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Evaluating Student Learning: Preparing to Report: Unit 7 Similarity and TransformationsThis unit provides an opportunity to report on the Shape and Space (3-D Objects and 2-D Shapes/Transformations) strand. Master 7.4 Unit Summary: Similarity and Transformations provides a comprehensive format for recording and summarizing evidence collected.

Here is an example of a completed summary chart for this Unit:Key: 1: Not Yet Adequate 2: Adequate 3: Proficient 4: Excellent

Most Consistent Level of Achievement*

Strand: Shape and Space (3-D Objects and 2-D Shapes/Transformations)

Conceptual Understanding

Procedural Knowledge

Problem-Solving Skills

Communication Overall

Ongoing observations 3 2 4 3 3

Work samples or portfolios; conferences

3 3 3 3 3

Unit Test 3 2 3 2 2/3

Unit ProblemDesigning a Flag 3 3 3 2 3

Achievement Level for reporting 3

* Use locally or provincially approved levels, symbols, or numeric ratings.

Recording How to ReportOngoing observations Use Master 7.2 Ongoing Observations: Similarity and Transformations to determine the most

consistent level achieved in each category. Enter it in the chart. Choose to summarize by achievement category, or simply to enter an overall level.Observations from late in the unit should be most heavily weighted.


Use Master 7.1 Unit Rubric: Similarity and Transformations to guide evaluation of collections of work and information gathered in conferences. Teachers may choose to focus particular attention on the Assessment Focus questions. Work from late in the unit should be most heavily weighted. You may choose to assign some or all of the questions in the Review. Master 7.1 Unit Rubric: Similarity and Transformations may be helpful in determining levels of achievement. See Assessment for Learning at the end of each lesson for specific data.

Unit Test Master 7.1 Unit Rubric: Similarity and Transformations may be helpful in determining levels of achievement. #4 provides evidence of Conceptual Understanding; #1 provides evidence of Procedural Knowledge; #2 provides evidence of Problem-Solving Skills.

Unit Problem Use Master 7.3 Performance Assessment Rubric: Designing a Flag. The Unit Problem offers a snapshot of students’ achievement. In particular, it shows their ability to synthesize and apply what they have learned.

Student Self-Assessment Note students’ perceptions of their own progress. This may take the form of an oral or written comment, or a self-rating. Use any of Master 7.5 or Program Masters 2, 3, 4, 5, 6, 7, or 8.

Comments Analyse the pattern of achievement to identify strengths and needs. In some cases, specific actions may be planned to support the learner.

Learning SkillsPM 9: Learning Skills ChecklistUse to record and report throughout a reporting period, rather than for each unit and/or strand.

Ongoing RecordsPM 11: Observation Record 1PM 12: Observation Record 2PM 14: Work Sample Records

The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2009 Pearson Education Canada

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Use to record and report evaluations of student achievement over several clusters, a reporting period, or a school year. These can also be used in place of the Unit Summary.

Unit 7 Evaluating Student Learning71

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Unit Rubric: Similarity and Transformations

Not Yet Adequate Adequate Proficient Excellent

Conceptual UnderstandingShows understanding of transformations by demonstrating and explaining:– scale factors– similar polygons– similar triangles– line symmetry– rotational symmetry

little understanding; may be unable to demonstrate or explain:– scale factors– similar polygons– similar triangles– line symmetry– rotational symmetry

some understanding; partially able to demonstrate or explain:– scale factors– similar polygons– similar triangles– line symmetry– rotational symmetry

shows understanding; able to demonstrate and explain:– scale factors– similar polygons– similar triangles– line symmetry– rotational symmetry

shows depth of understanding; in various contexts, demonstrates and explains:– scale factors– similar polygons– similar triangles– line symmetry– rotational symmetry

Procedural KnowledgeAccurately:– determines if two polygons

are similar– draws similar polygons– determines scale factor for

given diagram– draws scale diagrams (an

enlargement or reduction) – identifies a line of

symmetry – determines the angle of

rotation symmetry

limited accuracy; often makes major errors/omissions in:– determining if two

polygons are similar– drawing similar

polygons– determining scale

factor– drawing scale

diagrams – identifying a line of

symmetry – determining an

angle of rotation symmetry

partially accurate; makes frequent minor errors/omissions in:– determining if two







generally accurate; makes few errors/ omissions in:– determining if two







accurate and precise; rarely make errors/omissions in:– determining if two







Problem-Solving SkillsSelects and uses appropriate strategies to solve problems using:– properties of similar

polygons and triangles– scale diagrams– line and rotational

symmetry

does not select and use appropriate strategies to solve problems involving similarity and transformations successfully

with limited help, selects and uses some strategies to solve problems involving similarity and transformations with partial success

selects and uses appropriate strategies to solve problems involving similarity and transformations successfully

selects and uses appropriate strategies to solve problems involving similarity and transformations with a high degree of success

CommunicationRecords and explains reasoning and procedures clearly and completely, including appropriate terminology

does not record and explain reasoning and procedures clearly and completely

records and explains reasoning and procedures with partial clarity; may be incomplete

records and explains reasoning and procedures clearly and completely

records and explains reasoning and procedures with precision and thoroughness


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Master 7.1

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Ongoing Observations: Similarity and Transformations

The behaviours described under each heading are examples; they are not intended to be an exhaustive list of all that might be observed. More detailed descriptions are provided in each lesson under Assessment for Learning.

STUDENT ACHIEVEMENT: Similarity and Transformations*Student Conceptual

UnderstandingProcedural Knowledge


Communication

Demonstrates and explains: scale factors; similar polygons and triangles; line and rotational symmetry

Determines:– if polygons are

similar– scale factors– line of symmetry– angle of rotational

symmetry

Solves problems that involve similarity and transformations, including scale diagrams

Records and explains reasoning and procedures clearly and completely, including appropriate terminology

*Use locally or provincially approved levels, symbols, or numeric ratings.


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Master 7.2

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Performance Assessment Rubric:Designing a Flag

Not Yet Adequate Adequate Proficient Excellent

Conceptual UnderstandingDescriptions show understanding of line and rotational symmetry

shows very limited understanding; descriptions are omitted or inappropriate

shows partial understanding; descriptions are often incomplete or somewhat confusing

shows understanding; descriptions are appropriate

shows thorough understanding; descriptions are effective and thorough

Procedural KnowledgeAccurately:– classifies flags by numbers

of lines of symmetry– determines scale factor– draws a scale diagram

limited accuracy; major errors or omissions in:– classifying flags by

numbers of lines of symmetry

– determining a scale factor

– drawing a scale diagram

Partially accurate; some errors or omissions in: – classifying flags by




generally accurate; few errors or omissions in: – classifying flags by




accurate and precise; very few or no errors in:– classifying flags by


– determining scale factor


Problem-Solving SkillsUses appropriate strategies to design a flag at least 3 m by 2 m, with line and rotational symmetry, and explains its use

uses few appropriate strategies; does not design a flag that meets criteria and is appropriate for designated use

uses some appropriate strategies to design a flag that meets some criteria and is appropriate for designated use

uses appropriate strategies to design a flag that meets the criteria and is appropriate for designated use

uses effective and often innovative strategies to design a flag that meets the criteria and introduces additional complexity; flag is well-designed for designated use

CommunicationPresents diagrams and descriptions clearly, using appropriate mathematical terminology and symbols

does not present diagrams and descriptions clearly, uses few appropriate mathematical terms and symbols

presents diagrams and descriptions with some clarity, using some appropriate mathematical terms and symbols

presents diagrams and descriptions clearly, using appropriate mathematical terms and symbols

presents diagrams and descriptions precisely, using a range of appropriate mathematical terms and symbols


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Master 7.3

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Unit Summary: Similarity and Transformations

Review assessment records to determine the most consistent achievement levels for the assessments conducted. Some cells may be blank. Overall achievement levels may be recorded in each row, rather than identifying levels for each achievement category.

Most Consistent Level of Achievement*Strand: Shape and Space (3-D Objects and 2-D Shapes/Transformations)

Conceptual Understanding

Procedural Knowledge


CommunicationOverall

Ongoing observations


Unit Test

Unit ProblemDesigning a Flag

Achievement Level for reporting

* Use locally or provincially approved levels, symbols, or numeric ratings.

Self-Assessment:

Comments: (Strengths, Needs, Next Steps)


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Master 7.4

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Reflecting on Learning: Unit 7

1. Read the Learning Goals for similarity and transformations. Think about your learning. Rate your learning for each goal using these symbols:+ = I can understand and do this very well. = I can understand and do this all right.– = I am still having trouble with this. = This is a big problem for me.

Learning Goal My RatingInterpret scale diagrams.

Draw scale diagrams.

Apply properties of similar polygons.

Identify and describe line symmetry.

Identify and describe rotational symmetry.

2. Think about how you learn geometry best. a) When is it helpful for you to work with other students and when is it better for you to

work on your own?

________________________________________________________________________

________________________________________________________________________

b) How much do examples from real life help you?

________________________________________________________________________

c) How much do you use visualizing or seeing with your ‘mind’s eye’ when you are working with transformations? What helps you visualize the shapes and their transformations?

________________________________________________________________________

________________________________________________________________________

d) What should your teacher or others who help you understand about how you learn best?

________________________________________________________________________


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Master 7.5

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Additional Activity 1: Polygoncentration

Play with a partner.You will need Polygoncentration Game Cards (Master 7.6b), and scissors.

How to Play:

Cut out the cards. Shuffle the cards.Place them face down in a rectangular array on the table.

Player A picks a card from the array, and turns it face up. Player A then picks another card and turns it face up.If the two cards have polygons that are similar (matching cards), the cards are removed and kept by Player A. Players who pick matching cards get to choose again until their cards do not match.

If the two cards do not match, the cards are turned face down, and it is Player B’s turn to try to pick two matching cards.

Players take turns until all the cards have been picked. The player with more cards wins.

Take It FurtherPlay the game again, but this time, create your own pairs of similar polygons and add them to the game cards.


Master 7.6a

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Polygoncentration Game Cards


Master 7.6b

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Additional Activity 2: How Far Is It Across the Street?

Work in groups of three. Use Master 7.7b to set up.You will need a large piece of cardboard, scissors, ruler, metre ruler or tape measure.

What to Do:

Draw an isosceles right triangle on a piece of cardboard.Cut out the triangle.Go out to the street, or the area indicated by your teacher.

Person 1 stands at point A, directly across the street from an object such as a lamp post or fire hydrant. Person 2 moves to a point where she can hold the triangle horizontally at eye level so that she can sight along one side of the triangle to see the object, and along the other side of the triangle to see Person 1 at point A. Person 3 uses a tape measure or metre ruler to measure the distance AB between Person 1 and Person 2

Students exchange roles so that each person takes a sighting and there are three measures of AB. Calculate the mean measure of AB.This mean measure represents the distance across the street (BC).

Sighting Distance AB

1

2

3

Mean

Work as a group and use diagrams to explain why the distance across the street (BC) is the same as the distance AB between Person 1 and Person 2.

Take It FurtherDetermine a strategy to use similar triangles to determine a height that cannot be measured directly. Apply your strategy to determine the height of a flagpole or lamp post.


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Set-up Diagram


Master 7.7b

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Additional Activity 3: Star Creation

Work on your own.

You will need a Star Creation Grid (Master 7.8b), light cardboard, scissors, ruler, sharp pencil or compass point, and a felt pen.

What to Do:

1. Draw and cut out a triangle from cardboard. Pick any point on the triangle. Use a sharp pencil or compass point to make a small hole and line it up with the centre point on the Star Creation Grid. Position the triangle so that one of its vertices lies on Line 1.Trace the triangle.

2. Rotate the triangle clockwise until the chosen vertex lies on Line 2.Trace the triangle.

3. Repeat the procedure in Step 2 until you have rotated the triangle a full turn, tracing the triangle at Lines 3, 4, 5, and 6.

4. Use a felt pen to draw the outline of the star shape you created.Describe the symmetry of your star shape.

Take It FurtherRepeat this procedure with a Star Creation Grid of your own design to make new stars.


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Star Creation Grid


Master 7.8b

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Additional Activity 4: Symmetry Scavenger Hunt

Work in a group of three or four.

What to Do:

Your teacher will assign an area for your Scavenger Hunt.Find examples of objects with line and/or rotational symmetry. Sketch each object and indicate the number of lines of symmetry, and/or the order of rotational symmetry.

Points are scored as follows: 1 point for an object or design with 1 line of symmetry, 2 points for 2 lines of symmetry, and so on. 2 points for rotational symmetry of order 2, 3 points for rotational symmetry order 3, and so on. An object or design receives points for both rotational and line symmetry.

The team with the most points at the end of the Scavenger Hunt wins.

Take It FurtherFind an object or design with the greatest order of rotational symmetry or the greatest number of lines of symmetry.


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Step-by-Step 1

Lesson 7.1, Question 11

Step 1 Record the lengths and widths of each rectangle.

Length Width

Yellow

A

B

C

D

Step 2 Calculate the following ratios:

= _______ = _______

= _______ = _______

= _______ = _______

= _______ = _______

Step 3 Identify which rectangles have equal ratios for length and width.

Complete these sentences.

The ratios of the corresponding sides of the yellow rectangle and rectangle ______ are

equal. So rectangle ______ is an enlargement of the yellow rectangle with a scale

factor of ___________.

Step 4 Repeat Steps 1 to 3 for the shape in part b. Determine the ratios of all corresponding

sides.


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Step-by-Step 2


Step 1 Measure the dimensions of your room in centimetres.

Length = ____________________ Width = ____________________

Step 2 Measure the dimensions of the paper to be used for the scale diagram in centimetres.

Length = ____________________ Width = ____________________

Step 3 Divide the length of the paper by the length of the room. __________

Divide the width of the paper by the width of the room. _________

Use these quotients to help you choose a scale factor for the reduction.

The scale factor is _____________.

Step 4 Draw a scale diagram of the room using your choice of scale factor.

Step 5 Measure the lengths and widths of other items in the room.

Multiply these measurements by the scale factor.

Include these items on your scale diagram.


Master 7.11

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Step-by-Step 3


Use Master 7.12b. You will need 1-cm grid paper (PM 16).

Step 1 Write the ratios of the lengths of each pair of rectangles.

Write the ratios of the widths of each pair of rectangles.

Identify which rectangles have ratios of corresponding sides equal,

and are therefore similar.

Rectangles Ratio of lengths Ratio of widths

A and B

A and C

A and D

B and C

B and D

C and D

The pairs of similar rectangles are:

_____________________________________________________________________

Step 2 For the rectangle that is not similar, multiply its dimensions by two different scale

factors. Draw two rectangles with these new dimensions.

Step 3 To calculate the length of the diagonal of each rectangle that is similar to rectangle A,

use the proportion:

=


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Step-by-Step 3, Continued


Master 7.12b

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Step-by-Step 4


Step 1 Stand close to the object to be measured. Have your partner measure your height to

the nearest hundredth of a metre, and the length of your shadow.

Record these measurements.

Your height: ____________________ m

Length of your shadow: ____________________ m

Step 2 Measure the length of the object’s shadow. Record the measurement.

Length of object’s shadow: ____________________ m

Step 3 Mark these measurements on the diagram.

Mark the unknown height of the object as x on the diagram.

Step 4 Complete a proportion for the lengths of the sides of the two similar triangles.

Step 5 Solve the proportion for x.

The height of the object is: ___________________ m


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Step-by-Step 5


Use 1-cm grid paper (PM 16).

Step 1 Pick a point on the x-axis. Pick a point on the y-axis. Join these points to make a right

triangle with the origin (0, 0) and these two points as the vertices.

Step 2 Reflect the triangle in the x-axis. The original triangle and its reflection create a larger

shape. Write the coordinates of this shape.

The coordinates are: (___,___) (___,___) (___,___)

How many lines of symmetry does this shape have? _________________

Step 3 Reflect the triangle in the y-axis. Then repeat Step 2.

The coordinates are: (___,___) (___,___) (___,___)

How many lines of symmetry does this shape have? _________________

Step 4 Reflect the triangle in the hypotenuse of the triangle.

What new shape is created? _________________________

How many lines of symmetry does this shape have? ___________________

Step 5 Repeat Steps 2 to 4 for a different type of triangle.

Describe the shape formed after each reflection.

_________________________________________________________________________

_________________________________________________________________________

Step 6 Which types of triangles produce a shape that is a quadrilateral with line symmetry?

____________________________________________________________


Master 7.14

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Step-by-Step 6


Step 1 Rotate ABCD 180° about vertex A, and then about centre E.

Which of these rotations produces an image that coincides with ABCD?

_________________________________________________________________________

Complete the sentence:

The rectangle ABCD has ____________________ symmetry of order

______________ and an angle of rotation symmetry of ______° about the point

_______.

Step 2 Rotate FGHJ 90° about vertex F, and then about centre K.

Which of these rotations produces an image that coincides with FGHJ?

_________________________________________________________________________


The square FGHJ has ____________________ symmetry of order ______________

and an angle of rotation symmetry of ______° about the point _______.

Step 3 Rotate MNP 120° about vertex M, then about centre Q.

Which of these rotations gives an image that coincides with MNP?

_________________________________________________________________________


The triangle MNP has ____________________ symmetry of order ______________

and an angle of rotation symmetry of ______° about the point _______.

Step 4 How are the images in Steps 1 to 3 the same?

_________________________________________________________________________

How are these images different?

_________________________________________________________________________


Master 7.15

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Step-by-Step 7


Use Master 7.16b.

Step 1 Triangle CDE has vertices C(2, 3), D(−2, −1) and E(3, −2).

Move each vertex 1 unit right and 3 units up. Label the image C′D′E′.

List the coordinates of the vertices:

C′(__,__), D′(__,__), E′(__,__)

Step 2 Why does this translation not result in:

line symmetry?

_________________________________________________________________________

rotation symmetry?

_________________________________________________________________________

Step 3 Translate ΔCDE 2 units right and 2 units up. Label the image C″D″E″.

Describe any symmetry in the shape created by combining these two triangles.

_________________________________________________________________________

How do you know the diagram has symmetry?

_________________________________________________________________________

Step 4 Use the result of Step 3. Find another translation that results in symmetry.


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Step-by-Step 7 continued


Master 7.16b

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Scale Diagram of Room for Lesson 7.2 Practice Question 19


Master 7.17

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Results Table for Lesson 7.3 Investigate

Enlargement

Lengths of Sides (mm)

AB A′B′ BC B′C′ CD C′D′ DA D′A′

Measures of Angles (°)

A A′ B B′ C C′ D D′

Reduction

Lengths of Sides (mm)

AB A″B″ BC B″C″ CD C″D″ DA D″A″

Measures of Angles (°)

A A″ B B″ C C″ D D″


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Shapes for Lesson 7.5 Investigate


Master 7.19

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Wheel Covers for Lesson 7.7 Practice Question 4

a) b)

c) d)


Master 7.20

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Unit Test: Unit 7 Similarity and Transformations

1. These two pentagons are similar.

a) Calculate the length of PT.b) Calculate the length of DE.c) Draw a reduction of pentagon PQRST with a scale factor of .d) Draw an enlargement of pentagon ABCDE with a scale factor of 2.

2. Naomi wants to calculate the height of a tree. She is 1.5 m tall and casts a shadow of 2.5 m. At the same time, the shadow of the tree is 10.5 m long.a) Sketch a diagram that can be used to calculate the height of the tree.b) What is the height of the tree?

3. Plot these points on a grid: A(−2, 4), B(2, 4), C(2, 2), D(−2, 2)For each transformation below:i) Draw the transformation image.ii) Record the coordinates of its vertices.iii) Describe the symmetry of the diagram formed by the original shape and its image.a) rotation 90° clockwise about point E(0, 3)b) reflection in the horizontal line passing through (0, 2) c) a translation 4R, 2U


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Master 7.21a

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Unit Test, continued

4. Describe the symmetry of each shape. Draw any lines of symmetry and state the order of rotation and the angle of rotation symmetry. a) b)

c) d)

e)


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Unit 7 Test Sample Answers


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1. a) 15 cm b) 8 cm c)

d)

2. b) 6.3 m

3. a) i)

ii) A′(1, 5), B′(1, 1), C′(−1, 1), D′(−1, 5)iii) The y-axis is a line of symmetry; the

horizontal line through (0, 3) is a line of symmetry; the oblique line through (−1, 2) and (1, 4) is a line of symmetry; the oblique line through (−1, 4) and (1, 2) is a line of symmetry; the shape has rotational symmetry of order 4 about (0, 3).

b) i)

ii) A′(−2, 0), D(–2, 2), C(2, 2), B′(2, 0)iii) The y-axis is a line of symmetry; the

horizontal line through (0, 2) is a line of symmetry; the oblique line through (−2, 0) and (2, 4) is a line of symmetry; the oblique line through (−2, 4) and (2, 0) is a line of symmetry; the shape has rotational symmetry of order 4 about (0, 2).

c) i)

ii) A′(2, 6), B′(6, 6), C′(6, 4), D′(2, 4) iii) The shape has rotational symmetry of

order 2 about the point (2, 4).

4. a) 6 lines of symmetry, rotational symmetry of order 6 with angle of rotation symmetry 60°

b) 2 lines of symmetry, rotational symmetry of order 2 with angle of rotation symmetry 180°

c) 1 line of symmetry, no rotational symmetry

d) 2 lines of symmetry, rotational symmetry of order 2 with angle of rotation symmetry 180°

e) 5 lines of symmetry, rotational symmetry of order 5 with angle of rotation symmetry 72°


Master 7.22

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