MA.912.GR.2.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.GR.1.2
• MA.912.GR.1.6 
• MA.912.GR.3.2
• MA.912.GR.3.3
• MA.912.GR.6.5

Terms from the K-12 Glossary

• Coordinate Plane 
• Dilation 
• Origin 
• Reflection 
• Rigid Transformation 
• Rotation 
• Scale Factor 
• Translation

Vertical Alignment

Previous Benchmarks

• MA.8.GR.2

Next Benchmarks

• MA.912.F.2.2
• MA.912.F.2.3
• MA.912.F.2.5

Purpose and Instructional Strategies

• Instruction includes the comparison of a variety of geometric figures before and after a single transformation (including reflections, translations, rotations and dilations) to solidify that each of these transformations preserves angle measure. (MTR.4.1) 
• Instruction includes the student understanding that in order for a transformation based on stretching and shrinking to preserve angle measure, it must have a stretch or shrink of the same scale factor in both the $x$-direction and the $y$-direction. 
  • For example, the transformation ($x$, $y$) → (2.5$x$, 2.5$y$) would preserve angle measure, but the transformation ($x$, $y$) → (2.5$x$, 3.5$y$) would not. 
• While the intent of the benchmark is to focus on dilations, instruction includes the introduction of other non-rigid transformations to showcase that not all non-rigid transformations preserve angle measure. 
  • For example, “stretch” or “shrink” in the direction of one of the axes, such as ($x$, $y$) → (5$x$, $y$) or ($x$, $y$) → ($x$, $\frac{\text{<math><mi>y</mi></math>}}{\text{6}}$), does not preserve angle measure. Students can visualize this using geometric software to see how each affects angle measures of a triangle. 
• Transformations should be presented using words and using coordinates. 
• Instruction includes the use of folding paper (e.g., patty paper) for hands-on experiences, as needed, and interactive geometry software to allow more flexibility in exploration, when possible.

Common Misconceptions or Errors

• Students may incorrectly assume that dilations change angle measures. 
• Students may assume that dilations only occur in either the $x$-direction or the $y$-direction based on knowledge of function transformations. 
• Students may assume that every non-rigid motion preserves angle measures since the only non-rigid motions in the benchmark are dilations, which do preserve angle measures. But many non-rigid motions preserve neither length nor angle measure.

Instructional Tasks

• Penelope made the following statement in Geometry class, “Figure A is a rotation of Figure B about the origin.” Chalita disagreed because distance and angle measures are not preserved between the two figures. 
  • Figure A has the coordinate points (1, −1), (3, −1) and (1, −2). 
  • Figure B has the coordinate points (0.8, 1), (1, 3) and (2, 0.8). 
    • Part A. What does “distance and angle measures are not preserved” mean in relation to the two figures? 
    • Part B. Determine whether angle measures were preserved from Figure A to Figure B. 
    • Part C. Determine whether distance measures were preserved from Figure A to Figure B. 
    • Part D. Based on your answers from Part B and Part C, determine which student is correct. 

• Sort the following transformations into preserves distance and does not preserve distance.
  
  
  

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