Appendix A: Activity sheets

Activity Topic Cabri tools introduced Math concepts reviewed or introducedA1 Constructing a

parallelogramPolygon, Pointer, Segment, Parallel Line, Point, Hide/Show

Definitions: Quadrilateral, parallelogram.

A2 Properties of parallelograms

Angle measure, Distance measure.

Properties of parallelograms: Opp. angles congruent, Adj. angles supp., Opp. Sides congruent, Diagonals bisect each other, etc.

A3 Redefining “parallelogram”

Rotate, Circle, Midpoint, Translate, Vector

B1 Angle sums of triangles

Triangle, Polygon, Label, Angle measure, Distance, Calculate…, Area.

Triangle properties, triangle inequality, area & perimeter, angle sum

B2 Angle sums of quadrilaterals

Quadrilateral properties, angle sum

B3 Angle sums of polygons

Angle sum in general, number of diagonals in general

C1 Equidistance: Point Point, Label, Pointer, Distance, Trace, Circle

Circle as locus

C2 Equidistance: Two Points

Perpendicular Bisector Perpendicular Bisector as locus

C3 Equidistance: Two Lines

Angle Bisector Angle Bisector as locus

C4 Equidistance: Triangle

Segment, Triangle Special lines in triangles: Perpendicular bisectors and angle bisectors.

D1 Review of established knowledge

None Perpendicular Bisector as locus, circumcentre, Angle sum of polygons, number of diagonals of polygons, triangle midpoint theorem, Varignon parallelogram.

D2 Explanations: Perpendicular Bisector

None Perpendicular Bisector as locus, circumcentre.

D3 Explanations: Polygons

None Angle sum of polygons, number of diagonals of polygons

D4 Explanations: Varignon

None Triangle midpoint theorem, Varignon parallelogram.

E Perpendicular Bisectors of Quadrilaterals

None None

Page 1

Activity A1: Constructing a parallelogram

1. Construct a quadrilateral (Use the Polygon tool under the Line menu)

2. Make your quadrilateral look like a parallelogram (Use the Pointer tool). Notice that it doesn’t remain a parallelogram when you drag the points around. It doesn’t pass the Drag Test. For a shape to pass the Drag Test it has to stay the same shape even if you drag it.

4. A textbook definition of a parallelogram is:“A Parallelogram is a quadrilateral with two pairs of parallel sides.”

We will use this definition to construct a parallelogram that will pass the Drag Test. a. Start by constructing a segment (Use the Segment tool under the Line menu). b. Now construct another segment that shares an endpoint with the first one. c. Construct a line, parallel to one segment, passing through the loose endpoint of the other segment (Use the Parallel Line tool under the Construction menu).

d. Construct a second line, parallel to the other segment, passing through the loose endpoint of the first segment.e. Construct a point at the intersection of the two parallel lines (use the Point Tool)f. Construct a quadrilateral using the four points. Hide the lines (Use the Hide/Show tool).

Does the parallelogram pass the Drag Test?

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Activity A2: Properties of parallelograms

1. Measure the interior angles of your parallelogram. (Use the Angle tool under the Measure menu. Note: Press F1 to get help using the tools. For Angle measure Help tells you that you need to click on three points, and the second one should be the vertex of the angle).

2. Drag your parallelogram and look for properties of the angles. a. Are there any congruent angles (Equal measures)?b. Are there any supplementary angles (Measures add up to 180°)?c. Are there any complementary angles (Measures add up to 90°)?

3. Measure the sides of your parallelogram (Use the Distance tool). Drag your parallelogram and look for properties of the side lengths. What properties did you notice?

4. Construct the diagonals of your parallelogram (use the Segment tool). Measure anything about them you think might be useful. Drag your parallelogram and look for properties of the diagonals.

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Activity A3a: Re-defining “parallelogram”

1. Use the properties you observed in Activity A2 to construct a parallelogram without using the Parallel Line tool. (You may find these tools useful: Rotate, Point Symmetry, Circle, Midpoint, and Compass.) How many different constructions can you find?

2. Come up with a new definition of “Parallelogram” based on one of your constructions.

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A

B

C

D

Activity A3b: Re-defining “parallelogram”

1. Alex and Terry made a list of properties that a quadrilateral ABCD might have:

a. Side AB is the same length as side CDb. Side AD is the same length as side BCc. Angle ABC is congruent to angle CDAd. Angle BCD is congruent to angle DABe. Diagonal AC goes through the

midpoint of diagonal BDf. Diagonal BD goes through the

midpoint of diagonal ACg. Angle BCD is supplementary to angle

CDAh. Angle DAB is supplementary to angle

ABCi. Side AB is parallel to side CDj. Side AD is parallel to side BCk. Angle ABD is congruent to angle BDCl. Angle DCA is congruent to angle CAB

They claim that a parallelogram must have all of these properties. Do you agree? Why or why not?

2. Alex says that if a quadrilateral has any two of these properties, it must be a parallelogram. Terry says this isn’t true. Who do you agree with? Why?

3. Use the properties you observed in Activity A2 and those listed by Terry and Alex to construct a parallelogram without using the Parallel Line tool. (You may find these tools useful: Rotate, Point Symmetry, Circle, Midpoint, and Compass.) How many different constructions can you find?

4. Come up with a new definition of “Parallelogram” based on one of your constructions.

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Activity B1: Angle sums of triangles

1. Construct a triangle (Use the Triangle tool or the Polygon tool under the Line menu).

2. Label the vertices of the triangle A, B, C (Use the Label tool).

3. Measure all the interior angles of the triangle.(Use the Angle tool under the Measure menu. Note: Press F1 to get help using the tools. For Angle measure Help tells you that you need to click on three points, and the second one should be the vertex of the angle).

4. Measure all the sides of the triangle (Use the Distance tool).

5. Calculate the sum of the lengths of the sides and the sum of the angle measures (Use the Calculate… tool).

6. Measure the area of the triangle (Use the Area tool).

7. Drag the vertices of the triangle around and observe how the measures change.

a. When angle ABC is the largest angle, which side has the largest length? Does this always happen?

b. Can you make a triangle in which one side is longer than the other two combined?

8. Create a second triangle and measure the same things. (Hint: to do this quickly choose Select All from the edit menu, then Copy, the Paste, then drag the new triangle to one side.)

a. When two triangles have the same area, are their perimeters the same?

b. When two triangles have the same perimeter, are their areas the same?

c. When two triangles have the same area, are the sums of their angle measures the same?

9. List any special properties of triangles you have observed:

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Activity B2: Angle sums of quadrilaterals

1. Construct a quadrilateral (Use the Polygon tool under the Line menu).

2. Label the vertices of the quadrilateral A, B, C, D (Use the Label tool).

3. Measure all the interior angles and sides of the quadrilateral (use the Angle tool and the distance tool under the Measure menu).

4. Calculate the sum of the lengths of the sides and the sum of the angle measures (Use the Calculate… tool).

5. Measure the area of the quadrilateral (Use the Area tool).

6. Drag the vertices of the quadrilateral around and observe how the measures change.

a. When angle ABC is the largest angle, which side has the largest length? Does this always happen?

b. Can you make a quadrilateral in which one side is longer than the other three combined?

7. Create a second quadrilateral and measure the same things.

a. When two quadrilaterals have the same area, are their perimeters the same?

b. When two quadrilaterals have the same perimeter, are their areas the same?

c. When two quadrilaterals have the same area, are the sums of their angle measures the same?

8. List any special properties of quadrilaterals you have observed:

9. Construct one diagonal of the quadrilateral (Use the Segment tool to join two non-adjacent corners). Can you connect what you have observed about quadrilaterals to what you observed about triangles?

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Activity B3: Angle sums of polygons

1. Construct a polygon with five or more sides (Use the Polygon tool under the Line menu).

2. Label the vertices of the polygon.

3. Measure all the interior angles and sides of the polygon.

4. Calculate the sum of the lengths of the sides and the sum of the angle measures.

5. Measure the area of the polygon.

6. Drag the vertices of the polygon around and observe how the measures change.

7. List any special properties of polygons you have observed:

8. How many diagonals does your polygon have? In general how many diagonals does a polygon with n sides have?

9. Can you connect what you have observed about polygons to what you observed about triangles and quadrilaterals?

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Activity C1 Equidistance: Point

Cabri has tools that can help you find locations that have specific properties (“Locus” is the math word for them). In this set of activities you will explore how to use Cabri to find special locations on the plain.

1. Alex and Terry want to put lots of stones in their garden so that every stone is 3 metres from their birdbath.

Make a conjecture: What shape will the stones form? Why do you think so?

2. In Cabri, construct a point (Use the Point tool).

3. Label the point “Birdbath” (Use the Label tool).

4. Make a new point and label it “Stone”.

5. Measure the distance from the birdbath to the stone (Use the Distance or Length tool).

6. Drag the stone so that it is 3 cm from the birdbath (Use the Pointer tool).

7. Select the Trace On/Off tool and click on the stone. This tool will leave a track behind the stone when it is dragged. (Hint: You can use Ctrl-F to erase the track).

8. Drag the stone, trying to keep the distance from the birdbath about 3 cm. Where can you drag it? What shape does it form?

9. Find a Cabri tool that will produce the shape Alex and Terry should use to place their stones. What tool should they use?

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Birdbath

Stone

3.00 cm

Activity C2 Equidistance: Two Points

1. Alex and Terry want to build a fence in their garden to separate Alex’s vegetables from Terry’s flowers. There are two wells, and they agree to put the fence between the two wells, so neither of them has to walk further than the other to water their plants.

Make a conjecture: Where will the fence be with respect to the wells? What shape will the fence be? Why do you think so?

2. In Cabri, construct two points. Label them “Alex’s well” and “Terry’s well”

3. Make a new point and label it “Fence”.

4. Measure the distance from the fence to Alex’s well and from the fence to Terry’s well. Drag the fence so that it is the same distance from Alex’s well as it is from Terry’s well.

Alex's well

Terry's well

Fence

4.56 cm4.56 cm

5. Select the Trace On/Off tool and click on the fence. Drag the fence, trying to keep the distance from both wells the same. Where can you drag it? What shape does it form?

6. Find a Cabri tool that will let you construct the fence just by clicking on the two wells (Hint: What properties does the fence have?)

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Activity C3 Equidistance: Two Lines

1. Alex and Terry want to plant bushes in their garden between two paths. The paths are straight and meet at an angle. They agree to plant the bushes so that they are the same distance from both paths.

Make a conjecture: Where will the bushes be with respect to the paths? What shape will the bushes make? Why do you think so?

2. In Cabri, construct two lines through a point.

3. Make a point and label it “Bush”.

4. Measure the distance from the bush to each path. Drag the bush so that it is the same distance from each path.

Bush

1.66 cm

1.66 cm

5. Select the Trace On/Off tool and click on the bush. Drag the bush, trying to keep the distance from both paths the same. Where can you drag it? What shape does it form?

6. Find a Cabri tool that will let you construct the location where they should plant the bushes just by clicking on points on the two paths. (Hint: What properties does the location of the bushes have?)

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Activity C4 Equidistance: Triangle

Alex and Terry have bought a new piece of land for their gardening projects. It is in the shape of a triangle, with fences all around and gates at each corner.

1. They want to build a shed to store tools and materials. Because they are equally likely to bring in materials through any of the three gates, they want to build the shed the same distance from all three gates.

Make a conjecture: Where should the shed be? What shape is made by the possible locations for the shed? Why do you think so?

2. The fields around Alex and Terry’s land are used for grazing horses. They like to feed apples to the horses so they want to keep a large barrel of apples in their garden. Because they are equally likely to feed the horses in each of the three adjacent fields, they want to place the barrel so that it is the same distance from each of the three fences

Make a conjecture: Where should the barrel be? What shape is made by the possible locations for the barrel? Why do you think so?

3. Use Cabri to investigate these two situations.

4. Describe how to construct exactly the locations of the shed and the barrel using Cabri’s tools. (Make sure your construction would work even if someone dragged the triangle.)

5. When would the barrel and the shed be in the same location? Why?

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Gate A

Gate B

Fence AB

Gate C

Fence AC

Fence BC

Activity D

Part 1

For each of these statements indicate whether you think the statement is always true, sometimes true, never true, or you don’t know.

Statement Always true

Some-times true

Never true

Don’t know

a. Any point on the perpendicular bisector of segment AB (the line through the midpoint at a 90 degree angle) is the same distance from the two points A and B.

b. If the perpendicular bisectors of two sides of a triangle ABC meet at the point O, then there is a circle with its centre at O that passes through A, B and C.

c. A polygon with n sides can be divided into exactly n – 2 triangles.

d. The sum of the angles in a polygon with n sides is 180(n –2).

e. A polygon with n sides has n(n – 3)/2 diagonals.

f. If M and N are the midpoints of AB and BC, two sides of the triangle ABC, then the line segment MN is parallel to the line segment AC and half as long.

g. If you join the midpoints M, N, O, P of the sides of any quadrilateral ABCD then the quadrilateral MNOP will be a parallelogram.

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Activity D

Part 2

For each statement: If you think it is always true, explain why it is always true. If you think it is sometimes true, give an example where it is true and an example

where it is not true. If you think it is never true, explain why it is never true. If you don’t know, say nothing.

You can always use Cabri to help you in your thinking.

2a. Any point on the perpendicular bisector of segment AB (the line through the midpoint at a 90 degree angle) is the same distance from the two points A and B.

2b. If the perpendicular bisectors of two sides of a triangle ABC meet at the point O, then there is a circle with its centre at O that passes through A, B and C.

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Activity D

Part 3

For each statement: If you think it is always true, explain why it is always true. If you think it is sometimes true, give an example where it is true and an example

where it is not true. If you think it is never true, explain why it is never true. If you don’t know, say nothing.

You can always use Cabri to help you in your thinking.

3a. A polygon with n sides can be divided into exactly n – 2 triangles.

3b. The sum of the angles in a polygon with n sides is 180(n –2).

3c. A polygon with n sides has n(n – 3)/2 diagonals.

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Activity D

Part 4

For each statement: If you think it is always true, explain why it is always true. If you think it is sometimes true, give an example where it is true and an example

where it is not true. If you think it is never true, explain why it is never true. If you don’t know, say nothing.

You can always use Cabri to help you in your thinking.

4a If M and N are the midpoints of AB and BC, two sides of the triangle ABC, then the line segment MN is parallel to the line segment AC and half as long.

4b If you join the midpoints M, N, O, P of the sides of any quadrilateral ABCD then the quadrilateral MNOP will be a parallelogram.

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Activity E

Note: When working on this problem you may use any of the following facts:1. The sum of the angles in a polygon with n sides is 180(n –2).2. The diagonals of a parallelogram meet at a point that is the midpoint of both diagonals.3. Any point on the perpendicular bisector of segment AB is the same distance from the two

points A and B.4. If the perpendicular bisectors of two sides of a triangle ABC meet at the point P, then

there is a circle with its centre at P that passes through A, B and C. 5. Any point on the angle bisector of an angle ABC is the same distance from the two lines

AB and BC 6. If the angle bisectors of two angles of a triangle ABC meet at the point P, then the point P

is the same distance from all three sides of ABC.7. If M and N are the midpoints of AB and BC, two sides of the triangle ABC, then the line

segment MN is parallel to the line segment AC and half as long.You can also use anything else you know is true about geometry.

Problem:

Construct a quadrilateral ABCD using the Polygon tool. Label it.

Construct the perpendicular bisectors of its sides using the Perpendicular Bisector tool. Label the perpendicular bisector of AB “a”. Label the perpendicular bisector of BC “b”. Label the perpendicular bisector of CD “c”. Label the perpendicular bisector of DA “d”.

Construct these points and label them: H is the intersection of a and b. K is the intersection of a and d. L is the intersection of c and d. M is the intersection of c and b.

Construct the quadrilateral HKLM.

Investigate how HKLM changes when you drag ABCD.

Make conjectures like this:When ABCD is a parallelogram then HKLM is a ___________.(Write your conjectures on the back of this sheet.)

Try to explain why your conjectures are true.

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