Geometry of Plane-Shapes & Space-Shapes Unit 5

Farmington Public Schools Grade 9 Mathematics

Michele Hall & Christina Lepi

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Table of Contents Unit Summary

………………….….…………..page 3

Stage One: Standards

Stage One identifies the desired results of the unit including the broad understandings, the unit outcome statement and essential questions that focus the unit, and the necessary knowledge and skills. The Understanding by Design Handbook, 1999

…………………………….... pages 4-6 Stage Two: Assessment Package

Stage Two determines the acceptable evidence that students have acquired the understandings, knowledge and skills identified in Stage One.

……………………………… pages 7-8 Stage Three: Curriculum and Instruction

Stage Three helps teachers plan learning experiences and instruction that aligns with Stage One and enables students to be successful in Stage two. Planning and lesson options are given, however teachers are encouraged to customize this stage to their own students, maintaining alignment with Stages One and Two.

………………..……………… pages 9-12

Appendices

Michele Hall & Christina Lepi

….....………………………. pages 13-52

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Unit Summary

This unit on Geometry of Plane-Shapes and Space-Shapes is the 5th unit in the 9th grade Integrated Math I. This is the first unit in the high school sequence on geometry topics, and follows units on algebra (Linear Models) and statistics (Patterns in Data and Patterns of Change). This will take approximately 6-7 weeks to complete.

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Stage One: Standards Stage One identifies the desired results of the unit including the broad understandings, the unit outcome statement and essential questions that focus the unit, and the necessary knowledge and skills. The Understanding by Design Handbook, 1999

Essential Understandings and Content Standards Directions: List the essential understandings and content standards which the unit or course addresses. Asterisk the content standards that are addressed by the performance task. These can be found in the K-12 standards document for your discipline. Example: #6 Students will understand that people analyze and use spatial relationships and basic geometric concepts in order to construct, draw, describe and compare geometric models and their transformations. The student will 6a. (8th) identify, visualize, model, describe and compare properties of and relationships among 2- and 3-dimensional shapes; 6b. (8th) describe and use fundamental concepts and properties of….the Pythagorean theorem; 6b. (12th) deduce properties of, and relationships among, figures from given assumptions; 6d. (12th) analyze the geometry of three dimensional shapes; 6e. (12th) solve real-world and mathematical problems using geometric models. #5. Students will understand that people must appropriately apply customary and metric measurement units in order to approximate, measure and compute length, area, volume, mass, temperature, angle and time. The student will 5a. extend, apply and formalize understandings of measurement, including strategies for determining perimeters, areas and volumes, and the dimensionality relationships among them; 5c. choose appropriate tools and techniques to measure quantities to specified degrees of precision and accuracy. #3. Students will understand that people must use estimation and approximation in order to judge the reasonableness of results and to guide their mathematical thinking. The student will 3c. make reasonable estimates of the values of formulas, functions and roots. Writing standards #1. Students will understand a deliberate process- prewriting, drafting, revising, editing and publishing – is essential to effective writing. The student will identify, select and apply the most effective processes to create and present a written, oral or visual piece. #4. Students will understand that writing is a way to clarify thinking in all content areas. The student will use writing to communicate information, generalizations, theories, and interpretations. Michele Hall & Christina Lepi

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Unit Outcome Statement Consistently aligning all instruction with this statement will maintain focus in this unit.

As a result of this unit the students will understand how to use geometric shapes and their properties to make sense of real-world situations such as art, architecture, and manufacturing. The students will also understand how to: ™ differentiate and apply appropriate measures of plane- and space-shapes (perimeter, area, volume) ™ draw and construct both plane- and space-shapes ™ visualize space-shapes on a two-dimensional plane through different points-of-view Essential Questions These questions help to focus the unit and guide inquiry.

™ How is an understanding of geometry useful in real-world situations? ™ What is the relationship between plane-shapes and space-shapes? ™ How is geometry related to other disciplines?

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Knowledge and Skills The Knowledge and Skills section includes the key facts, concepts, principles, skills, and processes called for by the content standards and needed by students to reach desired understandings. The Understanding by Design Handbook, 1999

Knowledge ‰

‰

‰ ‰ ‰

‰ ‰ ‰

Space-shapes - names and characteristics -pyramid -prism -cylinder -cone Plane shapes - names and characteristics -triangles (acute, obtuse, equilateral, isosceles, scalene, and right) -quadrilaterals (square, rectangle, trapezoid, parallelogram, rhombus) -polygon (especially regular) -circle Identifying vertices and edges of plane-and space-shapes Identify space-shapes in real-world context Identify the correct measurement needed to describe a shape -perimeter/circumference -lateral area -surface area -volume Pythagorean theorem Tessellation Symmetry

Skills/Processes Example: How to use plane-shapes to create space-shapes ‰ Draw a sketch of a space-shape with and without isometric dot paper ‰ Applying formulas for perimeter, area and volume appropriately ‰ Applying Pythagorean theorem to find missing leg or hypotenuse ‰ Applying Pythagorean theorem to determine if a triangle is a right triangle ‰ Find the measure of interior and exterior angles in a regular polygon ‰ Find the total angle measure of any polygon ‰ Describe symmetry of a tiling pattern ‰

Thinking Skills ‰ ‰ ‰ ‰ ‰

Identify appropriate evidence Sorting and categorizing facts Infer from/evaluate ideas – draw conclusions Making generalizations Interpretation and synthesis

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Stage Two: Assessment Package Stage Two determines the acceptable evidence that students have acquired the understandings, knowledge and skills identified in Stage One.

Authentic Performance Task: Designing a Sports Bottle Container (Manufacturing) Goal: Your task is to design a drinking container to hold a specific amount of liquid. You will then determine the cost of making the container and the box to package the containers in to deliver to stores. Role: You have been hired to design a container for a company that makes sports drinks. Audience: You need to convince the head of packaging that the container you designed will hold the required amount of liquid. The cost of producing the container and box to transport it are also factors in the job. Situation: The container you design will serve as a means to contain the liquid and serve as a drinking container. The client wants the container to hold 20 oz. of liquid, with minimum empty space inside the container. The amount of material used to make the container needs to be minimized for price purposes. You also must take into account that the sports drink containers will be shipped in a cardboard box. Product: You will create an original design for the container using any of the shapes discussed in Integrated Math 1; a sketch must be included (Show all work used to verify the volume and surface area of the container to meet the company guidelines). The box used for delivery should hold 18 containers. Sketch and find the dimensions of the box needed for your design as well as the surface area, to determine the amount of material needed to make the box. Provide all formulas used and show the computations used as well as an explanation of all numbers. Information needed for the project: Conversion factor: Cost of plastic material: Cost of cardboard:

1 fluid oz. = 1 in 3 *1.804 $. 0025 per square inch $.10 per square foot.

Standard: Your design will be assessed on: Original Design including sketch Surface area of container including formulas Volume of container including formulas Dimensions of box including sketch/rationale and formulas Surface area of box including formulas Cost of container including work Cost of box including work Neatness Tests, Quizzes, and Other Quick and Ongoing Checks for Understanding Otter Lake Problem: Area and Perimeter of Irregular Shapes This progress check is a follow-up to an investigation from the text on finding the area and perimeter of an irregular shape. Students are asked to write about a method for solving, but not asked to find a solution. Quiz I: Area and Perimeter of Plane shapes, Pythagorean theorem This quiz is an assessment of the skills and formulas associated with finding the area and perimeter of basic plane-shapes, including finding missing sides of a right triangle.

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Test I: Plane shapes This test is an assessment of the processes involved in measuring plane-shapes. Besides basic skills of finding the area and perimeter, students are also asked to use problem-solving skills to find the area of an irregular shape (by dividing into rectangles) and determining whether 3 lengths guarantee a right triangle (via Pythagorean theorem). Quiz II: Circles and surface area This quiz is an assessment of the skills and formulas associated with measuring circles and finding the surface area of a space-shape. Test II: Space shapes This test focuses on identifying space-shapes, and finding missing information about plane and space-shapes. Students will also have to demonstrate understanding through applying their understanding of sketching, calculating and comparing different space-shapes.

Projects, Reports, Etc. Poster Project: Geometry, Geometry, Everywhere This poster project is an extension of the introductory activity from the unit, which asks students to find 2 examples that demonstrate some of the 4 major geometry topics in their community and either sketch or take pictures. After the 1st test, students will be asked to revisit their assignment, creating a poster using their pictures with more detailed descriptions of the geometry concepts demonstrated.

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Stage Three: Learning Experiences and Instruction Stage Three helps teachers plan learning experiences and instruction that align with Stage One and enables students to be successful in Stage Two.

Learning Experiences and Instruction The learning experiences and instruction described in this section provide teachers with one option for meeting the standards listed in Stage One. Teachers are encouraged to design their own learning experiences and instruction, tailored to the needs of their particular students.

Guiding Questions Day 1-2: Introduction Where can you find examples of geometry in your life? E.Q.: How is an understanding of geometry useful in realworld situations?

Instructional Strategies

Checking for Understanding

Think About the Situation p. 326; Hook: Students will look at pictures in the text of different art and architecture, identifying the geometric shape or idea. Design and test columns for “strength”

Geometry, Geometry Everywhere assignment.

Day 3-4: Recognizing space-shapes Students in groups of 3 will What are the similarities and build straw models of differences among spacedifferent space-shapes. Each shapes? group will build, identify, and compare the different spaceshapes created. Emphasis on vocabulary such as prism, pyramid, edges, vertex, etc. occurs during class discussion.

Lesson 1, Investigation 1 p. 327-328 #2,3 Checkpoint p. 328 (if students are struggling at this point, reinforce as needed.) Lesson 1, Investigations 2 p. 329-333 #1,2,3,4,5,8 Checkpoint, On Your Own (OYO) p. 332 (if students are struggling at this point, reinforce as needed.) Modeling #3, p. 335 Worksheet: Identifying spaceshapes Worksheet: Space-shape terms

Days 5: Visualizing space shapes Students will study the models How can you draw spaceshapes using different types of constructed in class and brainstorm ideas on how to graph paper? draw their models on graph paper. They will be given both regular and isometric E.Q.: What is the dot paper to do this. relationship between Students will need LOTS of plane-shapes and spacevisuals and practice. Pairing shapes? visual students up with students who are struggling may help. In addition, students can use blocks to build a 3-D model before transferring to paper. Michele Hall & Christina Lepi

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Lesson 1, Investigation 3 p. 341-344, #1-4 Modeling #3,4,5 p. 347-8 Extending #4, p. 353 Suggested check: have students identify models you hold up in class by name using models made in class. Then, have them recreate model on dot paper. This can be repeated as needed throughout the unit.

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Day 6-10: Area, Perimeter and Pythagorean theorem Hook: Think About the How can we use familiar Situation p. 355 formulas to determine area and perimeter of basic and Assumed that student have irregular plane-shapes? familiarity with finding area and perimeter of rectangles, How do we use the squares, triangles. Students Pythagorean theorem when will apply prior knowledge to working with plane-shapes? situations involving other shapes (which do not necessarily have a formula) where they will either develop a new formula or a strategy using existing formulas. Students review Pythagorean theorem and apply it to triangles and other shapes which need a height calculated to determine area. Skill practice on squaring and square roots is necessary to correctly use Pythagorean theorem.

Lesson 2, Investigation 1, p. 357 #4 OYO p. 359 Worksheets: -Perimeter/Area of Rectangles & Squares -Squares and Square Roots -Pythagorean theorem (graphing) -Pythagorean theorem -Area of Parallelogram (activity) -Area of parallelograms and triangles -Area of triangles (mixed review) Otter Lake Problem Progress check Quiz I: Area and Perimeter of Plane shapes, Pythagorean theorem

Also, demos in Geometer’s Sketchpad are available to use as in-class or as a lab assignment. (Pythag.gsp and TV.gsp) Day 11-14: Misc. Area and Perimeter Hands-on activity: Cut out a How do the formulas for triangle on paper. Measure area/perimeter of trapezoids base and height to determine and circles compare to other area. Then, draw a line area and perimeter formulas parallel to one side through that you are familiar with? the triangle and cut along the line. Find the area of the smaller triangle, and use the information to find the area of the trapezoid. From this information, develop a formula for the area of a trapezoid.

Worksheets: -Area of trapezoids -Circumference of circle/Area of circle -Circles- mixed practice -Test review Test I: Plane shapes (Students may use geometry formula sheet. Incentive for not using may be encouraged.)

Review of formulas for circles; emphasis now is not only on finding circumference and area, but using circumference and area to determine the radius and diameter of the circle.

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Day 15-18: Polygons, Symmetry and Tessellations Hook: How is geometry used How can you determine the in creating artistic patterns? measure of all the angles in a Show students examples of polygon? artwork such as Escher, vases, etc. How can we use the properties of plane-shapes to Activity: Students will create symmetric patterns develop the concept that a and tessellation? triangle has a sum of 180º. Using this fact, students will E.Q.: How is geometry then investigate how this related to other information can be used to disciplines? develop the formulas for interior & exterior angle sums of polygons. This information is extended to determine the measure of a single angle if the polygon is regular. Students will investigate how knowing the angle sums can help create tiling patterns (tessellations).

Lesson 3, Investigation 1 p. 383-88 #1, 3, 4, 5, 6 OYO p. 389 Lesson 3, Investigation 2 p. 390-391, #1, 2, 4 Lesson 3, Investigation 3 p. 402-404 #2, 3 Modeling #3 p. 409 Poster project: Geometry, Geometry Everywhere (These skills are an introduction to skills that are mastered in Integrated II. It is sufficient that students are exposed to this material)

Students will study strip patterns from different cultures to identify different types of symmetry (rotational, reflection, translational, glide reflection) Day 19-24: Surface Area of Space-shapes Hook: How much cardboard How do we use the area is needed to make a box of formulas from plane-shapes cereal (p. 394) or can of soup? to find the area of spaceshapes? Students will revisit the space-shapes they created E.Q.: What is the earlier in this unit. They will relationship between discuss what plane-shapes plane-shapes and spaceare used to create each spaceshapes? shape, then use this How is an understanding information to develop formulas and strategies for of geometry useful in determining lateral and total real-world situations? surface area. Students will also draw space-shapes as nets and vice versa to see the relationship between plane-shapes and space-shapes.

Lesson 2, Investigation 3, p. 373-374 #1-3 Lesson 3, Investigation 2, p. 393-394, #6, 7 OYO p 394 Modeling #5 p. 397 Worksheets: -Calculating Surface Area (activity) -Area of Rectangular prisms -Area of Pyramids -Area of Cylinders -What is cold and comes in cans (mixed review) -Review for Quiz Quiz II: Circles and surface area

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Day 25-31: Volume of Space-shapes Using their previously built How can you adapt the area models, students discuss how formulas for plane-shape to volume formulas for prisms calculate volume of a spaceand cylinders can be shape? developed (by thinking of layering). E.Q.: What is the relationship between Pyramids/Cones: Using plane-shapes and spaceplastic solids, students fill shapes? them with sand/water to How is an understanding compare how volume of a prism is similar and different of geometry useful in from the volume of a pyramid real-world situations? (by exactly 3 times). Information is used to develop How is geometry related formulas for pyramid and to other disciplines? cone volumes. Geometer’s Sketchpad is also available for demos/investigations. Also, Rhino 3D can be used to create and calculate measurements of spaceshapes, which may be used with struggling students.

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Lesson 2, Investigation 3, p. 374-376 #4-8 Modeling #2,3,4 p. 377-8 Worksheet: -Calculating volume (activity) -Volume of prisms -Volume of cylinders -Area/Volume mixed practice -Volume of pyramids -Volume of cones -Review for test -Unit 5 Summary (from teacher resources) Test II: Space-shapes (Students may use geometry formula sheet. Incentive for not using may be encouraged.) Authentic Performance Task: Designing a Sports Bottle Container

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Appendices Texts: Contemporary Mathematics in Context, Book 1B, Everyday Learning Corporation. Alternate resources (majority of worksheets cited are from these sources): Cognitive Tutor: Integrated Math I Middle School Math with Pizzazz Book D McDougall-Littell Geometry Resource book Instructional Fair Geometry Masters Houghton Mifflin Basic Geometry Practice Masters Geometry Formulas worksheet Miscellaneous worksheets (attached) Geometer’s Sketchpad (for demo purposes) Rhino 3-D (for demo/reinforcement purposes – talk to Bill Wright)

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Integrated Math 1

Name ____________________

In the following drawing of Otter Lake, a popular fishing spot, is covered with a grid of squares. Each side of each small square represents 10 meters.

On a beautiful day, Tina and her father rowed around the outer edge of Otter Lake, stopping to fish along the way. Explain how you might estimate the total distance they rowed around the lake if the began and ended at the same spot.

Explain how you could estimate the area of Otter Lake.

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Integrated Math 1 Area-Perimeter Quiz

Name ____________________________ Date _______________

Find the area and perimeter of each. Show the formula used and includes the units.

1

2.

2 ft. 3

1 in 2

3 ft.

3

1 in 2

Perimeter ___________

Perimeter _______________

Area _______________

Area ________________

3.

4. 3’

c 4’

4 ft.

2’

6’ 5ft. Perimeter ___________

c = _______________

Area _______________

Perimeter ________________

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5.

6 m.

6. 1 m.

a

5 cm. 8 m. 10 m. 3 cm. 5 m.

7 m. a= ___________

Perimeter _______________

Perimeter _______________

Area ________________

7. 25 ft.

10ftft. 10

1 ft.

12 ft. 3 ft.

3 ft.

Area of shaded region ___________________

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Integrated Math I Area/Perimeter TEST

Name_________________________

Find the perimeter of each figure. 1)

2)

1’

6’

5” 3’

1’

5”

4’

5” 4”

2’

3)

4) 2’

6’

2”

4’

3’

2’ 2’

5” 3’

3”

2”

5” 2”

2”

5’

3’

Part II. Find the area of each figure. 5)

6)

7)

6m 5m 6m

6m

7” 8m 5”

8)

9)

18”

.4 ft

10 cm

6 cm

22”

1.4 ft

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10)

30”

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Complete the following table about rectangles.

11) 12)

Base

Height

6m

3m 4m

Perimeter

Area 24 m2

Complete the following table about squares. Side 13) 14)

Perimeter

Area

8ft 16 ft2

15) Mr. Z must mow his lawn, 12 m by 6 m. However, his rectangular pool (2m by 3m) is right in the middle of the lawn. How many square meters must be mowed?

POOL

Find the total area of each figure. 16)

17)

15 cm 3 cm

4 in

7 cm

9 in

17 cm

10 in

10 in

2 cm 3 in

16 cm

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Find the area of the shaded region. 18)

19)

3 in 15 cm

12 in

7 in

4 cm 9 cm

10 cm

7 in

Use the Pythagorean Theorem to answer each of the following: 20)

21) 7 in 15’ 9 in 8’

Can a right triangle be constructed whose sides measure (show work): 22) 9m, 12m, 15m

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23) 5ft, 12ft, 13ft

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Bonus Question: Mass Pike (55 km)

John is traveling from Hartford to Boston.

Boston

Rte 91 (40 km) Rte. 84

Hartford a) He could travel north on Rte. 91 and then east on the Mass Pike into Boston. How far would he travel?

b) OR John could take Rte 84 into Boston. How far would he travel?

c)

Which is the shortest route? By how much?

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Poster Project “Geometry, Geometry Everywhere” At the beginning of the unit, you were assigned the task of finding examples of different geometric ideas in your environment. Now, using this information and what you have learned so far, create a poster or booklet that clearly illustrates your findings for someone who has not taken Integrated Math I. Poster/Booklet must include: • At least 2 different photos/drawings that illustrate the ideas of symmetry, measurement, tessellations and rigidity. One picture may illustrate all 4 very nicely, but you could also use one photo for each idea. • Write descriptions for each photo, describing in detail what idea is represented with supporting evidence. Indicate why the geometric idea is relevant to this real-world object (ex. is it essential to its structure, or does it just make it look nice?) • Pick one of your pictures to demonstrate your knowledge of perimeter and area. You must indicate your dimensions and units, and show the calculations you performed to find perimeter/area. Explain why these measurements might be needed. You will be assessed on not only your demonstration of math knowledge, but also on clarity of your explanations and overall visual presentation.

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Integrated Math 1 Circle and Surface Area Quiz

Name _________________ Date ________

1. Find the circumference of the circle, using the value of

π

as indicated and units.

Use 3.14 for

π

14 m

Circumference ____________

2. Find the area of the circle, using the value of

π

as indicated and units.

Use 3.14 for

π

10 cm.

Area

____________

3. A circle has a circumference of 10 π . What is its radius?

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4. Find the surface area, show all work, and include units.

3m

2m

12m

5. Find the surface area, show all work, and include units

3”

10 “ Use 3.14 for

π

Total surface area ___________

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6. Find the surface area, showing all work, include units.

?

3 ft. f t

7 ft

4 ft

Total surface area ____________________

7. Find the surface area, showing all work, include units.

8 ft

6 ft 6 ft

Total surface area ____________________

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TEST – Integrated 1 Area & Volume 1.

name:________________________

Name that shape! Match the shape with a name from the list below.

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A

trapezoidal prism

B

cylinder

C

cone

D

rectangular prism

E

square pyramid

F

cube (square prism)

G

triangular prism

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2.

Find the circumference/perimeter and the area (Show all work to receive full credit):

π

= 3.14 x ft.

3 cm 12 ft.

A=

A=

C=

P= 9 ft.

3.

Find the surface area and volume. (Show all work to receive full credit):

6 cm As =

15 ft.

V= As =

5 cm

V=

4 cm

12 ft. 12 ft.

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π

4 in.

7m As =

As =

V=

V=

9 in.

7m 8m

6m 5m 3. (cont.) Find the volume (Show all work to receive full credit):

πV == 3.14 13 cm

13 m

8m 9 cm 8m V=

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= 3.14

4.

What are the missing values?

(Show all work to receive full credit) Find Area

X= 45 cm 17 in

X= 9 ft. 6 ft.

16 in.

33 cm Area =

5.

C

D What is the sum of the interior angles in a heptagon and how many triangles can be drawn from one vertex?

E B F A

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Sum =

# of triangles =

( Draw the triangles. ) G

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6. Find the area of the shaded region (Show all work to receive full credit): a.

6 cm

5 cm

b.

2 in

6 in

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7.

Cans of peaches are placed upright in a shipping box in two snugly packed layers. Each layer contains 3 rows of cans with 4 cans per row. For each question below, show your work and give the appropriate units for your numerical answers. a.

When the box is opened by cutting off the top, the tops of the first layer of cans are all that is visible. Make a sketch of how this would look from directly above the box.

b.

If the diameter of the base of each can is 7.5 cm. find the smallest possible dimensions of the rectangular top. Explain how you arrived at your answer.

c.

If the height of each can is 11 cm. find the smallest possible height of the box. Explain how you arrived at your answer.

d.

Using the can dimensions in parts b and c find the volume of the smallest shipping box. Show calculations.

e.

Using the can dimensions above find the volume of one can of peaches. Show calculations and a diagram to clarify your thinking.

f.

Because cans are cylindrical and the box is a rectangular prism, there will be space between packed cans. What is the smallest possible amount of space in your box? Explain your method for determining this.

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g.

How much cardboard is used to make the shipping box? (ignore overlap in flaps). Draw a picture to illustrate where your calculations are coming from.

h.

A paper label is on the can of peaches. Determine the least amount of paper needed to make one such label. (ignore overlap of label)

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Integrated Math 1 Project - Geometry

Name ___________________ Date _______

You have been hired to design a container for a company that makes sports drinks. The container you design will serve as a means to contain the liquid and serve as a drinking container. The client wants the container to hold 20 oz. of liquid, with minimum empty space inside the container. The amount of material used to make the container needs to be minimized for price purposes. Also, you must design the box that will used to ship containers to stores. You are to come up with an original design for the container using any of the shapes discussed in Integrated Math 1 (sketch must be included). Show all work used to find the volume and surface area of the container. The box used for delivery should hold 18 containers. Find the dimensions of the box needed for your design as well as the surface area, to determine the amount of material needed to make the box. Once you have determined these calculations, find the cost per container for material and the cost of 1 box. Provide all formulas used and show the computations used as well as an explanation of all numbers. Conversion factor: Cost of plastic material: Cost of cardboard:

1 fluid oz. = 1 in 3 *1.804 $. 0025 per square inch $.10 per square foot.

Due Date: ________________________

Grade: Original Design including sketch Surface area of container including formulas Volume of container including formulas Dimensions of box including sketch/rational and formulas Surface area of box including formulas Cost of container including work Cost of box including work Neatness

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Geometry, Geometry Everywhere Assignment

Name ______________

Purpose: Geometry provides the language and concepts to describe the appearance, shape, size and patterns of objects in the real world. For this project you will have to take notice of geometry in your own surroundings, thereby helping you to better understand, appreciate and communicate the ideas of geometry. Directions: 1.

Some of the main ideas in this unit are symmetry, measures (such as area, perimeter, and volume), tessellations and rigidity. Find in your community at least two interesting examples of buildings, bridges, plane patterns (such as park designs, designs in nature or even designs in wallpaper), or other objects or structures which can be described using these important geometric ideas.

2.

Choose your examples so that you can write descriptions of them, you will make use of all four of the main geometric ideas given above. This mean that you may find one example whose description involves all four of the ideas. (ex. The super structure of a large building that is under construction may have various symmetries, some patterns in it you could describe as tessellation, and some area or perimeter measures and rigidity properties that may be important features to describe.

3.

Make photographs or sketches of your examples that illustrate the geometric ideas in your descriptions.

4.

This information that you have collected will be used at a later date.

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Integrated Math 1 Pythag Worksheet

Name _________________ Date ________

For each of the following pairs of points: a. Plot the points on graph paper b. Connect the two points with a line segment c. Draw a right triangle using the endpoints of the segment d. Determine the length of each leg of the triangle. e. Use Pythagorean theorem to find the length of the hypotenuse

1. A (3,5)

B (4,7)

Length of AB ____________

2. A (-1, 0)

B (5, 7)

Length of AB _____________

3. A (-3,-2)

B (3,-1)

Length of AB ______________

4. A (-1,-5)

B (-2,-7)

Length of AB ______________

5. A (8, 1)

B (4, 3)

Length of AB ______________

6. A (3, 0)

B (0, 8)

Length of AB ______________

7. A (4, 9)

B (-3, 2)

Length of AB ______________

8. A (-1,-3)

B (1, 3)

Length of AB ______________

9. A (3, 7)

B (7, 3)

Length of AB ______________

10. A (-1, 4)

B (-3,-2)

Length of AB ______________

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Review for QUIZ on Surface Area

name:___________________

Given the following circle: Use 3.14 for

π.

r

What would the following circle dimensions be if . . . 1.

radius = 5

diameter =

circumference =

2.

radius =

diameter = 6.6

3.

radius =

diameter =

circumference =

area = 314

4.

radius =

diameter =

circumference = 14.56

area =

5.

What are the formulas for: Area of square:

circumference =

area =

area =

Area of rectangle: Area of circle: Area of triangle:

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Find the surface area for each of the following. Show ALL work! 6.

7.

Use 3.14 for

π.

8.

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9.

10.

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Integrated Math 1 Angles of a Polygon

Name _____________________ Date _________

Draw a triangle using a straightedge. Measure the three angles of the triangle.

What is the sum of the three angles? Compare your sum with your partner. What do you notice?

Generalize your findings: The sum of the angles of a triangle is ______________ Draw a polygon for each of the ones in the chart. To find the sum of the interior angles (those angles that are at the vertices of the shape) of a polygon could be found by measuring and adding like we did for the triangle, however, there is a more efficient method. Choose a vertex and draw all the diagonals from that vertex. (A diagonal is a segment from one vertex connecting it to a non-adjacent vertex). Count the number of triangular regions. Find the sum of the interior angles by using this information along with what you know about the sum of the angles of a triangle. polygon triangle quadrilateral Pentagon Hexagon Octagon N-gon

Number of sides 3

Number of triangular regions 1

Sum of the interior angles

Measure of one interior angle

How do you find the sum of the interior angles of any polygon? _________ A regular polygon has all sides and angles equal. How would you find the measure of one angle?

Fill in the measure of one angle of a regular polygon above.

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Exterior angle

Exterior angle

Exterior angle

An exterior angle is one that is found by extending a side of the polygon. If each side is extended once, (as above) the sum of the exterior angles can be found. Take your original triangle and extend each side as above. The number of degrees at a point of a line is _________. What are the measures of the exterior angles of your triangle? What is the sum of the exterior angles in your shape? Do the same for the quadrilateral. __________ Do the same for the pentagon. ____________ Do you notice a pattern? What can you say about the sum of the exterior angles of a polygon?

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