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CCGPS

Frameworks Student Edition

7th Grade Unit 5: Geometry

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Mathematics

Georgia Department of Education Common Core Georgia Performance Standards Framework Student Edition

Seventh Grade Mathematics • Unit 5

MATHEMATICS GRADE 7 UNIT 5: Geometry Georgia Department of Education

Dr. John D. Barge, State School Superintendent May 2012 of 31

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Unit 5 GEOMETRY

TABLE OF CONTENTS Overview ....................................................................................................................................................... 3 Standards Addressed in this Unit .....................................................................................................3

• Key Standards & Related Standards ....................................................................................3 • Standards for Mathematical Practice ...................................................................................4

Enduring Understandings.................................................................................................................6 Concepts & Skills to Maintain .........................................................................................................6 Selected Terms and Symbols ...........................................................................................................7 Tasks

• Take The Ancient Greek Challenge .....................................................................................9 • Cross-Sections of a Cube ...................................................................................................11 • Similar Cross-Sections .......................................................................................................12 • What’s My Solid? ..............................................................................................................13 • It’s Easy as Pi .....................................................................................................................14 • Saving Sir Cumference ......................................................................................................15 • Circle Cover Up .................................................................................................................17 • I Have A Secret Angle .......................................................................................................23 • Stained Glass Design .........................................................................................................24 • Boxing Bracelets ................................................................................................................27 • Staircase .............................................................................................................................30





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OVERVIEW In this unit students will:

• draw geometric figures using rulers and protractor with emphasis on triangles, • write and solve equations involving angle relationships, • explore two-dimensional cross-sections of cylinders, cones, pyramids, and prisms, • know and use the formula for the circumference and area of a circle, and • solve engaging problems that require determining the area, volume, and surface area of

fundamental solid figures.

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources. STANDARDS ADDRESSED IN THIS UNIT

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. KEY STANDARDS Draw, construct and describe geometrical figures and describe the relationships between them.

MCC7.G.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. MCC7.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.





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Solve real‐life and mathematical problems involving angle measure, area, surface area, and volume. MCC7.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. MCC7.G.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. MCC7.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. RELATED STANDARDS

MCC7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. STANDARDS FOR MATHEMATICAL PRACTICE The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). 1. Make sense of problems and persevere in solving them. In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?” 2. Reason abstractly and quantitatively. In grade 7, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students





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contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. 3. Construct viable arguments and critique the reasoning of others. In grade 7, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always work?”. They explain their thinking to others and respond to others’ thinking. 4. Model with mathematics. In grade 7, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students explore covariance and represent two quantities simultaneously. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences, make comparisons and formulate predictions. Students use experiments or simulations to generate data sets and create probability models. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context. 5. Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 7 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. Students might use physical objects or applets to generate probability data and use graphing calculators or spreadsheets to manage and represent data in different forms. 6. Attend to precision. In grade 7, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students define variables, specify units of measure, and label axes accurately. Students use appropriate terminology when referring to rates, ratios, probability models, geometric figures, data displays, and components of expressions, equations or inequalities. 7. Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables making connections between the constant of proportionality in a table with the slope of a graph. Students apply properties to generate equivalent expressions,( i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12) by subtraction property of equality and determine that c = 6 by the division property of equality. Students compose and decompose two‐ and three‐dimensional figures to solve real world problems involving scale drawings, surface area, and volume. Students examine





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tree diagrams or systematic lists to determine the sample space for compound events and verify that they have listed all possibilities. 8. Look for and express regularity in repeated reasoning. In grade 7, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. They extend their thinking to include complex fractions and rational numbers. Students formally begin to make connections between covariance, rates, and representations showing the relationships between quantities. They create, explain, evaluate, and modify probability models to describe simple and compound events. ENDURING UNDERSTANDINGS • Coordinate geometry can be a useful tool for understanding geometric shapes and

transformations. • Cross-sections of three-dimensional objects can be formed in a variety of ways, depending

on the angle of the cut with the base of the object. • The area of irregular and regular polygons can be found by decomposing the polygon into

triangles, squares, and rectangles. • “Pi” (π) is the relationship between a circle’s circumference and diameter. • Parallelograms and rectangles can be used to derive the formula for the area of a circle. • Approximate volumes of simple geometric solids may be found using estimation • Formulas may be used to determine the volume of fundamental solid figures. • Approximate surface area of simple geometric solids may be found using estimation • Manipulatives and the construction of nets may be used in computing the surface area of

right rectangular prisms. • Formulas may be used to compute the surface area of right rectangular prisms.

CONCEPTS AND SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• number sense • computation with whole numbers and decimals, including application of order of

operations • addition and subtraction of common fractions with like denominators • measuring length and finding perimeter and area of rectangles and squares





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• characteristics of 2-D and 3-D shapes • angle measurement • data usage and representations

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary. Note – At the elementary level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. http://www.teachers.ash.org.au/jeather/maths/dictionary.html This web site has activities to help students more fully understand and retain new vocabulary (i.e. the definition page for dice actually generates rolls of the dice and gives students an opportunity to add them). http://intermath.coe.uga.edu/dictnary/homepg.asp Definitions and activities for these and other terms can be found on the Intermath website.

• Adjacent Angle: Angles in the same plane that have a common vertex and a common side, but no common interior points.

• Circumference: The distance around a circle. • Complementary Angle: Two angles whose sum is 90 degrees.

• Congruent: Having the same size, shape and measure. ∠A ≅∠ B denotes that ∠A is

congruent to ∠B.

• Cross- section: A plane figure obtained by slicing a solid with a plane.

• Irregular Polygon: A polygon with sides not equal and/or angles not equal.

http://www.teachers.ash.org.au/jeather/maths/dictionary.html

http://intermath.coe.uga.edu/dictnary/homepg.asp





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• Parallel Lines: Two lines are parallel if they lie in the same plane and they do not

intersect. AB || CD denotes that AB is parallel toCD .

• Pi: The relationship of the circle’s circumference to its diameter, when used in calculations, pi is typically approximated as 3.14; the relationship between the circumference (C) and diameter (d), 𝐶

𝑑≈ 3 1

7 or 3.14

• Regular Polygon: A polygon with all sides equal (equilateral) and all angles equal

(equiangular).

• Supplementary Angle: Two angles whose sum is 180 degrees.

• Vertical Angles: Two nonadjacent angles formed by intersecting lines or segments. Also called opposite angles.





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Take the Ancient Greek Challenge The study of Geometry was born in Ancient Greece, where mathematics was thought to be embedded in everything from music to art to the governing of the universe. Plato, an ancient philosopher and teacher, had the statement, “Let no man ignorant of geometry enter here,” placed at the entrance of his school. This illustrates the importance of the study of shapes and logic during that era. Everyone who learned geometry was challenged to construct geometric objects using two simple tools, known as Euclidean tools:

• A straight edge without any markings • A compass

The straight edge could be used to construct lines; the compass to construct circles. As geometry grew in popularity, math students and mathematicians would challenge each other to create constructions using only these two tools. Some constructions were fairly easy (Can you construct a square?), some more challenging, (Can you construct a regular pentagon?), and some impossible even for the greatest geometers (Can you trisect an angle? In other words, can you divide an angle into three equal angles?). Archimedes (287-212 B.C.E.) came close to solving the trisection problem, but his solution used a marked straight edge. We will use a protractor and marked straight edge (you know it as a ruler) to draw some geometric figures. What “constructions” can you create? Your 1st Challenge: Draw a regular quadrilateral. Your 2nd Challenge: Draw a quadrilateral with no congruent sides. Your 3rd Challenge: Draw a circle. Then draw an equilateral triangle and a square inside so that both figures have their vertices on the circle (inscribed). Your 4th Challenge: Draw a regular hexagon. Then divide it into three congruent quadrilaterals





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Your 5th Challenge: Draw a regular octagon. The divide it into two congruent trapezoids and two congruent rectangles Your 6th Challenge: Draw a triangle with side lengths of 5, 6, and 8 units. Your 7th Challenge: Draw a triangle with an obtuse angle. Your 8th Challenge: Draw an equilateral right triangle. Your 9th Challenge: Create some challenges of your own and pose them to a classmate.





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Cross-Sections of a Cube Try to make each of the following cross sections by slicing a cube:

a. Square

b. Equilateral triangle

c. Rectangle, not a square

d. Triangle, not equilateral

e. Pentagon

f. Regular hexagon

g. Hexagon, not regular

h. Octagon

i. Trapezoid, not a parallelogram

j. Parallelogram, not a rectangle

k. Circle

1. Record which of the shapes you were able to create and how you did it. If you can’t make the shape, explain why not.

2. Describe, name, and sketch any additional cross-sections that are possible and explain why they are possible.





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Similar Cross-Sections The Great Pyramid of Giza in Egypt is often called one of the Seven Ancient Wonders of the world. The monument was built by the Egyptian pharaoh Khufu of the Fourth Dynasty around the year 2560 BC to serve as a tomb when he died. When it was built, the Great pyramid was 145.75 m high. The base is square with each side 231 m in length. Construct a model of the pyramid, with a base that is 1 foot on each side. Be sure to make the height proportional to the base just as in the real pyramid.

1. Suppose the pyramid is sliced by a plane parallel to the base and halfway down from the top. What will be the shape of the top? What will the dimensions of the slice be? Justify your answer.

2. What if the slice is 15% of the way down from the top?

3. What if the slicing plane is not parallel to the base? What will the shape of the slice be under those conditions?





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What’s My Solid? Each of the following descriptions fit one or more solids (prism, pyramid, cone, cube, a cylinder). For each clue, describe what solid it may be and your justification for selecting that solid. If the description fits more than one solid, name and provide justification for each solid. Sketch the solid, and illustrate the properties described. a) A set of my parallel cross sections are squares that are similar but not congruent.

b) A set of my parallel cross sections are congruent rectangles.

c) A set of my parallel cross sections are circles that are similar but not congruent.

d) A set of my parallel cross sections are congruent circles.

e) A set of my parallel cross sections are parallelograms.

f) One of my cross sections is a hexagon, and one cross section is an equilateral triangle.

g) I can be made by sliding a rectangle through space.

h) I can be made by twirling a triangle through space.

i) My volume can be calculated using the area of a circle.

j) My volume can be calculated using the area of a rectangle.





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Name________________________________________ Date ____________________________

It’s as Easy as Pi

1. Carefully wrap string around the circumference of your circular object.

2. Cut the string when it is exactly the same length as the circumference. 3. Now take your “string circumference” and pull it across the diameter of your circular

object. Cut as many “string diameters” from your “string circumference” as you can. 4. Repeat steps 1-3 with at least 3 different circular objects. 5. Tape your string sets to the bottom of this paper and explain what you noticed.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

6. Compare your data with others. What do you notice?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

7. How are diameter and circumference related? How do you know?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

π π

π





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Name _________________________________________ Date _________________________

Saving Sir Cumference

Sir Cumference has been turned into a dragon! Help Radius and Lady Di of Ameter break the spell and save Sir Cumference. The answer to this problem lies in this poem. Can you solve the riddle? 1. Read “The Circle’s Measure”

• What do you think is meant by “measure the middle and circle around”?

• What two numbers should you divide? How

do you know?

2. Use a measuring tape to find the diameter and circumference of at least 5 different sized

circular objects. (You may want to measure the circumference with narrow ribbon and then measure the ribbon to find the measure of the circumference.) Discuss with your group how to record your data in a table, and then create the table below.





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3. Use the grid paper below to make a coordinate graph. Use the horizontal axis for diameter and the vertical axis for circumference. Plot your data for each object your group measured on the graph.

Cir

cum

fere

nce

Diameter A. Enter your diameter and circumference data on the class graph.

B. What do you notice about the points you plotted on your graph? How is your graph similar

to/different from the class graph that is being created? ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________





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Name _________________________________________ Date _________________________

Circle Cover-Up

1. Compare the areas of the square and circle below.

Which one has the larger area? Write to explain how you know.

2. Compare the areas of the two figures below.

3. Do you think one of the areas is larger than the other? Write to explain your thinking.





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4. If the radius of the circle is r, what is the length of the large square in terms of r?

5. What is the approximate area of the three smaller squares �3

4 of the large square� in

terms of r?

6. Follow the directions on the “Circle Cover-Up Cut and Cover” student recording sheet. Which figure has the larger area? Write below to explain your findings.

7. How do you think finding the area of a circle is related to finding the area of a square?

8. What role do you think pi plays in finding the area of a circle?





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Name _________________________________________ Date _________________________

Circle Cover-Up - Cut and Cover 1. Color the area of the squares that are outside the circle. 2. Cut out the circle, but save the colored area that was not inside the circle. 3. Paste the area outside the circle into the blank quadrant in a mosaic design. Try not to overlap

pieces. You may need to cut your colored pieces into smaller pieces so they will fit.





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Name _________________________________________ Date _________________________

Circle Cover-Up - Circles and Parallelograms 1. Cut out one circle from the “Circle Cover-Up, Circles to Cut” student sheet. 2. Cut the sectors of the circle apart and arrange them on the

grid paper as shown to form a parallelogram.

3. Use the grid paper to help you approximate the area of the “parallelogram” formed. What is

the approximate area of the “parallelogram”? __________ square units

4. Write the formula for the area of a parallelogram. _________________________________

5. Rewrite the formula for the parallelogram replacing the base (b) with 12 the length of the

circumference of the circle �𝐷2�. _______________________________________________

6. The length of 𝐷2

is equivalent to the length of what part of a circle? ___________________

7. Rewrite the formula for the parallelogram from step 5 above. Replace 𝐷2

with r.





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8. Rewrite the formula above, replacing the height (h) with the radius of the circle (r).

9. If the radius of the circle is 5 units, find the measure of its diameter.

10. Compare the area of the parallelogram you approximated above with the area of the circle you found using the formula. What do you notice?





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Performance Task: I Have a Secret Angle An 8th Grade Math Teacher saw the following job listing in a magazine: The teacher decided to ask her 8th grade students for help, since they are very creative. They will need to develop problems for the 7th grade standard:

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

She gave them craft sticks, rulers, protractors, and other related supplies and instructed them to design exercises similar to exercises that they solved last year. They designed worksheets about fashion design, cell phones, entertainment, etc. that contained problems with illustrations similar to the following examples:

• Write and solve an equation to find the measure of angle x.

• Write and solve an equation to find the measure of angle x.

We will use the information we have discovered about angle relationships to write and solve equations for the above examples





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Name _________________________________________ Date _________________________

Stained Glass Designs

Windows to the World is a locally owned company specializing in stained glass windows. You have just been hired to create stained glass designs for circular windows. As part of your agreement, you need to submit a design. You may include both regular and irregular shapes in your original stained glass window design. Windows to the World has certain requirements for their circular windows:

• All designs must fit inside windows with an area ranging from 200 in2 to 400 in2. • All designs must include at least three different sets of congruent figures. • Each design must include a supplies/materials list and cost of materials.

Windows to the World increases the selling price for their windows for resale. The selling cost for each window is the cost of the materials multiplied by 1.5.

• Create a formula to calculate the selling price, and then use the formula to find the selling price for your design.

Windows to the World pays for the supplies you used to create the window design. For each design, your commission is 3

4 the cost of the materials.

• Create a formula for finding the commission for each design. You need to submit an invoice with the formula you used to calculate your commission as well the total cost including the cost of the materials and the commission.

IMPORTANT: Sheets of glass are only sold in square foot sections (12 inches x 12 inches). Prices range from $6.12 to $11.46, depending on the color. A price list is included for your use.





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Stained Glass Supplies Sheets of Glass are sold by the square foot.

Color Cost per square foot

Clear/White $6.12 Blue $6.93 Pink $10.26 Black $7.14 Gray $6.12 Green $6.93 Orange $11.01 Purple $6.12 Red $10.74

Yellow $11.46 White $6.12





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INVOICE Designer’s Name Date

Item Description Quantity Price (each) Total Price

Total Price of Materials: ___________________

Formula for the selling price of each design: _____ Selling Price: ______ Formula for the total cost of each design: ______ _____ Total cost for the design (Cost of materials + Commission) ______

• Attach your complete design to the invoice. • On the back of the invoice, attach a sample of each figure used in your design and list the

area of each figure. Include a description of how you found the area of each figure using words and numbers.

Name _________________________________________ Date _________________________

Stained Glass Designs

Invoice





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Performance Task: Boxing Bracelets

You are the owner of a prestigious jewelry store that sells popular bracelets. They are packaged in boxes that measure 8.3 centimeters by 11 centimeters by 2 ½ centimeters.

Part I. a. Sketch a drawing of the box and label its dimensions. b. Estimate the volume of the bracelet box. c. Find the volume of the bracelet box. Be sure to show all of your work.





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Part II. Suppose the company that makes your boxes is out of the ones that you usually purchase. They have offered to send you another size box for the same cost. The three different boxes that you may choose from have two of the dimensions the same as your regular box, but increase one of the dimensions by exactly 1 centimeter.

d. Make a prediction of which box you would order if you wanted the largest possible increase

in volume. Explain with details how you could be certain of which dimension you should increase. Test your prediction. Was your prediction correct? Why or why not?

e. Make a sketch of the new box and label its dimensions. Find the volume of the new box. Be sure to show all of your work.

f. What is the difference of the volume of the original box and the volume of the new box? g. What is the ratio of the difference from part (f) and the original volume part (e)? h. What is the percent that is equivalent to the ratio found in part (g)? (This is known as the

percent of increase.)





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Part III. Some of your customers frequently like to have their bracelets gift wrapped. To determine the price for wrapping the new boxes, you wish to compare the surface area of the original box to the surface area of the box with the largest volume. (See part d.)

i. Estimate the surface area of the two boxes. j. Find the actual surface area for each box.

Were your estimates close to your actual results? Why or why not?

k. What is the difference between the two surface areas? l. What is the ratio of the difference from part (j) and the surface area of the original

box?

m. What is the percent that is equivalent to the ratio found in part (l) (percent of increase)? n. If you charge $2.00 to gift wrap the original box, how much should you charge to gift wrap

the new box? Justify your answer.





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Performance Task: Staircases Used with permission from: Kersaint, G. & Chappell, M.F. (2004). What do you see? A case for examining students’ work. Mathematics

Teacher, (97), 102. Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org All rights reserved.

A same-color staircase is made from Cuisenaire rods. Each time that a rod is added to the staircase, it is offset by the space of a white (unit) rod. The rods that are used to make the staircase are 3 units in length.

A. What is the surface area and volume of a staircase that is 3 units tall?

B. Predict the volume and surface area of a staircase that is 5 units high. Find the actual

surface area and volume, and compare them with your answer. Explain any discrepancies that you found.

C. What will the volume and surface area be when you add the hundredth rod? Explain how

you found your answer.

http://www.nctm.org/





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D. Develop a general method or rule that can be used to determine the volume and surface area for any number of rods. Explain your thinking.

Extension

E. How would changing the color of the rods affect your results? Verify your answer by completing A-D for a different color. Were the results what you predicted? If not, explain where your thinking was off.

F. How would changing the way the stairs were made affect your results? For example, allowing different colors of rods to be used to make the stairs. Describe how you would change the way the stairs were made and complete A-D with this new pattern. Were the results what you predicted? If not, explain where your thinking was off.

GLADIS KERSAINT, [email protected], teaches at the University of South Florida, Tampa, FL 33620. MICHAELE CHAPPELL,[email protected], teaches at Middle Tennessee State University, Murfreesboro, TN 37132. Both authors are interested in the professional development of teachers and the mathematics achievement of minority students.A direct link to this article is http://my.nctm.org/eresources/view_media.asp?article_id=6504.

http://my.nctm.org/eresources/view_media.asp?article_id=6504

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