Understand similarity in terms of similarity transformations
Verify experimentally the properties of dilations given by a center and a scale factor:
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Dilations
Dilation [a]
Dilation of segment [b]
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5 - Verify (the properties of dilations given by a center and a scale factor through experimentation with objects and line segments)
2 - Understand (dilation takes a line not through center of dilation to a parallel line [a])
2 - Understand (dilation does not change a line through center of dilation [a])
2 - Understand (dilation of a line segment is longer or shorter in the ratio given by the scale factor [b])
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5 - I can verify the properties of dilations given by a center and a scale factor through experimentation with objects and line segments.
2 - I can understand that dilation takes a line not passing through center of dilation to a parallel line [a].
2 - I can understand that dilation does not change a line through center of dilation [a].
2 - I can understand that the dilation of a line segment is longer or shorter in the ratio given by the scale factor [b].
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A. The definition of a dilation can be used to verify experimentally the properties of a dilations in figures and line segments.
B. If a line is passing through the center of the dilation, it will not change.
C. A line segment dilated will not be congruent to the original segment.
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A.1 How can the definition of dilation, coupled with experimentation, lead to verification of dilation properties in a figure?
B.1 Does dilation always change the figure being dilated?
C.1 Are figures ever congruent after dilation?
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Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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3 - Use (definition of similarity in terms of similarity transforma-tions)
5 - Decide (if two figures are similar)
4 - Explain (the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding sides, using similarity transformations)
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5 - I can decide if two figures are similar by using the definition of similarity in terms of similarity transformations.
4 - I can explain the meaning of similarity for triangles as the equality of all
1 - orrespond-ing pairs of angles and the proportionality of all corresponding sides, using similarity transformations.
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A. The definition of similarity in terms of similarity transformations can be used to decide if two figures are similar.
B. Similarity for triangles exists when all corresponding pairs of angles are equal and when all corresponding pairs of sides are proportional, and can be explained using similarity transformations.
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A.1 How can the definition of similarity(in terms of similarity transformations) be used to show that two figures are similar?
B.1 How is the meaning of similarity related to similarity transformations?
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Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Accuracy level
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None Available |
3 - I can choose a level of accuracy appropriate to limitations on measurements when I am reporting quantities.
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A. When solving problems and reporting quantities, the level of accuracy reported should be appropriate to stated or implied limitations on measurement.
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A.1 How can the appropriate level of accuracy be chosen with regard to limitations on measurement of quantities within a problem?
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Prove theorems about triangles.
Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Triangle Theorems
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3 - Prove (triangle theorems, including a line parallel to one side of a triangle divides the other two sides proportionally)
3 - Prove (if a line divides two sides of a triangle proportionally, then it is parallel to the third side)
3 - Prove (Pythagorean Theorem using triangle similarity)
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3 - I can prove that a line parallel to one side of a triangle divides the other two sides proportionally.
3 - I can prove that if a line divides two sides of a triangle proportionally, then it is parallel to the third side of the triangle.
3 - I can prove the Pythagorean Theorem using the concept of triangle similarity.
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A. Similarity and its properties can be used to prove theorems about triangles.
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A.1 How can similarity be used to prove theorems related to triangles, parallel lines, and proportional sides?
A.2 How can similarity be used to prove the Pythagorean Theorem?
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Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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3 - Use (congruence and similarity criteria for triangles)
3 - Solve (problems by using congruence and similarity criteria for triangles)
3 - Prove (relationships in geometric figures by using congruence and similarity criteria for triangles.)
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3 - I can use congruence and similarity criteria for triangles.
3 - I can solve problems by using congruence and similarity criteria for triangles.
3 - I can prove relationships in geometric figures by using congruence and similarity criteria for triangles.
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A. Congruence and similarity criteria for triangles can be used to solve a variety of problems.
B. Congruence and similarity criteria for triangles can be used to prove relationships in geometric figures?
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A.1 How can problems be solved through the use of congruence and similarity criteria for triangles?
A.2 What kind of problems can be solved through the use of congruence and similarity criteria for triangles?
B.1 How can congruence and similarity criteria for triangles be used to prove relationships in geometric figures?
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Define trigonometric ratios and solve problems involving right triangles
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Definitions of Trigonometric Ratios
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3 - Understand (similarity of triangles leads to definitions of trigonometric ratios, and that these side ratios are properties of the acute angles in the triangle)
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3 - I can understand that similarity of triangles leads to the definitions of the trigonometric ratios, and that these ratios are properties of the acute angles in the triangle.
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A. Similarity of triangles is the basis for the definitions of the trigonometric side ratios for acute angles.
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A.1 What are the trigonometric ratios?
A.2 How is similarity used to define the trigonometric ratios?
A.3 How do the trigonometric side ratios relate to the acute angles of the triangle?
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Explain and use the relationship between the sine and cosine of complementary angles.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Sine and Cosine Ratios
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5 - Explain (relationship between sine and cosine ratios for complementary angles)
3 - Use (relationship between sign and cosine ratios for complementary angles)
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5 - I can explain the relationship between the sine and cosine ratios for complementary angles.
3 - I can use the relationship between sine and cosine ratios for complementary angles.
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A. The relationship between the sine and cosine ratios of complementary angles can be used in a variety of problem solving situations.
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A.1 What is the relationship between the sine and cosine of complementary angles?
A.2 How can the relationship between sine and cosine of complementary angles be used?
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Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Right Triangle Application Problems
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3 - Use (trigonometric ratios to solve right triangles in applied problems.)
3 - Solve (applied problems)
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3 - I can use trigonometric ratios to solve right triangles in applied problems.
3 - I can use Pythagorean theorem to solve right triangles in applied problems.
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A. Trigonometric ratios and the Pythagorean theorem can be used, either together or separately, in applied problems to solve right triangles.
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A.1 What does it mean to solve a right triangle?
A.2 How can the trigonometric ratios and the Pythagorean theorem be used to solve right triangles in applied problems?
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(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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2 - Draw (auxiliary lines)
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6 - I can derive the area of a triangle formula A= ½ ab sin (C) by drawing and using an auxiliary line from a vertex perpendicular to the opposite side in a triangle.
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A. The formula for finding area of a triangle,
= ½ ab sin (C), can be derived with the use of a (drawn) auxiliary line.
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A.1 How can an auxiliary line be drawn from a vertex perpendicular to the opposite side in a general triangle?
A.2 How can the formula for finding area of a triangle,
= ½ ab sin (C), be derived?
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(+) Prove the Laws of Sines and Cosines and use them to solve problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Proofs
Problems
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3 - Prove (Law of Sines)
3 - Prove (Law of Cosines)
3 - Solve (problems using Law of Sines and Law of Cosines)
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3 - I can prove the Law of Sines.
3 - I can prove the Law of Cosines.
3 - I can solve problems using the Law of Sines and the Law of Cosines.
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A. The Law of Sines and the Law of Cosines relate the sides and angles of a triangle.
B. The Law of Sines and the Law of Cosines can be proven and used to solve a variety of problems.
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A.1 What is the formula for the Law of Sines?
A.2 What is the formula for the Law of Cosines?
A.3 How can the Law of Sines and the Law of Cosines be proven?
A.4 How can the Law of Sines and the Law of Cosines be used to solve problems?
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(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Right and Non-Right Triangle Problems
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2 - Understand (Law of Sines)
2 - Understand (Law of Cosines)
3 - Apply (Law of Sines and Law of Cosines)
3 - Find (unknown measurements)
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2 - I can understand the Law of Sines.
2 - I can understand the Law of Cosines.
3 - I can apply the Law of Sines and the Law of Cosines to find unknown measurements in real world problems such as surveying and resultant forces.
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A. The Law of Sines and the Law of Cosines can be applied to find unknown measurements in real-world problems.
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A.1 When is the Law of Sines applicable in a problem?
A.2 When is the Law of Cosines applicable in a problem?
A.3 How can the Law of Sines and Law of Cosines be applied in real-world problems to find unknown measurements?
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