HONORS MATH 7

[Pages:7]Skills Summary:

HONORS MATH 7

Phase 1 ? Week 2 April 20th ? 24th

In the Week 2 Packet, we will practice working with our knowledge of a circles to calculate volumes or unknown variables relating to cylinders, spheres and cones. We will also practice finding the surface areas of cylinders, pyramids and composite shapes as well as the volumes of cylinders, pyramids, cones, spheres and composite threedimensional shapes.

Unit 7 Review Idaho Core Standards:

Geometry

7.G - Draw, construct, and describe geometrical figures and describe the relationships between them.

? 7.G.3 - Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

7.G - Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

? 7.G.4 - Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle

? 7.G.6 - Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms

8.G - Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

? 8.G.9 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Phase Week Practice

P1

P2

1 2

P3

P4

Date (#) 0420

0421

0422

0423

Lesson

CC2 9.2.1 ? 9.2.4: Focus: Surface Area of Prisms

CCSS Standards

7.G.6 7.G.3

SelfReflection

Practice Sections Completed/ Total

/ 4

CC2 9.2.1 ? 9.2.4: Focus: Volume of Prisms

7.G.6 7.G.3

/ 4

CC3 10.1.2: Cylinders ? Surface Area and Volume

8.G.9

/ 4

CC3 10.1.3 Pyramids and Cones ? Volume

8.G.9

/ 4

P5

0424 CC3 10.1.4: Spheres ? Volume

8.G.9

/ 4

Phase 1 ? Week 2 ? P1

Name: ______________________________ Per.: ______ # 0420

CC2: 9.2.1 ? 9.2.4 Surface Area, Volume & Cross Sections of Prisms Points: _________ Reflection:

Warm-Up: Find the Area and Perimeter of Each Shape.

3

1 3

cm

3 cm

4.5 cm

5.25 cm

Notes: PRISMS ? SURFACE AREA

CC2 9.2.1 ? 9.2.4

The surface area is the sum of the areas of all faces, including bases. Surface area is expressed in square units.

For additional information, see the Math Notes box in Lesson 9.2.4 of the Core Connections, Course 2 text.

Example: Find the surface area of the triangular prism at right.

Step 1: Area of the 2 bases: 2 [ 1 (6 )(8 )] = 48 2

2

Step 2: Area of the 3 lateral faces Area of face 1: (6 )(7 ) = 42 2 Area of face 2: (8 )(7 ) = 562

Area of face 3: (10 )(7 ) = 70 2

Step 3: Surface Area of Prism = sum of area of bases and area of the lateral faces = 482 + 42 2 + 56 2 + 70 2 =

Practice: Find the surface area of each prism.

1.

2.

3.

Contextual Practice: A builder is pouring a concrete building footing as part of a foundation for a complex architectural design. The footing incorporates two cubic sections of foundation that are overlapping. How much concrete will the builder need to pour for this foundation if each cube is 6 yards across and the cubes are overlapping through each other's center-points? He will also need to apply a sealant to the surfaces to prevent the concrete from soaking up ground water. What is the surface area of the shape?

Activity: As many of us are relying on Amazon for deliveries of staple products we use in the home (especially now) , there is a high probability that we all have a cardboard shipping box laying around the house, in the garage or maybe you are fastidious and have already broken it down and placed it in your recycle bin. Either way, if you have a cardboard shipping box, calculate the surface area of the physical box (Don't worry about the overlapping flaps underneath.). If you don't have a cardboard shipping box, calculate the surface area of a cereal box instead.

Shipping boxes are made of corrugated cardboard (three layers with the middle layer fluted to create air pockets). The cool thing about corrugate cardboard is that it also makes a great insulator for both heat and sound. So let's say that your parents are tired of you blasting your TikTok videos, Music, Anime, and Video Games and you want to sound insulate your room. Awesome! Let's say you decide to use the discarded shipping boxes (like the one you just calculate the surface area of) to sound proof your room. Measure the surface area of your room, walls and ceiling only. Using the surface area you calculated from the shipping box and the surface area of your room, calculate how many of those shipping boxes it would take to sound proof your room using two layers. (To get about 40% sound reduction, you would need two layers!)

Phase 1 ? Week 2 ? P2

Name: ______________________________ Per.: ______ # 0421

CC2: 9.2.1 ? 9.2.4 Surface Area, Volume & Cross Sections of Prisms Points: _________ Reflection:

Warm-Up: Find the Area and Perimeter of Each Shape.

4 5 in

6

i n

3 1 in

4

5 8

6.6 cm 4.3 cm

5.5 cm

2

6 2 in

5

11.4 cm

Notes: PRISMS ? VOLUME

CC2 9.2.1 ? 9.2.4

Volume is a three-dimensional concept. It measures the amount of interior space of a three-dimensional figure based

on a cubic unit, that is, the number of 1 by 1 by 1 cubes that will fit inside a figure.

The volume of a prism is the area of either base (B) multiplied by the height (h) of the prism. = ( ) ? () or =

For additional information, see the Math Notes box in Lesson 9.2.4 of the Core Connections, Course 2 text.

Examples:

Find the volume of the square prism. Find the volume of the triangular prism. Find the volume of the trapezoidal prism.

The base is a square with area (B) 8 ? 8 = 64 2.

= ()

= 64(5) = 320 3

The base is a right triangle with

the area

1 (5)(7)

2

=

17.5 2.

= ()

= 17.5(9)

= 157.5 3

The base is a trapezoid with the

area

1 (7 + 15)(8)

2

=

88 2.

= ()

= 88(10)

= 880 3

Practice: Calculate the volume of each prism. The base of each figure is shaded.

1.

2.

3.

Contextual Practice: The gas tank is 20% full. Use the current price of gas in your community to find the cost to fill the tank. (1 gal = 231 in.3)

11 in

1.25 ft

1.75 ft

Activity: If you happen to have a small brick of cheese in the fridge, awesome. Cut a thick slice from the brick of cheese so that you can cut it into cubes at least a half inch in size (String Cheese sticks also work; cut half-inch chunks and just trim the round edges to make them flat.). They don't have to be perfect as long as each chunk is close to a cube. Try to cut through each cube to create a different cross section. See if you can create the following cross sections: triangle, parallelogram, trapezoid, pentagon and hexagon. *** SAFETY NOTE: Use a fork or toothpick to hold the cube of cheese while slicing through it. If you hold them with your fingers, you risk losing control of the cube and slicing through your finger. Please do not cut through your finger. If you are not comfortable using a knife, ask a parent to work with you on "cutting the cheese" so to speak.

Phase 1 ? Week 2 ? P3

Name: ______________________________ Per.: ______ # 0422

CC3: 10.1.2 Cylinders - Surface Area and Volume Points: _______

Reflection:

Warm-Up: Solve for the unknown in each shape (use 3.14).

=

= 25.12

Notes: CYLINDERS ? SURFACE AREA AND VOLUME

CC3 10.1.2.

The volume of a cylinder is the area of its base multiplied by its height: = ?

Since the base of a cylinder is a circle of area A = r2, we can write:

=

The surface area of a cylinder is the sum of the two base areas and the lateral surface area. The formula for the

surface area is: = + = +

For additional information, see the Math Notes box in Lesson 10.1.2 of the Core Connections, Course 3 text.

Examples: The soda can has a volume of 355 cm3 and a

height of 12 cm. What is its diameter? Use a

If the volume of the tank is 500 ft3, what is the surface area?

calculator for the value of .

= 2

355 = 2(12)

355 12

=

2

9.42 2

Radius = 3.07

Diameter = 2(3.07) = 6.14

= 2

500 = 2(5)

500 5

=

2

100 = 2

10 =

= 22 + 2 = 2(10)2 + 2(10)(5) = 200 + 100 = 300

The surface area of the tank is 300 or about 942.48 ft2.

Practice: 1. = 29 , = 13 Find the volume.

2. = 5,175 3 = 23 Find the diameter.

3. = 1000 3 = 25 Find the surface area.

Contextual Practice: 1. You are rolling out some dough to make sugar cookies using a rolling pin. The roller on the rolling pin is 15 inches long and has a diameter of 3 inches. As you are rolling out the dough, you forget to sprinkle some flour on the surface and the dough sticks to the rolling pin covering the whole rolling pin as you roll across the dough. If the thickness of the dough covering the rolling pin is about one quarter of an inch. What is the volume of the dough?

2. You cut a 30? wedge from a wheel of cheese. If the cheese wheel is 12 cm tall and has a diameter of 36 cm, what is the volume of the cheese wedge that you cut?

Activity:

Many of us have a favorite water bottle or drinking container that we keep around to help us stay hydrated throughout the day. Find your favorite water bottle or drinking container and calculate the approximate volume of the container (use metric measurements). Various sources recommend that you consume on average about 3.2 Liters of water per day (varies by sex and age). This daily fluid intake can come from food and beverages but plain drinking water is a good way of getting fluids as it has no calories. So, how many of your favorite drinking container would you have to drink to get the recommended daily amount? (Hint: 1 = 1000 3) Do you consume enough water? Interesting Fact: You can get water poisoning by drinking too much water. Curious? Do research on water toxemia.

Phase 1 ? Week 2 ? P4

Name: ______________________________ Per.: ______ # 0423

CC3: 10.1.3 Pyramids and Cones ? Volume Points: _______

Reflection:

Warm-Up: As you ride your bike, you hear the clicking sound of a rock stuck in your tire every time the wheel turns. After 6 clicks, the sound stops. The rock must have fallen out of the tire tread. If your bike wheel is 24" in diameter. How far did you travel before the rock dislodged from the tire?

Notes: PYRAMIDS AND CONES ? VOLUME CC3 10.1.3

The volume of a pyramid is one-third the volume of the

prism with the same base and height and the volume of a

cone is one third the volume of the cylinder with the

same base and height. The formula for the volume of the

pyramid or cone with base B and height h is:

=

.

Examples:

For the cone, since the base is a circle the formula may

also be written:

=

.

For additional information, see the Math Notes box in Lesson 10.1.4 of the Core Connections, Course 3 text.

Practice: Find the volume of each figure based on the given dimensions.

1. Cone = 4 = 10

2. Cone = 2.5 = 10.4

3. Pyramid

right triangle base

legs = 4 ft and 6 ft = 10 1

2

4. Pyramid rectangle base = 6 , = 8

= 5

Contextual Practice: Opening day for the Seattle Mariners was going to be on March 26th at the T-Mobile Park in Seattle. They were going to be offering a special on concessions. They were going to have various snacks you could get in a cone container or a square pyramid container. Both containers would cost the same, $8.00. The cone is 6 inches wide while the pyramid is 5.25 inches wide. If the both containers are 8 inches tall, which container would be the better deal?

Activity: It's springtime and many people are planting their gardens and potting plants and flowers. Most pots are shaped like the bottom portion of a cone or pyramid. This shape is referred to as a "Frustum". There's a formula for calculating the volume of conical and pyramid based frustums but you can also calculate the volume by simply using

what you know about cones or pyramids. Figure out the height of the cone or pyramid that the frustum is derived from and then calculate the volume of the top portion that is being cut off and subtract that smaller volume (smaller cone or pyramid) from the larger combined volume to get the volume of the frustum. You can figure out the height of the original cone or pyramid by putting two sticks, rulers, etc. ... anything straight on opposite sides of the frustum (the pot) and then measure the height from where the two sticks meet (the vertex). If you don't have a flower pot, you could try doing this with a cereal bowl or any bowl that is a frustum (has a flat bottom and angled sides). Give it a try. Figure out the volume of a flower pot or a bowl from the kitchen, whichever you have.

Phase 1 ? Week 2 ? P5

CC3: 10.1.4 Spheres ? Volume

Name: ______________________________ Per.: ______ # 0424

Points: _______

Reflection:

Warm-Up: You decided to get a couple of back-yard chickens this spring. You are trying to figure out the best layout for their pen. You want to give them the most area to run around but also use the least amount of fencing. Which chicken pen layout would be the best balance of area and minimal cost of chicken fencing: a 6 ft. by 12 ft. rectangular pen, a 12 x 16 x 20 ft. triangular pen in a corner of the yard or a round pen with an 11 ft. diameter in the middle of the yard?

Notes: SPHERES ? VOLUME CC3 10.1.4

For a sphere with radius r, the volume is found using:

=

For more information, see the Math Notes box in Lesson 10.1.5 of the Core Connections, Course 3 text.

Example 1

Find the volume of the sphere.

=

4 3

3

=

4 3

23

=

32 3

3

or approximately 33.49 ft3

Example 2

A sphere has a volume of 972 3. Find the radius. Use

the formula for volume and solve the equation for the radius.

= 4 3 = 972

3

43 = 2916

Substitution Multiply by 3

3 = 2916 = 729

4

= 3729 = 9

Divide by 4 to isolate r. Find cube root

Practice:

1. What is the volume of the sphere if the radius is 4.4 inches? (Use = 3.14. Round answers to nearest hundredth.)

2. What is the volume of the hemisphere if

the

radius

is

3 8

of

an

inch.

(Use

=

3.14.

Round answers to nearest hundredth.)

Contextual Practice:

1. If the shape below represents an ice-cream 2. The USS Idaho (SSN-799) is a Virginia-Class nuclear-

cone that is packed full of ice cream with a powered, cruise-missile, fast-attack submarine currently

half spherical scoop of ice-cream on top,

under construction. It has a hull width of about 34 ft. and a

what is the volume of all the ice-cream if the length of about 377 ft. Assuming the hull of the submarine

cone is 6 inches tall and the scoop has a

is shaped similarly to the diagram,

radius of 2 inches?

what volume of water does the

submarine displace?

Activity: Have you ever wondered how the size of the Moon relates to the size of the Earth? We can read about it on

the internet and in books and look up graphical representations to give us a visual, but it's different when you can see

it in person and touch it. No, we're not going to the moon, but we can make a model of the volumes though. The

earth has a radius of about 4 thousand miles. The moon's radius is a little over 1 thousand miles. Based on these

rough measurements, calculate the volume of the Earth and the Moon. Use the word "thousand" as a unit in your

calculations to avoid massive numbers. You will get thousand-miles3. To make a physical model of this volume

though, let's take the value of x and divide by 10 (to make it manageable; you'll still need a large container) and

replace our unit of thousand-miles3 with 3. We can use water to now simulate the volume. Each cubic inch is

roughly

2 3

of

a

cup.

So

using

a

measuring

cup,

pour

as

many

2 3

cups

of

water

into

a

container

as

you

calculated

cubic

inches ( 3) for our model of the Earth's volume. If you have a second, equivalent

container,

pour

as

many

2 3

cups

of

water

into

it

as

you

calculated

cubic

inches

(

3)

for our model of the Moon's volume. Now you can compare the models side by side to see just

how much more space the Earth takes up than the Moon.

Phase 1 ? Week 2 ? Answer Key

Unit 8: Surface Area and Volume

Honors Math 7

P1 Warm-Up

1. = 11 1 2; = 13 1

9

3

2. = 15.75 2; = 19.5

P1 Practice: 1. = 314 2

2. = 210 2

3. 408 2

P1 Contextual Practice: 1. = 405 3 ; . . = 378 2

P2 Warm-Up:

1. = 8 2 2; = 14 29

5

60

2. = 38.7 2; = 29

P2 Practice:

1. = 12 3

2. = 168 3

3. = 324 3

P2 Contextual Practice:

80% of the volume of the gas tank is 2,772 in3

Answers vary based on gas price used. Sample answer for $2.11 per gal (mid-grade at Fred Meyer) = $25.32

P3 Warm-Up: 1. = 63.585 2

2. = 4

P3 Practice: 1. 8586.76 3

2. = 16.93

3. = 640.50 2

P3 Contextual Practice: 1. 38.27 3

2. 1017.36 3

P4 Warm-Up: 1. 452.16 or 37.68

P4 Practice: 1. 167.55 3

2. 68.07 3

3. = 42 3

4. = 9.8 3

P4 Contextual Practice:

1.

75.36 3;

= 80 2 3

3

The pyramid container is a better deal. You get more volume for your $8.

P5 Warm-Up:

1. : = 72 ft2 ; = 36 .

Smallest Area but also not the cheapest.

2. : = 96 ft2 ; = 48 .

Best Area but highest cost.

3. : 94.985 ft2; 34.54 .

Nearly the same Area as Triangular Pen and smallest cost. (Winner)

P5 Practice: 1. 356.64 3

2. 0.22 3

P5 Contextual Practice: 1. 58.55 3

2. 331,828 3

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