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GRADE 10 MATH: A DAY AT THE BEACH
UNIT OVERVIEW This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The final task assesses student mastery of the geometry standards related to finding the volume of geometric figures.
TASK DETAILS
Task Name: A Day at the Beach
Grade: 10
Subject: Geometry
Depth of Knowledge: 4
Task Description: Students find the volume of geometric figures (cylinder, cone, pyramid, and sphere).
Standards:
G.MGD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Standards for Mathematical Practice:
MP.1: Make sense of problems and persevere in solving them.
MP.4: Model with mathematics.
MP.6: Attend to precision.
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TABLE OF CONTENTS The task and instructional supports in the following pages are designed to help educators understand and implement Common Core–aligned tasks that are embedded in a unit of instruction. We have learned through our pilot work that focusing instruction on units anchored in rigorous Common Core–aligned assessments drives significant shifts in curriculum and pedagogy. Callout boxes and Universal Design for Learning (UDL) support are included to provide ideas around how to include multiple entry points for diverse learners.
PERFORMANCE TASK: A DAY AT THE BEACH ………………………………………………………………….3
RUBRIC……………………………………………………………………………………………………………………………6
ANNOTATED STUDENT WORK…………………………………………………………………………………………9
INSTRUCTIONAL SUPPORTS ……………………………………………………………………………………………22
UNIT OUTLINE………………………………………………………………..……………………………………23
LEARNING PLAN…………………………………………………………………………………………………..27
VOLUME UNIT ACTIVITY………………………………………………………………………………………33
PRE-ASSESSMENT TASK……………………………………………………………………………………….46
FORMATIVE ASSESSMENT TASK…………………………………………………………………………..48
Acknowledgements: This bundle was developed by the teacher team of Hui Lu, Nadine Pearson, and Olga Gandlin at New Dorp HS. Additionally, the unit and task were reviewed for Common Core standards alignment by the NYC DOE Common Core Fellows Michelle Venditti, Brooke Nixon- Friedham and Jenny Kim.
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GRADE 10 MATH: A DAY AT THE BEACH
PERFORMANCE TASK
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Performance Task
A Day at the Beach
You are at the beach with your friends. You have brought some supplies to make sand castles. These supplies include a pail that has a base with a circumference of 6π inches, is 10 inches tall, and has an opening on top that is twice the diameter of the base. You also have a plastic pyramid mold that has a square base with an edge that measures 4 inches and is 5 inches tall, and an empty soup can with a diameter of 3.25 inches and is 4.5 inches tall. For each question, include correct units of measurement and round your answers to nearest thousandth or in terms of . Refer to the general rubric for a successfully completed task to ensure that you receive full credit.
1) Draw and label diagrams that represent the soup can and pyramid. Calculate how much sand you can fit into each object. 2) Draw and label a diagram that could represent the pail. Calculate how much sand will fit in the pail.
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3) Which object would fill the pail faster, the pyramid or the soup can? Justify your answer and include your algebraic thinking. 4) Your friends are bored making sand castles and decide to play with a beach ball they found in the car. The package says the beach ball has a diameter of 20 inches. While playing with the beach ball, the plug comes loose and the beach ball deflates by one quarter. How much air is left in the beach ball?
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GRADE 10 MATH: DAY AT THE BEACH
RUBRIC
The rubric section contains a scoring guide and performance level descriptions for Day at the Beach. Scoring Guide: The scoring guide is designed specifically to the culminating performance task. The points highlight each specific piece of student thinking and explanation required of the task and help teachers see common misconceptions. The scoring guide can then be used to refer back to the performance level descriptions.
Performance Level Descriptions: Performance level descriptions help teachers think about the overall qualities of work for the task by providing information about the expected level of performance for students. Performance level descriptions provide score ranges for each level, which are assessed using the scoring guide.
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High School Geometry: A Day at the Beach A Day at the Beach Rubric
The elements of performance required by this task are: • Visualizes and identifies the dimensions of geometric shapes • Determines the volume relationships of cylinders, pyramids, cones. and spheres • Justifies geometric arguments.
Possible Responses Points Section Points
1. a) Draws and labels each diagram accurately. b) Radius of the soup can = 1.625 inches c) Volume of the soup can equals 37.33 in3 or 11.8828π in3 d) Volume of the pyramid equals 26. in3 Partial credit
• Computational error(s) in parts (b), (c), and (d)
2 1 1
1 (1)
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2. a) Draws and labels a diagram for the pail. b) The lateral sides of the pail are extended to form a cone. c) Calculates the dimensions of the cone. d) Calculates the volume of both cones accurately. Volume large cone =240π in3 Volume of small cone = 30π in3 e) Volume large cone – Volume of small cone = 210 π in3 Partial credit
• Conceptual error, such as finding the volume of the pail in part( a)
• Computational error(s) in parts (d) and( e)
1 1 1 2
1
(1)
(1)
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3. a) Compares the volume of the pail with the can and the pyramid.
b) Justifies his or her answer, such as, "The soup can will fill the pail faster because it will require fewer scoops." Partial credit
• Computational error(s) in part 3(a) • Computational error(s) carried over from questions 1 and
2; however, the corresponding answer is accurate. *c) Student may justify his or her answer by comparing the volume of the soup can and the pyramid without any calculations.
2
1
(1) (2)
(3)
3
4. a) Calculates the dimensions of the sphere. b) Calculates the volume of the sphere accurately.
= = 4188.79 in3
c) Calculates the new volume of the sphere accurately.
= = 4188.79 in3 = 1047.20 in3
4188.79 − 1047.20 = 3141.59 in3 Partial credit
• Computational error(s)
1 1
(1)
4
Total Points 18
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High School Geometry: A Day at the Beach
Rubric
Performance Level Descriptions and Cut Scores Performance is reported at four levels: 1 through 4, with 4 as the highest.
Level 1: Demonstrates Minimal Success (0–4 points) The student’s response shows few of the elements of performance that the task demands as defined by the CCLS. The student’s work shows a minimal attempt and lack of coherence. The student fails to use appropriate tools, such as volume and circumference formula, strategically. The student is unable to make sense of the problem and apply geometric concepts in this modeling situation.
Level 2: Performance Below Standard (5–10 points) The student’s response shows some of the elements of performance that the task demands as defined by the CCLS. The student might ignore or fail to address some of the constraints of the problem. The student may occasionally make sense of quantities or relationships in the problem. The student attempts to use some appropriate tools, such as volume or circumference formula, with limited success. The student may recognize geometric shapes, but has trouble generalizing or applying geometric methods in this modeling situation.
Level 3: Performance at Standard (11–15 points) For most of the task, the student’s response shows the main elements of performance that the tasks demand as defined by the CCLS with few minor errors or omissions. The student explains the problem and identifies constraints. The student makes sense of quantities and their relationships in the modeling situation. The student uses appropriate tools, such as radius, circumference and volume. The student might discern patterns or structures and make connections between representations. The student is able to make sense of the problem and apply geometric concepts to this modeling situation.
Level 4: Achieves Standards at a High Level (16–18 points) The student’s response meets the demands of nearly all of the tasks as defined by the CCLS and is organized in a coherent way. The communication is clear and precise. The body of work looks at the overall situation of the problem and process, while attending to the details. The student routinely interprets the mathematical results, applies geometric concepts in the context of the situation, reflects on whether the results make sense and uses all appropriate tools strategically.
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GRADE 10 MATH: DAY AT THE BEACH
ANNOTATED STUDENT WORK This section contains annotated student work at a range of score points. The student work shows
examples of student understandings and misunderstandings of the task.
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Grade 10: A Day at the beach Annotated Student Work: Level 1
G.MGD.3
G.MGD.3
10
G.MGD.3
11
Grade 10: A Day at the beach Annotated Student Work: Level 2
G.MGD.3
G.MGD.3, G.MG.1
G.MGD.3
12
G.MGD.3
G.MGD.3
13
Grade 10: A Day at the beach
Annotated Student Work: Level 2
G.MGD.3
G.MG.1, G.MGD.3
14
G.MGD.3
G.MGD.3
15
Grade 10: A Day at the beach Annotated Student Work: Level 3
G.MGD.3
G.MGD.3
16
G.MGD.3
G.MGD.3
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Grade 10: A Day at the beach Annotated Student Work: Level 3
G.MGD.3
G.MGD.3, G.MG.1
18
G.MGD.3
G.MGD.3
19
Grade 10: A Day at the beach Annotated Student Work: Level 4
G.MGD.3
G.MGD.3, G.MG.1
G.MG.1
20
G.MGD.3
G.MGD.3
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GRADE 10 MATH: DAY AT THE BEACH
INSTRUCTIONAL SUPPORTS The instructional supports on the following pages include a unit outline with formative assessments and suggested learning activities. Teachers may use this unit outline as it is described, integrate parts of it into a currently existing curriculum unit, or use it as a model or checklist for a currently existing unit on a different topic.
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Unit Outline
Grade 10 Math: A Day at the Beach
UUNNIITT TTOOPPIICC AANNDD LLEENNGGTTHH:: The unit uses an investigation of real-world three-dimensional figures to teach students how to
calculate the volume of any object. Students will demonstrate mastery of the content by making sense of the performance task “A Day at the Beach” and persevering in solving the task. Suggested unit length is 7 school days.
CCOOMMMMOONN CCOORREE CCOONNTTEENNTT SSTTAANNDDAARRDDSS:
G.MGD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder). G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or
structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
SSTTAANNDDAARRDDSS FFOORR MMAATTHHEEMMAATTIICCAALL PPRRAACCTTIICCEE:: MP. 1: Make sense of problems and persevere in solving them. MP. 4: Model with mathematics. MP. 6: Attend to precision.
BBIIGG IIDDEEAASS//EENNDDUURRIINNGG UUNNDDEERRSSTTAANNDDIINNGGSS:: Recognize that real-world objects are
represented by a combination of different geometric figures.
The volume of real-world objects can be calculated by using multiple formulas.
EESSSSEENNTTIIAALL QQUUEESSTTIIOONNSS:: How can we identify geometric shapes from
real-world objects?
How do we apply the volume formulas to real-life situations?
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CCOONNTTEENNTT::
VVOOLLUUMMEE of cylinder
of cone
of pyramid
of sphere
SSKKIILLLLSS:: Identify the corresponding geometric
figures of real-world objects. Transform the real-world object into a
geometric diagram. Identify the appropriate formula(s) of
volume for each real-world object. Apply ratio, proportion, and Pythagorean
Theorem to find the missing dimensions of a three-dimensional figure.
Calculate the volume of geometric figures accurately.
Create and justify valid arguments.
PPRREE--RREEQQUUIISSIITTEE SSKKIILLLLSS:: Find the area of different polygons: rectangle, triangle, circle.
Find the volume of rectangular solids.
VVOOCCAABBUULLAARRYY//KKEEYY TTEERRMMSS:: Radius, diameter, circumference, cylinder, cone, pyramid, sphere, volume, square, Pythagorean theorem, ratio, proportion
AASSSSEESSSSMMEENNTT EEVVIIDDEENNCCEE AANNDD AACCTTIIVVIITTIIEESS::
IINNIITTIIAALL AASSSSEESSSSMMEENNTT:: The initial assessment will allow students to express their creativity and bring a unique perspective to the task by recognizing geometric figures in real-world objects. As the students learn new ideas or procedures, the students and the teacher can reflect upon how these new ideas might apply to the initial task. The assessment also gives opportunity for teachers to assess students’ prior skills in calculating area and volume as well as their conceptual understanding between area and volume.
FFOORRMMAATTIIVVEE AASSSSEESSSSMMEENNTT:: The first formative assessment will be used after the lessons on Volume of Cylinders and Cones
(see the Formative Assessment Task 1)
The second formative assessment task will be used after the lessons on Volume of Spheres (see the Formative Assessment Task 2)
The students will be able to apply volume formulas to complex geometric figures, use geometric concepts to solve non-routine problems, and justify their reasoning. The purpose of these formative assessments is to surface misconceptions, resolve them, and deepen students’ understanding. Based on the results of these formative assessments, the instruction will be adapted to meet the students’ needs. Consequently, students’ experiences in the classroom will help to improve learning, rather than waiting until the final assessment to uncover gaps in learning.
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FFIINNAALL PPEERRFFOORRMMAANNCCEE TTAASSKK:: AA DDAAYY AATT TTHHEE BBEEAACCHH
At the end of the unit, students will be given A Day at the Beach to determine how they have improved their thinking and mathematical skills over the instructional unit. This task assesses the students’ ability to identify three-dimensional geometric shapes in real-world objects and calculate the volume of these figures accurately by using volume formulas, ratio, and proportions. Students must justify their answers by using valid mathematical arguments.
LLEEAARRNNIINNGG PPLLAANN && AACCTTIIVVIITTIIEESS::
Students will discover the formulas for volume of three-dimensional solids through the exploration of real-world objects and by observing online animations (See Activities 1, 2, 3, and 6). Further deepening of understanding will occur during the lesson “Volume of a Plastic Cup,” where the students will experience finding the volume of an irregularly shaped object (See Activity 4 and its answer key). Students will also discover the formula for the volume of a pyramid by using cut-out nettings and fitting them together to create a cube (See Activity 5). All of these activities will help students recognize that real-world objects are represented by a combination of different geometric figures. Use the “General Rubric for a Successfully Completed Task” during each of the lessons to establish clear expectations and consistency of students’ work. For more details, please refer to the “Geometry Learning Plan (Volume).“
RREESSOOUURRCCEESS:: http://www.learner.org/courses/learningmath/measurement/session8/part_b/cylinders.html http://ww.CutOutFoldUp.com © Laszlo Bardos Materials
Calculators (TI83/TI84) Magazines or pictures as reference Different size cylindrical cans Rulers Cylindrical chalk Red plastic party cups Cylindrical pencil Pencil sharpeners Scissors Tennis ball Activities 1–6 The Sharpened Pencil Formative Task Sleeve of Golf Balls Formative Task A Day at the Beach Performance Task
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Learning Plan: A Day at the Beach (Volume) Teacher’s Guide Prior to starting the unit, have the students complete the pre-assessment. You can give it as a homework assignment if you wish.
Activity Topic Materials/Resources Questions/Challenges
Activity 1
Volume of Cylinder Objectives: -Recognize a real- life object as a cylinder. -Discover the formula for volume of a cylinder. -Recognize the geometric parts of a cylinder. -Calculate the volume of a cylinder accurately.
- Worksheet: Volume Unit Activity 1 - Ruler - Cans, and other cylindrical objects
• Students may identify the can as a cylinder and not the other geometric parts of the cylinder (such as two circles, radius, height, diameter).
Encourage students to use mathematical vocabularies to describe the object.
• Students may identify a part as “the width of the base.”
Ask them questions such as What shape is the base? What does “the width of the base” mean? Why? • Students may incorrectly identify the
volume formula of a cylinder as lwh. Ask them questions such as How can you draw a geometric diagram that represents lw? Does this match our object? Why? Which formula can you use to replace lw?
UDL Supports:
• You may consider providing a list of geometric terms for students to work with. • Use data from the pre-assessment to determine if students need reinforcement of
previously learned concepts (e.g., volume of rectangular solids). • You may consider providing alternate representation for the volume of a cylinder (e.g.,
quarters stacked up to make a cylinder).
Activity 2
Topic Materials/Resources Questions/Challenges Integrating three -dimensional objects in real-world applications Objectives: -Create a plan to
- Worksheet: Volume Unit Activity 2a,b: “Chalk in a Box” - Chalk/sidewalk chalk (make sure you change the dimensions of the box on the activity worksheet 2a to reflect any constraints)
• Students may find the volume of the rectangular prism and the chalk, then divide the two numbers.
Bring them together as a group and show them a picture of a box filled with chalk. Ask them questions such as,
What stands out in this picture?
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fit real-world objects inside one another.
- Ruler
(The students should mention the space between the pieces of chalk and that the chalk is lined up.) Ask them to describe in more detail what it means for the chalk to be lined up. What part(s) are lined up? (answer: diameter) Then tell them to re-calculate their answer.
• Distribute the “General Rubric for a Successfully Completed Task.” The students should check their work based upon this rubric. Did they meet all standards? (Use this rubric during each of the lessons that follow to establish consistency.)
UDL Supports: • This activity provides visual and tactile aid by providing actual chalk. You may consider
providing the actual box for some students. • As an extension to this activity, students may create their own box and/or items in the
box to stimulate their interest and increase their engagement level. • Using the “General Rubric for Successfully Complete Task” will provide options for self-
regulation.
Activity 3
Topic Materials/Resources Questions/Challenges Volume of a Cone Objectives: -Identify the relationship between the volume of a cone and a cylinder. -Discover the formula for volume of a cone.
- Worksheet: Volume Unit Activity 3: The Cone -Visual demonstration: http://www.learner.org/courses/learningmath/measurement/session8/part_b/cylinders.html
• Some students may not recognize the relationship of one third.
- Ask the question: Which parts can we compare to see the relationship between the cylinder and cone?
UDL Supports: • The website above provides alternatives for visualizing volume of a cone. You may also
consider using the alternate experiment on the website allowing students to engage in hands-on activity. This activity allows students to develop understanding of volume of cylinder to cone by using rice/water.
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Activity 4
Topic Materials/Resources Questions/Challenges Volume of Irregular Objects Objectives: -Recognize all geometric shapes that create an irregularly shaped three-dimensional object. -Calculate the volume of an irregular object successfully.
- Worksheet: Volume Unit Activity 4: “Volume of a Plastic Cup” -Answer Key
• Review the “General Rubric for a Successfully Completed Task” at the end of class.
UDL Supports: • The website above (Volume Unit Activity 3) provides a video clip of how engineers
calculate volume of irregular shapes in a real-world setting. • Consider assessing students’ understanding in similarity and proportion. • Prior to this activity, consider providing a context for this question. For example, "The
bottom two inches of an ice-cream cone is filled with chocolate. How much ice cream is needed to fill the remaining cone?"
Formative Assessment
1
Topic Materials/Resources Questions/Challenges Formative Assessment 1: “The Sharpened Pencil”
-Brand new cylindrical pencil -Ruler -Access to a pencil sharpener
Students will work independently on this task.
UDL Supports: • Consider adding scaffold questions to assist students in managing information. For
example, • Question 1 – Draw a diagram to represent the pencil.
Question 2 – Measure the dimensions of the pencil and label the diagram. Question 3 – Sharpen the pencil. Draw a sharpened pencil and find the new dimension and label it on the diagram. Question 4 – Use the formulas that you learned in a class to determine the amount material that is lost after sharpening the pencil.
Activity 5
Topic Materials/Resources Questions/Challenges Volume of a Pyramid Objectives: -Discover the formula for
- Worksheet: Volume Unit Activity 5: The Pyramid -Scissors -Tape
• Students may not be able to fit the
pyramids together to make a cube. - Encourage students to keep
trying different configurations until they make a cube.
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volume of a pyramid. -Calculate the volume of a pyramid successfully.
UDL Supports: Similar to the UDL Supports listed for Activity 3, you may also consider using an alternate experiment, which allows students to develop understanding of volume of square prism to pyramid by using rice/water to fill up plastic solids. This will offer students multiple representations of the same concept.
Activity 6
Topic Materials/Resources Questions/Challenges Volume of a Sphere Objectives: -Recognize the relationship between a cylinder, cone, and sphere. -Discover the formula for the volume of a sphere.
- Worksheet: Volume Unit Activity 6: The Sphere -Animation of Activity 6 http://www.learner.org/courses/learningmath/measurement/session8/part_b/cylinders.html -Tennis ball
• Most students will not be able to create the formula for volume of a sphere. Use the following questions to guide them through the derivation of the formula for volume of a sphere:
Show them a ball (e.g., tennis). What two-dimensional figure do you see when you look at this ball? (circle) What part of a three-dimensional object do we start with when we write the volume formula? (area of the base) What geometric shape represents the base? (circle) What is the formula for the area of a circle? What geometric shape is needed to make an object three dimensional? (height) What expression could represent the height? (diameter or 2r) What did we learn about the sphere in Activity 6? (2/3 the volume of a cylinder) As the students answer your questions, put all of these expressions together on the board: πr2
, 2r and (2/3). What mathematical procedure do we always use to find the volume of any object? (multiplication) Tell the students to simplify the expression.
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Write the formula on the board.
UDL Supports: • The website above provides alternatives for visualizing volume of a sphere. You may also
consider using the alternate experiment on the website allowing students to engage in hands-on activity. This activity allows students to develop understanding of volume of cylinder to sphere by using rice/water.
Formative Assessment 2
Topic Materials/Resources Questions/Challenges Formative Assessment 2: “Sleeve of Golf Balls”
Formative Assessment 2 • Students will work independently on this assessment.
UDL Supports: • In order to fully comprehend this task, some students may need pre-teaching or
background knowledge of what a “sleeve of golf balls” looks like. • Consider adding scaffold questions to assist students in managing information. For
example, Question 1 – Draw and label a diagram to represent the sleeve of golf balls. Question 2 – Explain in words the steps you would take to find the empty space inside the sleeve of golf balls. Question 3 – Use mathematical formulas and the dimensions from your diagram to verify your explanation from Part 2.
Reflection
Topic Materials/Resources Questions/Challenges Filling the instructional gaps
Students will get back their Formative Assessments and reflect on and correct their mistakes.
• Teacher will address the common misunderstandings, and the students will correct their own mistakes.
• Students will also receive a rubric for each formative assessment to compare with their work.
UDL Supports: • To challenge students and maintain interest for students who have shown mastery of the
standards through the formative assessments, you may consider developing more activities or problems that reinforce the relationships between a cone, cylinder, and sphere. Example 1: You have a cone and cylinder with the same diameter. If the liquid from the cone is transferred to the cylinder, how much of the cylinder is filled? Example 2: You have a cylinder and sphere. The diameter of the sphere is twice the diameter of the cylinder. If the cylinder is filled to capacity with water, how much of the sphere could be filled with the liquid in the cylinder?
• If the formative assessments show evidence that students are not mastering standard G.MGD.3 (using formulas), students may need some example problems to develop fluency in using the formulas.
• If the formative assessments show evidence that students are not mastering the G.MG.1 standards (modeling), students may need additional opportunities to visualize and experience the real-life situations through the use of physical objects, rather than only a description of the objects. For example, given physical objects (a sleeve of golf balls), create a word problem that could be used to represent the volume of the empty space.
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Topic Materials/Resources Questions/Challenges
Culminating Task
Performance Assessment Task: A Day at the Beach
-Culminating Task -Calculator
• Students should be given a Performance Assessment Task and a calculator.
Reflection
Filling the instructional gaps
Students will get back their Performance Task and reflect on and correct their mistakes
• Teacher will address the common misunderstandings, and students will correct their own mistakes.
31
Volume Unit Activity 1 Name(s):___________________________________
1) Use geometric terms to describe the object on your desk. 2) Draw a diagram that illustrates the object. Label and measure the parts you believe are most important. 3) What procedure would you use to generate the volume of the object?
32
Volume Unit Activity 2a: “Chalk in a Box” Name(s):___________________________________ You have been given a piece of sidewalk chalk. Calculate how many pieces of chalk will fit into a container that measures 9”L x 6” W x 4.5” H. Describe your mathematical plan and justify your answer. Include a diagram to illustrate the situation and show all calculations that lead to your answer.
33
Volume Unit Activity 2b: “Chalk in a Box” Name(s):___________________________________ You are given dimensions of a piece of cardboard. What are the dimensions of box you could create from this piece of cardboard to hold the greatest number of pieces of chalk (used in Activity 2a)?
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Volume Unit Activity 3: The Cone Name(s):___________________________________
1) Calculate the volume of each cylinder below. Estimate the volume of the
shaded region made of clay. Justify your answer by using mathematical reasoning.
35
2) The fist pressed the clay cone into the cylinder. Find the volume of the
clay cylinders. What does this volume represent?
3) What is the relationship between the volume of the glass cylinder and the volume of the cone? Justify your solution by using mathematical reasoning.
4) Write the formula for the volume of the cone. Calculate the volume of each cone in question 1. Compare these calculations with the ones you found in question 2. Use mathematical reasoning to justify why they are not equal.
36
Volume Unit Activity 4: “Volume of a Plastic Cup”
Name(s):___________________________________
1) Describe the geometric parts of the plastic cup. Draw a diagram of the cup
and label the parts with appropriate measurements. 2) Design a mathematical model to find the volume of this cup.
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Volume Unit Activity 4: “Volume of a Plastic Cup” Answer Key
Name(s):___________________________________
1) Describe the geometric parts of the plastic cup. Draw a diagram of the cup
and label the parts with appropriate measurements. 2) Design a mathematical model to find the volume of this cup.
(Make sure you have a picture of the plastic cup on the SMART board or screen in order to illustrate the plan. See sample image below.)
You should start out with just the picture of the cup by itself. On the SMART board, you can have the geometric parts “fade in” as you proceed through the pivotal questions of the lesson. Use these sample pivotal questions to guide you through the lesson:
• Which parts of the cup did you measure? • Why did you find these parts to be important enough to
measure? • What geometric shape(s) create this cup? • Why is this cup cylindrical? (it has two circular bases) • Why did you say this cup is similar to a cone? (the sides are
slanted or the two bases are different sizes) (at this point, draw in the lines to create the large cone)
• What does it mean if this cup is both a cylinder and a cone? (it is an irregular shape)
• The students will have the most difficulty seeing the two cones. To help illustrate this concept, you could use different colors to highlight the large and small cones.
• Describe the procedure we may use to find the volume of the cup (find the measurement of the large cone and subtract the measurement of the small cone that is underneath the cup).
• You can also describe the procedure in different ways,such as “subtract the large cone from the small cone,” “cut off the bottom piece of the large cone to get the red cup.” Using a color for the cup is helpful because it will stand out.
• What geometric measurement is missing? (height of the small cone)
• How can we find this measurement? (proportion) • The other difficult piece is the proportion. Use the last two
pivotal questions on the previous page to generate the correct response.
After the heights have been calculated, the students should be instructed to finish the problem on their own. You should collect their work to check their level of understanding.
38
1.937
1.2
4.75
x
x + 4.75
1.9375 4.751.25
1.25 5.9375 1.93755.9375 .6875
8.63
xx
x xx
x
+=
+ ==
=
( ) ( )
2large cone
2large cone
3large cone
131 1.9375 13.3863352.623
V r h
V
V in
π
π
=
=
=
( ) ( )
2small cone
2small cone
3small cone
131 1.25 8.6363314.1311
V r h
V
V in
π
π
=
=
=
plastic cup large cone small cone
plastic cup
3plastic cup
52.623 14.1311
38.4919
V V VV
V in
= −
= −
=
39
Volume Unit Activity 5: The Pyramid
Name(s):___________________________________ 1) Cut out the figure on the left side of your handout and fold it along the
dotted lines. What figure did you make? 2) Now fit three of these figures together. Name this geometric object. 3) Describe the procedure you would use to find the volume of the original
object. Justify your answer using mathematical reasoning.
40
Volume Unit Activity 5
3 ×
www.CutOutFoldUp.com © Laszlo Bardos 0971
Student Materials
41
Volume Unit Activity 6: The Sphere
Name(s):___________________________________ 1) Use a geometric term to name the shaded object. Estimate its volume and justify
your answer using mathematical reasoning.
42
2) The fist pressed the clay sphere into the glass cylinder. Find the volume of the clay cylinders. What does this volume represent?
3) What is the relationship between the volume of glass cylinder and the sphere?
Justify your answer using mathematical reasoning. 4) Compare your solutions from Activity 3, questions 1 and 2 with today’s activity
questions. Analyze the results and state your conclusions. Provide a mathematical justification.
43
(This checklist should be provided to students prior to the formative assessments)
General Rubric for a Successfully Completed Task
The following elements must be included in your solution to receive full credit.
□ Diagrams are drawn and labeled accurately.
□ Formulas with appropriate substitutions are shown.
□ Appropriate step-by-step algebraic solution is provided.
□ Calculations are precise.
□ Explanations include the use of correct mathematical vocabulary.
□ Correct units of measurement are indicated.
□ Justification of the solution with mathematical reasoning is provided.
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Geometry Volume Unit Pre-Assessment Task 1. Find the area of each figure. Include the unit measure and round your
answer to nearest thousandth place.
a. b.
10 inches c.
7 feet 2. Find the volume of the rectangular prism below. Include the unit measure
and round your answer to nearest hundredth place.
3 cm
4 inches
12 feet
5 in
3 in
4 in
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3. Explain the difference between area and volume. Give an example of when
you would use area or volume.
Your assignment is to find real-world objects that are shaped in an irregular manner. You can find them in your house or pictures in a newspaper or magazine. You must provide pictures or drawings of two irregularly shaped objects. Explain why you believe they are irregular, what shape(s) they represent, and how you believe they were constructed. You may use the space below to attach your pictures and write out your explanation.
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Formative Assessment Task 1
The Sharpened Pencil You have a brand new pencil. Design a mathematical model to determine how much material is lost after sharpening the pencil. Be sure to label the diagram(s) to illustrate your procedure and include the units of measurement.
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High School Geometry Rubric: The Sharpened Pencil
Elements of performance required by this task:
• Visualizes and identifies the dimensions of the pencil • Determines the volume relationships of cylinders and cones • Justifies geometric arguments.
Possible Responses Points Section Points
1. a) Draws and labels the dimensions on the diagram(s).
or b) Calculates the volume of the new pencil.
or Partial credit Computational error(s) in parts (a) and (b)
1 1 (1)
2
2. a) Calculates the volume of the cylindrical portion of the sharpened pencil. b) Calculates the volume of the cone portion of the sharpened pencil. c) Adds the solutions from parts (a) and (b). Answers will vary because each sharpened pencil will be unique. Partial credit Computational error(s) in parts (a), (b) and (c)
1 1 1 (1)
3
3. Calculates the lost material by subtracting the solution found in part 2c from part 1b.
1
1
Total Points 6
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High School Geometry: The Sharpened Pencil Rubric
Performance Level Descriptions and Cut Scores Performance is reported at four levels: 1 through 4, with 4 as the highest. Level 1: Demonstrates Minimal Success (0 or 1 point) The student’s response shows few of the elements of performance that the task demands as defined by the CCLS. The student’s work shows a minimal attempt and lack of coherence. The student fails to use appropriate tools, such as volume and circumference formula, strategically. The student is unable to make sense of the problem and apply geometric concepts in this modeling situation. Level 2: Performance Below Standard (2 or 3 points) The student’s response shows some of the elements of performance that the task demands as defined by the CCLS. The student might ignore or fail to address some of the constraints of the problem. The student may occasionally make sense of quantities or relationships in the problem. The student attempts to use some appropriate tools, such as volume or circumference formula, with limited success. The student may recognize geometric shapes but has trouble generalizing or applying geometric methods in this modeling situation. Level 3: Performance at Standard (4 points) For most of the task, the student’s response shows the main elements of performance that the tasks demand as defined by the CCLS with few minor errors or omissions. The student explains the problem and identifies constraints. The student makes sense of quantities and their relationships in the modeling situation. The student uses appropriate tools, such as radius, circumference and volume. The student might discern patterns or structures and make connections between representations. The student is able to make sense of the problem and apply geometric concepts to this modeling situation. Level 4: Achieves Standards at a High Level (5 or 6 points) The student’s response meets the demands of nearly all of the tasks as defined by the CCLS and is organized in a coherent way. The communication is clear and precise. The body of work looks at the overall situation of the problem and process, while attending to the details. The student routinely interprets the mathematical results, applies geometric concepts in the context of the situation, reflects on whether the results make sense, and uses all appropriate tools strategically.
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Formative Assessment Task 2
Sleeve of Golf Balls
Design a mathematical plan to calculate how much empty space is inside a sleeve of golf balls that is filled to capacity. Be sure to include diagram(s) and justify your solution using mathematical reasoning. (Note: A “sleeve” is a type of container.)
Information you may need: A golf ball has a diameter of 1.68 inches.
Sleeve of golf balls: Base: 32221 in x
32221 in
Height: 15 in8
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High School Geometry Rubric: Sleeve of Golf Balls
The elements of performance required by this task are:
• Visualizes and identifies the dimensions of the golf ball and the sleeve (box) • Determines the volume relationships of rectangular prism and sphere • Justifies geometric arguments.
Possible Responses Points Section Points
1. a) The given dimensions are labeled accurately on the diagram(s).
b) Calculates the volume of the sleeve.
Vsleeve = lwh =
815
32221
32221 = 14.5942 in3
c) Calculates the volume of the golf ball.
in3 Partial credit Computational error(s) in parts (b) and (c)
1
1
1 (1)
3
2. a) Determines the maximum number of golf balls that fit inside the sleeve. (answer: 3 balls)
b) Justifies the answer given in part (a).
0506.368.1815 =÷
Partial credit Incorrect answer given in part (a) but a conceptual error is made such as dividing the two volumes.
1
2
(1)
3
3. a) Calculates the volume of the three golf balls. in3
b) Calculates the amount of empty space. 14.5942 – 7.4481 = 7.1461 in3
Partial credit Computational error(s) in parts (a) and (b)
1
1
(1)
2
Total Points 8
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High School Geometry: Sleeve of Golf Balls Rubric
Performance Level Descriptions and Cut Scores Performance is reported at four levels: 1 through 4, with 4 as the highest. Level 1: Demonstrates Minimal Success (0 or1 point) The student’s response shows few of the elements of performance that the task demands as defined by the CCLS.
The student’s work shows a minimal attempt and lack of coherence. The student fails to use appropriate tools, such as volume and circumference formulae, strategically. The student is unable to make sense of the problem and apply geometric concepts in this modeling situation.
Level 2: Performance Below Standard (2–4 points) The student’s response shows some of the elements of performance that the task demands as defined by the CCLS. The student might ignore or fail to address some of the constraints of the problem. The student may occasionally make sense of quantities or relationships in the problem. The student attempts to use some appropriate tools, such as volume or circumference formula, with limited success. The student may recognize geometric shapes, but has trouble generalizing or applying geometric methods in this modeling situation. Level 3: Performance at Standard (5 or 6 points) For most of the task, the student’s response shows the main elements of performance that the tasks demand as
defined by the CCLS with few minor errors or omissions. The student explains the problem and identifies constraints. The student makes sense of quantities and their relationships in the modeling situation. The student uses appropriate tools, such as radius, circumference and volume. The student might discern patterns or structures and make connections between representations. The student is able to make sense of the problem and apply geometric concepts to this modeling situation.
Level 4: Achieves Standards at a High Level (7 or 8 points) The student’s response meets the demands of nearly all of the tasks as defined by the CCLS and is organized in a
coherent way. The communication is clear and precise. The body of work looks at the overall situation of the problem and process, while attending to the details. The student routinely interprets the mathematical results, applies geometric concepts in the context of the situation, reflects on whether the results make sense, and uses all appropriate tools strategically.
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