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Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Coordinate Plane
- Dilation
- Origin
- Reflection
- Rigid Transformation
- Rotation
- Scale Factor
- Translation
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students first learned to associate congruence and similarity with reflections, rotations, translations and dilations. In Geometry, students learn that these transformations provide an equivalent alternative to using congruence and similarity criteria for triangles.- Instruction includes the connection to the Angle-Angle, Side-Angle-Side, Hypotenuse-Leg and Side-Side-Side similarity criteria and to justifying congruence criteria.
- For example, if one wants to justify the Angle-Angle criterion, one method is as follows: Start with triangles ABC and PQR, with ∠A ≅ ∠P and ∠B ≅ ∠Q, as shown below.Students should be able to realize the need of a dilation, in this case with a scale factor such that 0 < < 1 and = . After this dilation, triangle A’B’C’ is obtained, such that the length of A’B’ equals the length of PQ. Since dilations preserve angle measures, ∠A ≅ ∠A′ and ∠B ≅ ∠B′. Using the transitive property of congruence, if ∠A ≅ ∠P and ∠A ≅ ∠A′, then ∠P ≅ ∠A′, and if ∠B ≅ ∠Q and ∠B ≅ ∠B′, then ∠Q ≅ ∠B′. With PQ ≅ A′B′, ∠P ≅ ∠A′ and ∠Q ≅ ∠B′, we can prove ΔPQR ≅ ΔA′B′C′ by Side-Angle-Side Congruence Criterion. Additionally, since ΔABC~ΔA′B′C′ and ΔPQR ≅ ΔA′B′C′, it can be concluded that ΔABC~ΔPQR. Justification of the other criteria can be done in a similar manner.
- For example, if one wants to justify the Angle-Angle criterion, one method is as follows: Start with triangles ABC and PQR, with ∠A ≅ ∠P and ∠B ≅ ∠Q, as shown below.
Common Misconceptions or Errors
- When determining the scale factor of a dilation, students may misidentify the preimage and image, leading to an incorrect scale factor.
Instructional Tasks
Instructional Task 1 (MTR.3.1, MTR.4.1)- Triangle XYZ has the coordinates (0,2), (2,4) and (6,0) and triangle DEF has the coordinates (4, −4), (8,0) and (16, −8).
- Part A. How can ΔACB~ΔLMN be proved using one of the similarity criteria?
- Part B. How can ΔACB~ΔLMN be proved using rigid and non-rigid transformations?
Instructional Items
Instructional Item 1- Shown below are two triangles where ∠X = ∠R, ∠Y = ∠S, and ∠Z = ∠T. Determine a dilation that maps ΔXYZ onto ΔRST.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Problem-Solving Task
MFAS Formative Assessments
Students are given the definition of similarity in terms of similarity transformations and are asked to explain how this definition ensures the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Student Resources
Problem-Solving Task
In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Task
In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.
Type: Problem-Solving Task
