Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Overview of Instructional Task: Calculating
Volumes of Compound Objects—Glasses
This task helps students develop strategies for solving problems involving volume.
Task should be implemented after students have begun work on volume in Chapter 10, but before complete
mastery (after Lesson 10.3, February 16–23, 2012).
One or Two Days Before Lesson
•
•
•
Students work independently on Glasses task (pages 2–3) (15 minutes). Provide calculators.
Collect student work.
Analyze responses and write questions on each student paper. Do not score at this time.
Lesson
•
•
•
•
•
•
Return each student’s Glasses work with your questions.
Students work independently revising Glasses task, guided by your questions (15 minutes).
Students work in small groups to produce common solutions on a fresh Glasses task paper (10 minutes).
Students work with same small groups to analyze flawed student work samples (pages 11–13) (15 minutes).
Conduct whole class discussion of student work samples (15 minutes), using projector resources (pages 11–17).
Students individually revisit and revise their original work on Glasses task (10–15 minutes at end of lesson or
during next class).
After Lesson
•
•
Analyze student responses to identify next instructional steps.
Task may be graded if desired (see DPS Rubric, page 18).
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
1
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Name:
Geometry, Semester 2
Date:
Glasses
Glass 1
Glass 2
Glass 3
This picture shows three glasses.
The measurements are all in centimeters.
The bowl of Glass 2 has a cylindrical top and a hemispherical base.
The bowl of Glass 3 goes down into the stem.
1. Calculate the volume of liquid that would fill the bowl of each glass. Show all your calculations and
explain your reasoning.
a. Glass 1
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
b. Glass 2
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
c. Glass 3
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
2
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
2. Find the height of liquid in Glass 2 when it is half full (half the amount of liquid of a full glass).
Show how you figure it out.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
3. Glass 4 is shown in the diagram. One of the following formulas gives the volume of Glass 4. Which is it?
1
πd2h
6
1
πdh
6
1
πdh2
6
1
πd2h2
6
Explain how you can tell by just looking at the form of these expressions.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
3
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Glasses: Extension Questions
1. Explain how you can tell when a formula represents a length, an area, or a volume.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
2. Show step by step how a formula for the volume of glass 4 may be derived.
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
Glass 4
3. Find the volume of Glass 3 when it is half full. Explain your answer.
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
Glass 3
4. On a separate sheet of paper, make up your own Glass question and solve it.
Try to make your problem challenging, but solvable!
Now give it to someone else in your class to solve.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
4
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Instructional Task: Calculating Volumes of
Compound Objects—Glasses
Mathematical Goals
This instructional task helps assess how well students:
•
•
•
Compute measurements using formulas.
Decompose compound shapes into simpler ones.
Use right triangles and their properties to solve real-world problems.
Common Core State Standards
This instructional task emphasizes the following Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
4. Model with mathematics.
6. Attend to precision.
This instructional task also asks students to select and apply mathematical content from the Common Core State
Standards.
G-SRT-8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-GMD-3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Required Materials
•
•
•
•
•
Copies of task, Glasses, for each student (pages 2–3)
Copy of task, Glasses, for each small group (pages 2–3)
Copies of student work samples for each small group (pages 11–13)
Copies of extension task, Glasses: Extension Questions (page 4), if necessary, for students
Projector resources to help with whole class discussion (pages 11–17)
Time
Times given are only approximate. Exact timings depend on class needs.
•
•
One or two days before lesson: 15 minutes
Lesson: 60 minutes
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
5
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Before Lesson
Instructional Task: Glasses (15 minutes)
Have students do this task in class a day or more before the instructional task, so you can assess the work, find
out the kinds of difficulties students are having, and more effectively focus the follow-up lesson.
Distribute copies of the task, Glasses (pages 2–3) and calculators.
Introduce the task briefly and help the class understand the task
and its context.
Read the questions and try to answer them as carefully
as you can. Show all your work so I can understand your
reasoning. Use another paper if you need more room.
In addition to trying to solve the task, I want to see if you
can present your work in an organized and clear manner.
It is important that students answer the questions without
assistance, as far as possible.
Students who sit together often produce similar answers, then
when they come to compare their work, they have little to
discuss. For this reason, you might have students do the task
individually and move them to different seats. Then at the
beginning of the instructional task session, allow them to return
to their usual seats. Experience has shown that it produces more
profitable discussions.
Students should not worry too much if they cannot understand or do everything, because there will be a lesson
using this task that should help them. Explain to students that by the end of that lesson, they should expect to
answer questions such as these with confidence, which is their goal.
Provide Student Feedback
Collect students’ responses to the task. Make some notes on what their work reveals about their current levels
of understanding and their different problem-solving approaches.
Do not score students’ work. Research shows that it is counterproductive, as it encourages students to compare
their scores and distracts their attention from what they can do to improve their mathematics.
Instead, help students to make further progress by summarizing their difficulties as a series of questions, such as
the suggestions that follow. Write questions on each student’s work that will advance his or her thinking. You
may also note students with particular issues, so you can monitor their work during the lesson.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
6
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Common Issues
Suggested Questions and Prompts
Student has difficulty getting started.
•
•
•
•
Student does not discriminate between length, area,
and volume formulas.
For example: Student multiplies too many or too few
numbers together to calculate volume or uses
incorrect formula.
Student has difficulty identifying values to substitute
for variables in the formula.
For example: Student does not match variables in
formula with measures on the figure when applying
formula or uses diameter rather than radius or uses
slant height rather than Pythagorean Theorem to find
perpendicular height (question 1c).
Student makes calculation errors.
•
•
•
What is the task about?
What do you know?
What do you need to find out?
What is the difference between length, area, and
volume formulas?
Compare formulas for the cylinder and cone.
What is their common base area? How does it
show in the formulas?
What measures do variables in your formula stand
for? Draw these measures on the diagram.
Are there any measurements you need that you do
not have? How could you find them?
•
How can you check your answers?
Student works unsystematically.
•
•
Student presents work poorly.
•
Can you organize your work in a systematic way?
Have you labeled your answers and made clear
what you were calculating?
Would someone unfamiliar with your type of
solution easily understand your work?
Have you explained how you arrived at your
answer?
Would labels or diagrams help make your process
clearer?
What is the difference between prisms and
pyramids?
What is the difference between cones and cylinders?
Imagine you can take this three-dimensional shape
apart. What pieces would you make to calculate
the volume using formulas you know?
For example: Student makes numerical errors in
calculations, such as doubling rather than squaring.
•
•
Student chooses wrong formula.
Student has difficulty decomposing threedimensional shapes.
•
•
•
For example: Student does not calculate volumes of
compound shapes using appropriate formulas for
constituent parts.
Student assumes proportionality.
•
Look back at question 1b. Are the volumes of the
two parts of the figure the same? How does it
affect your answer?
•
Provide extension questions (page 4) for student
to work.
For example: Student assumes that halving the volume
also halves the height, giving a response of 3 cm
(question 2).
Student answers all questions correctly.
Student needs extension task.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
7
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Suggested Lesson Outline
Improve Individual Solutions to Instructional Task (15 minutes)
Return students’ instructional task papers. Make calculators available.
Recall what we looked at previously. What was the task?
Ask students to reread the Glasses task and their own work. Draw their attention to your questions about
their work.
I have read your solutions, and I have some questions about your work. I would like you to work on your
own to answer my questions for about ten minutes. Write your answers on your mini-whiteboard,
so I can see what you’re thinking as I walk around.
The aim of this activity is to encourage students to re-engage with their work and reflect on what they have
done. Often students comment that they find even their own reasoning hard to follow, which is a sure sign that
they need to work on the quality of their explanations.
Small Group Collaborative Work (10 minutes)
Organize the class into small groups of two or three students. You may want to provide them with a fresh copy
of the task.
Now I would like you to work in small groups. Take turns to explain what you did on the problems and
where you got stuck. Your task is to produce a solution that is better than your individual solutions.
While students work in small groups, note different student approaches to the task and support student
problem solving.
Note Different Student Approaches to Task
Listen to students and identify issues they are discussing. In particular, see if they address difficulties outlined in
the Common Issues table above. Which parts of the task do they struggle with most? Use this information to
focus the whole class discussion later in the lesson.
Support Student Problem Solving
Periodically, intervene and ask questions to help students clarify their thinking. Try not to help students too
much by “taking over” their work. Instead, encourage students to explain to one another. If several students
struggle with the same issue, write a relevant question on the board.
For students who succeed with the task, suggest that they try to find the height of liquid in Glass 3 when it is half
full, since it is a more demanding problem.
Collaborative Analysis of Student Work Samples (15 minutes)
Give each small group of students copies of the student work samples (pages 11–13).
None of the sample responses shows the correct answer for the volume. Work together to find and
explain the errors each student made. Explain what the student could do to complete his or her solution
correctly.
Sample responses show some common errors students make on questions 1c and 3. This step gives students an
opportunity to address those common misconceptions without providing a complete solution strategy.
Whole Class Discussion of Student Work Samples (15 minutes)
Organize a whole class discussion. Focus on getting students to understand different methods of working out
answers, to articulate common misconceptions, and to share their reasoning, rather than to check numerical
solutions.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
8
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Let’s stop to talk about the different errors the students made.
•
•
•
•
•
•
•
Look at Logan and Isaac’s solutions to question 1c.
Which student produced the better solution? Explain your response.
Which height do you need to calculate this volume?
[Zsa-Zsa,] your group wrote an explanation. Could you explain it for us?
Put Logan's or Isaac's explanation into your own words.
Look at Yasmin’s and Brianna’s solutions to question 1.
Which student produced the better solution? Explain your response.
Individually Review Original Solutions to Task (5 minutes)
Next Lesson: Review Individual Solutions to Instructional Task (10–15 minutes)
Unless you have time during the lesson, schedule this task for the next lesson.
Return students’ individual papers from the Glasses task, along with a fresh copy of the task sheet.
Read your original responses and think about what you learned in this lesson. Suppose you were to work
on another volume task tomorrow. What advice would you give yourself? Using what you learned, try to
improve your solution.
If students are satisfied they completed the task satisfactorily, ask them to try some questions on the extension
sheet, Glasses: Extension Questions. Question 3 on this sheet is much more difficult that previous questions, so
it should provide an appropriate challenge for students who succeeded on the other parts of this task.
Solutions
Instructional Task: Glasses
1a. The volume of Glass 1 = π × 3 × 3 × 6 = 54π = 170 cm3.
1b. The volume of the hemisphere = (4π × 33) ÷ 6 = 18π = 56.5.
The volume of the cylinder = π × 32 × 3 = 27π = 84.8.
Total volume of Glass 2 = 45π = 141 cm3.
1c. Using the Pythagorean Theorem, the height of the cone is 7 2 – 32 = 40 = 2 10 .
The volume of Glass 3 = (π x 32 x 2 10 ) ÷ 3 = 6 10 π = 60 cm3.
2. The volume of liquid in the half-full Glass 2 is 141 ÷ 2 = 70.5.
The volume of liquid in the cylinder = 70.5 − 56.5 = 14.
14 = π × 32 × height in cylinder.
Height in cylinder = 14 ÷ 9π = 0.5.
The total height = 3 + 0.5 = 3.5 cm.
3. Glass 4 is composed from a cylinder and cone. While we do not yet have enough information to deduce the
formula, it is possible to rule out three of the formulas on grounds that the dimensions of the formulas are
incorrect:
1
πdh
Only two lengths are multiplied, so this has the dimension of an area.
6
1
πd2h2
Four lengths are multiplied, so this is not a volume either.
6
1
1
1
Both πd2h and πdh2 involve multiplying together three lengths, but πdh2 involves the square of the
6
6
6
height and so, cannot be correct.
The correct formula is therefore
1
πd2h.
6
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
9
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Glasses: Extension Problems
1. This question is intended to encourage the discussion of dimensional analysis.
When lengths are combined by addition, we obtain another length.
If two lengths are multiplied, we obtain an area.
If three are multiplied, we obtain a volume.
2. The volume of the Glass = volume of cylinder + volume of cone
2
d   h
2 2
= π    +
2
1 d   h
π   
3 2 2
2
4 d   h
π   
3 2 2
1
= πd2h
6
=
3. When Glass 3 is half full, it will hold 30 cm3 (from question 1c).
If the height of liquid is h and the radius of the top of the liquid is r, then
1 2
π h = 30.
3
So πr2h = 90.
(1)
By similar triangles:
The ratio height of bowl : radius of bowl = h : r = 2 10 : 3
This means that r = 0.47h (2)
Substituting (2) in (1):
0.225 h3 = 29, and so h = 5.1 cm.
The height of liquid will be 5.1 cm
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
10
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Student Work Samples
Here is some work on Glasses done by other students.
Neither Isaac nor Logan has found the correct volume for Glass 3.
Neither Brianna nor Yasmin has found the correct height of liquid in
Glass 2 when it is half full.
1. Find and explain the errors each student made.
2. Explain what the student needs to do to complete his or her
solution correctly.
For example, you might write sentences beginning with:
The student has substituted an incorrect measure…
This is the wrong measure because...
The correct measure is…
The correct solution is…
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
11
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Sample 1: Logan
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Sample 2: Isaac
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
12
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Sample 3: Yasmin
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Sample 4: Brianna
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
13
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Glass 1
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
14
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Glass 2
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
15
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Glass 3
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
16
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Geometry, Semester 2
Glass 4
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
17
Instructional Task: Calculating Volumes of Compound Objects—Glasses
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
Geometry, Semester 2
18