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Course Profile (for a locally developed course)

Essential Mathematics, Grade 9

Unit 6: Investigating Three-Dimensional Figures

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 | Activity 6 | Activity 7

Time: 14 hours

Unit Description

Students are engaged in a variety of activities dealing with three-dimensional geometry that allows them to solve measurement problems in real-life contexts. With or without the aid of technology, students construct three-dimensional models. Through the use of concrete materials, drawings, and technology, students investigate the effect that varying one dimension has on surface area and volume. They also investigate optimal values of various measurements of three-dimensional figures. Students communicate their findings and apply them to identify and solve problems in familiar settings.

Strand(s) and Expectations

Number Sense Strand Specific Expectations: NS 1.01, .15, .16.

Relationships Strand Specific Expectations: RE1.04, .06, .07.

Measurement and Geometry Specific Expectations: MG1.02, .03, .04, 2.02, .03, .04, .05, .06.

Activity Titles

What follows is a suggested sequence, with timing, for teaching Unit 6. This profile develops activities that encourage skill development. These activities are designed to help students make sense of surface area and volume. *Time has been allotted for remediation and further development of skills prior to certain activities.

This unit relies heavily upon concepts developed in the two-dimensional measurement unit.

Activity 1	Constructing 3-D Shapes	90 minutes
Activity 2	Volume of a Prism	150 minutes
Activity 3	Looking at Volume and Surface Area	150 minutes

* Time has been allotted for students to practise solving problems in a context that deal with the surface area and volume of three-dimensional figures. 75 minutes

Activity 4	Building A Box	75 minutes
Activity 5	Building a Better Pop Can	75 minutes
Activity 6	Keeping Costs Down	75 minutes
Activity 7	Summative Assessment	150 minutes

Unit Planning Notes

· This unit helps students understand the concepts of surface area and volume through the use of manipulatives, concrete materials, and technology.

· The teacher could also gather examples of other activities, in different contexts, that address the same expectations as outlined in each activity to use these as follow-ups, warm-ups, or substitutes.

· This unit lends itself to raising the awareness of students to environmental issues (e.g., packaging, efficient house design) and related career possibilities (e.g., product design, advertising, architecture, manufacturing). Activities 4, 5, 6, 7, and 8 provide opportunities for further related explorations as time allows.

Teaching/Learning Strategies

· This unit requires flexibility of timing while at the same time requires structure so that students are engaged in meaningful tasks. Teachers work diagnostically with students to determine what type of support each student requires. Time has been built into the activities to allow for these opportunities and to further develop skills within a context.

· Initially, students should be given measurement problems that include diagrams. As they gain confidence in their abilities they are then given diagrams and asked to enter information in order to solve the problem. Ultimately, they have to construct simple diagrams from written text and problem solve.

Resources

Coxford, Arthur Jr. Geometry from Multiple Perspectives. National Council for Teachers of Mathematics, 1991.

Ebos, F., D.W. McKillop, E. Milne, B.J. Morrison, B. Robinson, and K. Whelan. Math In Context 7. Nelson Canada, 1992.

Ebos, F., D.W. McKillop, E. Milne, B.J. Morrison, B. Robinson, and K. Whelan. Math In Context 8. Nelson Canada, 1992.

Flewelling, G., J. Routledge, J. Clark, and T. Brown. Making Mathematics 8. Gage, 1991.

Elchuck, L., J. Hope, B. Scully, J. Scully, M. Small, and S. Tossell. Interactions 9. Prentice Hall Ginn, 1996.

Kenney, M.J., S.J. Bezuszka, and J.D. Martin. Informal Geometry Explorations. Dale Seymour, 1992.

Lunney, J., P. Rae-Dion, B. Tuck, and B. Walters. Math Sense Book 1. Nelson Canada, 1991.

Reak, C., K. Stewart, and K. Walker. 20 Thinking Questions for GeoBoards. Creative Publications, 1995.

Woodward, E. and T. Hamel. Visualized Geometry - A Van Hiele Level Approach. J. Weston Walch, 1990.

Activity 1: Constructing 3-D Shapes

Time: 90 minutes

Description

In this introductory activity, students build prisms given a set of nets. They identify characteristics of each net they construct.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships and Measurement and Geometry

Specific Expectations: NS1.01, .15, RE1.06, .07, MG2.02, .05.

Planning Notes

· Provide scissors and tape.

· Provide nets for two rectangular-based prisms, (one tall and thin and the other short and squat), two square-based prisms, (one tall and thin and one a cube), two triangular-based prisms, (one with a right angle triangle for the base and the other with an equilateral triangle for the base) and two cylinders, (one with a small circular base and one with a large circular base).

Teaching/Learning Strategies

Student Activity

· Students may work in pairs to complete the task but each student submits a written assignment.

· Students cut, fold, and tape the nets to form three-dimensional figures.

· Students examine the shapes and complete a table on a worksheet listing the characteristics of the different shapes.

· Students examine a series of nets and determine, without constructing the shapes, which ones could form rectangular prisms. Students could check their results by constructing the figures.

Teacher Facilitation

· The teacher may wish to choose the learning partners or students may be allowed to make their own choice to complete the activity. A group of three students may be helpful if there are students who have problems with fine motor control.

· Provide each group with the nets and materials for construction of the shapes. It is a good idea to draw the nets with grids on the one side of the paper.

· These constructions need to be saved for use in Activity 3.

· Explain some terminology before the chart is completed.

· If students work co-operatively to construct the shapes, they should each record the information onto their own chart.

· Circulate throughout the class to provide support and encouragement as students work.

Sample Chart

CONSTRUCTING 3-D SHAPES

Name of Shape	Shape(s) used for the faces	Number of Faces	Number of Vertices	Number of Edges
Rectangular-based prism #1
Rectangular-based prism #2
Square-based prism #1
Square-based prism #2
Triangular-based prism #1
Triangular-based prism #2
Cylinder #1
Cylinder #2

· Discuss with the students the particular characteristics of each figure. Compare the like shapes. Also look for patterns in the chart.

· Distribute a worksheet that shows a series of nets. Students try to determine which ones make rectangular-based prisms. They could check their answers by cutting and folding the shapes, if time permits.

Assessment/Evaluation

Collect charts at the end of class to ensure task completion and accuracy.

Activity 2: Volume of a Prism

Time: 150 minutes

Description

This activity introduces the concept of volume. Students build various cubes and prisms utilizing multi-link cubes and record data on length, width, height, and volume. They determine the area of the base and volume using concrete materials. Students discover the formula for calculating the volume of a rectangular based prism.

Strand(s) and Expectations

Strand(s): Number Sense and Measurement and Geometry

Specific Expectations: NS1.01, .15, MG2.02, .04, .05.

Planning Notes

· Provide worksheets for each student.

· Provide about 84 multi-link cubes for each student.

· Have a number of triangular and circular pieces of cardboard cut and use them to develop the formulas for volume of a triangular prism and a cylinder.

Teaching/Learning Strategies

Student Activity

· Students look at diagrams of 3-D figures and build them using multi-link cubes.

· Students build layers using multi-link cubes, count the cubes to determine volume, and record the results. They also add multiple layers, one at a time, calculating the volume after each addition.

· Through discussion and additional teacher demonstration, students discover the method for calculating volume of a right-rectangular prism using volume = area of base x height.

· Through demonstration this approach is extended to develop the method for calculating volume of a triangular based prism and cylinder.

Teacher Facilitation

· This is an opportunity for the students to manipulate geometrical solids to recognize the three dimensions of width, length, and height.

· As students use the concrete materials, the teacher promotes the use of appropriate mathematical language, helping students expand their thinking.

· Provide each student or pair of learning partners with multi-link cubes and worksheet 1, which shows a variety of 3-D shapes that were built from cubes. Students reproduce each shape using their cubes.

· Instruct students to build a 3 x 4 layer of cubes. Count the cubes to determine the volume of the layer and the faces and areas of each face to determine the surface area of the layer. Record the results in a chart like the one below.

· This lesson is largely teacher-directed. It is important to circulate as students are working to provide assistance and support where required.

Sample Diagram and Chart

Volume: (Count the number of cubes used to build the figure)
Number of Cubes in 1 Layer	Height (number of layers)	Volume (total number of cubes used)
3 x 4 = 12	1	12
3 x 4 = 12	2	24
etc.

· Through discussion, lead students to the result that volume = area of base x height

· Use a series of same-sized circles, cut from cardboard, to show students that the same formula applies for cylinders when they are viewed as a number of circular layers. Use the formula for area of a circle to calculate area of one circle. Each time another layer is added, calculate the volume as A2 layers of circles or 2 x area of the base@. Repeat the steps for each layer.

· Repeat the same process for a series of same-size triangles.

As students will be called upon to create drawings of various solids that they build, time could be spent on the following activities; copying geometric drawings of solids, building a solid from drawings, and drawing solids on isometric dot paper. The emphasis is on moving from a 3-D to a 2-D model.

This is also an opportunity for students to practice calculating the volume of prisms from diagrams.

Assessment/Evaluation

Worksheets are marked for completion and accuracy.

Activity 3: Looking at Volume and Surface Area

Time: 150 minutes

Description

Using the three-dimensional figures constructed in Activity 1 and a variety of empty containers, students determine the surface area and volume of some solids.

Strand(s) and Expectations

Strand(s): Number Sense and Measurement and Geometry

Specific Expectations: NS1.15, .16, MG2.02, .03, .04, .05, .06.

Planning Notes

· Provide each group with worksheets, rulers, scissors, and their constructions from Activity 1 and a variety of empty containers that are rectangular-based prisms, triangular-based prisms, and cylinders (e.g., cereal boxes, candy packages, oatmeal containers, etc.).

Teaching/Learning Strategies

Student Activities

· Students determine the volume of the shapes they constructed in Activity 1 by using the grids on the outside of the shape or a ruler to measure the dimensions.

· Students cut the shapes apart, lay them flat, determine the area of each face, and then calculate the total surface area of the figure (students can also be given the original nets to complete this part of the activity).

Teacher Facilitation

· Students could work in pairs.

· Students determine the volume of the shapes that were built in Activity 1. The teacher leads the students through the exercise for one figure of each type and then has students complete the assignment on their own. Data can be recorded in a chart similar to the one included in this activity.

· If the original nets were prepared using grid lines on the paper that are 1 cm apart, students can quickly determine the measurements and spend the allotted time doing the calculations. It allows students to complete many more calculations during the same period of time. Rulers are used in the second part of the activity to measure packages.

· When determining volume, count the grid lines to determine the measurements of the base and height or measure using a ruler. Use volume = area of base x height. Calculate and record the answer in the chart.

· When determining surface area, cut apart the shape, measure the area of each face, and record the area of each face by writing directly on the net. Total the areas of all of the faces and record the total surface area in the chart.

· Circulate throughout the class to ensure accuracy in measuring and to provide support and encouragement where required.

Sample Chart

3-D Figure	Area of Base (cm^2)	Height of Figure (cm)	Volume = Area of Base x Height (cm^3)	Total Surface Area (total the areas of each of the faces in the figure) (cm^2)
example: cylinder (radius is 4 cm and height is 6 cm)	pr^2 = 3.14 x 4 x 4 = 50.2 cm^2	6 cm	vol = 50.2 x 6 = 301.2 cm^3	2 circles + 1 rectangle = 2 x 3.14 x 4 + 2 x 3.14 x 6 Ñ 100.4 + 226 Ñ 325.4 cm^2
etc.

· Repeat the above process using the empty packages and containers. Extend the length of the table so that students could continue to record their results.

Assessment/Evaluation

Assess completed charts for accuracy and completion. Learning skills could be observed using the rubric in Appendix 1.

Activity 4: Building a Box

Time: 75 minutes

Description

Students explore the surface area and volume of cubes by building a series of cubes, each time increasing the side length by one unit. They record their findings on a chart as they progress through the activity. They also determine different relationships involving volumes and surface areas by using graphs.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships and Measurement and Geometry

Specific Expectations: NS1.01, .15, .16, RE1.04, .06, MG1.02.

Planning Notes

· Provide multi-link cubes.

· Provide worksheets.

· Provide labelled and scaled grids for graphs.

Teaching/Learning Strategies

Student Activity

· Students construct cubes. Each one has a side length one unit greater than the previous cube.

· For each cube, students record the dimensions, area of the base, volume and total surface area.

· Students graph side length vs. area of base, side length vs. volume and side length vs. total surface area. They look for patterns, similarities and differences in the graphs. A discussion should include the fact that since the cubes were built using multi-link cubes the points should not be connected but in real life, since you can have fractional measurements, the points could be connected.

Time could be spent reviewing plotting points. Students could plot a list of points, joining them as they go, in order to draw a diagram.

Teacher Facilitation

· Distribute multi-link cubes.

· The teacher may work with the students to complete the first few examples and then circulate throughout the classroom to provide support and encouragement.

Sample Worksheet

1. Start with one cube. Determine the area of its base, its volume and total surface area. Record your results in the chart.

1. Now build a 2x2x2 cube (a cube with a side length of 2 units). Determine the required values and record them in the chart.

2. Keep repeating this process, each time increasing the side length by one cube until a 6x6x6 cube is made. (You may have to share cubes with other students to have enough).

Dimensions of Cube	Area of Base	Volume	Surface Area
1x1x1	1	1	1
2x2x2	4	8	6x4=24
......
6x6x6

3. Use the grids, provided by the teacher, to construct graphs displaying side length vs. area of base, side length vs. volume and side length vs. surface area.

4. What patterns do you see? How are the graphs different from one another? How are the graphs similar to one another?

· Students may use multi-link cubes for up to 4x4x4 and then use patterns to complete the chart.

· The teacher can highlight the patterns involving perfect squares and cubes.

· Finish the lesson with a discussion of the graphs that were drawn and the observations made by the students, based on their answers to question 5 on the worksheet.

Assessment/Evaluation

Mark the worksheet for completeness and accuracy. Use the criteria from the rubric in Appendix 1 to observe a few students. Use rubric for assessing graphs from Unit 1 - Activity 3.

Activity 5: Building a Better Pop Can

Time: 75 minutes

Description

Students explore the relationship between the surface area and volume of a cylinder. Students use a spreadsheet to help them collect data and draw conclusions about optimal measurements. They ultimately decide whether the current shape of a standard soda pop can is optimal.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships and Measurement and Geometry

Specific Expectations: NS1.01, .15, .16; RE1.04; MGV1.01, .02, .04.

Planning Notes

· Provide two sheets of standard size letter paper for students to construct models.

· Provide worksheets for this activity.

· This activity uses a spreadsheet and thus a computer resource lab is required.

· Depending upon the comfort level of students with spreadsheets, the teacher may choose to set up a template of the activity in a file and have students load it from their computer station.

Teaching/Learning Strategies

Student Activity

· Students construct two cylinders by folding two sheets of letter paper in each direction.

· For each of the cylinders students record the dimensions, area of the base and volume. Students can determine the diameter of the base by tracing the bottom of the cylinder onto another sheet of paper.

· Students complete a spreadsheet to calculate the least amount of material required to construct particular cylinders for a given volume. They use this information to determine any relationships and conclude if a standard pop can has been built in the most efficient manner.

Teacher Facilitation

· Distribute sheets of paper and worksheets.

· The teacher works with the students to complete the first example and circulate throughout the classroom as students complete the second.

· The teacher works with students to complete the spreadsheet exercise.

· It may be necessary to highlight the fact that all of the minimized surfaces have heights that are very close to their diameters. A discussion around symmetry could ensue. Connections can be drawn to the discussions involving minimization in the 2-D unit.

· This discussion can finally focus on soda pop cans. This is an opportunity to discuss why cans could be designed to minimize area and why they may not (aesthetics, ergonomics). This can be extended to a discussion concerning environmental concerns.

Sample Worksheet 1

5. You will need 2 same-size, rectangular sheets of paper to complete this investigation.

6. Measure the length and width of the paper in centimetres. Record the results.

7. Roll the paper to make 2 different cylinders without overlapping the edges. Tape them.

8. What do you notice about the surface area of each cylinder if it is open at each end?

9. Record the height and circumference of each cylinder in the chart below.

10. Calculate the diameter and then the radius of the base.

Labelled sketch of the flat piece of paper	Labelled sketch of the cylinder	Dimensions Height (cm) and Circumference (cm)	Diameter of the Base (circumference ÷π)	Radius of the Base (diameter ÷ 2)	Volume of the cylinder (Area of Base x Height = πr^2 x height) (cm^3)
example:		Height is 30 cm Circumference is 20 cm	20 ) 3.14 = 6.4	6.4 ) 2 = 3.2	3.14 x 3.2 x 3.2 x 30 = 964.608
		Height is 20 cm Circumference is 30 cm
Use these for question 8.

11. Which cylinder has the larger volume?

12. Repeat the exercise using rectangles of paper that are twice as long as they are wide.

13. What would happen if you used a square piece of paper?

14. Why would a packaging company want to know this information?

Sample Worksheet 2

BUILDING A BETTER POP CAN

Using a spreadsheet calculate the dimensions of the can that will minimize the surface area. Record your observations in the chart below.

	A	B	C	D	E	F
1	Volume	Radius	Diameter	Area of base	Height	Surface Area
2	355	1	2xB2	3.14 xB2xB2	A2 /D2	2xD2+2x3.14xB2xE2
	$ $ $	Etc.
10	355	10

Test the following volumes for the most "efficient" dimensions.

· The teacher instructs students to test for the most "efficient" dimensions of other sized containers by substituting the amount of volume in their spreadsheets with the values below. Other values in the spreadsheet should be recalculated automatically by the program.

Observations

Volume (ml)	Diameter (cm)	Height (cm)
355
500
750
1000

What is the relationship between the diameter and the height?

· Once students have tested the various sized containers it is important that the relationship between diameter and height is discussed. At this time, connections can be drawn to the discussions involving minimization in the 2-D unit.

Assessment/Evaluation

Mark worksheets for completion and accuracy.

Students may write a short summary in their journals of what they determined through this investigation and how it can be applied in a real-life context.

Activity 6: Keeping Costs Down

Time: 75 minutes

Description

Students determine the minimum surface area for a given volume within a real life context using manipulatives.

Strand(s) and Expectations

Strand(s): Number Sense and Measurement and Geometry

Specific Expectations: NS1.01, .15, .16, MG1.03, .04, 2.05, .06.

Planning Notes

· Prepare worksheets for each student.

· Provide multi-link cubes.

Teaching/Learning Strategies

Student Activity

· Students determine the possible surface areas of rectangular prisms built from a total of 64 multi-link cubes by following the instructions in the sample worksheet.

· Students could record their answers in a chart.

Teacher Facilitation

· Provide each student with a worksheet and 64 multi-link cubes. If there are not enough cubes, students may need to work in groups of two or three.

· The teacher may choose to direct the first few calculations. Students must build as many rectangular prisms as possible from the 64 cubes (one prism at a time). Point out that a 2x4x8 cube is the same as a 4x2x8 cube, etc..

· The teacher may choose to circulate while the students are working to provide support and encouragement where necessary.

Sample Worksheet

PACKAGING

A group of Canadian farmers have decided to send a shipment of wheat overseas to a third-world country. They wish to send the wheat in a container that has the shape of a rectangular-based prism. They want it to have the smallest surface area possible but still hold all of the wheat.

You will simulate this problem using 64 multi-link cubes. The cubes represent the total amount of wheat to be shipped. You must use all 64 cubes for each rectangular-based prism. Note that any side could be the base so the order of the dimensions does not matter.

Build any rectangular-based prism using all 64 cubes. Fill in the chart below for your prism.

Cube Number	Length	Width	Height	Area of Base	Volume	Surface Area
example: #1 1x1x64	1	1	64	1x1=1	Area of base x height = 1 x 64 = 64 cubes	1 + 1 + 64 + 64 + 64 + 64 = 258 squares
#2 2x8x4	2	8	4	2x8 = 16	Area of base x height = 16 x 4 = 64 cubes	16 + 16 + 32 + 32 + 8 + 8 = 112 squares
etc.

· There are 7 possible prisms to consider. 1x1x64, 1x2x32, 1x4x16, 1x8x8, 2x2x16, 2x4x8, 4x4x4. Depending upon the comfort level with this work, students may be given these possibilities.

· Examine their results in the chart and determine which container size has the smallest surface area. They also notice that each time the volume was 64 (They knew this because they always used 64 cubes to build the prism. They also calculated this result.)

· Discuss other cases when it is important to minimize surface area for a fixed volume. It is often a consideration in packaging or when something is to be constructed using a very costly material (e.g., a mahogany box, a silver jewellery case, etc.).

Assessment/Evaluation

Mark worksheets to assess for accuracy and completion. Use the rubric in Appendix 1 to observe the work habits and initiative of a few students.

Activity 7: Summative Assessment

Time: 150 minutes

Description

The assessment component of this unit should be completed in three parts. During the first and second parts, students prepare two summary sheets and complete a performance-based assessment task. Students use the previously prepared summary sheets to complete a pencil and paper test.

Strand(s) and Expectations

Strand(s): Number Sense and Measurement and Geometry

Specific Expectations: NS1.01, .15, .16, MG1.02, .03, 2.02, .03, .04, .05, .06.

Planning Notes

· Prepare two summary templates that include diagrams of a square-based prism, a rectangular-based prism, a triangular-based prism, and a cylinder. One template is for volume and another for surface area (landscape the paper).

· Prepare overheads of the summary templates.

· Provide copies of the rubric for the Designing a Shipping Carton performance-based activity and the worksheet.

· Teachers have to develop a pencil and paper test.

Teaching/Learning Strategies

Student Activity

· Students complete the first example in the summary template with teacher direction. They complete the summary sheet using their notebook and/or with teacher support.

· Students check their work with an overhead acetate that is prepared by some of the faster workers. (See Teacher Facilitation.)

· Students complete a performance-based task assessment.

· Students complete a pencil and paper test.

Teacher Facilitation

· Lead the students through the first question of a summary template, like the one shown, and then support and encourage their efforts as they complete the rest. As it is verified that they have completed a section or question accurately, you may choose to direct some students to write their work on the prepared summary acetate.

· Display the completed acetate so students may check their work. Have a few extra summary sheets available for those students who may wish to rewrite their sheets.

Sample Summary Templates

Volume Summary (Template #1)

Name and Diagram	Area of Base	Height	Volume
Triangular-based Prism
Square-based Prism
Rectangular-based Prism
Cylinder

Surface Area Summary (Template #2)

Name and Diagram	Net	Surface Area
Triangular-based Prism		Total surface area = 2 rectangle +1 rectangle + 2 triangles = 2 ( 5 x 6 ) + 1 ( 4 x 6 ) + 2 ( 4 x 3 ) 2 ) = 2 x 30 + 1 x 24 + 2 x 6 = 60 + 24 + 12 = 96 cm^2
Square-based Prism
Rectangular-based Prism
Cylinder

· Introduce and explain the rubric that the teacher will be using to observe the students as they complete the Design A Shipping Carton activity.

· Circulate throughout the classroom while the students complete the performance task. Provide support and encouragement where required.

· It should be noted that the investigation was designed so that only two possible containers satisfy all of the conditions. This reduces the amount of repetition in the problem.

Sample Performance-Based Assessment Task

DESIGNING A SHIPPING CARTON

Your company is trying to win a contract from a U.S. company, the Stay-Puffy Marshmallow Company, who ships marshmallows around the world. This company ships in bulk and typically sends their marshmallows in containers that are 24 cubic feet in size. They have asked for different companies to provide the shipping containers for their product. They have also asked that all measurement be whole numbers.

15. What are the possible dimensions of the container? Use the table below to list and draw them.

volume	depth	length	width	sketch
24
24
24
etc.

16. To avoid any long or flat boxes, that are difficult to transport, the company has asked that all boxes must have dimensions greater than 1 foot. Which of the above boxes fit this criterion?

17. Draw a sketch of each possible container and determine its surface area.

18. Which box would you recommend to the Stay-Puffy Marshmallow company? Why?

Sample Pencil and Paper Test

· This test should include questions that allow for demonstration of Level 4 performance. Teachers may wish to include questions such as;

19. A cube has 6 faces. Which of the following nets can be folded to make a cube?

Assessment/Evaluation

Assess the worksheets for completeness and accuracy. Observe some students for initiative and independent work habits using the rubric in Appendix 1. Use the problem-solving rubric included with this activity to assess student performance on the Designing a Shipping Carton task.

Designing a Shipping Container Rubric

	Level 1	Level 2	Level 3	Level 4
Understanding the problem	- shows partial understanding of the problem, needs teacher assistance to clarify	- shows a good understanding of the problem, with minor misconceptions	- shows complete understanding of the problem with little or no assistance	- shows complete understanding of the problem and has insight beyond the problem
Formulating a plan	- identifies most combinations	- identifies combinations in a random manner	- identifies combinations in a systematic manner	- identifies combinations in a systematic a manner and identifies a general method for determining combinations
Communication	- requires assistance to write and organize the solution - communicates with limited clarity and limited justification of reasoning	- requires minimal prompting to write and organize the solution and to draw the sketches - communicates with some clarity and some justification of reasoning	- completes the chart and draws the sketches independently - communicates with clarity and clear justification of reasoning	- completes the chart and draws the sketches with considerable effectiveness - communicates concisely with a high degree of clarity and full justification of reasoning
Analysis	- makes superficial inferences from data	- makes correct inferences from data, with some minor errors	- makes correct inferences from data	- makes correct inferences and identifies other relevant variables beyond the scope of the problem
Calculations	- computations are rarely accurate - needs step by step help to calculate minimum surface area. - is able to build the package with step-by-step instructions.	- computations are frequently accurate - regularly seeks help to calculate minimum surface area. - needs direction to build the package.	- computations are usually accurate - can find minimum surface area with little or no assistance. - can build package, may require teacher reassurance.	- computations are completely accurate - able to use formulas to find minimum surface area without assistance. - builds the package without assistance.

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