Area and Perimeter 
of
Compound Shapes

6.2  Perimeter and area of rectangle, triangle, and compound shapes derived from these. Area of trapezoid and parallelogram.

6.5  Areas and volumes of compound shapes. Involving combinations of the shapes in section 6.4

Area and Perimeter
of
Compound Shapes

Context:  There are a lot of great mathematical tools that students can develop in this topic.  This topic can help make the following concepts and abilities accessible to students:

• Dimension and units, area is a unit-squared, and perimeter is just the unit.
• Unit conversion from distance to area … 1 m² = 10,000 cm², even though 100 cm = 1 m
• Deriving formulas
• Writing equations in math
• Problem solving strategies

Big Idea

Students need to understand that area is “space in two dimensions.”  For shapes with curves we get to rely on formulas that we do not have to derive or even understand at this level.  For polygons, we have some variation and portion of the product of two perpendicular distances.

They also need to understand why the units in area are also squared.

Key Knowledge

Prerequisite knowledge includes basic arithmetic skills and the knowledge of basic two-dimensional geometric shapes.

In this unit students need to come away with the ability to find the area and perimeter of compound shapes by breaking those shapes into smaller pieces.

Pro – Tip

When confused, draw each shape that combines to make the compound shape.  Write formulas on each drawing.  Then plug in the numbers, then calculate.

Click on the PDF icon to download a Lesson Guide that will pace the PowerPoint.  Click on the PowerPoint icon to download the PowerPoint Lesson.  

The lessons are intentionally over-built with more material than the time will allow.  This way you have more options on how to best serve the needs of your students.

This is a two-day lesson, complete with Lesson Guide which will help you to pace the lesson and understand the objective of each slide in the PowerPoint.  There are two homework assignments for the lesson.

Below is a copy of one of the slides from the lesson.

There are two homework assignments for this lesson.  The first is pictured below. 

To download the first assignment, click here.

To download the second assignment, click here.

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Sector Area & Arc Length

Circumference and area of a circle.  Arc length and area of sector. From sector angles in degrees only.

Sector Angles, Area and Arc Length

Note:  Credit for this original idea goes to Dan Meyer.  If you’re not familiar with Dan Meyer’s work, please visit his wonderful website.  It is full of excellent resources.  His creative approach to making practical problems into interesting learning experiences for students is a treasure for educators! 

Context for Teaching:  Geometry is particularly difficult to teach, and this is an unusual place to start, with sector angles, area and perimeter.  However, the key to success, for a Geometry student, is the ability to recognize what conceptual understanding they have that might play a role in the issue they’re facing.

Geometry requires many cognitive leaps from the students!  They need to draw connections between various pieces of knowledge, without prompt, in order to be excellent problem solvers.  That is perhaps what learning Geometry really does for a student.  It helps develop their ability to think “outside the box.”  (See what I did there?)

Big Idea

We want students to take a few known facts and develop a mathematical approach to answering a question.  The question is something near and dear to their hearts … How to get the most pizza?

In doing so, students will develop an understanding of sectors and area, and may even stumble upon some understanding of dimension (area versus length).  This will all happen on their own, before you help them to formalize their thinking.

Key Knowledge

Prerequisite:  Students need to know how many degrees are in a circle, and what the radius, diameter and circumference are.  They need to recognize that a circle is a nice approximation for a pizza.

New Information:  Through review and summary, you will help students formalize their understanding of sectors and arc length.  This will begin in the lesson, and continue with homework review the next day.

Pro – Tip
(for students)

Record your questions and realizations throughout the lesson.  This lesson is as much about learning a successful approach to Geometry as it is about learning a few basic Geometric facts and formulas.

More practice problems coming soon.

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Surface Area, Volume, Modeling

6.4  Surface area and volume of a prism and a pyramid (in particular, cuboid, cylinder, and cone).  Surface area and volume of a sphere.  Formulae will be given for the lateral surface area of a cylinder and a cone, the surface area of a  sphere, and the volume of a pyramid, a cone, and a sphere.

6.6  Use geometric shapes, their measures, and their properties to describe objects. e.g., modeling a tree trunk or a human torso as a cylinder.

Geometric Shapes
and Formulas

Context:  This lesson deals with combining various shapes that will collectively make a volume together.  Students will deal with this happening in two ways.  First, solids added to a fluid, and second, solid removed from a fluid.  They’ll also be given a volume and a shape and be asked to determine particular dimensions of the object.

It is important to encourage, and allow, students to struggle with a problem solving approach on this topic.  The most difficult part is typically figuring out where to start.  The second most difficult part is keeping track of your intention and not losing what numbers represented as a student works through the problem.

With those two ideas in mind, it will be important to allow the time needed for students to work through both issues.  It is only after they recognize the usefulness of an efficient method that they will latch onto it.

The latter portion of this 2 to 3 day lesson is all problems.  Have students try each problem on their own, or in small groups.  Check for quality of thought and consistent follow through, not final answer.  If students get “help,” but don’t understand, their work will not math their answer.  In this case, the student has probably fooled themselves into thinking they understand.  It is important to encourage them to make sense of the entire process.

When modeling how to do these problems, encourage students to NOT perform calculations on the calculator until the last possible step.  This will keep them from making as many mistakes, they won’t round, their work will be neat, and it will be faster. 

Big Idea 

Volume is, roughly speaking, how much space a 3-dimensional object occupies.  When solid objects are added to a fluid, like in a cylinder, the total volume inside the cylinder increases by the volume of the object added.  The reverse has the same concept. 

Key Knowledge

Prerequisite Knowledge:  Students need to understand how to manipulate formulas Algebraically, basic calculator functions, and the names and basic properties of common 3D shapes.

Current Facts:  Students need to know how the formulas for volume of common 3D shapes.

Pro – Tip

(for students)

There are two keys to success here.  (1)  Draw a picture of your idea when applicable.  (2)  Manipulate all formulas Algebraically, saving the arithmetic for the last possible step.  Write your un-rounded answer on your paper before rounding.  This can save you a lot of time and points from simple mistakes.

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Surface Area, Volume & Cross Sectional Area

6.4  Surface area and volume of a prism and a pyramid (in particular, cuboid, cylinder, and cone). Surface area and volume of a sphere. Formulae will be given for the lateral surface area of a cylinder and a cone, the surface area of a sphere, and the volume of a pyramid, a cone, and a sphere.

6.7  Identify the shapes of two-dimensional cross sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Surface Area and Volume

Context:  Teaching Surface Area and Volume lends itself to easy, tangible connections.  This can help make learning fun and easily accessible to students.  However, the following concepts and skills can be developed when students are learning this topic:

• Use of formulas
• Calculator proficiency
• Rounding to given significant figures
• Accurate constructions
• Unit conversion
  • 100³ cm³ = 1 m³
• Recognizing what 3D shapes can approximate real-life objects
• Relationship between the growth of squared and cubed units

When teaching this unit, students should be challenged to track their thinking.  A good strategy for students is to sketch their ideas in diagrams, then add formulas to those diagrams, then plug the numbers in (on paper), then use the calculator to evaluate.  They should write down their unrounded number and then round to the appropriate place (typically three significant figures at this level).

Students need to understand that for prisms and cylinders, volume is the product of the base’s area (which is a cross section) and the perpendicular “height.”  For other 3D shapes, formulas will be provided.

To help them visualize the volume of prisms use a large stack of printer paper.  When holding up one sheet, it is practically a two-dimensional plane.  It is unlikely that the volume of one sheet could be measured in a High School classroom.  However, when there are 500 sheets, they create enough dimension in a 3^rd direction that volume could be calculated.  Side Note:  After a stack of 500 sheets has been calculated, students could figure out the volume of one sheet.  This could be a good exercise!

Students also need the ability to see how a net can make a 3D shape.  They should be able to calculate the volume of a prism from its net.  They should also be able to draw a net from seeing a 3D shape.

Big Idea

Surface area is the outward facing surface.  Like the skin on a person.  Finding surface area is similar to finding the area of a compound shape.

Volume is how much three-dimensional space is occupied by an object.  The units are three-dimensional. 

Key Knowledge

Prerequisite:  Students need to be proficient with finding the area of common 2D shapes.

New: How to draw a net.  The volume of prisms, the formulas for surface area and volume of spheres, cones and rectangular pyramids.

Pro-Tip
(for students)

When finding the surface area of a 3D object, be sure to account for ALL sides.  This is the easiest thing to over-look.

Click on the PDF icon to download a Lesson Guide that will pace the PowerPoint.  Click on the PowerPoint icon to download the PowerPoint Lesson.  

The lessons are intentionally over-built with more material than the time will allow.  This way you have more options on how to best serve the needs of your students.

This is a week-long lesson.  The direct instruction is only one day.  For that day you will be provided with a Lesson Guide which will help you to pace the lesson and understand the objective of each slide in the PowerPoint.  There is a homework assignment for that day.  The rest of the week is a project where students will build two different prisms that must contain the same volume.  The process of administering this project, along with documents for the project, are linked in the lesson plan, which can be downloaded by clicking the PDF icon above.

Below is a copy of one of the slides from the lesson.

There is one homework assignment for this unit.  The rest of the learning will be done through the project outlined in the lesson plans.

To download the homework, click here.

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