Grade 10 Essential Math Unit 6: Similarity of Figures

Learning Goal(s)/Topics:

1)demonstrate an understanding of transformations on a 2-d shape, including:

  1. translations
  2. rotations
  3. Reflections
  4. Dilations (similarity)

** Translations, rotations, and reflections are not covered in the textbook. Do the worksheets indicated to review these concepts.**

2)demonstrate an understanding of primary trigonometric ratios (sine, cosine, tangent) by:

  1. applying similarity to right triangles
  2. generalizing patterns from similar right triangles
  3. solving problems

Learning Activities:

Complete the following questions, and make sure to check your answers at the back of the textbook. There is an answer key there. Make sure to ask questions in class if you are not getting the right answers.

____ Read “Transformation Review Sheet”

____ Translation, Rotation, Reflection worksheet 1

____ Translation, Rotation, Reflection worksheet 2

____Section 6.1 Similar Polygons (work book)

____Section6 .2 Determining if Two Polygons are Similar (work book)

____Section 6.3 Drawing Similar Polygons (work book)

____Section 6.4 Similar Triangles (work book)

____Review pp. 265-269 #1-10

Assessment:

This unit will have two assessment pieces – a project and a test. The test will be based on the review. You will hand in the project by the date of the test. The test and the project will each be worth 5% of your final grade.

______Unit Test date ______Unit Project due

Transformation Review Sheet

Transformations

Plane figures and solids can be changed (or transformed) by translation, reflection and dilation. The symmetry of shapes is related to translation, reflection and rotation.

Reflection

Triangle ABC and its reflectionA'B'C'have the same size and shape, that is they are congruent.

Translation

A translation is defined by specifying the distance and the direction of a movement. For example, triangle ABC is translated by 2 units to the right.

Triangle ABC and its image A'B'C' have the same size and shape. That is, they are congruent.

Rotation

Rotation is defined by stating the centre of rotation, amount of turning in degrees and the direction of rotation (clockwise or anticlockwise). For example, triangle ABC is rotated about O through 90º in an anticlockwise direction.

Triangle ABC and its image A'B'C' have the same size and shape, that is they are congruent.
We notice that:The reflection, translation and rotation are congruent transformations.

Dilation

If a figure is enlarged or reduced and retains its shape, then it is said to be dilated. This is an aspect of similarity as shown below.

Note that the stretching (or shrinking) of a shape is called a dilation. It is clear that dilation is not a congruent transformation, because the size of the shape is changed.
In general:
  • Lengths and areas are not preserved under dilation.
  • If the dilation factor is the same for each side of the figure, then the figures are similar.

(Source: )

Unit 5 Project: Finding the Height of a Tree (not the textbook project)

Objective:

Your assignment is to find the height of tree on campus. You will calculate the height using two different methods, and then compare the answers that you get using the two methods.

Method

Method 1) Shadow Reckoning:

Go to your tree and measure the shadow cast by the tree. Then measure the length of the shadow of a student from your group, who stands by the tree (but not in it's shadow). Measure the student's height. Then draw a sketch of the tree and the student, label the sketch with the measurements, and calculate the height of the assigned object.

Draw the tree, the student, and both shadows on a piece of poster board. Explain every step of your work, and write these explanations clearly and completely, including all of equations and any theorems that you use. This explanation will be part of your "poster".

Method 2) Mirror Method:

Use a small hand mirror (a locker mirror will do). Place the mirror on the ground about 10 or 15 feet from the base of the tree. Then, looking in the mirror, walk forwards or backwards until you can see the top of the tree centered in the mirror, Measure the distance from where you are standing to the center of the mirror, and the distance from the center of the mirror to the base of the tree. Use similar triangles to calculate the height of the tree.

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In each of the two methods, explain every step of this project. Your explanation must contain labelled diagrams, measurements, and a complete explanation of the steps in this project, including any and all theorems used.

Comparison:

Compare the answers you got. If you found somewhat different heights from you the two methods, explain why this might have happened. Explain why the triangles are similar, in each case. How might these methods be useful in other situations? Do you think this might be how people in earlier times might have measured objects too tall to measure directly?

(Based on: )

Gr.10 Essential Math Unit 5 Project Rubric: Finding the Height of a Tree

Criteria / 0 / 1 / 2 / 3
Method 1 sketch / Not submitted / Sketch includes few of the requirements / Sketch includes some of the requirements / Sketch requirements:
  • Tree, student and shadows drawn
  • labelled measurements for all

Method 1 calculation / Not submitted / Calculation has many errors and/or hard to follow steps / Calculation has few errors and easy to follow steps / Calculation has:
  • easy to follow steps
  • correct answer based on measurements
  • Clear explanation/ connection to math theorems from this unit (similar triangle connections)

Method 2 sketch / Not submitted / Sketch includes few of the requirements / Sketch includes some of the requirements / Sketch requirements:
  • Tree, student , mirror
  • labelled measurements for all

Method 2 calculation / Not submitted / Calculation has many errors and/or hard to follow steps / Calculation has few errors and easy to follow steps / Calculation has:
  • easy to follow steps
  • correct answer based on measurements
  • Clear explanation/ connection to math theorems from this unit (similar triangle connections)

Comparison / Answer all of the following questions: (4 possible marks)
1) If you found somewhat different heights from you the two methods, explain why this might have happened.
2)Explain why the triangles are similar, in each case.
3)How might these methods be useful in other situations?
4)Do you think this might be how people in earlier times might have measured objects too tall to measure directly?

_____ 16 = ______%