©2013 Judo Math Inc.

VOLUME AND SURFACE AREA

©2013 Judo Math Inc.

6th grade Equations and Geometry Discipline: Black Belt Training

Black Belt training – Volume/Surface Area

You’re almost there! You have worked very hard through all the disciplines and you have finally

made it to your black belt in the Equations/Geometry Discipline and I am very proud of you. In

this final scroll of knowledge, you will spend some more time with Geometry, but this time you

will be thinking of three-dimensional shapes instead of two like you did in the last belt. The

good news is, you live in a three-dimensional world so you are quite familiar with everything you

see. For example, the room you’re sitting in right now is probably a rectangular prism:

Or maybe it’s even a triangular pyramid?: Or possibly a dodecahedron?! That would be an interesting building! I hope you enjoy this last belt. You will be awesome!

Good Luck Grasshopper.

Order of Mastery: Volume/Surface Area

1. Volume of a prism with fraction unit cubes

2. Volume formula and real world application

3. Intro to nets

4. Using nets to find surface area

Standards Included:

6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of

the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by

multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right

rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to

find the surface area of these figures. Apply these techniques in the context of solving real-world and

mathematical problems.

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1. Cubes! You are trying to build a cube with 4 cm sides using 1 cm cubes. How many cubes do you need? Follow up questions: a) How many cubes are visible along the outside? i. How many cubes have three sides showing? ii. How many cubes have two sides showing?

iii. How many cubes have one sides showing? b) How many cubes are invisible from the outside (no sides showing)? c) How many 1 cm cubes would she need to make a 6x6 cube? What about 7x7? Is there a shortcut for figuring these out?

1. Volume of a prism with fraction unit cubes

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From 2-D to 3D: Shade each of the 2D shapes on the left in their corresponding 3D shape on

the right.

2. Based on this information, draw a few more prisms:

Pentagonal Prism Hexagonal Prism Octagonal Prism

3D shapes like this are defined by their base shape.

This bottom one is called a “triangular prism” because it’s base is a triangle and the sides come

directly up from it to form another triangle.

The second shape up is a “rectangular prism” for the same reason.

The top shape here has a special name: cylinder, but it could also be called a “circular prism”

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3. Find the volume of a rectangular prism with the dimensions 8 units x 5 units x 2 units. One

great way to do this would be to think about how many unit cubes would fit inside of it?

Can you find dimensions of a different rectangular prism that would have the same volume?

4. A square prism has a special name: cube. Find the dimensions of a cube that would be able to fit 1000 unit cubes into it.

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5. How could we pack unit cubes into a triangular prism to find the volume? Try to find the volume here:

6. Now try this one… How can we pack unit cubes here?

In this problem, did you try to use unit cubes that had fractional lengths? For example, a unit cube that was 1

2x1

2𝑥1

2

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7. The baking dilemma: The other day you were baking a cake with your mom. Your batter

fills the 81

2 inch by 11 inch by 1

3

4 inch pan to the very top, but when it bakes it spills over

the side. You search the cupboards and fine another pan that is 9 inches by 9 inches by 3 inches. Your recipe says that you should leave about an inch between the top of the batter and the rim of the pan. Should he use this pan? Hints: Try creating a shape that is the size of this pan You could use a unit cube to try to find volume.

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.

You may have noticed in the last section that packing 3D shapes with unit cubes is handy, but once you

get edges that aren’t whole numbers, it gets a little bit tricky. In this section, we are going to try to

determine some shortcuts and even develop a formula for ourselves to make finding volume easier!

Let’s try looking for patterns:

Volume = 4x4x4 Volume = 1

2 (3x4) x 2 V= 13.5 x 5 x 18.5

Perhaps you were thinking something like this?...

Volume of Prism = Area of base x height

2. Volume formula and real world application

Inspecting the figures above that you worked hard in the last section to find the volume of, do you see any

shortcuts or ideas for possibly creating a formula? List ideas here:

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In each of the figures below, shade the base and label the height with an H:

Now practice with your new formula by trying to find the volume of the trapezoidal prism above.

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1 2

3 4

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5. You have a fish tank that is the shape of a rectangular prism that is 18 cm by 20

cm by 12 cm.

a. Draw a picture of this tank and find the volume.

b. What if you only fill the tank up 3

4 of the way? What is the volume now?

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6. Your sister has a rectangular tank that she does science experiments in! It is 24 cm

wide, 30 cm long. It contains a stone (she thinks it’s worth millions of dollars) and is

filled with water to a height of 8 cm. You pull the stone out and take it to your room

(because it was quite beautiful!) The height of the water in the tank drops to 6cm.

a. Draw a picture of this scenario. Label all lengths.

b. What was the volume of the stone?

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7. (The ultimate!) A rectangular fish tank is 50cm wide by 60cm long. It can hold 126 liters of water when

full. If you fill it 2/3 of the way full, find the height of the water in cm (remember, 1liter=1000cm3)

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8. (mini project) You have decided that you want to get a fish or a turtle or a

reptile. You want to use your math skills and design the perfect tank for them.

a. Decide which animal you want to get.

b. Research to see how much space that animal needs

c. Draw your own tank that has enough space for your new pet.

d. Draw a beautiful drawing of the tank, labeling all dimensions and showing how you found the volume.

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2. Draw all of the shapes that make up the faces of this rectangular prism (you should end up with 6

shapes) :

3. Intro to nets

1. Quick-write: How can we represent three-dimensional shapes in only two dimensions?

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Sometimes there is more than one net that can represent the same shape!

3. Here is a cube and it’s net. Draw at least 2 more nets that would represent this cube. You may want to

cut out a few nets from a big sheet of paper. Ask your teacher for supplies to do this!

Net #1

Definition: NETS

A net is a 2 dimensional pattern of a 3 dimensional figure that can be folded to form that

figure. Here is an example of a rectangular prism and a net that could be folded to form it.

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4. Draw a net for the following shapes. In order to check your answer, draw the net on a separate piece of

paper and cut and fold it.

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1. Bedroom Problem: Suppose your room is 9.5 feet wide, 11 feet long, and 8 feet tall.

a. Draw the net for this room and use it to calculate the surface area of all of the

walls.

b. If one quart covers about 120 square feet, how many gallons do you need?

4. Using nets to find surface area

Beyond being fun to draw and cut and fold, nets are a very helpful tool. They can help us to find the surface area of a shape. Surface area is the total area of the faces and curved

surface of a solid figure.

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c. Wait! You may have overestimated on the paint! Did you remember that you don’t have to paint the floor, doors, and windows of your bedroom? Continue your work on this bedroom problem by estimating size of windows/door (3’x8’). Draw pictures and write explanations for come to a more precise solution for how much paint she needs.

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2. Below are the same shapes that you drew nets for in the previous section. Use the nets to find the surface area for each shape.

8.54 m

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3. Draw and label a triangular prism and a rectangular prism that have the same surface area. How many different solutions can you find for this?

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4. Milk Carton Problem/mini-project:

Your school is looking to use less waste and change the design of their milk carton. They heard

about your amazing geometry skill and they have selected you to be the designer!

1. First do some research on dimensions and draw a net to figure out the current surface area of

the milk carton. If possible, find a milk carton to help you in your research!

2. Now go to this site and read how the milk carton is created:

http://www.madehow.com/Volume-4/Milk-Carton.html

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3. Finally, start brainstorming some new designs for the milk carton. First think about the things that the

design would need to have and then how you could meet everyone’s needs while creating less waste in

the landfill. Be prepared to present your design to the class, both as a net and a 3D figure. Also be

prepared to share all the reasons why your design is better!