accuracy with a meter stick or a calibrating wheel. Use the now known measures of length and width to show students the number of square meters that are in the room s floor. Display a grid with a scale drawing of the classroom floor. Demonstrate how to figure the perimeter of the floor. Give students various measuring tools and a list of objects to measure. It is important to include irregularly shaped objects to measure. Activity 4: Exploring Area of a Tangram Square (CCSS: 6.G.1) Materials List: Tangram BLM, scissors, pencil Distribute two Tangram BLMs to each pair of students. Have students cut apart one Tangram BLM. Present the following problem to the students. If the area of the small square tangram piece is 6 square units, what is the area of each tangram piece and what is the area of the original tangram square using all seven pieces? As students find the area of each tangram piece, have them trace the piece on the second copy of the BLM and label its area inside. When the students have found the area of each piece and the original square, discuss the areas as a class. 48 square units 12 square units 6 square units 3 square units 6 square units 6 square units Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-4

5 Make sure that students understand that the square can be composed of two of the small triangles, so it has an area of 3 square units + 3 square units or 6 square units. Continue to connect the areas of the other pieces to the areas of the small triangles. Have students create a trapezoid using the small square and one small triangle. What is the area of the trapezoid? 9 square units Have students add the other small triangle to create another trapezoid. What is the area of the new trapezoid? 12 square units Have students work with their partners to build additional trapezoids and find the area. Have groups share their trapezoids with the class. Discuss how to find the area of large or unusually shaped objects by decomposing them into small pieces in which the areas are easy to find. Have students use the tangram pieces to create unusually shaped polygons and find the area. Have groups share their polygons with the class. Present the following problem to the students: If the original square, made up of all seven tangram pieces, has an area of 80 square units, what is the area of each polygon? 80 square units 20 square units 10 square units 5 square units 10 square units 10 square units If the 2 large triangles make up ½ of the large square, the area of each has to be 20 sq. units. If 2 small triangles make up the square and the medium triangle, the area of the square has to be 10 sq. units and the area of the medium triangle has to be 10 sq. units. Activity 5: Decomposing Figures (CCSS: 6.G.1) Materials List: Triangle BLM, Area BLM, pencil Display the Triangle BLM Part A. Have students count the shaded squares to determine the area of the triangle. Students should clearly see, and be able to prove, that the shaded area is half of Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-5

6 the rectangle, A = ½ (b h). The area of the rectangle is 5 units 5 units or 25 square units and the area of the triangle is ½ of 25 square units or 12 ½ square units. altitude base Display Part B of the Triangle BLM. Tell students they are going to find the area of XYZ. Have them discuss how they might find this area. X Y Z X X The altitude is 6 units 6 units Y Z Y 8 units Z Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-6

7 Tell students that one way to find the area is the following: XYZ can be decomposed (separated) into two right triangles by drawing the altitude and determining the height. The triangles are congruent. Have students find the area of both triangles and add the areas together or draw a rectangle around the triangle, find the area and divide it by 2. Special quadrilaterals and polygons are more challenging and should be approached after students have good understanding of rectangles, squares and triangles. Distribute the Area BLM. Have students work with a partner to complete the problems and then discuss the answers as a class. Activity 6: Calculating Perimeter and Area of Triangles, Parallelograms, and Trapezoids (GLEs: 14, 16; CCSS: 6.G.1) Materials List: Quadrilaterals BLM, ileap Reference Sheet BLM, Area and Perimeter BLM, scissors, pencil Have students review how to calculate the area of a rectangle. (They may say that they could count the square units, count the number of squares in 1 row and multiply that number times the number of rows, or multiply the length times the width.) Provide students with the Quadrilateral BLM. Have students estimate the area of the parallelogram. (They can do this by counting squares and parts of squares.) Tell students that they are going to find the area of this parallelogram. Have them discuss how they might do this. Allow time for students to grapple with the problem. If they have done Activity 5 and the triangle part of Activity 5, they can probably come up with some good ideas. This is one way to find the area. Have students cut off the triangular part formed when a line segment is drawn from a corner of the parallelogram perpendicular to the opposite side. This cut is along the height of the parallelogram. Next, have them flip this triangular shape over and reattach it to the opposite end of the parallelogram to form a rectangle (see diagram below). Ask students to indicate how they could find the area of the newly formed rectangle, and thus the parallelogram. Have students estimate the area of the isosceles trapezoid. (They can do this by counting squares and parts of squares.) Tell students that they are going to find the area of this trapezoid. Have them discuss how they might do this. Allow time for students to discuss the problem. This is one way to find the area. After allowing students to share ideas, have them find the midpoints of the Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-7

8 legs, and draw a line segment connecting them. Have them cut along the mid segment, and then rotate the top around one of the midpoints to make the trapezoid into a parallelogram. Now you have a parallelogram. Have students use their previous knowledge of parallelograms to find the area. Lead a discussion about the formula for finding the areas of the parallelogram and the trapezoid. Discuss with the students that length width in a rectangle is the same as base height. Students should see that the height is the perpendicular distance between the base and the side opposite the base. To find the area of a trapezoid, multiply the sum of its bases (b 1 + b 2 ) by ½ of the height. Refer students to their work with the trapezoid earlier in the activity. Remind them that they cut the trapezoid into 2 pieces and formed a parallelogram. Each piece had a new height that was ½ of the old height. Have students cut a rectangle along one of its diagonals to form two right triangles. Lead a discussion about how they could determine the area of the resulting triangles. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-8

9 Parallelograms and trapezoids can be decomposed into a rectangles and triangles to find the area. 6 units 4 units 4 units 4 units Area of rectangle= 4 2 = 8 square units Area of square = 2 2 = 4 square units Area of Parallelogram = = 12 square units Parallelograms can also be decomposed into two triangles by drawing a diagonal. 6 units Divide the parallelogram into two congruent triangles. 6 units Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-9

10 Calculate the area of each triangle and then add them together. 6 units To find the area of a triangle, multiply the base times the height and divide by two. (6 2) 2 = 6 Each triangle has an area of 6 square units. Add the areas together to find the area of the parallelogram. 6 square units + 6 square units = 12 square units² Finish by asking questions about finding the perimeter of these shapes. Distribute the ileap Reference Sheet BLM and the Area and Perimeter BLM. Have students use the area and perimeter formulas on the ileap Reference Sheet to complete the problems. Discuss the solutions as a class. 6 units Activity 7: Estimating Area and Perimeter of 2-D Shapes (GLEs: 22) Materials List: 2-D Shapes BLM, Grid Paper BLM, string, pencil Distribute the 2-D Shapes BLM, have the students engage in discussion (view literacy strategy descriptions) using the Think Pair Square Share approach. This form of discussion helps improve student learning and remembering of a particular topic. Give students the problem of determining the lengths of segments which do not follow a grid line. Ask them to think alone for a short period of time, and pair up with someone to share their thoughts. Then have pairs of students get together with other pairs, forming small groups of four students and continue the discussion. Finally, have groups share what they discussed with the whole class. Some strategies student may suggest are using a piece of string, measuring with their pencil, cutting off a row of the grid to use as a measurement tool, etc. Have the students estimate the area and perimeter of each of the shapes. Have the students guess the shape that has the largest area. To explore irregular shapes, have students trace one of their hands onto the Grid Paper BLM and estimate the area of their hands by counting the squares and partial squares and record their estimates. Finally, distribute a piece of string to each group to use to help estimate the perimeter of their hand. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-10

11 Activity 8: Getting Ready for Nets: 3-D Figures (CCSS: 6.G.4) Materials List: three-dimensional objects, paper, pencil Have students observe three-dimensional objects and record their properties in a split-page notetaking (view literacy strategy descriptions) format. Model the approach by placing on the board or overhead an example of how to set up their paper for split-page notetaking similar to the example below. Explain the value of taking notes in this format by saying it logically organizes information and ideas; it helps separate big ideas from supporting details; it allows inductive and deductive prompting for rehearsing and remembering the information. Tell students to draw a vertical line approximately 2 to 3 inches from the left edge on a sheet of notepaper. They should try to split the page into one third and two thirds. Have students count the number of faces, edges, vertices and determine the shape of each face for each of the following three-dimensional figures: triangular pyramids, square pyramids, rectangular pyramids, triangular prisms, cubes, and rectangular prisms. After the students examine the three-dimensional figures and record their observations in the 2/3 column of the notes page, have students compare their notes with a partner. Show them how they can bend the sheet of notes so that information in the right or left columns is covered, and then use the uncovered information as prompts to recall the covered information. Continue to periodically model and guide students as they use split-page notetaking and increase their effectiveness with this technique. Assessments should include information that students recorded in their split-page notetaking. In this way, they will see the connection between the taking notes in this format and achievement on quizzes and tests. Example: Date: Period: Topic: Three-Dimensional Figures Rectangular Prism Triangular Prism --Number of faces: --Number of edges: --Number of vertices: --Shape of faces: --Number of faces: --Number of edges: --Number of vertices: --Shape of faces: Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-11

12 Activity 9: 3-D Construction (CCSS: 6.G.4) Materials List: toothpicks or straws, clay balls, miniature marshmallows or gumdrops, ruler, paper, pencil Once students show an understanding of the polygon faces, the edges and vertices of various polyhedra, have students create individual models. Students may work alone or in pairs to plan and implement the construction of the solid figures using toothpicks or straws, clay balls, and miniature marshmallows or gumdrops. Have students draw nets to represent the 3 dimensional figures created and use a ruler to measure the dimensions of each face. A net is a flat pattern that can be folded together to create a three dimensional solid. Label each net with the names of the shapes of each face and the dimensions of each figure. Figures to be constructed by the class should include but are not limited to triangular pyramids, square pyramids, rectangular pyramids, triangular prisms, cubes, and rectangular prisms. Activity 10: Cube It! (CCSS: 6.G.4 ) Materials List: Cube Net BLM, 1 Inch Grid Paper BLM, Net BLM, tape, scissors Cut out the Cube Net BLM and create the cube prior to the lesson. Display the cube to the students. Ask, How many vertices, edges, and faces does the cube have? What is the shape of each face? Introduce the term unit cube and explain that a unit cube is a cube with all edges that are 1 unit long. Remove the tape from the cube and show students the net. Explain that a net is a two-dimensional pattern that can be folded to create a three-dimensional figure. Discuss that when the net is folded to create the three-dimensional figure, none of the sides overlap. Distribute the 1 Inch Grid Paper BLM, and have the students work in groups of 2 or 3 to draw as many nets as possible that can be folded together to make a unit cube. Have the students cut the nets out and fold them together to see if they create a unit cube. Have students count the number of squares needed to make each net. Have groups share their nets. There are 11 nets that can fold into a cube. The nets are displayed on the Net BLM. Have the class compare the nets they made to the 11 possible nets. Discuss that each net is 6 square units. The cube has 6 faces and each one has an area of 1 unit. Activity 11: Wrap It Up! (CCSS: 6.G.1, 6.G.4) Materials List: cereal Box, 1 Inch Grid Paper BLM, Surface Area BLM, scissors Display a cereal box and a cube from the previous lesson. Ask the students to compare the two three-dimensional figures. What is the same and what is different? They both have the same Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-12

13 number of faces, edges, and vertices. The cube has square faces that are all congruent. The cereal box has different sized rectangular faces. Distribute the Grid Paper BLM and have the students draw two different nets for a rectangular box with dimensions of 1 unit 1 unit. Have students cut each net out and fold it into a box to see if the net works. Have students count the number of squares needed to make each net. Have students share their nets and the area of each net. Discuss that all of the nets made are Rectangular Prisms. A rectangular prism is a threedimensional figure that has 6 rectangular faces. A cube is a special type of rectangular prism in which all of the faces are squares. Tell students that they are going to use nets to find the surface area of some 3-dimensional figures. Surface area is the total area of the surface of a solid. It is the sum of the areas of all of the faces of a solid. Display the following three dimensional figure. Ask, what is the name of the three dimensional figure? Rectangular prism Discuss what the net would look like for the rectangular prism. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-13

14 Ask the following questions about the figure: 1. What is the length and width of rectangle A? 2 feet by 3 feet What is the area of rectangle A? 6 square feet 2. What other rectangle has the same area as A? F 3. What is the length and width of rectangle B? 2 feet by 6 feet What is the area of rectangle B? 12 square feet 4. What other rectangle has the same area as rectangle B? E 5. What is the length and width of rectangle C? 3 feet by 6 feet What is the area of rectangle C? 18 square feet 6. What other rectangle has the same area as rectangle C? D 7. What is the surface area of this solid? To find the surface area, add the areas of all 6 faces = 72 square feet Display the following three dimensional figure. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-14

15 Ask, What is the name of the three dimensional figure? Triangular prism Discuss what the net would look like for the triangular prism. Ask the following questions about the triangular prism: 1. What is the base and height of triangle A? 6 inches by 5.2 inches What is the area of triangle A? 15.6 square inches 2. What other triangle has the same area as A? E 3. What is the length and width of rectangle B? 8 inches by 6 inches What is the area of rectangle B? 48 square inches 4. What other rectangles have the same area as rectangle B? C and D 5. What is the surface area of this solid? To find the surface area, add the areas of all 5 faces = square inches Display the following three dimensional figure. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-15

16 Ask, What is the name of the three dimensional figure? Square Pyramid would look like for the square pyramid. Discuss what the net Ask the following questions about the square pyramid: 1. What is the base and height of triangle A? 6 feet by 8 feet What is the area of triangle A? 24 square feet 2. What other triangle has the same area as A? B, C, and D 3. What is the length and width of rectangle E? 8 feet by 8 feet What is the area of rectangle B? 64 square feet 4. What other rectangles have the same area as rectangle B? none 5. What is the surface area of this solid? To find the surface area, add the areas of all 5 faces = 160 square feet Distribute the Surface Area BLM. Have students work with a partner to complete the problems and then discuss the answers as a class. Activity 12: Picture It! (CCSS: 6.G.4, 6.W.4) Materials List: magazines, digital camera (optional) Assign small groups of students a 3-dimensional figure. Have groups find a picture or create a display of their assigned figure. This could be done in a variety of ways. Students could use a digital camera to take pictures and display them using a slide show, or students could create a collage of magazine pictures on a poster. Have students identify and explain the shapes and their properties, including such things as the name of the shapes and their faces, parallel and perpendicular lines, types of angles, and properties of the figures. Have students create a net to represent their 3 dimensional figure and calculate the surface area. Once students have acquired their information, ask them to demonstrate their understanding of the figure by completing Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-16

17 RAFT writing (view literacy strategy descriptions). This form of writing gives students the freedom to project themselves into unique roles and look at content from unique perspectives. From these roles and perspectives, RAFT writing is used to explain processes, describe a point of view, envision a potential job or assignment, or solve a problem. It s the kind of writing that when crafted appropriately should be creative and informative. For this activity, use the following RAFT: R - Role: Three dimensional figure A- Audience: The class F- Form: Poem, song T- Topic: Properties of three dimensional figures including the shapes used to create the net and surface area In their RAFTed song or poem, students must include the properties to describe their figure but not name the figure. The small groups will present their song or poem to the class and the class will identify the described figure. The group will then reveal its figure and share the picture or display of their figure Activity 13: What s my Volume? (CCSS: 6.G.1, 6.G.2, 6.G.4) Materials List: cubes, Exploring Volume BLM, Volume BLM, paper, pencil Distribute cubes to the students and have students build figure 1. If cubes are not available, distribute the Exploring Volume BLM. Figure 1: What are the dimensions of figure 1? How many cubes were used to create figure 1? 18 cubes If you add a second layer, how will the dimensions change? It will be 2 layers tall Figure 2: Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-17

18 What are the dimensions of figure 1? How many cubes were used to create figure 1? 36 cubes Predict what will happen if a third layer is added. The rectangular prism will be 3 layers tall and will be made using 54 cubes. Figure 3: Ask students if their prediction was correct. Have students create rectangular prisms with various dimensions and record the dimensions and volume of each figure. Students should begin to make a connection between the dimensions of the rectangular prisms and the formula used to find volume. The number of cubes it took to create each figure is the volume of the figure. In figure 1, the length of the figure was 6 units and the width was 3 units. The area of the base (6 3) is 18 square units. Figure 1 was one layer high, so the volume was 18 cubic units. In figure 2, the base remained 18 square units while the height increased to two layers. The base (18) times the height (2 layers) equals a volume of 36 cubic units. Students should begin to see that the base times the height equals the volume, V = b h. Discuss again with the students how the base was found. Length times width. Lead students to see that the b in the volume formula can be replaced with l w, V = l w h. Present the following problem to the students and have them work with a partner to solve. Jacob has several boxes to mail. The dimensions for each box are as follows: Box 1 2 inches long by 3 inches wide by 5 inches high Box 2 2½ inches long by 3 inches wide by 5 inches high Box 3 3 inches long by 3 inches long by 5 inches high Box 4 2½ inches long by 1½-inch wide by 5 inches high A. What is the volume of each box? Box 1 30 in³, Box in³ or 37 ½ in,, Box 3 45 in³, Box in³ or 18 ¾ in³ B. What is the total volume of all the boxes combined? in³ or 131 ¼ in³ Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-18

19 C. Determine the dimensions of the smallest possible box that Jacob could use to send all 4 boxes in one shipment Students could make models of the boxes to manipulate the boxes to help determine the smallest possible box or draw the boxes. Some students may realize if you add all of the length,s you get 10 inches long, the largest width is 3 inches and all of the heights are 5 inches. All of the boxes would fit in a box that is 10 inches long 3 inches wide 5 inches high. Draw a net for the box used to ship all of the boxes together. Distribute the Volume BLM. Have students work with partners to solve the problems. Discuss the answers as a class Activity 14: The Candy Box (CCSS: 6.G.1, 6.G.2, 6.G.4) Materials List: Centimeter Graph Paper or the Grid Paper BLM, cubes, scissors, tape SPAWN writing (view literacy strategy descriptions) prompts can be used to expand students understanding of dimensions. SPAWN is an acronym that stands for five categories of writing prompts. These prompts can be given to students before, during, and after a lesson to stimulate higher levels of thinking about the topic. The SPAWN prompts below are designed to cause students to reflect on what has been learned about dimensions and demonstrate applications of their new understandings. Present the following SPAWN prompts to the class and have them work in groups of 2 to complete each prompt. Allow students to use cubes to model the problem. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-19

20 S- Special Powers: The Candy Shop wants to repackage their Chewy Cubes candy. You have been selected by your company to design a new consumer friendly box to present to the Candy Shop. Create a net for your design and give the dimensions of the package. Answers will vary. Sample answer. P- Problem Solving: The dimensions of the Chewy Cubes are 2 cm 2 cm 2cm. How many Chewy cubes will fit in your new candy box? Answers will vary. 6 chewy cubes would fit in the sample box. A- Alternative Viewpoints: The Candy Shop is considering Going Green, Can your design be changed to an eco-friendly design that uses fewer materials but still holds the same number of Chewy Cubes? Answers will vary. If the dimensions of the design were changed to , it would still hold 6 chewy cubes but use less material. W- What If? : What if the Candy Shop changes the dimensions of the Chewy Cubes candy to 3 cm 3 cm 3 cm? How many Chewy Cubes would fit in your original design and the eco-friendly design? Should you create a new design to fit the new size of Chewy Cubes? Answers will vary. If the dimensions of the Chewy Cubes were changed, no Chewy Cubes would fit in the original design or the eco-friendly design. If the dimensions of the design were changed to , the package would hold 4 Chewy Cubes without any wasted space. N- Next: Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-20

21 Design a prototype of the container you think the Candy Shop should use. If the packaging material costs $0.005 per square centimeter, calculate the cost of your box. Write a report to the owner of the Candy Shop explaining which design you selected, why you selected it, and the cost to produce it. You are trying to persuade the Candy Shop that your design is the best and they should select your work. Your report should be neat, well organized and easy to read. Answers will vary. I selected the package. I selected this size because I think 6 Chewy Cubes is a good number to have in the package. The surface area of my package is 104 cm². To find the cost of the packaging I multiplied , so the cost per package is $0.52. Have groups share their designs with the class. Encourage students to ask questions and for clarification. Be sure to correct any misconceptions. As a class, compare the designs and decide which design the Candy Shop should select. Sample Assessments General Assessments Have students create portfolios containing samples of his/her work. Facilitate during small group discussions to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to promote problem solving might be: o What do you need to find out? o What information do you have? o What strategies are you going to use? Ask probing questions to help students learn to reason mathematically. Possible questions to ask may be o Is that true for all cases? Explain o Can you think of a counter example? o How would you prove that? Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-21

22 o What assumptions are you making? o How would you convince the rest of us that your answer makes sense? Have students create learning logs (view literacy strategy descriptions) using such topics as these: o The most important thing I learned in math this week was o The easiest thing about today s lesson Activity-Specific Assessments Activity 4: Present the following problem to the students: If the original square, made up of all seven tangram pieces, has an area of 120 square units, what is the area of each polygon? Have students choose their own value for the little square and then find the value of each piece. Activity 7: Draw an irregular figure on a grid and present it to the student with the following task: How would you estimate the area and perimeter of an irregular figure such as the one given. Could you find the actual perimeter? If so, what method would you use? Explain your method thoroughly. What is a reasonable estimate for the area of the figure? Explain your reasoning. Activity 11: Draw the net and find the surface area of the figure below. Grade 6 Mathematics Unit 6 Geometry, Perimeter, Area and Measurement 6-22