Cluster 2: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (Additional Cluster)Archived

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

You are a robotic spider tangled up in an angular web. Use your knowledge of angle relationships to collect flies and teleport through wormholes to rescue your spider family!

Type: Educational Game

This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

Type: Educational Software / Tool

Students are asked use knowledge of angle relationships to write and solve an equation to determine an unknown angle measure.

Type: Formative Assessment

Students are asked to solve a problem involving the circumference of a circle.

Type: Formative Assessment

Students are asked to write the formula for the circumference of a circle, explain what each symbol represents, and label the variables on a diagram.

Type: Formative Assessment

Students are asked to write the formula for the area of a circle, explain what each symbol represents, and label the radius on a diagram.

Type: Formative Assessment

Students are asked to solve a problem involving the area of a circle.

Type: Formative Assessment

Students are asked to complete and explain an informal derivation of the relationship between the circumference and area of a circle.

Type: Formative Assessment

Students are asked to determine the surface area of a right triangular prism and explain the procedure.

Type: Formative Assessment

Students are asked to find the area of a composite figure.

Type: Formative Assessment

Students are asked to calculate the volume and surface area of a cube.

Type: Formative Assessment

Students are asked to find the surface area of a composite figure.

Type: Formative Assessment

Students are asked to find the area of a composite figure.

Type: Formative Assessment

Students are asked to solve a problem involving the volume of a composite figure.

Type: Formative Assessment

Students are asked to write and solve equations to determine unknown angle measures in supplementary angle relationships.

Type: Formative Assessment

Students are asked to write and solve equations to determine unknown angle measures in supplementary and complementary angle pairs.

Type: Formative Assessment

Students are asked to use knowledge of angle relationships to write and solve equations to determine unknown angle measures.

Type: Formative Assessment

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will support you in identifying and helping students who have the following difficulties with solving problems related to the interior or exterior angles of polygons.

Type: Formative Assessment

The context: There is 24,400 in 1 bills under a mattress. How far will the mattress lower if the money is deposited in the bank? Complete with worksheets, student examples, student mistakes for analysis. This lesson unit is intended to help you assess how well students are able to:

• Model a situation.
• Make sensible, realistic assumptions and estimates.
• Use assumptions and estimates to create a chain of reasoning, in order to solve a practical problem.

Type: Formative Assessment

This lesson unit is intended to help you assess how well students are able to:

• Model a situation.
• Make sensible, realistic assumptions and estimates.
• Use assumptions and estimates to create a chain of reasoning in order to solve a practical problem.

Type: Formative Assessment

This lesson unit is intended to help you assess how well students are able to:

• Recognize and use common 2D representations of 3D objects.
• Identify and use the appropriate formula for finding the circumference of a circle.

Type: Formative Assessment

This large collection of clipart contains images of angles that can be freely used in lesson plans, worksheets, and presentations.

Type: Image/Photograph

Students will help a volunteer coordinator choose cleanup projects that will have the greatest positive impact on the environment and the community.  They will apply their knowledge of how litter can impact ecosystems along with some math skills to make recommendations for cleanup zones to prioritize.  Students will explore the responsibilities of citizens to maintain a clean environment and the impact that litter can have on society in this integrated Model Eliciting Activity.  

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.

Type: Lesson Plan

This set of geometry challenges focuses on creating a variety of polygons using the coordinate plane as students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor.

Type: Lesson Plan

This set of geometry challenges focuses on creating circles and calculating area/circumference as students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor

Type: Lesson Plan

This lesson addresses the topic of limiting factors looking specifically at maintaining wildlife populations. There is an engineering design challenge included in which students will have to take on the role of a wildlife conservation officer and create a container for the egg of an endangered species that will protect it in the field until he/she can get it back to the lab.

Type: Lesson Plan

Students will use nets of prisms to find the surface area of composite 3-D figures. Students will learn to identify the faces of 3-D figures that are needed to find the surface areas, and those that are not needed.

Type: Lesson Plan

This is a STEM-Engineering Design Challenge lesson. Students will go through the process of creating an organ transport container using their knowledge of human body systems, heat flow, and volume.

Type: Lesson Plan

Students will calculate unit rate & circumference, compare & order decimals, convert metric units, and round decimals. Bubble Burst Corporation has developed some chewing gum prototypes and has requested the students to assist in the selection of which gum prototypes will be mass produced by using both quantitative and qualitative data to rank the prototypes for Bubble Burst Corporation.

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.

Type: Lesson Plan

7th grade math/science lesson plan that focuses on the concept of circumference and rotation relationship. Culminates in a problem-solving exercise where students apply their knowledge to the "rotations" field in programming a LEGO/NXT robot to traverse a set distance.

Type: Lesson Plan

In this Lab, students create their own definition for the term density and calculate the densities of different substances- solids and liquids. Students will learn that every substance has its own unique density, depending on how tightly atoms or molecules of the materials are packed. Students gather data about known samples to infer the identity of an unknown sample.

Note: This lesson will only cover the density portion of benchmark SC.8.P.8.4

Type: Lesson Plan

In this lesson, students will explore and apply the use of nets to find the surface area of pyramids.

Type: Lesson Plan

This lesson is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will support you in identifying and helping students who have the following difficulties:

• Solving problems relating to using the measures of the interior angles of polygons.
• Solving problems relating to using the measures of the exterior angles of polygons.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students are able to solve simple problems involving ratio and direct proportion, choose an appropriate sampling method, collect discrete data, and record their data using a frequency table.

Type: Lesson Plan

This lesson allows students to discover complementary and supplementary angles on their own. Students will measure the degrees for sets of angles. They will then use their knowledge of straight, right, obtuse, and acute angles to make connections to complementary, supplementary, adjacent, and vertical angles. This lesson will complement all different learning styles because it provides kinesthetic, auditory, and visual learning.

Type: Lesson Plan

Students explore how the formulas for surface area and volume were derived and apply this knowledge to solve problems. Students will be presented with a problem-solving task that incorporates finding the surface area and volume when designing an aquarium.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students are able to recognize and use common 2D representations of 3D objects, as well as identify and use the appropriate formula for finding the circumference of a circle.

Type: Lesson Plan

This lesson is designed to be a fun and creative way to introduce math vocabulary (radius, diameter, and circumference) related to circles. Students will create a story board (comic strip) to retell the story using targeted vocabulary, and then demonstrate understanding of the relationships among radii, diameter, and circumference by completing the worksheet.

In my experience, lower performing students struggle to not confuse radius, diameter, and circumference. My learning objective in this lesson is to have them correctly remember the "parts" of a circle and that a diameter is comprised of 2 radii. I frequently refer back to the story in later lessons when they are having difficulty identifying radius or diameter.

Type: Lesson Plan

Students will learn what a "net" is, draw nets of three dimensional shapes, accurately calculate the surface area of their nets, and put them together to create an original person or pet.

Type: Lesson Plan

Students create surface area nets with graph paper and work with manipulative cubes to decide if there is a relationship between surface area and volume in rectangular prisms.

Type: Lesson Plan

This is a lesson designed to teach students how to calculate volume and surface areas of rectangular prisms. It provides an interactive lesson where students get to learn hands-on with cereal boxes and on the computer with a GeoGebra activity.

Type: Lesson Plan

This MEA requires students to formulate a comparison-based solution to a problem involving finding the best plan for installing tile floor considering different aspects. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

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.

Type: Lesson Plan

This is the final lesson of the unit , "Makeover, Home Edition." This lesson is designed to teach students the applicability of finding area of composite figures as well as understanding the importance of ratios in the real world. Part I (#48705) dealt with determining backyard dimensions for fence installation. Part II (#48967) concentrated on inserting a pool and patio into this backyard. Part III (#49025) dealt with creating a scale drawing of the backyard.

Type: Lesson Plan

This is the second part of the Unit Lesson, "Makeover, Home Edition". This lesson will continue focusing on unit prices, but also incorporates area and volume as well. Part I (Makeover, Home Edition #48705) is based on creating backyard dimensions for fencing. Part III (Makeover, Home Edition #49025) will deal with creating a scale drawing of this backyard. Part IV (Makeover, Home Edition Final #49090) focuses on inserting a window and painting walls inside the house.

Type: Lesson Plan

In this lesson, students will design two outdoor gardens, 1) a raised garden bed and 2) a ground level garden (traditional). Students will, with help of the teacher, till the ground with removal of ground cover, build border for garden, add soil, attach poles with string to create a life size graph all so they can grow tomatoes and plot the data easily in their survival journals.This is Part 2 of a 4-Part Project on Survival.

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

In this lesson, students will develop an understanding of the meaning of pi and how it relates to the relationship to circumference and diameter. This is a hands-on activity where students measure circular objects and calculate the relationship of the circumference to the diameter.

Type: Lesson Plan

In this lesson, students will explore the relationship between volume and surface area through real world problem solving. They will work with a partner as they are in charge with the task of finding the least expensive packaging (smallest surface area) for a given number of caramels (volume). Students will justify their packaging strategy in a group discussion.

Type: Lesson Plan

"Students will learn the relationship between the radius and diameter of a circle and the relationship of the diameter and the circumference of a circle. Students will also find the approximate circumference of a circle, given either the diameter or radius." (from MATH Can Take You Places)

Type: Lesson Plan

Students hear a story about a middle-school student who wants to redecorate his bedroom. They measure the classroom wall dimensions, draw a scale model, and incorporate measurements for windows and doors to determine the area that could be covered by wallpaper. Students then hear more about the student's redecorating adventure and learn about expenses, budget constraints, and tradeoffs.

Type: Lesson Plan

Using volume in the real world

Type: Lesson Plan

This lesson allows a hands-on approach for students to use real- life problem solving. Students will apply their measurement skills to the concept of surface area. This lesson provides opportunities for students to work cooperatively with others as a team.

Type: Lesson Plan

This lesson is designed for 7th grade students and is best suited for advanced students. It can be used (with modifications) in the general education classroom for 7th grade or in an advanced 6th grade classroom. In this MEA, students select jars for candles based on a variety of factors and then design boxes to contain the jars.

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.

Type: Lesson Plan

Students relate their body parts to six cylinders. They adjust the surface area of a cylinder formula, and then measure, record, and compute the approximate surface area of their skin. Students will develop a better understanding of how much skin covers up their body when they calculate how much surface area their skin takes up.

Type: Lesson Plan

This lesson uses a paper cutting activity to teach vertical angles. The lesson provides examples in which students must solve equations to find missing measures.

Type: Lesson Plan

Students will recognize that while the surface area may change, the volume can remain the same. This lesson is enhanced through the multimedia CPALMS Perspectives Video, which introduces students to the relationship between surface area and volume.

Type: Lesson Plan

This lesson includes measurement activities that help students understand where the value of pi originated, as well as activities that help students apply the value of pi to real-life scenarios.

Type: Lesson Plan

Students, through a hands-on-activity and guided practice, will explore parallel lines intersected by a transversal and the measurements and relationships of the 8 angles created. They will solve for missing measurements when given a single angle's measurement. They will also use the relationships between angles to set up equations and solve for a variable.

Type: Lesson Plan

Explore complementary and supplementary angles around the playground with Jacob in this interactive tutorial.

This is Part 1 in a two-part series. Click HERE to open Playground Angles: Part 2.

Type: Original Student Tutorial

Help Jacob write and solve equations to find missing angle measures based on the relationship between angles that sum to 90 degrees and 180 degrees in this playground-themed, interactive tutorial.

This is Part 2 in a two-part series. Click HERE to open Playground Angles: Part 1.

Type: Original Student Tutorial

Explore the origins of Pi as the ratio of Circumference to diameter of a circle. In this interactive tutorial you'll work with the circumference formula to determine the circumference of a circle and work backwards to determine the diameter and radius of a circle.

Type: Original Student Tutorial

Explore how to calculate the area of circles in terms of pi and with pi approximations in this interactive tutorial. You will also experience irregular area situations that require the use of the area of a circle formula.

Type: Original Student Tutorial

Learn to solve problems involving the circumference and area of circle-shaped pools in this interactive tutorial.

Type: Original Student Tutorial

A math teacher describes the relationship between area and circumference and gives examples in nature.

Type: Perspectives Video: Expert

Michael McKinnon of Gaines Street Pies explains how when making pizza the volume is conserved but the surface area changes.

Type: Perspectives Video: Professional/Enthusiast

How many times larger is the area of a large pizza compared to a small pizza? Which pizza is the better deal? Michael McKinnon of Gaines Street Pies talks about how the area, circumference and price per square inch is different depending on the size of the pizza.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.

Type: Perspectives Video: Professional/Enthusiast

An architect discusses how he uses circumference and area calculations to accurately create designs and plans.

Type: Perspectives Video: Professional/Enthusiast

A dance costume designer describes how she uses circumference and area calculations to make clothing for the stage.

Type: Perspectives Video: Professional/Enthusiast

What do you do if you don't have room for all your gear on a solo ocean trek? You're gonna need a bigger boat...or pack smarter with math.

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

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

A math teacher presents an idea for a classroom activity to engage students in measuring diameter and circumference to calculate pi.

Type: Perspectives Video: Teaching Idea

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]

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Teaching Idea

A science teacher demonstrates stepwise calculations involving multiple variables for designing robots with desired characteristics.

Type: Perspectives Video: Teaching Idea

Students derive the formula for area of a circle using worksheet in the attachment section of this resource or using the provided URL.

Type: Problem-Solving Task

Before the lesson, students attempt the Gold Rush task individually. You then look at their responses and formulate questions for students to think about as they review their work. At the start of the lesson, students reflect on their individual responses and use the questions posed to think of ways to improve their work. Next, students work collaboratively in small groups to produce, in the form of a poster, a better solution to the Gold Rush task than they did individually. In a whole-class discussion students compare and evaluate the different methods they used. Working in small groups, students analyze sample responses to the Gold Rush task, then, in a whole-class discussion, review the methods they have seen. Finally, students reflect on their work.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to find the area of a shaded region using a diagram and the information provided. The purpose of this task is to strengthen student understanding of area.

Type: Problem-Solving Task

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

Type: Problem-Solving Task

In this problem solving task, students investigate Shamu Stadium at Sea World. They will use knowledge of geometric shapes to solve problems involving area and volume and examine as well as analyze a diagram making calculations. Students will also be challenged to design an advertising poster using the measurements they mind.

Type: Problem-Solving Task

In this activity, students adjust the dimensions of either a rectangular or triangular prism and the surface area and volume are calculated for those dimensions. Students can also switch into compute mode where they are given a prism with certain dimensions and they must compute the surface area and volume. The application keeps score so students can track their progress. This application allows students to explore the surface area and volume of rectangular and triangular prisms and how changing dimensions affect these measurements. This activity also includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Problem-Solving Task

This resource guides the learner step-by-step in creating a scale floor plan of a classroom. The instructions include sample drawings of student work. The activity includes sketching a map of the classroom, measuring the room and calculating the area and perimeter, creating a scale drawing, and drafting a CAD (computer-aided design) floor plan. The lesson provides students with hands-on, real world practice solving problems of measurement, ratio, and scale.

Type: Project

This lesson unit is intended to help you assess how well students are able to:

• Interpret a situation and represent the variables mathematically.
• Select appropriate mathematical methods.
• Interpret and evaluate the data generated.
• Communicate their reasoning clearly.

Type: Teaching Idea

In this engineering task, students will apply concepts of volume to decide what they will need to take on a 2-1/2 year journey to Mars. Then plan how to fit everything into a 1-cubic-meter box, using only a measuring tape, pencil and paper, and math.

Type: Teaching Idea

The video will use algebra to find the measure of two angles whose sum equals 90 degrees, better known as complementary angles.

Type: Tutorial

Watch as we use algebra to find the measure of two complementary angles. 

Type: Tutorial

Watch as we use algebra to find the measure of supplementary angles, whose sum is 180 degrees.

Type: Tutorial

This video shows how the area and circumference relate to each other and how changing the radius of a circle affects the area and circumference.

Type: Tutorial

In this video, students are shown the parts of a circle and how the radius, diameter, circumference and Pi relate to each other.

Type: Tutorial

This video shows how to find the circumference, the distance around a circle, given the area.

Type: Tutorial

This video uses knowledge of vertical angles to solve for the variable and the angle measures.

Type: Tutorial

This video uses facts about supplementary and adjacent angles to introduce vertical angles.

Type: Tutorial

This video demonstrates solving a word problem involving angle measures.

Type: Tutorial

In this video, watch as we find the area of a circle when given the diameter.

Type: Tutorial

Find the volume of an object, given dimensions of a rectangular prism filled with water, and the incremental volume after the object is dropped into the water.

Type: Tutorial

This video involves packing a larger rectangular prism with smaller ones which is solved in two different ways.

Type: Tutorial

This video will show to find the volume of a triangular prism, and a cube by applying the formula for volume.

Type: Tutorial

The video will demonstrate the difference between supplementary angles and complementary angles, by using the given measurements of angles.

Type: Tutorial

This 5 minute video gives the proof that vertical angles are equal.

Type: Tutorial

This resource will allow students to have a good understanding about vertical, adjacent and linear pairs of angles.

Type: Tutorial

In this interactive, self-guided unit on 3-dimensional shape, students (and teachers) explore 3-dimensional shapes, determine surface area and volume, derive Euler's formula, and investigate Platonic solids. Interactive quizzes and animations are included throughout, including a 15 question quiz for student completion.

Type: Unit/Lesson Sequence

This video dynamically shows how Pi works, and how it is used.

Type: Video/Audio/Animation

This manipulative allows you to create shapes using colorful blocks to explore the relationship between perimeter and area. The game screen challenges you to build shapes or find the area of figures.

Type: Virtual Manipulative

This interactive lesson introduces students to the circle, its attributes, and the formulas for finding its circumference and its area. Students then perform a few calculations to practice finding the area and circumference of circles, given the diameter.

Type: Virtual Manipulative

This lesson is designed to develop students' knowledge of surface area and introduce them to calculating the surface area of a triangular prism. This lesson provides links to discussions and activities related to surface area as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Virtual Manipulative

This applet allows students to investigate the relationships between the area and circumference of a circle and its radius and diameter. There are three sections to the site: Intro, Investigation, and Problems.

• In the Intro section, students can manipulate the size of a circle and see how the radius, diameter, and circumference are affected. Students can also play movie clip to visually see how these measurements are related.
• The Investigation section allows students to collect data points by dragging the circle radius to various lengths, and record in a table the data for radius, diameter, circumference and area. Clicking on the x/y button allows students to examine the relationship between any two measures. Clicking on the graph button will take students to a graph of the data. They can plot any of the four measures on the x-axis against any of the four measures on the y-axis.
• The Problems section contains questions for students to solve and record their answers in the correct unit.

(NCTM's Illuminations)

Type: Virtual Manipulative

This lesson is designed to introduce students to the concept of surface area and how to find the surface area of a rectangular prism. This lesson provides links to discussions and activities related to surface area as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Virtual Manipulative

This lesson is designed to develop students' understanding of volume and ability to find volumes of triangular prisms. It provides links to discussions and activities related to volume as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: WebQuest