Standard #: MA.912.GR.4.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.GR.1.6 
• MA.912.GR.2.4
• MA.912.GR.2.8

 

Terms from the K-12 Glossary

• Area 
• Scale Factor 
• Scale Model

 

Vertical Alignment

Previous Benchmarks

• MA.7.GR.1.5 
• MA.8.GR.2.2
• MA.8.GR.2.4 
• MA.912.AR.2.1

Next Benchmarks

 

Purpose and Instructional Strategies

In middle grades, students learned about scale drawings and scale factors. In Geometry, students use that previous knowledge to learn about how changes in the dimensions of a figure due to a dilation will affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures in a way they can predict. (MTR.2.1) This understanding will be valuable to students in science courses.  

• Instruction includes exploring the effect of changing the dimensions of two-dimensional and three-dimensional figures using different factors. It may be helpful to begin exploring through specific problems working with a table of values or with algebraic formulas. 
  • For example, have students explore what happens to the area of a rectangle if the height is doubled and the length is tripled. Additionally, have them explore what happens to the volume of a cylinder if the height is multiplied by 0.5 and the radius is multiplied by 4. 
• Instruction includes reviewing that the area of the image of a dilation with scale factor $k$ is $k$² times the area of the pre-image for any two-dimensional figure (as this was done in grade 7). 
• Instruction includes the student understanding that the surface area of the image of a dilation with scale factor $k$ is $k$² times the surface area of the pre-image, and the volume of the image of a dilation with scale factor $k$ is $k$³ times the volume of the pre-image for any three-dimensional figure.

 

Common Misconceptions or Errors

• Students may multiply the area, surface area or volume by the scale factor instead of thinking about the multiple dimensions. 
• Students may believe the scale factor has the same effect on surface area and volume. To help address this, discuss the effects on surface area using two-dimensional nets of simple figures and then compare to the effects on volumes.

 

Instructional Tasks

Instructional Task 1 (MTR.4.1, MTR.5.1) 

• Use the table below to answer the following questions.
  
  
  

• Part A. Determine the surface area and volume of the square pyramid. 
• Part B. Given the three different dilations, or scale factors, determine the new surface areas and volumes. 
• Part C. Compare each of the new surface areas to the original surface area. Compare each of the new volumes to the original volume. 
• Part D. Predict the surface area and volume of the square pyramid resulting from a dilation with a scale factor of 5? Explain the method you choose..

 

Instructional Items

Instructional Item 1 

• The perfume Eau de Matimatica is packaged in a triangular prism bottle. The dimensions of the travel size are 1/3 the dimensions of the standard bottle. How does the volume of the standard bottle compare to the travel size?

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.