# MA.912.GR.4.3

Extend previous understanding of scale drawings and scale factors to determine how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.

### Examples

Example: Mike is having a graduation party and wants to make sure he has enough pizza. Which option would provide more pizza for his guests: one 12-inch pizza or three 6-inch pizzas?
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Area
• Scale Factor
• Scale Model

### Vertical Alignment

Previous Benchmarks

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 $k$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 $k$times the surface area of the pre-image, and the volume of the image of a dilation with scale factor $k$ is $k$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.

• 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.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.4.AP.3: Select the effect of a dilation on the area of two-dimensional figures and/or surface area or volume of three-dimensional figures.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plans

Cape Florida Lighthouse: Lore and Calculations:

The historic Cape Florida Lighthouse, often described as a conical tower, teems with mathematical applications. This lesson focuses on the change in volume and lateral surface area throughout its storied existence.

Type: Lesson Plan

St. Pi Day construction with a compass & ruler:

St. Pi Day construction with compass

This activity uses a compass and straight-edge(ruler) to construct a design. The design is then used to complete a worksheet involving perimeter, circumference, area and dimensional changes which affect the scale factor ratio.

Type: Lesson Plan

## Perspectives Video: Professional/Enthusiasts

Reflections, Rotations, and Translations with Additive Printing:

Transform your understanding of 3D modeling when you learn about how shapes are manipulated to arrive at a final 3D printed form!

Type: Perspectives Video: Professional/Enthusiast

Scale and Proportion for Bird Photography:

Mathematics plays a role in what we perceive as beautiful! Learn more about it while you learn about bird photography! Produced with funding from the Florida Division of Cultural Affairs.

Type: Perspectives Video: Professional/Enthusiast

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Perspectives Video: Professional/Enthusiasts

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Perspectives Video: Professional/Enthusiasts

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast