[PDF] [PDF] Three-Dimensional Geometry - Houston ISD

View - Download [PDF] Three-Dimensional Geometry - Houston ISD

Previous PDF

Next PDF

© 2010 College Board. All rights reserved.

Unit 301
6

Three-Dimensional

Geometry

Essential Questions

How are two- and three-

dimensional “ gures related?

How do changes in

dimensions of a geometric

“ gure affect area, surface

area, and volume?

Unit Overview

In this unit you will investigate 2- and

3-dimensional gures, and apply formulas to

determine areas and volumes of these gures.

Your study will include right triangles and the

Pythagorean eorem, polygons, prisms, and cylinders. You will relate geometric concepts to real world situations.

Academic Vocabulary

Add this word to your vocabulary notebook.

solid

This unit has two Embedded

Assessments. These embedded

assessments allow you to demonstrate your understanding of the Pythagorean Theorem and its uses, and lateral area and volume.

Embedded Assessment 1

Right Triangles p. 315

Embedded Assessment 2

Area and Volume p. 337

EMBEDDED

ASSESSMENTS

301-302_SB_MS3_6-0_SE

.indd 301301-302_SB_MS3_6-0_SE.indd 3012/19/10 11:16:32 PM2/19/10 11:16:32 PM

© 2010 College Board. All rights reserved.

302 SpringBoard

Mathematics with Meaning

TM

Level 3

Write your answers on notebook paper.

Show your work.

1. Simplify the following squares, cubes, and

square roots. a. 6 2 b. (1.2 ) 2 c. 3 __ 5 2 d. 4 3 e. 5 3 f. ___ 49
g. ____ 225

2. Write the ratio of the following:

a. 3 inches to 1 foot b. 13 days to 40 days

3. A square and a rectangle both have an area

of 16 square units and dimensions that are whole-number units. Find the dimensions of the square and the rectangle.

4. Draw a representation of each of the

following triangles and describe the characteristics of each. a. Scalene b. Isosceles c. Equilateral d. Right

5. Name each ? gure.

6. Find the perimeter or circumference of

each of the gures below. a. 5.3 2.7 b. h = 2 8 3 7 c. 3 d. 4 9 5

7. Find the area of each ? gure in Item 6.

8. Explain using speci? c formulas how you

could nd the area of the shaded area of the gure below.

UNIT 6

Getting

Ready

301-302_SB_MS3_6-0_SE.indd 302301-302_SB_MS3_6-0_SE.indd 30212/30/09 12:06:20 PM12/30/09 12:06:20 PM

© 2010 College Board. All rights reserved.

ACTIVITY

My Notes

Unit 6 € Three-Dimensional Geometry 303

6.1 SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Visualization, Interactive Word Wall Cameron is a catcher trying out for the school baseball team. He has played baseball in the community and is able to easily throw the ball from home plate to second base to throw out a runner trying to steal second base. However, the school baseball diamond is a regulation size eld and larger than the eld he is accustomed to. Will he be able to consistently throw out runners trying to steal second if he is able to throw the baseball 130 feet? e distance between each consecutive base on a regulation baseball diamond is 90 feet and the baselines are perpendicular. e imaginary line from home plate to second base divides the baseball diamond into two right triangles. ere is a relationship between the lengths of the three sides of any right triangle that might be helpful for determining if Cameron can throw across a regulation baseball diamond.

Pitcher

Home

3rd1st

2nd

90 ft90 ft

90 ft
90 ft

1. Use the terms hypotenuse and leg to identify and label the parts

of the right triangle below. a. b. Explain the di? erences between the hypotenuse and the legs of a right triangle.

Diamond in the Rough

The Pythagorean Theorem

CONNECT TO APAP

In AP Calculus, the Pythagorean

theorem is useful when solving related rates problems.

303-310_SB_MS3_6-1_SE.indd 303303-310_SB_MS3_6-1_SE.indd 30312/30/09 12:06:49 PM12/30/09 12:06:49 PM

© 2010 College Board. All rights reserved.

304 SpringBoard

Mathematics with Meaning

TM

Level 3

My Notes

The Pythagorean Theorem

ACTIVITY 6.1

continued

Diamond in the RoughDiamond in the Rough

? e Pythagorean theorem describes triangles containing a right angle.

2. In a right triangle, let c be the length of the hypotenuse and let

a and b be the lengths of the legs of the triangle. a. Draw and label a right triangle using a, b, and c. b. Write an equation using a, b and c to represent the

Pythagorean theorem.

3. Use the Pythagorean theorem to ? nd the length of the

hypotenuse in each of the following triangles. a. c 3 4 b. c 15 8

4. Use the Pythagorean theorem to ? nd the length of the missing

leg in each of the following triangles. a. b 25
7 b. a 10 6

SUGGESTED LEARNING STRATEGIES: Questioning the

Text, Create Representations, Think/Pair/Share, Group

Presentation, Work Backwards

The Pythagorean theorem

states that: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs of the triangle.

MATH TERMS

303-310_SB_MS3_6-1_SE.indd 304303-310_SB_MS3_6-1_SE.indd 30412/30/09 12:06:55 PM12/30/09 12:06:55 PM

© 2010 College Board. All rights reserved.

My Notes

Unit 6 € Three-Dimensional Geometry 305

ACTIVITY 6.1

continued

The Pythagorean Theorem

Diamond in the RoughDiamond in the Rough

5. Use the Pythagorean theorem to ? nd p in terms of r and t.

p r t

6. Sketch a diagram of a regulation

baseball diamond showing the baselines and the imaginary line from home plate to second base.

Identify and label the hypotenuse

and legs of any right triangles. What are the lengths of the legs of the triangles?

7. Write an equation that can be used to ? nd the distance from

home plate to second base.

8. Can the distance from home plate to second base be found

without a calculator? Why or why not? Because it may be di cult to nd some distances without a calculator, estimation is o en a useful problem solving tool.

9. Use the Pythagorean theorem to ? nd the exact length of the

hypotenuse for each of the right triangles described in the table below. en estimate the value of the square root to nd the estimated length of the hypotenuse. SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share, Group Presentation, Group Discussion

Triangle

Length of

Leg 1

Length of

Leg 2

Exact Length

of Hypotenuse

Estimated Length

of Hypotenuse

1 1 unit 2 units

2 1 unit 3 units

3 2 units 2 units

4 2 units 3 units

5 9 units 9 units

If you take the square root

of a number that is not a perfect square, the result is a decimal number that does not terminate or repeat and is called an irrational number. The exact value of an irrational number must be written using a radical sign.

Decimal approximations of

irrational numbers are found using technology such as a calculator.

TECHNOLOGY

303-310_SB_MS3_6-1_SE.indd 305303-310_SB_MS3_6-1_SE.indd 30512/30/09 12:06:59 PM12/30/09 12:06:59 PM

© 2010 College Board. All rights reserved.

306 SpringBoard

Mathematics with Meaning

TM

Level 3

My Notes

The Pythagorean Theorem

ACTIVITY 6.1

continued

Diamond in the RoughDiamond in the Rough

SUGGESTED LEARNING STRATEGIES: Simplify the

Problem, Think/Pair/Share, Group Presentation,

Activating Prior Knowledge

10. Which of the triangles in the table in Item 9 could be used to

help Cameron estimate the distance from home plate to second base? Justify your choice.

11. Use the Pythagorean theorem and the information in the

table in Item 9 to nd the distance from home plate to second base. Show all your work.

12. If Cameron can throw the baseball 130 feet, will he be able

to consistently throw out a runner trying to steal second base?

Explain your reasoning.

13. On a regulation so ball diamond, the distance between

consecutive bases is 60 feet and the baselines are perpendicular. a. Sketch and label a scale drawing of a so ball diamond. b. Use the Pythagorean theorem and your sketch to estimate the distance from home plate to second base on a so ball eld. Show all your work.

303-310_SB_MS3_6-1_SE.indd 306303-310_SB_MS3_6-1_SE.indd 30612/30/09 12:07:03 PM12/30/09 12:07:03 PM

© 2010 College Board. All rights reserved.

My Notes

Unit 6 € Three-Dimensional Geometry 307

ACTIVITY 6.1

continued

The Pythagorean Theorem

Diamond in the RoughDiamond in the Rough

SUGGESTED LEARNING STRATEGIES: Shared Reading, Visualization, Create Representations, Think/Pair/Share Identify a Subtask During summer vacation, Cameron"s parents take him to see his favorite baseball team, the Anglers, play. On their last day of vacation, he discovers that he will not be able to carry the autographed bat he won home on the plane. His dad suggests that he speak to the concierge at the hotel about options for shipping the bat home. e concierge only has one box that he thinks might be long enough. A er measuring the dimensions of the box to be 16 × 16 × 27, the concierge apologizes for not having a box long enough for the 34 bat. Cameron thinks he might still be able to use the box. His idea is to put the bat in the box at an angle as shown in the diagram below. He wonders if the bat will t in the box.

27 in.

16 in.

16 in.

14. ? e diagonal of the box is the hypotenuse of a right triangle.

Outline this triangle in the diagram above.

15. What are the lengths of the legs of this right triangle?

Show any work needed to nd these lengths.

16. Find the length of the diagonal of the box. Show any

necessary calculations.

CONNECT TO TRAVELTRAVEL

In a hotel, a concierge is a person

who helps guests with various tasks ranging from restaurant reservations to travel plans.

303-310_SB_MS3_6-1_SE.indd 307303-310_SB_MS3_6-1_SE.indd 30712/30/09 12:07:07 PM12/30/09 12:07:07 PM

© 2010 College Board. All rights reserved.

My Notes

308 SpringBoard

Mathematics with Meaning

TM

Level 3

The Pythagorean Theorem

ACTIVITY 6.1

continued

Diamond in the RoughDiamond in the Rough

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,

Group Presentation, Create Representations

17. Will Cameron be able to use the box to ship his bat? Justify

your response.

18. When Cameron returns to school, he tells his math teacher

how he applied the Pythagorean eorem while on vacation. She is excited and tells him they will do some investigation into why the Pythagorean eorem works. a. On centimeter grid paper or graph paper, draw right triangles having legs with each of the following lengths.

Use one piece of paper for each triangle.

Triangle 1: 3 units and 4 units

Triangle 2: 5 units and 12 units

Triangle 3: 8 units and 15 units

b. Use the Pythagorean ? eorem to nd the length of the hypotenuse in each of the triangles.

Hypotenuse of Triangle 1:

Hypotenuse of Triangle 2:

Hypotenuse of Triangle 3:

c. On each leg of each of the right triangles, draw a square with sides the same length as the leg of the triangle. Find the area of each of these squares and complete the table below. d. How does the area of each square relate to the length of a leg of the triangle?

Triangle

Length of

Leg 1

Area of Square

on Leg 1

Length of

Leg 2

Area of Square

on Leg 2

303-310_SB_MS3_6-1_SE.indd 308303-310_SB_MS3_6-1_SE.indd 30812/30/09 12:07:10 PM12/30/09 12:07:10 PM

© 2010 College Board. All rights reserved.

My Notes

Unit 6 € Three-Dimensional Geometry 309

ACTIVITY 6.1

continued

The Pythagorean Theorem

Diamond in the RoughDiamond in the Rough

e. Cut out each of the triangles and the squares drawn on each of the legs. f. Use the small squares drawn on the legs of each right triangle to build a large square on the hypotenuse of that triangle. You may cut and rearrange the small squares any way necessary to create one larger square having side length equal to the length of the hypotenuse of the triangle. g. Find the area of the square created on the hypotenuse of each triangle.

Area of square on the hypotenuse of Triangle 1:

Area of square on the hypotenuse of Triangle 2:

Area of square on the hypotenuse of Triangle 3:

h. What is the relationship between the areas of the squares drawn on each of the legs of the triangle and the large square built on the hypotenuse? Explain your reasoning. i. How does this relationship illustrate the Pythagorean eorem in the form c 2 = a 2 + b 2 SUGGESTED LEARNING STRATEGIES: Group Presentation,

Create Representations

303-310_SB_MS3_6-1_SE.indd 309303-310_SB_MS3_6-1_SE.indd 30912/30/09 12:07:14 PM12/30/09 12:07:14 PM

© 2010 College Board. All rights reserved.

310 SpringBoard

Mathematics with Meaning

TM

Level 3

The Pythagorean Theorem

ACTIVITY 6.1

continued

Diamond in the RoughDiamond in the Rough

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper.

Show your work.

Find x in each of the following.

1. x 14 10 2. x 4 2 3. x 24
10

4. A painter uses a ladder to reach a second-

story window on the house she is painting.

π e bottom of the window is 20 feet

above the ground. π e foot of the ladder is 15 feet from the house. How long is the ladder?

5. Estimate the square roots of each of the

following: a. 12 b. 17 c. 40 d. 99

6. Find the distance from point (2, 6) to

point (8, 3) to the nearest tenth of a unit. 7.

MATHEMATICAL

REFLECTION

In what type of triangle

is c 2 > a 2 + b 2 ? In what type of triangle is c 2 < a 2 + b 2 ? Explain your answers.

You can also use the Pythagorean

theorem to nd the distance between two points on a coordinate plane. Just think of the two points as the endpoints of the hypotenuse of a right triangle. π en draw the legs and use their lengths in the formula.

19. Find the distance from point (1, 2) to point (7, 6) to the

nearest tenth of a unit.

20. Find the distance from point (3, 1) to point (5, 5) to the

nearest tenth of a unit.quotesdbs_dbs21.pdfusesText_27

[PDF] finding complex solutions of polynomial equations practice and problem solving a/b answers

[PDF] finding interval of definition

[PDF] finding interval of validity

[PDF] finding the inverse of a 2x2 matrix

[PDF] finding the inverse of a 3x3 matrix

[PDF] finding the inverse of a function

[PDF] finding the inverse of a function calculator

[PDF] finding the inverse of a matrix

[PDF] finding the inverse of a quadratic function

[PDF] fine arts classes uva

[PDF] fine for calling 911 falsely ontario

[PDF] fingerprint technology read theory answers

[PDF] finite automata examples

[PDF] finite automata examples pdf

[PDF] finite automata examples with solution