# MA.7.GR.1.5

Solve mathematical and real-world problems involving dimensions and areas of geometric figures, including scale drawings and scale factors.

### Clarifications

Clarification 1: Instruction focuses on seeing the scale factor as a constant of proportionality between corresponding lengths in the scale drawing and the original object.

Clarification 2: Instruction includes the understanding that if the scaling factor is k, then the constant of proportionality between corresponding areas is k² .

Clarification 3: Problem types include finding the scale factor given a set of dimensions as well as finding dimensions when given a scale factor.

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
• Constant of Proportionality
• Scale Factor

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students solved problems relating to the perimeter or area of a rectangle as well as the area of composite figures by decomposing them into triangles or rectangles. In grade 7, students solve mathematical and real-world problems involving dimensions and areas of geometric figures, including scale drawings and scale factors. In grade 8, students will continue to work with scale factor and apply it to dilations before moving to high school and determining how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.
• Scale drawings of geometric figures connect proportionality to geometry, which leads to future work in similarity and congruence. Initially, students explore scale drawings as an enlargement or reduction of one object to obtain a similar object by using a scale factor. Begin with whole number measurements, progressing to rational numbers as students deepen their understanding.
• Instruction focuses on seeing the scale factor as a constant of proportionality between corresponding lengths in the scale drawing and the original object. Use manipulatives such as Geoboards/pegboards, dot paper, centimeter grid paper, etc. to enlarge and reduce shapes by simple scale factors (MTR.2.1). Discuss whether multiplication or division may be used, reminding students that division can be represented by multiplication, and reinforcing that multiplication by a factor between 0 and 1 will be a reduction in size.
• Geoboards
green square has a scale factor of 3 from the original red square

• Dot or Grid Paper
green rectangle has a scale factor of 2 from the original red rectangle

• Have students construct scale drawings of the classroom, school, their homes and/or backyards or other familiar places where they can take measurements (MTR.7.1).
• Instruction includes the understanding that if the scaling factor is $k$, then the constant of proportionality between corresponding areas is $k$². Once students have become comfortable with scaling dimensions, extend their knowledge to solving problems with area. Provide several figures where students will determine new dimensions based on a given scale factor. Have students then calculate the original and new perimeters, as well as the original and new areas. Then analyze/compare the scale factors used in scaling the perimeters versus the scale factors used for area (MTR.1.1, MTR.4.1).
• Instruction supports flexibility in the variable used for the constant of proportionality.

### Common Misconceptions or Errors

• Students may not understand how to read a map. To address this misconception, practice map reading skills, using familiar areas when possible.
• Students may incorrectly scale area in the same way they scale side length. To address this misconception, have students calculate areas of similar figures prior to determining the scale factor between the figures, then make comparisons. Interactive software can also be used to demonstrate.
• Students may incorrectly set up their proportions.
• Students may believe the scale factor is always greater than 1.
• For example, students may respond the scale factor is 2 when it is $\frac{\text{1}}{\text{2}}$.

### Strategies to Support Tiered Instruction

• Teacher provides instruction utilizing different types of maps to familiarize students with how to read a map and the key features of a map. Teacher can choose maps that are familiar to students within their region.
• Instruction includes the use of geometric software to allow students to explore the area of an original figure versus its scale and draw conclusions on the impact of scale factor.
• Teacher co-creates a graphic organizer with students containing examples of applying a scale factor to a length or to an area.
• Teacher provides instruction focused on color-coding and labeling the different units when setting up a proportional relationship to ensure corresponding units are placed in corresponding positions within the proportion.
• Teacher has students calculate areas of figures where the side lengths of one figure is a constant multiple of the corresponding side lengths of the other figure prior to determining the scale factor between the figures. Students can then make comparisons between the areas of the figures. Interactive software can also be used to demonstrate.

Many supersonic jet aircraft in the past have used triangular wings called delta wings. Below is a scale drawing of the top of a delta wing.
• Scale: 2 centimeters (cm) in the drawing equals 192 cm on the actual wing.

• Part A. What is the length of the actual wing? Explain how you found your answer.
• Part B. What is the area of the actual wing? Explain how you found your answer.

Mariko has an 80:1 scale-drawing of the floor plan of her house. On the floor plan, the dimensions of her rectangular living room are 1$\frac{\text{7}}{\text{8}}$ inches by 2$\frac{\text{1}}{\text{2}}$ inches. What is the area of her real living room in square feet?

### Instructional Items

Instructional Item 1
The triangle below needs to be recreated using the scale factor that produced Figure 2 from Figure 1. What is this scale factor?

Instructional Item 2
Andrew needs to repaint the side of his building white to prepare for a new mural that will be painted there. He measured the actual wall to be 26.25 feet long but he cannot easily measure the height. On his blueprints of the building, the wall measures 3.5 inches long and 4 inches tall. To determine how much paint to buy, calculate the area of the wall Andrew needs to cover.

*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.
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.7.GR.1.AP.5: Use a scale factor to draw a scale drawing of a real-world two-dimensional polygon on graph paper.

## Related Resources

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

## 3D Modeling

Wind Farm Design Challenge:

In this engineering design challenge, students are asked to create the most efficient wind turbine while balancing cost constraints. Students will apply their knowledge of surface area and graphing while testing 3D-printed wind farm blades. In the end, students are challenged to design and test their own wind farm blades, using Tinkercad to model a 3D-printable blade.

Type: 3D Modeling

## Formative Assessments

Space Station Scale:

Students are asked to find the ratio of the area of an object in a scale drawing to its actual area and then relate this ratio to the scale factor in the drawing.

Type: Formative Assessment

Flying Scale:

Students are asked to find the length and area of an object when given a scale drawing of the object.

Type: Formative Assessment

## Lesson Plans

Guiding Grids: Math inspired self-portraits:

Students will create a proportional self portrait from a photo using a gridded drawing method and learn how a grid system can help accurately enlarge an image in a work of art. Students will use the mathematical concepts of scale, proportion and ratio, to complete their artwork.

Type: Lesson Plan

Fish Kribs:

In this lesson, students create a fish tank for a fish supply company for a future sales campaign. They will use scale drawings and proportions to design the perfect fish tank.

• First, students have to complete a ranking activity of items that will be included in their scale drawing along with three types of fish.
• Next, students will conduct a pH lab activity to gain knowledge about how pH levels will affect population and the ecosystem within the tank.
• Finally, students will adjust their item selection and re-engineer their tank drawing to support their findings and additional information provided by the client. Students must determine what objects would be beneficial to the living things that the students chose in relation to available space and pH balance.

Type: Lesson Plan

Discovering Dilations:

This resource is designed to allow students to discover the effects of dilations on geometric objects using the free online tools in GeoGebra.

Type: Lesson Plan

Makeover, Home Edition Part III:

This is the third part of the lesson, "Makeover, Home Edition". This lesson is designed to teach students how to put ideas into reality by creating and using scale drawings in the real world. In Part I (#48705) students determined backyard dimensions for fence installation. Part II (#48967) concentrated on inserting a pool and patio into the backyard. In Part III (#49025) students will create a scale drawing of the backyard. Part IV (#49090) will focus on inserting a window and painting walls inside the house.

Type: Lesson Plan

Raising Your Garden MEA provides students with a real world engineering problem in which they must work as a team to design a procedure to select the best material for building raised garden beds. The main focus of this MEA is to recognize the importance of choosing the correct material for building a raised garden bed, what information is needed before starting a gardening project, and to consider the environmental and economic impact the garden will have on the school. Students will conduct individual and team investigations in order to arrive at a scientifically sound solution to the problem.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations.

Type: Lesson Plan

Boxing Candles:

In this MEA, students select jars for candles based on a variety of factors and then design boxes to contain the jars.

Type: Lesson Plan

Fastest Route:

In this Model Eliciting Activity, MEA, students will interpret and analyze a scale drawing to provide possible routes from a teacher’s home to the school. Students will consider factors including traffic patterns, construction zones, and wait times to recommend the best route including the total distance, in miles, and estimated delay times.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

How does scale factor affect the areas and perimeters of similar figures?:

In this lesson plan, students will observe and record the linear dimensions of similar figures, and then discover how the values of area and perimeter are related to the ratio of the linear dimensions of the figures.

Type: Lesson Plan

Students will apply their knowledeg of scale drawings, scale models, scale factors, and scale to create their own enlargement of an image they choose.

Type: Lesson Plan

## Original Student Tutorial

Scale Round Up:

Learn to use architectural scale drawings to build a new horse arena and solve problems involving scale drawings in this interactive tutorial. By the end, you should be able to calculate actual lengths using a scale and proportions.

Type: Original Student Tutorial

## Perspectives Video: Teaching Idea

KROS Pacific Ocean Kayak Journey: Kites, Geometry, and Vectors:

Set sail with this math teacher as he explains how kites were used for lessons in the classroom.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set [.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth [.KML]

Type: Perspectives Video: Teaching Idea

Floor Plan:

The purpose of this task is for students to translate between measurements given in a scale drawing and the corresponding measurements of the object represented by the scale drawing. If used in an instructional setting, it would be good for students to have an opportunity to see other solution methods, perhaps by having students with different approaches explain their strategies to the class. Students who can only solve this by first converting the linear measurements will have a hard time solving problems where only area measures are given.

## STEM Lessons - Model Eliciting Activity

Boxing Candles:

In this MEA, students select jars for candles based on a variety of factors and then design boxes to contain the jars.

Fastest Route:

In this Model Eliciting Activity, MEA, students will interpret and analyze a scale drawing to provide possible routes from a teacher’s home to the school. Students will consider factors including traffic patterns, construction zones, and wait times to recommend the best route including the total distance, in miles, and estimated delay times.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Fish Kribs:

In this lesson, students create a fish tank for a fish supply company for a future sales campaign. They will use scale drawings and proportions to design the perfect fish tank.

• First, students have to complete a ranking activity of items that will be included in their scale drawing along with three types of fish.
• Next, students will conduct a pH lab activity to gain knowledge about how pH levels will affect population and the ecosystem within the tank.
• Finally, students will adjust their item selection and re-engineer their tank drawing to support their findings and additional information provided by the client. Students must determine what objects would be beneficial to the living things that the students chose in relation to available space and pH balance.

Raising Your Garden MEA provides students with a real world engineering problem in which they must work as a team to design a procedure to select the best material for building raised garden beds. The main focus of this MEA is to recognize the importance of choosing the correct material for building a raised garden bed, what information is needed before starting a gardening project, and to consider the environmental and economic impact the garden will have on the school. Students will conduct individual and team investigations in order to arrive at a scientifically sound solution to the problem.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations.

## MFAS Formative Assessments

Flying Scale:

Students are asked to find the length and area of an object when given a scale drawing of the object.

Space Station Scale:

Students are asked to find the ratio of the area of an object in a scale drawing to its actual area and then relate this ratio to the scale factor in the drawing.

## Original Student Tutorials Mathematics - Grades 6-8

Scale Round Up:

Learn to use architectural scale drawings to build a new horse arena and solve problems involving scale drawings in this interactive tutorial. By the end, you should be able to calculate actual lengths using a scale and proportions.

## Student Resources

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

## Original Student Tutorial

Scale Round Up:

Learn to use architectural scale drawings to build a new horse arena and solve problems involving scale drawings in this interactive tutorial. By the end, you should be able to calculate actual lengths using a scale and proportions.

Type: Original Student Tutorial

Floor Plan:

The purpose of this task is for students to translate between measurements given in a scale drawing and the corresponding measurements of the object represented by the scale drawing. If used in an instructional setting, it would be good for students to have an opportunity to see other solution methods, perhaps by having students with different approaches explain their strategies to the class. Students who can only solve this by first converting the linear measurements will have a hard time solving problems where only area measures are given.