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Course Profile Foundations of Mathematics, Grade 9 applied, Public

Unit 3: Measurement and Dynamic Geometry

Activity 1 | Activity 2 | Activity 3 | Activity 4 |

Activity 5 | Activity 6 | Activity 7 | Activity 8

Time: 30 hours

Unit Description

Formulas for two- and three-dimensional shapes are used to solve problems involving composite plane figures. Students investigate relationships between area and perimeter and surface area and volume, to determine optimal values within a given context. A project and a summative assessment activity provide rich contexts for applications of measurement skills.

Geometric relationships involving two-dimensional figures are explored in a variety of ways. Conjectures are made and tested using dynamic geometry software. Teachers bring closure to investigations and guide students to form generalizations of their findings. Students’ knowledge is affirmed through the application of geometric relationships in new situations.

Strand(s) and Expectations

Number Sense and Algebra Specific Expectations: NA3.04, .05; NA4.01, .02, .03.

Relationships: RE1.03, .04, .05, .06; 2.04; 3.04.

Measurement and Geometry Specific Expectations: MG1.02, .03, .04; MG2.01, .02, .03, .04, .05; MG3.01, .02, .03, .04.

Activity Titles

What follows are suggested packages of activities for teaching Unit 3, with timing for each activity and timing for skill development. The measurement activities could be done before or after the geometric relationships involving two-dimensional figures, allowing different Grade 9 classes to use computer facilities at different times. This profile develops only the activities that depart from traditional pencil and paper skill development. These activities are designed to develop students’ ability to visualize and analyse geometric shapes and relationships, using numeric and algebraic skills as well as skills in the appropriate use of technology. Skills developed through the Activities are indicated in [brackets].

*Up to 525 of the 1800 minutes for this Unit may be used as needed for consolidating skills and further evaluation.

Activity 1	Lawn Sprinklers [two-dimensional measurement formulas, minimizing cost, estimation and approximation, numeracy skills]	225 minutes
Activity 2	Fence Me In - Introduction to The Geometer’s Sketchpad™ [maximizing area for a given perimeter, perimeter and area relationships]	150 minutes
Activity 3	The Size Is Right! [comparing volumes and surface areas for square-based prisms and cylinders, constructing a variety of prisms]	150 minutes
Activity 4	Why Do Elephants Have Big Ears? [surface area: volume ratios]	75 minutes
Activity 5	Thinking Out of the Box [a summative assessment activity that uses the skills and knowledge from Activities 1 to 4]	150 minutes
*Time for: practising the use of geometric formulas, substitution into formulas to solve, presentation of projects.		225 minutes
Activity 6	Exploring with The Geometer’s Sketchpad™ [Review of Angle Relationships]	150 minutes
Activity 7	If...then... [Investigating Geometric Properties Using The Geometer’s Sketchpad™]	300 minutes
*Time for: further investigations, assessments.		150 minutes
Activity 8	Script It! [Creating The Geometer’s Sketchpad™ Scripts; suggestions for summative assessment activities]	225 minutes

Prior Knowledge Required

· Measurement and Geometry and Spatial Sense skills from elementary school

· Unit 1: determine the relationship between two variables by collecting and analysing data, solve multi-step problems requiring numerical answers

Unit Planning Notes

· The teacher should check timing of the availability of a computer lab before starting Unit 3. Most of approximately eight 75-minute classes would be best spent having students work in pairs on a computer. It is possible to rearrange the order of some of the activities within this Unit. If computer lab time is not available, it would be possible to capture the power of dynamic geometry software using: one computer attached to a projection panel or TV, or one TI-92 calculator attached to a view screen.

· Each pair of students working on The Geometer’s Sketchpad™ can work at their own pace, but the teacher needs to ensure that all pairs get the help they need to complete the required investigations. Pairs who complete the investigation more quickly could be challenged with extensions, or to pose “What if....” questions; their investigations could be reported to the class.

· If a class cannot use a computer lab, it is possible to carry out the investigations in Activities 6 and 7 using careful constructions and measurements, by hand. If this is the case, the teacher must ensure that students understand that many more examples are needed than they, individually, have time to create, to test each hypothesis adequately. The pooling of results of the entire class has to be orchestrated. It is important that each pair of students understands what extreme cases need to be investigated for each hypothesis. Using technology, each pair of students is responsible for covering all of the special cases and a large number of examples. Textbooks suggest ways to teach this material if your department does not yet have computer access for your class.

· The class gathers tin cans, or other cylindrical shapes, in various sizes, plus a large quantity of plastic pellets, rice, or other small hard objects before Activity 3.

· The sixth activity is intended to introduce the use of The Geometer’s Sketchpad™ and review relationships from Grade 8. Subsequent activities introduce new relationships using Sketchpad as an investigative tool.

· Teachers should ensure that they know what mathematics needs to be drawn out of each of the activities and plan enough time at the end of each class to help bring closure to the investigations that students have performed.

Teaching/Learning Strategies

Focus the students’ previous graphing and identifying relationship experience to connect the measurement activities of this Unit back to Units 1 and 2. This brings the course around full circle and prepares the students for Grade 10.

Pairing works well as students learn to use the dynamic geometry software. Make sure that each student spends enough time using the keyboard to become comfortable with the software.

Move from pair to pair as the students carry out their investigations on the computer. By listening to what the students are discussing, ensure that they are forming good hypotheses, and drawing appropriate conclusions. By asking probing questions, be certain that each pair of students is getting as much out of each investigation as they should.

The teacher can test the computer constructions for each pair of students by asking students to create and save scripts or by performing a drag test on their screen. (If the diagram has been properly constructed, the structure stays connected when one of the elements is moved by clicking on the mouse and dragging it.)

There are some activities that are different in the Applied and Academic versions to reflect different expectations.

Assessment/Evaluation Techniques

As in Units 1 and 2, a variety of assessment tools and strategies is recommended. Performance assessments may be used to effectively assess Thinking/Inquiry/Problem Solving, Communication, and Application Categories of the Achievement Chart when students do open-ended tasks. Learning Skills can be assessed using teacher and peer observation, and self-reflection. Rubrics and rating scales are useful when a wide range of performance is expected and when many complex criteria are to be judged. Checklists and marking schemes are used for more traditional tasks with predictable solutions.

Resources

The Geometer’s Sketchpad Archive

www.forum.swarthmore.edu/sketchpad/sketchpad.html and

www.forum.swarthmore.edu/sketchpad/gsp.gallery/gallery.html

Graphing Calculator Activities for Enriching Middle School. Texas Instruments.

Exploring Geometry With the Geometer’s Sketchpad. Key Curriculum Press.

Visualized Geometry: A van Hiele Level Approach. Portland, Maine: J. Weston Walch, 1990.

Activity 3.1: Lawn Sprinklers

Time: 225 minutes

Description:

In this activity students review their area and perimeter work from Grade 8 including the measurement of irregular figures. Students apply their knowledge calculating area of various shapes to design the layout of a sprinkler system for a homeowner’s lawn. They also use their numeric skills in calculating percentages.

Strand(s) and Expectations

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

Specific Expectations: MG2.01, .03, .04, .05, NA1.01, NA1.02, .04, .05, .06.

Planning Notes

Time can be spent reviewing basic area and perimeter formulas. This can be done by students working in groups, working on problems, and researching formulas with which they need to be reacquainted. It may be beneficial to have students create a chart in their notebooks with all the formulas for perimeter and area. After students have become familiar with the formulas, they are ready to begin The Lawn Sprinklers group activity. Teachers may change the shapes of some of the figures (e.g., the pool could be a combination of a rectangle and three quarters of a circle; the patio could be a trapezoid, etc.). Students work in groups of three to recreate an enlarged version of the lot layout on large grid paper. They also need markers, compasses, etc.

Prior Learning Required

Students should be familiar with the formulas and methods of calculating the area of rectangles, triangles, trapezoids, circles, and composite shapes, as well as working with scale diagrams.

Teaching/Learning Strategies

Student Activity: The Lawn Sprinklers

The Lawn Sprinklers Inc. has just hired you to help custom design the layout for underground sprinkler systems. Your first job is to design a plan for the yard of the Howard family. The layout of their yard is shown below:

Special requests from the Howard family:

· The dog run should not get wet because the dog could catch a cold or get very muddy paws.

· The children do not want to play in a muddy sandbox. They like their sand dry.

· The pool already has enough water in it.

· There is no need to water the patio or driveway.

You are to choose from the following list of supplies and design a layout for a sprinkler system that will be inexpensive and efficient.

List of Supplies

Large sprinkler head (coverage radius of 5 m) $20.00

Small sprinkler head (coverage radius of 3 m) $12.00

Pipe (price per m) $ 2.50

Your final report should include:

· A layout of the placement of sprinkler heads and pipes. Note that sprinkler heads do not need to make a full circle but could be set up to spray a fraction of a circle rather than a full circle.

· A financial report showing quantity of and total cost of sprinkler heads and piping (a spreadsheet could be used).

· The percentage of the yard that receives water, the percentage of the lawn that does not receive water, the surface area that receives water but should not (e.g., the patio, deck, driveway, etc.) or that receives double watering. A full solution is required to show how these calculations were arrived at.

Teacher Facilitation:

Before beginning this project, have a whole class discussion about sprinkler systems. Perhaps a guest speaker could be brought in to explain how the systems are designed or actual price lists for necessary products could be obtained. Since this activity requires calculations involving formulas, teachers may find it necessary to review the order of operations, the use of π, and evaluating expressions using the scientific calculator.

Assessment/Evaluation Techniques:

Assessment of Communication and Inquiry skills (applying the steps of the inquiry process and posing extending questions) should be obtained through teacher observations as students are working in groups.

A rubric for assessing specific expectations associated with this activity should be created and used with the students’ written submissions. This should be built to include the assessment of:

· Knowledge/Application skills (calculation of cost and area are accurate; diagrams are accurate, well constructed and clearly labeled)

· Communication skills (report is written clearly and distinctly; uses mathematical terminology appropriately)

Activity 3.2: Fence Me In

Time: 150 minutes

Description

In this activity students explore the relationships between perimeter and area of a figure when one of the measures is fixed. They also gain experience using the computer. When the students are minimizing the perimeter for a given area in Part B they use their algebra skills to communicate with the technology.

Strand(s) and Expectations

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

Specific Expectations: NA4.01, .03; RE1.03, .04, .05, .06; MG1.01, .04; MG2.03, .04.

Planning Notes

· This activity requires students to have a rudimentary knowledge of The Geometer’s Sketchpad™. Time has been allotted (75 minutes) for the teacher to introduce the use of this program prior to the Fence Me In activity. Specifically, the students should be familiarized with the toolbar, the names of the tools and the menus. Resources for this are easily found by conducting a web search using key words The Geometer’s Sketchpad.

· The activity Fence Me In (Appendix 3.2) should be photocopied for each student in the class.

· The teacher reserves the computer lab for this activity.

· While this activity has been designed to meet curriculum expectations regarding the use of technology, students can perform this investigation using grid paper or geoboards in place of The Geometer’s Sketchpad™. The spreadsheet capability of the graphing calculators could be used for the data analysis.

· The activity sheet has been written for use with Excel version 7.0 for Windows 95. Teachers may need to modify the instructions to fit the spreadsheet that they have available or provide appropriate instructions if using graphing calculators. See “Building a Garden Fence,” TI-80/82/83 Graphing Calculator Activities for Enriching Middle School Mathematics.

Prior Knowledge Required

The students should be proficient in creating a spreadsheet.

Teaching/Learning Strategies

Teacher Facilitation: The teacher guides the students through a discussion of both problems and the formulation of their hypotheses for each. Adequate time should be given so the students can record their hypotheses before they begin their investigations. The teacher ensures that the students have selected the appropriate program settings for The Geometer’s Sketchpad™ prior to starting the activity (steps one and two). The teacher circulates amongst the students and assist as needed.

Student Activity: Students work in pairs for the activity Fence Me In. See Appendix 3.2.

Assessment/Evaluation Techniques

As students are working through the activity, their Knowledge/Understanding skills (using The Geometer’s Sketchpad™ to collect data and a spreadsheet to organize their data) may be assessed using a checklist. The teacher collects the students’ written responses and assesses Application skills (the correct analysis of data; identification of applications for maximizing area and minimizing perimeter). Their Communication skills (clarity; justification of reasoning; description of their applications) can also be assessed.

Accommodations

The teacher could assign partners to assist students who encounter difficulties when reading instructions or using the computer.

Resources

TI-80/82/83 Explorations. Graphing Calculator Activities for Enriching Middle School Mathematics.

Appendix 3.2: Fence Me In

This activity uses The Geometer’s Sketchpad™ and Excel to explore different ways of finding the optimal area or perimeter of a rectangular object when the other measure is fixed. You submit the written responses to questions that are asked in this investigation. You will be evaluated on the quality of your work. Your writing should be clear, concise, and accurate and should use appropriate units and correct sentence structure.

A. Finding the Maximum Area

Suppose you had 20 m of fencing and wanted to fence in part of your backyard for a pet rabbit. You want the pen to be rectangular and to have the largest area possible. This activity answers the question, “What is the largest area possible for the rabbit?”

Thinking About the Problem

Answer these questions in your notebook:

1. Sketch a diagram of the rabbit pen. How are length and width related in this problem? (Hint: what will their sum be?)

· What dimensions do you think give the maximum area? Explain your thinking.

Setting Up The Geometer’s Sketchpad™

2. Open The Geometer’s Sketchpad™.

· Select Preferences from the Display menu. Set Distance Unit to cm and Distance Unit Precision to units. The scale for this investigation is 1 cm = 1 m.

· Choose Show Grid from the Graph menu. Then choose Hide Axes from the Graph menu to show only the grid.

The Investigation

· Using the straightedge tool, starting on one of the grid points, construct a rectangle on the grid.

· Label the vertices using the text tool.

· Highlight the width of the rectangle using the selection arrow. Choose Length from the Measure menu to display the width measurement. Repeat this procedure to display the length measurement.

· Select all of the vertices by using the selection arrow while holding the Shift key down. Then choose Polygon Interior from the Construct menu. You can change the colour of the interior if you like from the Display menu.

· Make sure the interior is selected. Then select Perimeter and Area of the rectangle from the Measure menu.

· Use the selection arrow to highlight one of the sides and then drag the side in or out until the perimeter equals 20.00 cm. (Be careful not to drag the vertices of the rectangle at any time - this will change its shape. If you do move one of the vertices, select Undo from the Edit menu, or press ctrl-z).

· Set up a table in your notes for your rough work like this:

Width	Length	Area	Perimeter
			20

· Copy into the table the width, length and area of your rectangle.

· Drag the sides of the rectangle until you have a different size rectangle, but with perimeter still 20.00 cm. Continue to record different measurements until there are no more possibilities (you should have 9 rows).

Setting Up a Spreadsheet

· Open Excel and create the following worksheet:

	A	B	C	D
1	Finding Maximum Area
2
3	Width	Length	Perimeter	Area
4			20

· Fill out the worksheet from the table you created in your notes.

· Select the Sort option from the Data menu. Click OK, and the values in the first column will be sorted in ascending order.

Create a Scatterplot Relating Rectangle Width to Area.

· Highlight cells A4-A12 with the mouse (these are all the width measurements). Then while holding down ctrl, use the mouse to highlight cells D4-D12 (the area measurements).

· With these two columns highlighted, choose Chart and As new sheet from the Insert menu.

· A Chart Wizard box will open. Click Next.

· Select the chart type as XY-Scatterplot (centre graph), and click on Next.

· Select 2 from the next box, click on Next.

· Click Next again, to get to Step 5 of 5. Choose the following options (use tab or the mouse to move between options):

· no legend

· title the graph "Finding Maximum Area"

· title the x-axis "Width" and the y-axis "Area"

· Click Finish.

Questions (Continue to put your answers in your notebook.)

3. What is the maximum area for your rabbit pen? Was your hypothesis correct?

· Determine a formula that finds

(a) the length, l, if perimeter is P and width is w.

(b) the area, A, if perimeter is P and width is w.

· Choose a set of values for length and width and check your formulas with those numbers. Do the formulas work?

· Describe how to find the maximum area from:

(a) a table (spreadsheet);

(b) a graph.

· If decimal values were permitted for the length and width would your result be different? Explain how you would use your existing data to justify your response.

B. Finding the Minimum Perimeter

In the first part of this activity you generated data and entered it into a spreadsheet. In this investigation the spreadsheet does much of the work for you.

Suppose you had some seeds to plant a vegetable garden. Following the directions on the packets for spacing the seed you realize the area of the garden will be 100 m². You want to put up a decorative wooden lattice around the rectangular garden. Since the wood is expensive you would like to design the garden so that you have to buy the least amount of lattice possible (the smallest perimeter). In this investigation you will find the minimum perimeter for an area of 100 m².

Getting Ready

Answer these questions in your notebook:

4. Sketch a diagram of the garden. What do you think is the minimum amount of lattice needed to fence in the garden? Explain your thinking.

· If the width of the rectangular garden is w metres and its area is A metres squared create an algebraic expression for the length using w and A.

· If the length of the same rectangular garden is l metres create an algebraic expression for the perimeter using l and w.

Setting Up a Spreadsheet

5. Open Excel, and create the following worksheet:

	A	B	C	D
1	Finding Minimum Perimeter
2
3	Width	Length	Perimeter	Area
4	2	= D4/A4	= 2*A4+2*B4	100
5	= A4+2

Note: Hit Enter after entering each formula above. The value of the formula appears in the cell.

· Highlight A5 and while still holding the mouse button down, drag the mouse until row 50 is also highlighted. Release the mouse button and select Fill and Down from the Edit menu. Notice the values that have been entered in column A.

· Highlight B4, C4, and D4 and while still holding the mouse button down, drag the mouse until row 50 is also highlighted. Select Fill and Down from the Edit menu. Notice the values that have been entered in columns B, C, and D.

Create a Scatterplot Relating Rectangle Width and Perimeter

· Create a chart on a new sheet (as in Part A), labelling the title and axes appropriately.

Questions

Continue to put your answers in your notebook.

6. What is the minimum amount of lattice that you need to buy? Was your hypothesis correct?

· Explain why we would want to calculate maximum area when the perimeter is fixed or a minimum perimeter when the area is fixed.

· What are some real-life examples of situations where it might be necessary to find the maximum area with a fixed perimeter or the minimum perimeter with a fixed area?

Challenge Question

· Could an area of 100 m² be enclosed by a smaller perimeter if the shape did not have to be a rectangle? Explain your thinking.

Activity 3.3: The Size Is Right!

Time: 150 minutes

Description

In this activity each student pair works with a square-based prism and a cylinder, and fills each container with the same volume of material. They then measure the depth of the fill in each and imagine the depth as the height of the container. They then calculate the surface area of each container. Class results are gathered so that relationships between volume and surface area can be analysed.

Strand(s) and Expectations

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

Specific Expectations: NA1.02, .04; NA3.04, .06; NA4.01, .02, .03; RE1.03, .04, .06, .07; MG1.01, .02; MG2.02, .03, .04; RE 1.06.

Planning Notes

The class gathers a large variety of cylindrical containers, each having a volume of at least 250 mL. Students use stiff construction paper to build a square-based prism for homework the night before this activity is to be done in class.

Prior Knowledge Required

Knowledge of distinguishing characteristics of a square-based prism and of a cylinder as well as how to calculate the volume and surface area of each

Teaching/Learning Strategies

Teacher Facilitation: Establish student pairs for this activity in the previous class.

Give pairs about 5 minutes of time working together at the end of the previous class to plan a model for an open-topped, square-based prism which has a volume of between 300 and 500 mL. For homework, one student in each pair should construct the model, while the other student draws the net for their model and computes the surface area of a closed container like their model.

The teacher should have a couple of appropriate square-based prisms (e.g., one cracker box could be cut into two prisms) in case a model-building partner is away ill or forgets to bring the model to class.

Provide each pair of students with a different radius cylindrical container having a volume between 250 and 500 mL (e.g., cans for soup, tomato paste, drinking glasses, shampoo bottles).

Provide each pair of students with exactly 250 mL of material that can be poured into the different containers. Small beads, rice, Styrofoam pellets, or similarly small-sized, hard, dry material would be appropriate. Each pair needs a ruler marked in millimetres.

After all class data has been entered, the teacher may need to lead the discussion about minimum surface area. The degree of teacher direction depends on the class and on the variety of containers used.

Interpolation may be needed in questions 6 and 7 if no pair built the square-based prism that results in a cube when holding 250 mL or if the teacher did not provide the cylinder that results in height equal to diameter.

Student Activity: Student pairs should carry out the following:

7. Check your partner’s homework on the square-based prism. Discuss and fix any problems or discrepancies.

· Pour your 250 mL of material into your square-based prism. Shake it gently to ensure that the top surface is flat. Measure the height to which your material fills the prism. Record this height in the table below.

· Pour your material into the cylindrical container provided. Shake it gently to ensure that the top surface is flat. Measure the height to which your material fills the cylinder. Record this height in the table below.

· Complete the rest of the table. Record height measurements accurate to the nearest mm. Use the volume of fill and the volume formula for each shape to compute the base length or radius measures for each shape. Verify these calculations by measuring the actual containers with a ruler. Show all calculations.

Note: *When you calculate the surface area for a closed container, you should imagine the container that uses the bottom and side walls of the actual container you poured your material into, but has a top right on top of the upper surface of your material. This closed container would have a volume of exactly 250 mL.

Height in Square-based Prism

Length and Width of Prism Base

Surface Area for a Closed Prism*

Height in Cylinder

Radius of Cylinder

Surface Area of Closed Cylinder*

· Enter your final measures only on the class summary.

· By referring to the class data, what is the minimum surface area for a closed square-based prism that has a volume of 250 mL? What are the dimensions of this prism?

· Graph the data for the square-based prism from the table by setting height as the independent variable and surface area as the dependent variable. Is this relation linear? Does your graph help you to confirm your answers to #6? Explain.

· By referring to the class data, what is the minimum surface area for a closed cylinder that has a volume of 250 mL? What are the dimensions of this cylinder?

· Describe what is special about the shapes of the 250 mL containers that have minimal surface area? Justify your reasoning.

· Describe situations in which it is important to know the minimum surface area for a given volume.

Teacher Facilitation: Encourage pairs of students to check each other’s work. Observe the students’ accuracy in their work and give mini-lessons as needed.

Manage the accumulation and displaying of data from all pairs of students. This could be done using a spreadsheet and projected for the class or in a table on the blackboard.

Circulate amongst the students and use a colourful pen to prompt appropriate thinking on questions 6-9, as needed.

Student Extension Activity:

8. Construct a cone having the same radius and height as your 250 mL cylinder.

· Hypothesize how the volume of this cone compares to the volume of the cylinder.

· Fill this cone as many times as you can until you have used your 250 mL of material. Does this confirm or deny your hypothesis?

· Compare your results with two other groups. What affect does the shape of a cone (e.g., short and wide vs. tall and narrow) have on the relationship between the volume of the cone and the volume of the cylinder with the same radius and height?

· What would be an appropriate formula for the volume of a cone having radius r and height h?

· Construct a cylindrical container with a different radius that holds exactly 250 mL.

· Construct a different square-based prism that holds exactly 250 mL.

· When your containers are built and your calculations are written up in good form, ask your teacher to watch you pour your 250 mL of material into each of your new containers.

Assessment/Evaluation Techniques

Students could hand in a “log of problem-solving ideas” that leads them to their discoveries. This could be assessed using a rubric focussing on Thinking/Inquiry and Communication.

Students could design a similar problem in a real-world context, solve the problem, and present the problem and solution to the class. This could be peer-assessed for communication and teacher-assessed for application of knowledge.

Accommodations

The teacher could use any of the following accommodations: help individual students to get started; walk student through “small chunks” of tasks; provide both written and oral instructions; increase time lines; modify or provide additional structure for the extension component.

Activity 3.4: Why Do Elephants Have Big Ears?

Time: 150 minutes

Description

In this activity, students investigate volume and surface area of square prisms in order to explain the significance of optimal surface area and volume to heat loss. They gather and analyse data to model the relationship between an elephant’s surface area and volume. They form ratios that lead them to answer the question “Why do elephants have large ears?”, as well as other relevant questions such as “Why shouldn’t dogs or babies be left in cars in the summer heat?” Students also investigate the relationship between surface area and volume of rectangular prisms.

Strand(s) and Expectations

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

Specific Expectations: MG1.01, .02, .03, MG2.01, .02, .03, RE1.01, .04, .05, .06, .07, NA1.01, .03, .04, .05.

Planning Notes

· Copies of the student worksheet Why Do Elephants Have Big Ears? (from Math Mania magazine) must be copied or downloaded from the web site given in the Resources. Alternatively, a teacher creates a worksheet that has students examine the relationship between surface area and volume of a cube as the dimensions are varied.

· Copies of the Can You Do Better worksheet for each student.

· Students use a graphing calculator or spreadsheet to do the Can You Do Better activity.

Prior Learning Required

· Ratio and rates

· Formulas for surface area and volume

· Using a spreadsheet or graphing calculator

Teaching/Learning Strategies

Teacher Facilitation: Model the size and shape of an elephant using a cube to represent the shape of an elephant. Discuss the setup of the surface area to volume ratio that is examined. Prompt the students as they complete the activity.

Student Activity: Complete the Why Do Elephants Have Big Ears investigation

Follow-up Activity

Teacher Facilitation: Students can use spreadsheets or the LISTS functions on a graphing calculator to calculate their data for the chart below. If using LISTS, students may need assistance to enter l in L₁ as a sequence using LIST,OPS,seq(A, A, 1, 8, 1), w in L₂ using the formula “64/(4xL₁)”, and surface area in L₃ as “2xL₁xL₂+8I(L₁+L₂)”. They do a scatterplot graph on L₁ and L₃. Whole class discussion is needed after students have answered Part A to confirm their findings. Before the students start Part B assign them a different fixed height (1 to 8). Ensure each fixed height is done by at least two students.

Student Activity: Can You Do Better?

Students create a chart similar to the one given below, and answer the questions.

Part A: Using a fixed volume of 64 cm³, calculate each surface area by varying the dimensions of the base where the height is fixed at 4 cm.

V	h	l	w	Surface Area
64	4	1
64	4	2
. .	. .	. .
64	4	8

9. Examine the values for the surface area. What is the relationship between the surface area and the various lengths?

· Graph the relationship between length and surface area.

· For which length is the surface area the smallest? Why might this be important?

· Is there anything special about this dimension?

Part B: Can you do better if you changed the fixed height?

10. From your teacher obtain your new fixed height to investigate. Redo the calculations and questions in Part A.

· What was your smallest surface area? Compare it to the optimal surface area in Part A. Did you do better? (i.e., Was it smaller?)

· Compare your results to the results from other students.

· Make a hypothesis about the dimensions of a rectangular prism and optimal surface area.

Part C: The Big Chill Drink Company has hired you to design a new drink container to hold 216 mL. What dimensions would you suggest they use? Justify your answer. Is it realistic? Why or why not?

Homework

· Other optimization problems involving surface area and volume of other shapes as found in the textbook.

· Ratio and rate problems.

Assessment/Evaluation Techniques

Use a rubric to observe the students using their Inquiry and Thinking skills. A written report for Can You Do Better? can also be evaluated using a rubric to assess Knowledge, Application, and Communication skills. Responses to Part C. lend themselves to the possibility of Level 4 if part of the student’s answer is “Realistically, a cube is not the best shape because...”. Paper and pencil tasks can be evaluated as well.

Resources

Math Mania magazine (Feb, 1999)

Math Mania web site

http://www.mathadventures.com

Activity 3.5: Thinking Out of the Box Assessment Activity

Time: 150 minutes

Description

In this activity students develop and apply their knowledge of surface area and volume of rectangular and triangular prisms, cylinders, and cones through a package design activity.

Strand(s) and Expectations

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

Specific Expectations: MG 1.04, MG 2.02, MG2.03, MG2.04, MG2.05; NA1.03, .04, .05, .06; NA2.02; NA3.03, .05.

Planning Notes

Students work in groups of three. Each group should initially be given (or be responsible for bringing in) a cereal or cracker box. Students should have adequate supplies such as construction paper, markers, compasses, rulers, etc. Access to computer spreadsheet software would also be helpful.

Prior Learning Required

Before beginning this activity students need to know how to draw nets, and calculate volume and surface area for three-dimensional figures particularly rectangular and triangular prisms, and cylinders. Students should also be familiar with substitution into formulas, order of operations, and the use of scientific calculators for evaluating expressions.

Teaching/Learning Strategies

Teacher Facilitation: Before beginning this activity, the teacher may want to set the stage by talking about the kinds of products come in rectangular prisms and those products that do not (e.g., tennis balls, Toblerone chocolates, etc.).

Student Activity: Thinking Out of the Box

Your quality control team has been given the task to redesign your given container. Your new design must be a different shape (not a rectangular prism) yet must have the same volume as the current container.

Your final submission must include:

i) the current container, drawings of the net of the current container as well as a 3-D drawing;

ii) full calculations of the surface area and volume of the current container;

iii) 3-D drawings, and drawings of the net of the new container;

iv) full calculations of the surface area and volume of the new container;

v) a print-out of an investigation of using different dimensions for your new container. This includes calculations of surface area and volume. The volume must be constant and match the original container.

vi) a model of the new container complete with appropriate graphic design;

vii) a written proposal as to why your new design is worthy of consideration;

viii) a presentation to the Board of Directors of the company (the class fills this role) convincing the Board that your new design should win you a contract.

Extension:

Both your current and new packages are shipped to grocery stores in cartons that hold 24 of your containers. Investigate what shape you think the current shipping carton looks like. Design a carton that efficiently holds 24 of your re-designed containers. Which shipping carton uses less corrugated cardboard - the one for the old container or for the re-designed container?

Assessment/Evaluation Techniques

While students are working on this activity, teachers could be circulating and gathering observational data on Learning Skills such as: Teamwork, Organization, and Initiative.

Once the activity is completed, groups could trade their submissions with a “buddy group” and check one another’s calculations. Each group could then present their new design proposal to the class and put them on display for peer assessment. A checklist could be used for this peer assessment which includes:

· accuracy of calculations (checked by buddy group) of volume and surface area for both the new and old packages;

· clear, well-labelled drawings and diagrams;

· maintenance of required volume while decreasing the surface area;

· appropriate logo and shape of container for product.

The teacher reviews the written proposals as well as the peer assessment of calculations. When reviewing the written proposals the teacher should be looking for the appropriate use of measurement terminology (volume and surface area) and units, and the quality of the investigation by referring to the variety of solutions that were tested (with a spreadsheet or by hand).

Activity 3.6: Exploring with The Geometer’s Sketchpad™

Time: 150 minutes

Description

In this activity, students explore and review angle relationships as they become familiar with dynamic geometry software. Using the student worksheet Angle Relationships, students sketch angles relationships, make hypotheses about the relationships, and verify or refute the hypotheses using dynamic geometry software. This worksheet also places the angle work in contexts such as the design of bridges and roads, the path of a billiard ball, and Mexican blanket designs to demonstrate the application of geometric relationships.

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Specific Expectations: MG3.01, MG3.04, MG3.05.

Planning Notes

· A computer lab is required to complete this exploration. Teachers should familiarize themselves with the software so that technical difficulties can be anticipated.

· This activity is best done individually or in pairs, with each student maintaining an individual journal of diagrams and angle relationships.

Prior Knowledge Required

Geometry and Spatial Sense Grade 8: identify the angle properties of intersecting, parallel and perpendicular lines by direct measurement (interior, corresponding, opposite, supplementary and complementary angles); explore the relationship to each other of the internal angles in a triangle; solve angle measurement problems involving properties of intersecting line segments, parallel lines and transversals; create and solve angle measurement problems for triangles.

Teaching/Learning Strategies

Discuss the following uses of mathematics in career situations to the class, in order to build a context for the exploration:

A civil engineer designs bridges, buildings and networks of roads. A draftsman makes scale drawings of the designs. Angles and triangles are very important to both the engineer and the draftsman as they model situations for the structures that they are designing. The special properties and relationships found in triangles and angles help in the design and stability of visually appealing buildings and safe roads. Computer Assisted Design (CAD) programs are used to draw geometrically-exact drawings. We use The Geometer’s Sketchpad™ to construct models that mimic some of the engineering design work as we review angle relationships.

Teacher Facilitation: Show students aerial photographs of road patterns, maps of roads and buildings with intricate geometric designs as you discuss the use of angles and shapes in construction. Discuss drafting and design as done with and without the aid of computers. You may wish to demonstrate some of the capabilities of The Geometer’s Sketchpad™ for your students, or simply let them investigate and learn the program on their own, under your guidance.

Student Activity: Students use the dynamic geometry software and the Angle Relationships worksheet to explore angle relationships, form hypotheses, verify or refute the hypotheses, and communicate the results.

Extension:

11. Draw the path of a billiard ball on a pool table for 6 bounces off the side of the table. Remember that the angle the ball hits the wall equals the angle it bounces off of the wall.

· Draw a map using angle relationships and skills from The Geometer’s Sketchpad™. Include as many different angle relationships as possible.

· Many different cultures use geometric designs in their crafts. Find examples of Mexican blanket designs, Native Canadian beadwork and basket work, etc and make a geometric design similar to one of them using a variety of different angle and triangle relationships.

· Determine angle measures in multi-step problems from diagrams found in the textbook.

Assessment/Evaluation Techniques

· Teacher Observation of Students at Work in Problem Solving (see rubric in Appendix - Phase 1)

· Written report on angle relationships

· Paper and pencil assessment tasks determining angle measures in individual and multi-step problems

Resources

Exploring Geometry with The Geometer’s Sketchpad. Key Curriculum Press.

Appendix: Angle Relationships Student Worksheet

Use The Geometer’s Sketchpad to draw each angle relationship below. Make a conjecture about each type of angle. Use Sketchpad’s calculator to confirm or deny your conjecture, then test by dragging points on the diagram. Copy each diagram into your notebook and complete a statement about the angle relationship. (Some construction hints are provided.)

Supplementary Angles:

Make a hypothesis about the relationship between the 2 supplementary angles, ÐDCB and ÐACD.

Check your hypothesis by using The Geometer’s Sketchpad calculator, following this method: From the Measure menu, choose Calculate and a calculator appears on the screen. Click on the workspace where you see m ÐDCB and it will appear on the calculator screen. Click the + sign, and then click the work space where m ÐACD appears. The calculator should now display the sum m ÐDCB + m ÐACD. Click OK. You return to the workspace and the sum will be calculated on the screen for you.

Test your hypothesis in a variety of different cases by dragging the point D to the right and left. Watch the angle measurements change as you drag the point. Examine the sum as the angle sizes change.

Complementary Angles:

Make and check a hypothesis about the complementary angles, ÐCDE and ÐEDB.

To construct the perpendicular line, select point C and line AB. Choose Perpendicular line from the Construct menu. Use the point tool to place a point at D. Drag point E to test all cases of your hypothesis.

Vertically Opposite Angles:

Use the line segment tool to draw two intersecting lines. Select both lines. From the Construct menu, choose Point of Intersection.

Make a conjecture and then measure all 4 angles to verify the conjecture. Drag point D to test your conjecture for all cases.

Angle Relationships in Parallel Lines:

When houses and other buildings are built, parallel beams are used to support the building. The crossbars or transversals provide strength to the vertical beams. Three special angle relationships exist within parallel lines. Let’s investigate!

To draw parallel lines, select both a point and a line and choose Parallel Line from the Construct menu. Draw a transversal KL.

Write a hypothesis about a pair of angles that might be equal. Verify your hypothesis by measuring the angles and then dragging the transversal to ensure the accuracy of your hypothesis.

Can you hypothesize and verify 3 different angle patterns? Hint: The angles appear in patterns that resemble F (corresponding angles), Z (alternate angles) and C (interior angles).

An easier way to make parallel lines is by using transformations: Drag a line. From the Transform menu, select Translate. Click OK. (You may alter the magnitude if you wish.)

Sum of the Angles in a Triangle:

Write a hypothesis about the sum of the angle measures of the triangle.

Verify your hypothesis by determining the measure of each angle of the triangle and using Sketchpad’s calculator.

Drag a vertex of the triangle and watch the angle measures change. Does the sum change?

Exterior Angle of a Triangle:

Measure the external angle ÐBAD above and compare this measure to the sum of the measures of the two interior angles of the triangle that are not adjacent to ÐBAD. Make a hypothesis about the exterior angle of a triangle. Test the hypothesis by dragging various points in the diagram.

Angles of an Isosceles Triangle:

To construct an isosceles triangle, draw a line segment AB. Click on point A. From the Transform menu, Mark Center “A” as the center of rotation. To rotate the line segment AB, select AB. From the Transform menu, select Rotate and choose an angle of 80°. A rotated segment appears. Place a point at C and then add the third side of the triangle.

Make a hypothesis about the angles in the isosceles triangle. Measure, drag vertices, and verify your hypothesis.

If you were to draw an equilateral triangle, what angle of rotation must be used? Make an equilateral triangle, and check your accuracy by measuring all sides and angles.

There are three different ways to construct an isosceles triangle:

12. using rotations;

· using reflections (under the Construct menu);

· using a circle. (Remember that radii of a circle are equal. When you draw the triangle inside the circle, use Hide Circle from the Display menu.)

Construct an isosceles triangle using each of the methods listed above. Determine situations for which each method is best suited and least suited.

Activity 3.7: If ... Then ...

Time: 300 Minutes

Description

Using the worksheets provided in the Appendix, students develop and extend “If... then...” hypotheses for a variety of geometric figures. To do this, they explore geometric properties, form an hypothesis related to their exploration, test the hypothesis using dynamic geometry software, generalize the results, and communicate the results using “If... then...” format.

The specific topics of the student investigation worksheets are:

· Investigating Midpoints and Medians

· Investigating Perpendicular Bisectors

· Investigating Angle Bisectors

· Investigating Polygons

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Specific Expectations: MG3.01, .02, .03, .04, .05.

Planning Notes

· A context is provided for these activities through a variety of different methods:

a) Complete a variety of paper folding activities that leads to hypotheses about geometric properties that can be verified using geometry software

b) Provide a pictorial display of geometrically-based shapes and designs from the world around us. Through discussion, develop curiosity about geometric characteristics and relationships in the shapes: Are the lines the same length? Are the angles equal? Do they always intersect? Are they bisected? What if...?

c) Make paper and pencil constructions using compass and protractor. Form hypotheses as well as illustrate the accuracy, time, and skill required in comparison to the ease of geometry software.

d) Use MIRA reflections to formulate hypotheses and refute or verify using geometry software.

· A computer lab with dynamic geometry software is required to complete the activities. Teachers should familiarize themselves with the geometry software that technical difficulties can be anticipated.

· These activities are best done in pairs, with each student maintaining an individual journal of diagrams, hypotheses and generalizations in the “If… then...” form.

· Homework assigned each day could be explorations using paper-folding, compass/protractor, or MIRAs to develop hypotheses which they investigate the following day using dynamic geometry software. At the end of each computer lab session, some journal writing and paper/pencil tasks could be assigned from the textbook.

Prior Learning Required

Geometry and Spatial Sense Grade 7 and 8: Identify, describe, compare, and classify geometric figures. Identify congruent and similar figures. Identify and investigate the relationships of angles. Construct and solve problems involving lines and angles.

Teaching/Learning Strategies

One sample activity, Midpoints of Quadrilaterals, found in the investigations provided on the student worksheets in the Appendix, is examined in detail, to model the teacher set-up and facilitation that can be followed for the other investigations. The steps Explore, Form a Hypothesis, Test the Hypothesis, and Communicate the Results should be an integral part of each investigation. Students could pair/share their hypotheses and homework exploration results before investigating on the computer. The teacher should ensure closure and consolidation of newly developed concepts by leading class discussions on the concepts.

Teacher Facilitation: (This is a sample introduction to lead to the investigation of midpoints of a quadrilateral on the student worksheet in the Appendix.) Use a paper cutter to create a variety of quadrilaterals to distribute to the students. Draw to the students’ attention the fact that they all have different quadrilaterals.

Explore:

Hold two adjacent vertices of your quadrilateral, one in each hand. Bring your hands together so that the vertices are together. Crimp the paper at the midpoint between the vertices. Repeat step 1 for each pair of adjacent vertices, until all four midpoints have been located. Fold your paper between adjacent midpoints. You should have four such folds, creating a quadrilateral. Label this quadrilateral QUAD.

Form a Hypothesis: Make a hypothesis about the quadrilateral QUAD.

Test Your Hypothesis: Identify measurements that are needed to confirm or refute your hypothesis.

Using Sketchpad, draw any quadrilateral as illustrated below. Construct Point at Midpoint on each of the four sides. Label the midpoints, in order, Q, U, A, and D. Construct a segment between adjacent midpoints to create quadrilateral QUAD. Do a drag test of your construction.

Appendix: Investigating Midpoints and Medians:

A midpoint of a line segment is a point that divides a line segment into two equal parts. To construct the midpoint of a line segment, select the line and choose Point at Midpoint from the Construct menu.

A median is a line that joins a midpoint to the vertex directly across from the midpoint. To construct a median, determine the midpoint first, and then draw the median, connecting the midpoint to the opposite vertex.

In the following activities, we investigate midpoints and medians.

Midpoint Lines of a Triangle:

Draw a scalene triangle and construct the midpoints of two of the sides. Join the midpoints with a line.

a) Examine the angles created where the midpoint line intersects the two sides of the triangle. Compare these angles to the angles at the base of the triangle.

Measure the length of the midpoint line and compare it to the length of the third side of the triangle.

b) Draw an appropriate diagram in your notebook to illustrate the statement “If a line joins the midpoints of two sides of a triangle, then...”

c) Determine the third midpoint. Construct the midpoint triangle by joining the midpoints. Make a hypothesis about the ratio of the area of the midpoint triangle to the area of the entire triangle.

To measure the area of the midpoint triangle, select all three vertices of the triangle and choose Polygon Interior from the Construct menu.

When the interior is shaded, its area can be determined by choosing Area from the Measure menu. Verify your hypothesis about the ratio of the areas of the midpoint triangle and the original triangle.

d) Prove that the midpoint triangle is congruent to rEAG and rDEF and rFGB using rotations: Determine the midpoint of line segment EG. Set this midpoint as the center of rotation by choosing Mark as Center from the Transform menu. Select the midpoint triangle and determine the angle of rotation that will place it directly on top of rEAG, thereby proving that the two triangles are congruent. (Use the Undo command found under the Edit menu if you make an error.) Find other midpoints and rotations to prove the remaining triangles congruent to the midpoint triangle. Explain your answer to part c) above using what you just discovered about the congruent triangles.

Midpoints of Quadrilaterals:

a) Draw a quadrilateral. Determine the midpoints of each of the 4 sides. Join the midpoints in order to construct another quadrilateral. Make a conjecture about the type of quadrilateral formed by joining the midpoints. Verify your conjecture. Complete and illustrate the statement: “If the midpoints of a quadrilateral are joined, then...”

b) Determine if there is a relationship between the areas of the midpoint quadrilateral and the original quadrilateral. If a relationship exists, write an “If...then...” statement about the relationship.

c) Repeat the midpoint quadrilateral investigation but alter the original quadrilateral so that it is a:

1) parallelogram 2) a rectangle 3) a square 4) a rhombus.

Make “If... then...” statements for each case and draw the corresponding diagram in your notebook.

Median of an Isosceles Triangle:

Construct an isosceles triangle. Determine the midpoint of the base of the isosceles triangle. Draw the median by joining the midpoint to the vertex of the vertical angle. Conjecture and verify if any angle equalities exist in the diagram.

Complete the statement: “If the median to the base of an isosceles triangle is drawn, then...”

Explain why the safest position for a step ladder is standing spread open on flat ground.

Describe other situations where isosceles triangles, midpoints, and medians are used in the world around us.

Medians of a Triangle:

a) Draw a scalene triangle. Construct the medians to each of the three sides. The medians intersect at a point called the centroid. The centroid is the center of area and volume, and acts as the centre of gravity of the triangle.

b) Print out the triangle with the centroid located. If you cut out the triangle and glue it to a piece of cardboard, you will be able to balance it on the tip of a pencil using the centroid’s property as the center of gravity.

c) The centroid (G) divides each median into a longer and a shorter segment. Make a conjecture comparing the length of the shorter segment to the longer segment. How many smaller segments will fit into the larger segment? Verify your prediction. Write the relationship as a ratio.

Appendix: Investigating Perpendicular Bisectors

A perpendicular is a line that intersects another line segment at a 90° angle.

Given a line segment and a point not on the line, construct a perpendicular to the line from the point. (Select both the point and the line. Choose Perpendicular from the Construct menu.)

If the perpendicular passes through the midpoint of the line, it is called the perpendicular bisector. To construct the perpendicular bisector of a line, determine the midpoint and construct the perpendicular through it. Measure to ensure that the line segments are equal and the angles measure 90°. In your notebook, illustrate and define a perpendicular line and a perpendicular bisector.

We will now investigate several useful properties of the perpendicular bisector.

A Point Located on the Perpendicular Bisector:

Construct a point X anywhere on the perpendicular bisector. Construct line segments AX and BX to measure the distance from the point X to the endpoints of the line segment. Does a relationship occur between the lengths of AX and BX? Make and verify a conjecture. Test your conjecture by dragging X along the perpendicular bisector, and examining what occurs as the lengths of AX and BX change.

In your notebook, draw a diagram and complete the following statement: “If a point lies on the perpendicular bisector of a line segment, then...”

Perpendicular Bisectors in Triangles:

a) Draw an acute triangle ABC. Construct the perpendicular bisectors of each of the sides BC, AB and AC. What do you notice?

b) Name the point of intersection O. Construct a circle, center at O, and radius OA. The circle should also have radii OB and OC too! The point O is called the circumcenter of the triangle.

c) Explain why O is equidistant from the three points A, B, and C. Include your conclusions from the previous question in your explanation.

d) Repeat the investigation for an obtuse-angled triangle. Are the results the same as for the previous investigations?

e) Repeat for a right-angled triangle. What interesting phenomenon occurred with the circumcenter?

f) A fire hydrant must be located the same distance from three different houses in a subdivision. Explain how you would determine the location of the hydrant.

g) Describe other situations where you would use the circumcenter to solve a problem.

The Perpendicular Bisector of the Base of an Isosceles Triangle:

Construct an isosceles triangle rABD. Construct the perpendicular bisector of the base.

If the perpendicular bisector of the base of an isosceles triangle is drawn, then it will pass through the vertical angle and bisect the vertical angle.

What would you measure to verify the statement? Measure and verify.

In your notebook, draw an appropriate diagram and label all of the equal measures on the diagram.

For what other types of triangles is this statement true? Construct a right-angled triangle and one other triangle for which the statement is true.

Perpendicular Bisectors as Diagonals of Quadrilaterals:

Draw a line segment AB. Determine C, the midpoint of AB. Mark C as the center of rotation and rotate the line AB 90° using the Transform menu. The new line DE is the perpendicular bisector of AB, and, AB is the perpendicular bisector of DE. Join the 4 endpoints to form a quadrilateral AEBD with diagonals AB and DE. Make a hypothesis about the type of quadrilateral formed by the two diagonals. Verify your hypothesis. In your notebook, draw an appropriate diagram and complete this statement: “If the diagonals of a quadrilateral are equal in length and perpendicular bisectors of each other, then...”

If the diagonals of the quadrilateral are perpendicular bisectors of each other, but the diagonals are not equal in length, what type of quadrilateral is formed? Make a hypothesis, verify your hypothesis, and place a diagram and statement in your notebook.

Perpendicular Bisectors in Circles:

Construct a circle. Draw a chord in the circle. Construct the perpendicular bisector of the chord. Examine the diagram and make a hypothesis about the perpendicular bisector of a chord. Construct a second chord to verify your hypothesis.

Complete this statement: “If the perpendicular bisectors of 2 chords of a circle are drawn, then...”. Find a situation for which this statement is not true for two chords of a circle.

If you were given a circle cut out of paper, explain how you would find the centre of the circle by paper folding.

A carpenter has cut a circular table top for a pedestal (one-legged) table. Explain how the centre of the tabletop can be determined in order to attach the leg.

Describe a realistic situation where you would need to find the centre of a circle.

Appendix: Investigating Angle Bisectors

An angle bisector is a line that divides an angle into two equal parts. To bisect an angle ABC, select the vertices in order and then choose Angle Bisector from the Construct menu. Verify that two equal angles were constructed, by measuring each angle.

The following activities provide opportunities investigate bisectors of angles, while making and verifying hypotheses.

A Point on the Angle Bisector:

Construct the angle bisector of an acute angle. Place a point D on the angle bisector. To measure the distance from the point to the line BC, construct a perpendicular from D to the line. (The shortest distance is always the perpendicular distance.) Measure this distance. Determine the distance from D to the other arm of the angle.

Make and test a hypothesis about the distance from any point on the angle bisector to the arms of the angle. Test your hypothesis by dragging the point D along the angle bisector.

Test your hypothesis on obtuse, right, and straight angles. Draw an appropriate diagram in your notebook and complete the statement: “If a point lies on the bisector of an angle, then...”

Bisecting the Vertical Angle of an Isosceles Triangle:

Construct an isosceles triangle ABC. Construct the bisector of the vertical angle. Examine the point where the angle bisector intersects the base of the triangle. Make a hypothesis about the angles created at the base of the triangle. Test your hypothesis for isosceles and non-isosceles triangles.

Draw an appropriate diagram in your notebook and complete the statement: “If the vertical angle of an isosceles triangle is bisected, then...”

Angle Bisectors of a Triangle:

Construct a scalene triangle. Bisect two of the angles of the triangle. Select both bisectors and find the point of intersection by using the Construct menu. Bisect the third angle. All three of the angle bisectors intersect at a point called the incenter.

An inscribed circle is a circle drawn inside a triangle, touching each side of the triangle at only one spot. The incenter is the center of the inscribed circle. Use the intersection point of the angle bisectors to draw the inscribed circle using the circle tool. Repeat the construction to investigate whether an inscribed circle can be drawn in obtuse-angled, right-angled and isosceles triangles. Draw an appropriate diagram in your notebook and complete the statement: “If an incircle is to be constructed, then...”

A landscape designer must determine the location of the largest possible circular pond that will fit into a triangular piece of lawn. Explain how the location can be determined. Use a diagram with your explanation.

Appendix: Investigating Polygons

Construct a pentagon by rotating a line segment five times. What angle of rotation must be used?

Place points at the tip of each line segment and at the center. Join the endpoints to make the pentagon, then hide the construction lines.

From the center, construct perpendicular lines to each side in order to measure the distance to each side. Record the distance. Calculate the sum of the distances from the center to the sides.

Pick another point P anywhere on the interior of the pentagon. Construct perpendiculars to each of the 5 sides. Measure the total distance from point P to all of the sides. Compare your answer to the previous question.

In a pentagon, draw line segments joining alternate vertices to form a star. What figure is formed by the inner part of the star? Repeat this process and examine the resulting pattern.

Investigating the Hexagon:

Construct a hexagon by rotating a line segment 6 times. What angle of rotation must be used? What type of triangles were formed in making the hexagon? (This might suggest another method to make a hexagon!) The hexagon has been divided into 6 congruent sections!

Investigate the hexagon to determine if it has the same characteristics discovered above for the pentagon. Write the results of your investigations in your journal.

The diagrams below illustrate other methods of dividing the hexagon into 6 congruent figures. Use angle theorems, midpoints, and parallel lines to draw line segments to divide the hexagon into 9 congruent figures, 12 congruent figures, and 24 congruent figures.

Exterior Angles of Polygons:

Plot a point A and draw a ray. Drag the point B along the ray AB and draw a ray from B. Select points C and A and use the Construct menu to construct ray CA.

Select ray AB and use the Construct menu to construct point D on the ray, dragging the point to create exterior angle ÐDBC. Similarly, construct exterior angles ÐECA and ÐFAB.

Measure each exterior angle. Calculate the sum of the angle measures using the Measure, Calculate menu.

Create a table to display the angle measures and sum.

Drag one of the external points to alter the angle measures. Add at least ten entries to the table, including all extreme and special cases. Record your results in your journal before going on.

Use the steps above to construct and explore the sum of the exterior angles of a quadrilateral.

Make a hypothesis about the exterior angles of a polygon.

Test your hypothesis by constructing and analysing a:

a) pentagon;

b) hexagon;

c) another polygon.

Generalize the results, stating your conclusions in “If... then...” format.

Sum of the Interior Angles of a Polygon.

Design an investigation to determine the sum of the interior angles of a polygon. Use the steps Explore, Form a Hypothesis, Test the Hypothesis, Generalize, and Communicate the Result to prepare a convincing argument. Develop two different approaches and clear instructions for each approach, in order to explain your generalization to a fellow student.

Sum of the Interior and Exterior Angles of Regular Polygons

Design, carry out and report on an investigation to determine the sum of the interior angles of a regular polygon, and the sum of the exterior angles of a regular polygon.

Activity 3.8: Script It

Time: 225 minutes

Description

Students create scripts using dynamic geometry software to demonstrate and review geometric properties that they have discovered in previous activities.

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Expectations: MG 3.02, .03, .04.

Planning Notes

There are two different methods for making a script in The Geometer’s Sketchpad™. One way is to create a sketch and then make a script afterwards. Another method is to create the script as the sketch is being made. To start, it would probably be better to do the former so that students have completed a satisfactory sketch before they make a script. It is recommended that the teacher work through this activity in advance to familiarize themselves with the process, if needed. It is relatively easy but the “bugs” should be worked out first.

Teaching/Learning Strategies

Student Activity: Script It

In this activity you will be reviewing material that you learned in earlier activities while you create a script to record the commands used for your constructions.

13. Using The Geometer’s Sketchpad™, construct a triangle. Then construct an inscribed circle.

· You are now going to save the construction that you just made as a script. A script is a set of instructions that can be saved and replayed to demonstrate the construction process. Sketchpad writes the instructions for the construction you just made. Here is what you need to do:

a) Use the point tool to point to all of the elements of your construction or, to make it easier, and to make sure that you don’t forget anything, you can draw a rectangle around the entire sketch and all of the elements will be automatically selected.

b) Click on Work in the upper tool bar. Highlight Make Script.

c) You will see a script appear on the right hand side of the screen.

d) To play the script you need to start another Sketch. Start the sketch with three points and select those three points. Notice that the script lists three points as “Given”. This means that those must be given initially before running the script.

e) Now, click on Play in the Script window. Watch the construction begin.

f) You can save your script (rename it if you want) so that it can be replayed again.

· Now, create a script to construct a circumscribed circle

Teacher Facilitation: An advantage of creating scripts is that students can save the script and demonstrate their constructions for the teacher (or a peer), at any later time. This way, the teacher can assess a student’s constructions by having the student play the script when the teacher circulates to that student’s workstation.

Once students know how to create a script, they can go back over previous work and create scripts to illustrate various geometric properties that they have learned and test some of their conjectures. Teachers may want to select specific activities for students to revisit as needed.

Assessment/Evaluation Techniques

This final assessment activity should include the opportunity to use The Geometer’s Sketchpad™ scripts to demonstrate fluency with geometric properties and relationships as well as the ability to explore and conjecture.

Several ideas for assessment tasks, from which teachers can choose, are listed or referenced below:

14. Students use geometric constructions to complete a circle in “Oops! Glass Top” in the Assessing Mathematical Understanding and Skills Effectively: Harvard Assessment Tasks (See reference below). Assessment criteria and exemplars of written (rather than software) solutions are provided in “Glass Top”, Advanced High School Assessment Package I: Balanced Assessment for the Mathematics Curriculum (See reference below).

· Students use angle properties to write a set of instructions for hitting one billiard ball to rebound to hit another given certain criteria and a diagram of a billiard table with two billiard balls, A and B, located on opposite sides of the table. “Bouncing Off The Walls”, Harvard Project: Using the diagram of the billiard table, write a set of instructions (script) for:

· Hitting ball A so that it bounces exactly once off the north or south wall before hitting ball B.

· Hitting ball A so that it bounces exactly once off both the north and south walls before hitting ball B.

· Hitting ball A so that it bounces exactly once off the north or south wall and exactly once off the east or west wall before hitting ball B.

· Circle geometry can be assessed through the task of “Circling Trains”, Harvard Balanced Assessment in Mathematics Project (see Resources). In this activity, students design the placement of amusement park attractions and pathways by placing attractions on circles and creating pathways between the attractions. They then continue these constructions given certain criteria.

· “Mirror, Mirror II”, Harvard Balanced Assessment in Mathematics Project, requires students to use angle properties to determine the placement of a mirror so that spotlights can illuminate a statue even though there is a wall between the light and the statue.

These activities can be assessed for Knowledge/Understanding of the geometric properties, Thinking/Inquiry as teachers observe students solving the problems, Application as students apply their understanding of geometric properties to new contexts, and Communication as students explain or justify their solutions.

Resources

Berkeley, Harvard, Michigan State, and Shell Centre. Advanced High School Assessment Package 1: Balanced Assessment for the Mathematics Curriculum. White Plains, N.Y.: Dale Seymour Publications. 1999. (Assessment criteria and exemplars of written (rather than software) solutions are provided.)

Harvard Balanced Assessment in Mathematics Project. Assessing Mathematical Understanding and Skills Effectively. 1996.

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