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Math 6

Unit 4:

Surface Area &

Volume

Tasks Only

Math 6 Name: Date: Period:

Unit 4 – Surface Area & Volume — Entry Task

Squares & Cubes (Part 1) 1. Take out an orange rod.

a. How many of the squares below would it take to cover the entire rod? Explain how you decided (give enough details that I know you are right!).

b. How many of these cubes would fit inside the orange rod? Explain how you decided.

When everyone has finished, have your Resource Manager call the teacher over to check your answers and tell you how to fill in the table below.

2.

Checkpoint!

Math 6

Unit 4 – Surface Area & Volume — Entry Task

Squares & Cubes (Part 2) 1. Work together to decide which vocabulary word goes with which description:

surface area ! How many cubes fit inside a 3-D shape volume ! The length, width or height of a shape area ! How far it is around a 2-D shape perimeter ! How many squares cover a 2-D shape dimensions ! How many squares cover a 3-D shape

2. Marlon reached in a bag of Cuisenaire rods and pulled out one that has a surface

area of 62 square centimeters. How long is Marlon’s Cuisenaire rod? Explain how you know.

Unit 4 – Surface Area & Volume — Entry Task 2!

3. Build a block T using a yellow and a light green rod. Then find the surface area and volume of the block T. Show exactly how you decided. Be sure to organize your ideas carefully, including labels and color-coding.

4. Build a block baby using a purple, two red and three tan/white rods.

Then find the surface area and volume of the baby. Show exactly how you decided. Be sure to organize your ideas carefully, including labels and color-coding.

Unit 4 – Surface Area & Volume — Entry Task 3!

5. Build a rod dog using a yellow and three red rods. Then find the surface area and volume of the rod dog. Show exactly how you decided. Be sure to organize your ideas carefully, including labels and color-coding.

6. Build a block P using a yellow and three red rods.

Then find the surface area and volume of the P. Show exactly how you decided. Be sure to organize your ideas carefully, including labels and color-coding.


Unit 4 – Surface Area & Volume —Lesson Series 1

No Naked Numbers! Poor Pete was trying to find the surface area of a big P built from an orange (10cm) and 3 yellows (5cm). Somewhere in all those calculations, he made a mistake. And he has naked numbers. Your job is to label Poor Pete’s work – tell what each of his numbers mean. See if you can find his mistake!

5 x4 20 +2 22

22 x3 66

10 x4 40 +2 42

66 +42 108 cm2 Wait – what?


Unit 4 – Surface Area & Volume – Lesson Series 1

Cuisenaire Pairs Check

Work with your partner to find the surface area and volume of the Cuisenaire configurations below. For each problem, one partner writes and the other checks that all numbers and calculations are labeled. Both partners contribute to solving all problems. As you’re working, ask each other the following questions:

• What does that number represent? How should we label it? • How do you know to add/subtract/multiply? • How can we count it? Is there another way to count? • How do we know we’ve counted everything?

Writer: __________________________ Label Checker: ______________________


1.

Surface Area: Volume:

2.


4 cm

10 cm

8 cm

5 cm

Unit 4 – Surface Area & Volume – Lesson Series 1 2!

As you’re working, ask each other the following questions: • What does that number represent? How should we label it? • How do you know to add/subtract/multiply? • How can we count it? Is there another way to count? • How do we know we’ve counted everything?



3.


4.

5. 6.

6 cm

4 cm

6 cm

8 cm 10 cm

5 cm

20 cm

Surface Area: Volume:!



© University of Cambridge

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Unit 4 – Surface Area & Volume — Lesson Series 1 Adapted from “Changing Areas, Changing Volumes,” NRICH Maths, U. of Cambridge

Changing Surface Areas, Changing Volumes Part 1: Have your Resource Manager call your teacher for a set of cards with different Cuisenaire Rod configurations. Place your cards on the chart below, accoridng to the following rules:

! As you go from left to right, the surface area of the shape increases. ! As you go from top to bottom, the volume of the shape increases. ! All the shapes in the middle column (") have the same surface area. ! All the shapes in the middle row (#) have the same volume.

Surface Area

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Part 2: Take a look at the extended chart below: !

! Surface Area ! ! "! #! =! #! +!

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The check marks represent the nine cards you've already placed. Can you create cards with new Cuisenaire rod configurations that could that could go in the four blank spaces? Remember the rules:

! As you go from left to right, the surface area of the shape increases. ! As you go from top to bottom, the volume of the shape increases. ! All the shapes in the middle column (") have the same surface area. ! All the shapes in the middle row (#) have the same volume.

If needed, you can make up your own Cuisenaire rods that have lengths different from the ones in our Cuisenaire set.

BONUS:

Design your own set of nine cards with new Cuisenaire configurations that can be arranged on the chart.


3!

!Changing Surface Areas, Changing Volumes

Cards !!!A B C

D E F

G H I

4cm 4cm

8 cm

13 cm

6 cm

5 cm

7 cm

7 cm

7 cm


Unit 4 – Surface Area & Volume — Lesson Series 1

Cuisenaire Puzzles Today is your team’s opportunity to dig deeper with Cuisenaire rods, surface area, and volume. The problems are tricky, so you will need everyone in your team to make sense of them. Show ALL work on separate paper. Play your team role as you work together!

Recorder/Reporter: Make sure everyone records complete, labeled work on their indiviudal paper. “Don’t forget to draw the shape!” “What does that number represent?”

Resource Manager: Have the team build with Cuisenaire rods. Make sure all building—and math talk—happens in the middle of the table. “Can you move your arm so we can see what you’re pointing to?”

Facilitator: Make sure every calculation is backed up by a reason. “Why did you multiply?” “Where did the 6 come from?”

Team Captain: Help your team stick together. “Are we ready to move on?” “Let’s get back to work!”

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1. Get 4 purple rods.

a. Come up with three different arrangements (touching or not touching) with

three different surface areas. Draw the arrangements. Showm, label, and color-code all calculations to explain how you found their surface areas.

b. What is the volume of each of your arrangements from part (a)? Why didn’t the

volume change when the surface area did?

c. What arrangementof 4 purple rods would have the smallest surface area? The largest? Draw them and label all dimensions.

2. Use at least two Cuisenaire rods to build a shape that has a surface area larger

than its volume. Draw your shape, and justify that its surface area is larger.

Unit 4 – Surface Area & Volume — Lesson Series 1 2!

3. Get the 4 smallest rods (white/tan, red, light green, purple) and make an arrangement that has a surface area of 26 cm2. You do not have to use all 4 rods. a. Draw your arrangement and label all dimensions. Show calculations to find its

volume and tojustify that its surface area is 26 cm2.

b. Is there more than one way make an arrangement with a surface area of 26cm2 (using only the 4 smallest rods)? How many arrangements are possible?

4. A mystery rod has a surface area of 42 cm2.

a. How long of a rod would you have to connect in a row so that the surface area

grows to 112 cm2?

b. What happens to the surface area if you connect the mystery rod and a rod that has a volume of 20 cm3?

5. If I double the length of a rod, explain why the surface area does not also double.


Unit 4 – Surface Area & Volume — Formative Task

Building Giants For this task you will need:

! One Cuisenaire rod, as assigned by your teacher. ! Centimeter paper ! Scissors ! Tape

Your Team’s Task: Build two giant Cuisenaire rods. Then calculate their surface area and volume.

Part 1: Building the Giants

Build your giant rods in two different ways:

1. A rod that becomes giant by adding 3cm to all of the dimensions

2. A rod that becomes giant by multiplying all dimensions by 3

Before you can get centimeter paper, scissors, and tape, describe below exactly what each giant rod will look like. Call your teacher to check when everyone agrees. +3 Giant x3 Giant

Unit 4 – Surface Area & Volume — Formative Task 2!

Part 2: Finding Surface Area and Volume 1. Discuss together how to find the surface area of your +3 Giant and your x3 Giant.

Make a plan before you start your calculations. When everyone understands the plan, find the surface area and show your ideas on a separate piece of paper. Show and label all your work as carefully as you can! Be sure to organize your calculations (don’t write them all over the paper), and use diagrams to help you explain.

2. Discuss together how to find the volume of your +3 giant and your x3 giant. Make

a plan before you start your calculations.

When everyone understands the plan, find the volume and show your ideas on a separate piece of paper. Show and label all your work as carefully as you can! Be sure to organize your calculations (don’t write them all over the paper), and use diagrams to help you explain.

When your team finishes, have your Resource Manager call your teacher to check in!



Multilink Puzzles Work together to build each object described below. Your team needs to build one of each, in the middle, and then use it to answer the questions. Call the teacher over to grade your team’s work after each problem. Your grade will be based entirely on how well you organize and label your calculations. 1. A box like this is called a rectangular prism.

“Prism” is just a fancy work for a box, and “rectangular” just means that the bottom of the box is a rectangle. Use multilink cubes to build a rectangular prism that has a volume of 60 un3 (cubic units). One slice of your prism must have a volume of 12 un3. Make that slice all one color. When you’ve built it, find the surface area. Show your work below.


2. Build a rectangular prism that has a slice (again, all one color) with a volume of 6 un3. Your prism should have four slices. Find the volume of the whole prism.

3. Build a rectangular prism that has 5 slices. The volume must be 40 un3.

How many cubes are in one slice?

4. Build a shape (it doesn’t have to be a rectangular prism) that has 4 slices and a

volume of 20 un3. Make one slice all the same color.


Unit 4 – Surface Area & Volume — Lesson Series 2 Adapted from CPM Core Connections 1, Lesson 9.1.1

Building Buildings Carly and Stella are studying to become architects. They have been asked to design an office building. They have come up with 5 the possible floor plans, shown below.

They have not decided yet how many stories (floors) high they want to make each building, but they would like all of the buildings to hold a similar number of offices. 1. Use cubes to build a model of the first floor of each building. If each office takes up

one cubic unit, how many offices will fit on the first floor?

Building 1: Building 4:

Building 2: Building 5:

Building 3:

2. Your teacher will assign your team one building to examine further. For your

building, add cubes to make it 3 stories high. How many offices will it fit now? Explain how you know.

3. What is the relationship between the volume of the models you’ve built and the

number of offices that the building can fit?

1 2 3 4 5

Unit 4 – Surface Area & Volume — Lesson Series 2 Adapted from CPM Core Connections 1, Lesson 9.1.1

2!

4. Add more floors to your building model, and calculate the volume after each new floor. Organize your data into a table showing the relationship between the number of floors and the volume of your building (in un3, or cubic units).

# of floors Volume (un3)

Write a rule that describes the relationship between the # of floors and the volume of your model.

5. Carly and Stella have learned that their building can have no more than 195 offices.

a. For your floorplan, what is the highest number of stories they can build? Show how you know.

b. Will teams with other floorplans have the same answer to (a)? Why or why not? Which floorplans?

6. Jorge was working with Building 1. He built the first floor out of cubes and found that the volume was 10 cubic units. “Wait,” he said, “Looking at the floor I can see that the area is 10 square units. Why are the area and the volume both 10?”

Is this a coincidence? Will the area of the floor plan always have the same numerical value as the volume of the first floor? Justify your reasoning.



Folding a Cube

Valeria was wrapping presents during the holidays and needed a bunch of cube-shaped boxes. She decided to make her own cube-shaped boxes, but got sick of having to cut out so many squares to tape together. “I wonder if there’s a way to just cut one piece of paper,” Valeria said. “Then, instead of taping every edge, I could fold up the paper and tape just a few edges.” Valeria cut the following shapes and folded them along the dotted lines.

Which shape(s) will fold into a cube? Which will not?

Be sure you can explain how you decided. !

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Slicing a Cube

If you start with a cube, how many edges do you need to cut in order to create a net? That is, how many edges must be cut before the faces of the cube lie completely flat (nothing overlapping)? Work with your team to determine the minimum number of slices needed. Justify how you know that the number you found is the minimum. Discuss together, then make a poster of your team’s thinking.


Unit 4 – Surface Area & Volume — Lesson Series 3 Select problems adapted from Connected Mathematics “Filling and Wrapping: Three-Dimensional Measurement”

Wrapping and Filling 1. Which of the following nets could be folded along the dotted lines to form a closed

rectangular prism? If you’re not sure, draw the net onto grid paper so you can cut it out and experiment.

2. For the nets above that did make rectangular prisms, sketch the prisms below and

label their dimensions. Sketching 3D shapes can be tricky, so help each other out. Then calculate the volume of your prisms. Show and label all calculations!

10cm

10cm

10cm 20cm

10cm 10cm

4.5cm

4.5cm

4.5cm

18cm

18cm

12cm

1.5cm

1.5cm

1.5cm

1.5cm

10.5cm

3cm 3cm

8.5cm

5.5cm

5.5cm

6cm

3cm


3. Shantal was finding the surface area of the prism built from one of the nets in problem #1. Her calculations are below. Label and color-code her work to show what each number represents.

(1.5)(1.5)(2) = 4.5 cm2 (10.5)(1.5)(2) = 31.5 cm2

(10.5)(10.5) = 110.25 cm2 110.25 + 31.5 + 4.5 = 146.25 cm2 4. Gilman found the surface area another way.

Here are his calculations. Label and color-code his work to show what each number represents.

(13.5)(10.5) = 141.75 cm2

(1.5)(1.5)(2) = 4.5 cm2

141.75 + 4.5 = 146.25 cm2

12cm

1.5cm

1.5cm

1.5cm

1.5cm

10.5cm

12cm

1.5cm

1.5cm

1.5cm

1.5cm

10.5cm


5. Here’s a rectangular prism:

a. Find the volume of the prism. Be sure to label all numbers in your calculations!

b. Sketch two different nets for the prism on the grid below.

c. Use each net to find the surface area of the prism. Color-code and label your nets

and calculations to show connections between them. Make sure you get the same surface area both times!

3 un

7 un

5 un


Unit 4 – Surface Area & Volume — Expert Task

Foil Prank – Group Task

How much aluminum foil will it take to cover our classroom? How much will it cost? Some rules:

• We only want to cover objects, not the walls.

• When covering an object, we don’t want pieces of foil to overlap—use as little foil a possible.

• Be precise in your measurements—do not round!

--------------------------------------------------------------------------

Make a plan with your team for how you will collect measurements from your section of the room. Call the teacher over when everyone in your team agrees with the plan. Consider the following questions as you develop your plan:

! What objects will you measure? Who will measure what? ! What materials and supplies will you need? ! How will you record and keep track of your measurements? ! What will you do with the measurements once you have them?


Unit 4 – Surface Area & Volume — Expert Task

Foil Prank - Personal Poster Make a personal poster about one object from your group’s section of the room. Your poster should explain how much aluminum foil is needed to cover the object, and the cost of aluminum foil for your object. Be sure to include in your explanation:

! A diagram of your 3D object, with all important dimensions labeled.

! A diagram of the net that would wrap around your object. Again, label all important dimensions.

! Complete calculations for finding how much aluminum foil is needed. Use technical writing tools to connect your calculations and diagrams.

! Complete calculations for finding the aluminum foil cost. Use technical writing tools to explain your calculations.

An A+ PERSONAL POSTER should include:

" A unique title

" A problem statement describing the situation and the problem you were trying to solve.

" Solution sentences stating your final answers and what they mean about the problem.

" Detailed and labeled calculations

" Technical writing tools to help you explain your thinking and show connections between your models.

• Arrows • Big diagrams • Mini-diagrams • Highlighting

• Callouts • Different size letters • Labels • Color-coding

!


Unit 4 – Surface Area & Volume — Lesson Series 3 Adapted from NRICH Maths, “Cuboid Challenge,” University of Cambridge

Prism Challenge From a square sheet of 20cm by 20cm paper, we can make a box without a lid. We do this by cutting a square from each corner and folding up the flaps.

Part 1: Investigating Volume Will you get the same volume no matter what size squares you cut out? Investigate what volumes are possible for different sizes of cut-out squares. Record your data in an organized table. What is the maximum possible volume and what size cut produces it? Part 2: Investigating new squares Now try different sized square sheets of paper. Can you find a relationship between the size of paper and the size of cut that produces the maximum volume?

Student Materials Optimizing: Packing It In S-1 © 2014 MARS, Shell Center, University of Nottingham

Packing It In Stefan is packing boxes into the back of a truck.

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The empty space in the back of the truck is 245 cm by 250 cm by 890 cm.

The boxes are all identical and measure 50 cm by 60 cm by 80 cm. They can be arranged in any way in the back of the truck.

Give instructions to Stefan on how to pack the truck so that the maximum number of boxes will fit in. State how many boxes will fit, if he packs the truck according to your instructions.

This drawing is not to scale.


Sample Responses to Discuss: Leillah

1. Try to explain what Leillah has begun to do.

2. Do you agree with her conclusion? Why / Why not?


Sample Responses to Discuss: Faridah

1. Try to explain what Faridah has done.

2. What are the advantages of using decimeters rather than centimeters?

3. Do you agree with her reasoning? Why / Why not?


Sample Responses to Discuss: Moses

1. Try to explain what Moses has begun to do.

2. What are the strengths and weaknesses of his approach?


Unit 4 – Surface Area & Volume — Lesson Series 3 Adapted from NRICH Maths, University of Cambridge

Plutarch’s Boxes According to Plutarch, an ancient Greek philosopher, the Greeks found all the rectangles with integer (whole number) sides whose areas are equal to their perimeters. Can you find at least three? Draw them below and label their side lengths. What rectangular boxes, with integer sides, have their surface areas equal to their volumes? One example is a box with dimensions 4 by 6 by 12. Find as many as you can. Organize your findings in a table or list.


Unit 4 – Surface Area & Volume — Lesson Series 3 Adapted from “Christmas Presents,” NRICH Maths, University of Cambridge

Wrapping Presents Have you ever seen your baby brothers, sisters, or cousins unwrap a present? Sometimes they like the unwrapping more than the present itself! Today we will explore different ways to wrap presents. We should probably try to save paper, though, and use no more than is necessary. So let's do some wrapping up and never have one piece of paper overlapping another. Part 1: Wrap a Present Use multilink cubes to make a present for your teacher. Your teacher is kind of boring, so make a rectangular prism (but to make it a little more interesting, don’t make a cube). Sketch your prism below and label its dimensions. How many different wrapping paper shapes could you use to cover your present? Try to find them all! Sketch them below or on grid paper. Label all important dimensions on your nets.

Unit 4 – Surface Area & Volume — Lesson Series 3 Adapted from “Christmas Presents,” NRICH Maths, University of Cambridge

2!

Part 2: Another Present Don’t you want to give your teacher another present? Make another rectangular prism with different dimensions you made in Part 1. This time, make sure that your prism has no square faces. Sketch your prism below. How many different wrapping paper shapes could you use to cover your present? Try to find them all! Sketch them below or on grid paper. Label all important dimensions on your nets.

Math 6


Getting Good with Surface Area & Volume You know a lot of things about surface area and volume. But today, expect to be confused again because confusion leads to conversation, conversation leads to ideas and ideas lead to deeper understanding. For each problem your group should: Feel Confused ! Share Ideas (no calculations) ! Ask Questions ! Deepen Understanding Work on one problem at a time, in any order you choose. Show all work on a separate sheet of paper, carefully labeling all diagrams and calculations. Make sure you have done at least one volume problem and one surface area problem by the end of class. Facilitator: get your group started by asking, “Should we try a volume problem or a surface area problem first? --------------------------------------------------------------------------------------------------------------- Volume Problems 1. Gerardo’s recipe for banana bread won't fit in his favorite pan. The batter fills the

8.5 inch by 11 inch by 1.75 inch pan to the very top, but when it bakes it spills over the side. He has another pan that is 9 inches by 9 inches by 3 inches, and from past experience he thinks he needs about an inch between the top of the batter and the rim of the pan. Should he use this pan?

Recorder/Reporter: Ask “How can we draw a picture of this situation?” 2. A rectangular tank is 50 cm wide and 60 cm long. It can hold

up to 126 liters of water when full. If Amy fills 23

of the tank

as shown, find the height of the water in centimeters. (Note: 1 liter =1000 cm3)

Team Captain: Ask, “How can we figure out how much 23

of the tank is? 3. Kelly’s family is going go go camping this weekend, but Kelly and her sister are in a

big fight. “Mom,” said Kelly, “I don’t think that tent is big enough for the two of us—I need at least 100 cubic feet if I have to share with her!” The tent that Kelly and her sister share is pictured below. Is it big enough to make Kelly stop whining?

60cm 50cm

5’!

6’

7’

Unit 4 – Surface Area & Volume — Lesson Series 3 Select problems adapted from Illustrative Mathematics; Connected Mathematics “Filling and Wrapping: Three-Dimensional Measurement”

2!

Surface Area Problems 4. Sweet-Tooth Chocolates is marketing a special assortment of caramels. Each

caramel is an individually wrapped 1-inch cube. The company wants the caramels to completely fill the box.

a. What arrangements of caramels would require the most cardboard for a box? Make a sketch of the box and label its dimensions. Then draw the net for the box.

b. Which arrangements of caramels would require the least amount of cardboard for a box? Make a sketch of the box and label its dimensions. Then draw the net for the box.

c. How much more cardboard does your box from part (a) use compared to your box from part (b)?

5. Look at the net to the right.

a. What 3D shape would be formed from folding the net? Write a detailed description of it would look like or draw a picture. Include dimensions in your description or drawing!

b. What is the surface area of the 3D shape?

6. Draw two different nets of a prism with a surface area of 100 ft2. Label all

important dimensions in the net. BONUS: (make sure you have done at least 2 problems from each section before trying these) 7. Find a 3D prism that has a surface area with a larger numerical value than its

volume. Challenge yourself more: find one that is not a rectangular prism. Draw its 3D picture and its net.

8. Find a 3D prism that has a surface area with a numerical value equal to its volume.

Draw its 3D picture and its net.


Unit 4: Surface Area & Volume — Summative Task Adapted from Smarter Balanced Assessment Consortium Practice Test

Cereal Boxes Geometric Groceries, Inc. sells its cereal, XY-Bites, in boxes that are rectangular prisms. The cereal boxes have the following dimensions: 12 inches high, 8 inches wide and 2 inches deep. The managers at Geometric Groceries are planning to sell a new cereal, Super Shape Crunch, and want to sell it a different size cereal box. In this task, you will evaluate one box design proposed by the managers, then create your own proposal for a new box design. 1. Determine the volume of the XY-Bites cereal box with dimensions 12 inches high, 8

inches wide, and 2 inches deep. Show your work and include units in your final answer.

2. Below is a net of the cardboard used for the XY-Bites box. Label the missing

dimensions in the blank spaces. Then find the surface area of the XY-Bites box. Show all work and include units in your answers.

(x,y) (x,y) (x,y)

(x,y)

Unit 4: Surface Area & Volume — Summative Task Adapted from Smarter Balanced Assessment Consortium Practice Test

2!

The new cereal box for Super Shape Crunch will also be a rectangular prism, and should also must meet the following requirements:

• The new box must use less cardboard than the old XY-Bites boxes • The new box must hold the same or greater volume of cereal as the XY-Bites box.

3. One of the product designers proposed a design for a new Super Shape Crunch box

that is rectangular prism with dimensions 10 inches high, 7.5 inches wide, and 4 inches deep. Does this box meet all of the requirements for the new box? Explain why or why not.

4. Your turn: design a new box for Super Shape Crunch. Draw a sketch of the

rectangular prism OR its net. Label all dimensions, including units.

5. Justify that your box meets all of the requirements for the new design.

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