CPALMS Logo Generated on 9/17/2025 at 3:05 PM
The webpage this document was printed/exported from can be found at the following URL:
https://www.cpalms.org/PreviewIdea/Preview/1551

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Standard 1 : Apply geometric concepts in modeling situations. (Geometry - Major Cluster)Archived
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MAFS.912.G-MG.1.1 : Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★
  • MAFS.912.G-MG.1.2 : Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★
  • MAFS.912.G-MG.1.3 : Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★
Cluster Information
Number:
MAFS.912.G-MG.1
Title:
Apply geometric concepts in modeling situations. (Geometry - Major Cluster)
Type:
Cluster
Subject:
Mathematics - Archived
Grade:
912
Domain-Subdomain
Geometry: Modeling with Geometry
Cluster Access Points

This cluster includes the following Access Points.

  • MAFS.912.G-MG.1.AP.1a : Describe the relationship between the attributes of a figure and the changes in the area or volume when one attribute is changed.
  • MAFS.912.G-MG.1.AP.3a : Apply the formula of geometric figures to solve design problems (e.g., designing an object or structure to satisfy physical restraints or minimize cost).
  • MAFS.912.G-MG.1.AP.2a : Recognize the relationship between density and area; density and volume using real-world models.
Cluster Resources

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

Formative Assessments
  • The Duplex: Students are asked to solve a design problem in which the length of wall used in a rectangular floor plan is minimized.

  • Softball Complex: Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

  • Size It Up: Students are asked to name geometric solids that could be used to model several objects.

  • Land for the Twins: Students are asked to solve a design problem in which a triangular tract of land is to be partitioned into two regions of equal area.

  • How Many Trees?: Students are asked to determine an estimate of the density of trees and the total number of trees in a forest.

  • Estimating Volume: Students are asked to model a tree trunk with geometric solids and to use the model to estimate the volume of the tree trunk.

  • Camping Calculations: Students are asked to find the measure of an angle formed by the support poles of a tent using the properties of geometric shapes.

  • The Sprinters’ Race: Students are given a grid with three points (vertices of a right triangle) representing the starting locations of three sprinters in a race and are asked to determine the center of the finish circle, which is equidistant from each sprinter.

  • Population of Utah: Students are asked to determine the population of the state of Utah given the state’s population density and a diagram of the state’s perimeter with boundary distances labeled in miles.

  • Mudslide: Students are asked to create a model to estimate volume and mass.

  • Estimating Area: Students are asked to select appropriate geometric shapes to model a lake and then use the model to estimate the surface area of the lake.

Lesson Plans
  • The Seven Circles Water Fountain: Students will apply concepts related to circles, angles, area, and circumference to a design situation.

  • Building Graduation Caps: Students will apply skills from the Geometry Domain to build graduation caps for themselves using heavyweight poster paper. They will also apply some basic mathematical skills to determine dimensions and to determine minimum cost. Some of the Geometric skills reinforced in Building Graduation Caps: Cooperative Assignment are finding area, applying the concept of similarity, and the application of the properties of parallelograms. Other skills also involved in this application are measuring, and statistical calculations, such as finding the mean and the range. In addition to the hands-on group project that takes place during the lesson, there is the Prerequisite Skills Assessment: Area that should be administered before the group activity and a home-learning activity. Building Graduation Caps: Individual Assignment is the home-learning assignment; it is designed to reinforce the skills learned in the group activity.

  • NASA Space Shuttle Mission Patches: Students apply geometric measures and methods, art knowledge, contextual information, and utilize clear and coherent writing to analyze NASA space shuttle mission patches from both a mathematical design and visual arts perspective.

    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. Click here to learn more about MEAs and how they can transform your classroom.

  • 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.

  • Interchangeable Wristwatch Band: Students use measures and properties of rectangular prisms and cylinders to model and rank 3D printable designs of interchangeable wristwatch bands that satisfy physical constraints.

    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. Click here to learn more about MEAs and how they can transform your classroom.

  • It’s Not Waste—It’s Matter!: It's Not Waste—It's Matter is an MEA that gives students an opportunity to review matter, their physical properties, and mixtures. The MEA provides students to work in teams to resolve a real-life scenario creating a design method by which recyclable products are separated in order to further process.

    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. Click here to learn more about MEAs and how they can transform your classroom.

  • Propensity for Density: Students apply concepts of density to situations that involve area (2-D) and volume (3-D).

  • Solving Quadratic Equations: Cutting Corners: This lesson unit is intended to help you assess how well students are able to solve quadratics in one variable. In particular, the lesson will help you identify and help students who have the following difficulties; making sense of a real life situation and deciding on the math to apply to the problem, solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring, and interpreting results in the context of a real life situation.
  • Building Blocks of Geometry: Students will be introduced to the undefinable concepts of points, lines, and planes that are the building blocks of geometry and recognize that these three terms become the basis for many other geometric definitions. Students will participate in a Building Block Scavenger Hunt, using cameras to photograph examples of specified terms that they find outside of the math classroom. The students will compose a power point to display their photographs of the required terms.
    This lesson is adapted from a presentation that included an activity by Dianne Olix, 1995.

  • Olympic Snowboard Design: This MEA requires students to design a custom snowboard for five Olympic athletes, taking into consideration how their height and weight affect the design elements of a snowboard. There are several factors that go into the design of a snowboard, and the students must use reasoning skills to determine which factors are more important and why, as well as what factors to eliminate or add based on the athlete's style and preferences. After the students have designed a board for each athlete, they will report their procedure and reasons for their decisions.

  • Modeling: Rolling Cups: This lesson unit is intended to help you assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables and develop experimental and analytical models of a physical situation.

  • The Grass is Always Greener: The lesson introduces area of sectors of circles then uses the areas of circles and sectors to approximate area of 2-D figures. The lesson culminates in using the area of circles and sectors of circles as spray patterns in the design of a sprinkler system between a house and the perimeter of the yard (2-D figure).

  • Poly Wants a Bridge!: "Poly Wants a Bridge" is a model-eliciting activity that allows students to assist the city of Polygon City with selecting the most appropriate bridge to build. Teams of students are required to analyze properties of bridges, such as physical composition and span length in order to solve 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. Click here to learn more about MEAs and how they can transform your classroom.

  • A Pi-ece of Florida History: A Pi-ece of Florida History discovers significant dates in Florida History in the first 8 digits after the decimal of the number Pi. Historical people associated with those dates are identified and described. Students then use body measurements to approximate volume.

  • Quadrilaterals using Unit Origami - Sonobe Cube: This lesson can be used as an introduction - unit attention grabber- or as a final review on quadrilaterals. As the class forms a Sonobe cube, the different quadrilaterals are formed with each new fold. Included is a Power Point introduction with instructions, a video I made demonstrating how to fold the unit origami design, a video of my lesson I use as a middle or high school introduction to quadrilaterals and 2 worksheets which accompany the lesson. I enjoy doing this activity each year and am amazed at all the prior knowledge the students have retained as the lesson proceeds.

  • Turning Tires Model Eliciting Activity: The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

    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. Click here to learn more about MEAs and how they can transform your classroom.

  • A Pi-ece of Florida History: A Pi-ece of Florida History discovers significant dates in Florida History in the first 8 digits after the decimal of the number Pi. Historical people associated with those dates are identified and described. Students then use body measurements to approximate volume.

  • Concurrent Points Are Optimal: Students will begin with a review of methods of construction of perpendicular bisectors and angle bisectors for the sides of triangles. Included in the review will be a careful discussion of the proofs that the constructions actually produce the lines that were intended.

    Next, students will investigate why the perpendicular bisectors and angle bisector are concurrent, that is, all three meet at a single meet.

    A more modern point of currency is the Fermat-Torricelli point (F-T). The students will construct (F-T) in GeoGebra and investigate limitations of its existence for various types of triangles.

    Then a set of scenarios will be provided, including some one-dimensional and two-dimensional situations. Students will use GeoGebra to develop conjectures regarding whether a point of concurrency provides the solution for the indicated situation, and which one.

    A physical model for the F-T will be indicated. The teacher may demonstrate this model but that requires three strings, three weights, and a base that has holes. A recommended base is a piece of pegboard (perhaps 2 feet by 3 feet), the weights could be fishing weights of about 3 oz., the string could be fishing line; placing flexible pieces of drinking straws in the holes will improve the performance.

    The combination of geometry theorems, dynamic geometry software, a variety of contexts, and a physical analog can provide a rich experience for students.

  • Quadrilaterals using Unit Origami - Sonobe Cube: This lesson can be used as an introduction - unit attention grabber- or as a final review on quadrilaterals. As the class forms a Sonobe cube, the different quadrilaterals are formed with each new fold. Included is a Power Point introduction with instructions, a video I made demonstrating how to fold the unit origami design, a video of my introduction lesson  to quadrilaterals and 2 worksheets which accompany the lesson. I enjoy doing this activity each year and am amazed at all the prior knowledge the students have retained as the lesson proceeds.

Perspectives Video: Experts
Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
  • Toilet Roll: The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

  • Coins in a circular pattern: Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

  • The Lighthouse Problem: This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

  • Solar Eclipse: This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

  • Seven Circles III: This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

  • Running around a track II: The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

  • Running around a track I: In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

  • Paper Clip: In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

  • Ice Cream Cone: In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

  • How thick is a soda can? (Variation II): This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.

  • How thick is a soda can? (Variation I): This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

  • How many leaves on a tree? (Version 2): This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

  • How many leaves on a tree?: This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

  • How many cells are in the human body?: This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

  • Hexagonal pattern of beehives: The goal of this task is to use geometry to study the structure of beehives.

  • Global Positioning System II: Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

  • Eratosthenes and the circumference of the earth: This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

  • Archimedes and the King's Crown: This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

  • Use Cavalieri’s Principle to Compare Aquarium Volumes: This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

  • Tennis Balls in a Can: This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

  • Regular Tessellations of the Plane: This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Teaching Idea
Text Resource
  • Math for Hungry Birds: This informational text resource is intended to support reading in the content area. A new study indicates that the flying patterns of hunting albatrosses may resemble mathematical designs called fractals. This article describes the basics of fractals and why scientists think the albatross may hunt in such patterns. As it turns out, many animals may use math to find food!

Video/Audio/Animation
  • MIT BLOSSOMS - Using Geometry to Design Simple Machines: This video is meant to be a fun, hands-on session that gets students to think hard about how machines work. It teaches them the connection between the geometry that they study and the kinematics that engineers use -- explaining that kinematics is simply geometry in motion.