deriving the formula for the volume of a prism 4disme.org/all-files/volume4-lesson.pdf · 2014. 12....

Volume Unit 4

Deriving the Formula for the Volume of a Prism

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Mathematical Concepts • We call the space occupied by a 3-dimensional structure a volume.

• A polyhedron is a 3-dimensional structure formed by polygons

enclosing a single region of space (the interior of the solid).

• A prism is a polyhedron with two congruent, parallel faces called

bases. The bases can be any polygon (e.g., triangle, square, rectangle,

pentagon, etc.). The bases are joined by parallelograms (e.g., rectangle,

square, rhombus, parallelogram).

• The bases of a rectangular prism are rectangles or squares.

• A right prism has lateral faces that are rectangles. The bases can be

any polygon.

• The volume of the prism is the product of the area of a base and its

height (altitude).

• The volume of a cylinder is the product of the area of a base and its

height.

Unit Overview Students first find the volumes of right rectangular prisms by partially structuring an array of cubes, visualizing the completion of the array, and then counting the number of cubes that fill the prism. The formula for the volume of a rectangular prism as the product of the area of the base and height is developed and tested with right rectangular prisms, some of which have non-whole number areas and/or altitudes. Sweeping is used as a way of envisioning a transition between layers to a continuous product. By Cavaleri’s principle, and a stack of cards to visualize it, the formula applies also to oblique rectangular prisms. The unit concludes with an extension of the formula for volume measure to triangular prisms, and of the process of sweeping area through height to find the volume of a cylinder.

U N I T

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Contents Mathematical Concepts 1 Unit Overview 1 Materials & Preparation 2 Academic Vocabulary 2 Instruction 6 Rectangular (Square) Prism Rectangular Prism Rectangular Prism w/Fractional Meas. Developing a Formula Cavaleri’s Principle for Oblique Prisms Extension to Volume of Cylinder

6 8

10 12 13 15

Formative Assessment 16 Formative Assessment Record

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Volume Unit 4

Materials and Preparation Volume Unit 4

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Prepare

§ Every pair of students receives 3 rectangular prism boxes and a

number of 1-inch cubes that varies with the problem posed.

§ The dimensions of the rectangular prism boxes (open so that

students can insert cubes) are:

o 4 𝑖𝑛 × 4 𝑖𝑛 × 4𝑖𝑛

o 8 𝑖𝑛 × 6 𝑖𝑛 × 4 𝑖𝑛

o 5 𝑖𝑛 × 5 𝑖𝑛 × 3.5 𝑖𝑛

§ Every pair of students receives a triangular prism with a 2 inch

base and a height of 18.25 inches, or some other triangular prims

as determined by the teacher. A net of a different triangular prism

is included, and this can be used instead.

§ Print-outs of net of a triangular prism

§ Deck of playing cards or a stack of index cards or a stack of square

crackers.

Academic Vocabulary

§ Area

§ Volume

§ Cubic inch

§ Prism

§ Base

§ Lateral Face

§ Rectangular Prism

§ Triangular Prism

§ Cylinder

Mathematical Concepts Unit Overview

Materials and Preparation Academic Vocabulary

Instruction Rectangular (Square) Prism

Rectangular Prism Rectangular Prism with Fractional Measure

Developing a Formula Cavaleri’s Principle for Oblique Prisms

Triangular Prism Extension to Volume of Cylinder

Formative Assessment Formative Assessment Record

Volume Unit 4

Academic Vocabulary Volume Unit 4

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Surface Area

The two dimensional regions of a three-dimensional structure

enclose/occupy space. We call this a surface area to distinguish it from the

space contained by the three-dimensional structure.

Volume

The space enclosed/occupied by a three dimensional structure is called a

volume.

Volume Measure

Volume measure is a ratio of the space enclosed by a three dimensional

structure and a three-dimensional unit of measure. Units of volume

measure, typically cubes, tile the volume. Hence, counting cubic units,

including fractional cubic units, measures volume. For example, a

structure with a measure of 50 !! cubic units has a volume that is 50 !

! times

that of the volume of the cubic unit.

Net

A net is a 2-dimensional representation of the surface of a 3-dimensional

structure.








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Prism A polyhedron is a solid formed by polygons enclosing a single region of

space (the interior of the solid). A prism is a polyhedron with two

congruent, parallel faces called bases. The bases can be any polygon (e.g.,

triangle, square, rectangle, pentagon, etc.). The bases are joined by

parallelograms (e.g., rectangle, square, rhombus, parallelogram). A right

prism has lateral faces that are rectangles. The bases can be any polygon.

Right Rectangular Prism

Oblique Rectangular Prism

Triangular Prism

Oblique Pentagonal Prism

• The volume of the prism is the product of the area of a base and its

height (also called altitude). For a rectangular prism, this product is

represented by 𝑙𝑒𝑛𝑔𝑡ℎ×𝑤𝑖𝑑𝑡ℎ × ℎ𝑒𝑖𝑔ℎ𝑡.








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Cavaleri’s Principle The volume of an oblique prism is the same as it is for the corresponding right prism, because very thin slices or layers of volume translated parallel to the base do not change their volume under this transformation. This can be visualized with a stack of playing cards, where each stack has a volume and the cards can be configured to take on different configurations with the same base.

Cylinder A (right) cylinder is the 3-D structure enclosed by congruent circles, its bases, separated by another length, its height. It can be generated by all the points a fixed distance from an axis (a line segment), along with the bases perpendicular to this generated surface. The surface area of the cylinder is the product of the circumference of the base swept through its height, and the areas of each base. The volume of the cylinder is the product of the area of its base and its height.

Right Cylinder








Volume Unit 4

Instruction Volume Unit 4

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Finding the Volume of a Rectangular (Square) Prism

Pairs

Provide each pair with a box of dimension 𝟒 𝒊𝒏 × 𝟒 𝒊𝒏 × 𝟒 𝒊𝒏, and ten cubic inch blocks.

Say: Find the number of these cubes (hold up one of the cubes) that will fill this box. You can use any strategy that you like. About how many do you think it will take? Write that number down and then use the cubes to show why you think it will be that number.

Teacher Note If some pairs seem to be unproductively struggling with the problem, suggest using the cubes to partially structure the volume as a row of 4 cubes, a width of 4 cubes, and a height of 4 cubes, as shown. If that seems insufficient, provide the pair with another 9 cubes and suggest that they create a

bottom layer of cubes (4 × 4) and a height of 4 cubes. Then encourage them to think about the number of layers that are 4 × 4.








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Whole Group

The teacher elicits student solutions. The purpose of the conversation is to help students develop strategies for visualizing the structure of the volume as a lattice of cubes, and the volume as skip counting by layers of 16, (16 + 16 + 16 + 16 = 64 cubes) or as 16 columns of 4 cubes.

Pairs

The teacher asks students to visualize what would happen to the volume if the height changed to 𝟑 𝟏

𝟐 inches. Now that we have found that

this box holds 64 cubes, talk with your partner and predict the volume of the box if its height changes to 3 !

! inches.

Whole Group

The teacher elicits student solutions and justifications. The purpose of

the conversation is to extend thinking about whole layers to include

fractional parts of a layer.

Teacher Note A physical model of the 64 cubes with a strip of paper or other indicator of 3 !! inches for the third dimension may help students visualize the resulting

volume of 56 cubes.








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Finding the Volume of a Rectangular Prism

Pairs

Provide each pair with a box of dimension 𝟖 𝒊𝒏 × 𝟔 𝒊𝒏 × 𝟒 𝒊𝒏, and sixteen cubic inch blocks.

Say: This is called a rectangular prism. It has a rectangular base (point) and top face (point). Notice that the base and the top are congruent and parallel. The lateral faces (point to them) are rectangles that join the base and the top. Cubes are rectangular prisms too, because we can think of squares as a kind of rectangle. There are other prisms with different bases, like triangles (show a triangular prism). But for now, we will focus on the rectangular prism.

Say: Find the number of these cubes (hold up one of cubes) that will fill this box. You can use any strategy that you like. About how many do you think it will take? Write that number down and then use the cubes to show why you think it will be that number.

Teacher Note If some pairs seem to be unproductively struggling with the problem, suggest using the cubes to partially structure the volume as a row of 8 cubes, a width of 6 cubes, and a height of 4 cubes, as shown. If students seem to be

struggling, provide enough cubes so that they can visualize 4 layers of 48 cubes.








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Whole Group

The teacher elicits student solutions. The purpose of the conversation is to help students develop strategies for visualizing the structure of the volume as a lattice of cubes, especially as 4 layers of 48 cubic inches, for a total of 192 cubic inches.

Teacher Note Note the opportunities for mental arithmetic, as in 4 × 50− 4 × 2 to generate the volume measure. These opportunities can be exploited throughout this unit.

Pairs

The teacher asks students to visualize and predict what would happen

to the volume if the height changed to 𝟑 𝟏𝟒𝐢𝐧𝐜𝐡𝐞𝐬. Now that we have

found that this box holds 192 cubes, talk with your partner and predict the

volume of the box if its height changes to 3 !!inches.

Whole Group

The teacher elicits student solutions and justifications. The purpose of

the conversation is to extend thinking about whole layers to include

fractional parts of a layer.

Teacher Note A physical model of the 192 cubes with a strip of paper or other indicator of 3 !

!inches may help students visualize the resulting volume as

comprised of 3 layers, each of 48 cubes, comprising 144 cubes, and a fourth partial layer consisting of 48 !

!cubic inches, or 12 cubic inch cubes

in all, for a volume of 156 cubic inches.








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Volume of a Rectangular Prism with Fractional Measure

Pairs

Provide each pair with a box of dimension 𝟓 𝒊𝒏 × 𝟓 𝒊𝒏 × 𝟑.𝟓 𝒊𝒏, and 12 cubes.

Say: Find the number of these cubes (hold up one cube) that will fill this box. You can use any strategy that you like. Teacher Note

If some pairs seem to be unproductively struggling with the problem, suggest using the cubes to partially structure the volume as a row of 5 cubes, a width of 5 cubes, and a height of 3 !

! cubes, as

shown. If that seems insufficient, provide the pair with enough cubes to create a bottom layer of cubes

(5 × 5) and a height of 4 cubes (which will be too high). Then encourage them to think about the number of 5 × 5 layers of cubes.








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Whole Group

The teacher elicits student solutions. The purpose of the conversation is to help students develop strategies for visualizing the structure of the volume as a lattice of cubes, especially as 3 layers of 25 cubes, and a fourth layer of !

! of 25 cubes, for a total volume of 87.5 cubic inches.

Pairs The teacher prompts students to visualize thinner and thinner layers. Say: Predict what would happen to the volume if the height changed from 3 !! in to 3 !

! in.

Whole Group The teacher elicits student solutions and justifications. The purpose of the conversation is to extend thinking about whole layers to include fractional parts of a layer. Teacher Note

Student strategies should include a fractional layer of 25 × !! cubic inches,

so that the new volume is 80 cubic inches. To extend this thinking, ask students to predict the volume for a height of 3 !

!" in which would result in

a fractional layer of 25 × !!"

cubic inches for a total volume of 77 !! cubic

inches.








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Developing a Formula

Whole Group

The teacher holds up a deck of playing cards. This is another rectangular prism. Notice that now the layers have been sliced to be very thin. Take out one card and gesture sweeping its area through the height of the deck of cards to simulate how the volume could be generated.

Pairs

Formula Invention. With your partner, use the idea of a rectangular prism sliced into many, many very, very thin slices to make up a formula for its volume. Try it out to see if it works for some of the boxes for which we have found the volumes.

Whole Group

The teacher elicits student inventions and justifications. The purpose of the conversation is to help students understand volume of a prism as the product of the area of a base and the height: 𝑉 = 𝐴!"#$ × 𝐻 Teacher Note Some students will invent length × width × height. This is acceptable but a more general way of thinking is to remind students that the measure of the area of a rectangle is length × width. Be sure that students understand that length unit × width unit results in 𝑢! and that 𝑢! × 𝑢 = 𝑢!.








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Cavaleri’s Principle for Oblique Prisms

Whole Group/Pairs

Could we use our formula for an oblique prism, meaning that it was tilted so that its lateral faces (gesture) were not rectangles? (Show a model of an oblique rectangular prism). To help you think, consider what happens if I move the cards on this deck (or a stack of index cards or a stack of square crackers) like this (create an oblique rectangular prism with parallelogram lateral faces from the right prism formed by the deck of cards.) Talk with your partner about this.

Whole Group

The teacher elicits student opinions and justifications with the goal of helping students draw upon their visual regard to understand that the formula still works because there is compensation of the space taken up by each slice, each slice can be as thin as we like, and the height of the prism has not changed. Teacher Note A generalization is that you could re-arrange the stack of cards as shown below, and this structure will have the same volume as the right rectangular prism. This is Cavaleri’s principle.








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Finding the Volume of a Triangular Prism

Whole Group Generalizing to any prism. We have found the volume of a rectangular prism as the product of the area of a base and its height. Let’s think about how to find the volume of a triangular prism, like this one.

Pairs Students each work with a triangular prism. Tell students: Work you’re your partner. Find the volume of this prism: How many cubic inches (Hold up a cubic inch) will it take to fill this triangular prism? Teacher Note Some students may object that the triangle does not resemble the square face of the cube, so it makes no sense to pack the volume with cubes. Remind them that they have already used fractional parts of cubes to fill the space. Remind students that they know how to find the area of a triangle by measuring a base and the height (shortest distance) from that base to its opposite vertex.

Whole Group The teacher elicits student solutions and justifications. The purpose of the conversation is to extend thinking about the volume of a prism to include prisms whose bases are not rectangles or squares.








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Extension to the Volume of a Cylinder Whole Group Hold up a cylinder. Now that we have investigated the volume of prisms as a product of area of the base and the height, let’s go ahead and see if we can use that kind of thinking to find the volume of this cylinder (hold up model cylinder). Work with your partner to see if we can find a way to figure out how many of these cubic inches (hold up) can fit inside, without any gaps, this cylinder.

Pairs Each pair works with a cylinder provided. Whole Group The teacher selects students to describe and justify their approach. Teacher Note Remind students that they know how to find the area of a circle as a product of its radius and 𝜋, Area = 3.14 × 𝑟!. Alternatively, have students estimate the area of the circle with grid paper. Look to make sure that the area measure and height measure are in the same units, or if not, that students have the appropriate conception of the resulting non-cubic unit of volume measure.








Volume Unit 4

Formative Assessment Volume Unit 4

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Formative Assessment NAME: ______________________________________ 1. The rectangular prism shown below has a length of 6 inches, a width of

4 inches, a height of 10 inches. What is the measure of its volume? Draw below what one unit of volume measure looks like.








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2. The rectangular prism shown below has a length of 5 inches, a width of 4 inches and a height of 2 !

! inches. What is the measure of its

volume?








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3. The bases of the prism shown below each have a length of 10 inches and a width of 4 inches. The height of the prism is 10 inches. What is the measure of its volume? Explain why.








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4. Each side of a hexagonal prism has a length of 7 !!𝑐𝑚. The apothem

(the distance from the center of the hexagon to one of its sides) is 6 !!𝑐𝑚. The area of the base is 292 !

!𝑐𝑚!. The height of the prism

is 10 cm. What is the measure of its volume? Show all work.








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5. The area of a cylinder’s base is 25 𝑖𝑛! and the measure of its volume is 500 𝑖𝑛!. How tall is the cylinder?








Volume Unit 4

Formative Assessment Record Volume Unit 4

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NAME_______________________________________ Indicate the levels of mastery demonstrated by circling those for which there is clear evidence: Item Level

Circle highest level of performance Description Notes

Item 1 Volume rectangular prism, 6 × 4 × 10

ToMV 4 Find and compare volumes of right rectangular prisms by employing partial structuring strategies

240 cubic inches, or 240 in3 with diagram of a cubic inch, not a square inch or inch.

ToMV 3c Find and compare volumes of right rectangular prisms by counting unit cubes (no hidden cubes).

240 but lack of indication of unit

Item 2 Volume rectangular prism, 5 × 4 × 2

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ToMV 5 Find and compare volumes of right rectangular prisms (including some with fractional dimensions) using units with partial structuring strategies.

Either the product 45in3 with diagram of a cubic inch or a product strategy, such as 20 × 2 + 20 × !

!

ToMV 4 Find and compare volumes of right rectangular prisms using partial structuring strategies

A response that indicates use of whole numbers only, such as 40.

Item 3 Volume of oblique rectangular prism, 6 × 4 × 10

ToMV 6d Explain and use Cavalieri’s principle to find the volume of different non-right prisms and cylinders.

240 in3 with an explanation that refers to slicing, layering, or otherwise invoking the idea that sliding the slice or layer does not affect its volume. May include idea of translation parallel to base.

ToMV 4 Find and compare volumes of right rectangular prisms using partial structuring strategies.

Treats oblique cylinder as a right cylinder so finds a volume of 240 in3 but does not justify.

Formative Assessment Record continued on next page:

Volume Unit 4

Formative Assessment Record Volume Unit 4

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Item 4 Volume of hexagonal prism with fractional dimension

ToMV 6c Find and compare volume of a variety of prisms.

𝐴𝑟𝑒𝑎 × 𝐻𝑒𝑖𝑔ℎ𝑡 = 29212𝑐𝑚!

Note if student multiplies all dimensions, which indicates a lack of understanding of generating volume by sweeping area through a length.

Item 5 Find height of a cylinder given its area and volume measures.

ToM 6b Find and compare volumes of right cylinders using sweeping of area through height.

Thinks of volume as product of area and height, and so 500 in3 /25 in2

Volume Unit 4

Net of Prism Volume Unit 4

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deriving the formula for the volume of a prism 4disme.org/all-files/volume4-lesson.pdf · 2014. 12....

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