Revised 2/02/05
Learning Activity: Not Enough Cubes!
Grades 7, 8
Topic: Deriving Volume Formulas of a Right Rectangular
Prism and a Cylinder
Lesson Components
Lesson Plan Outline
Student Log
SOLVE-IT: Real World Volume Problems
Practice Short-Response Assessment Problem
Notes for the Teacher (with answer keys)
Blank template for SOLVE-IT: Real World Problems
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Lesson Plan Outline for “Not Enough Cubes!”
Florida DOE Sunshine State Standards
Strand B: Measurement, Strand C: Geometry and Spatial Sense
Grade 7
Benchmark MA.B.1.3.1: The student uses concrete and graphic models to derive
formulas for finding perimeter, area, surface area, circumference, and volume of two- and
three-dimensional shapes, including rectangular solids and cylinders.
Grade Level Expectations (GLE) for Mathematics Grade 7: 1. uses concrete or
graphic models to create formulas for finding volumes of solids (prisms and cylinders).
Grade 7, Benchmark MA.C.1.3.1: The student understands the basic properties of, and
relationships pertaining to, regular and irregular geometric shapes in two-and threedimensions.
Grade Level Expectations (GLE) for Mathematics Grade 7: 6. knows the properties
of two-and three-dimensional shapes.
Grade 8
Benchmark MA.B.1.3.1: The student uses concrete and graphic models to derive
formulas for finding perimeter, area, surface area, circumference, and volume of two- and
three-dimensional shapes, including rectangular solids and cylinders.
Grade Level Expectations (GLE) for Mathematics Grade 8: 1. uses concrete and
graphic models to explore and derive formulas for surface area and volume of threedimensional regular shapes, including pyramids, prisms, and cones. 2. solves and
explains real-world problems involving surface area and volume of three-dimensional
shapes.
Grade 8, Benchmark MA.C.1.3.1: The student understands the basic properties of, and
relationships pertaining to, regular and irregular geometric shapes in two-and threedimensions.
Grade Level Expectations (GLE) for Mathematics Grade 8; 3. draws and builds
figures from various perspectives (for example, flat patterns, isometric drawings, nets); 4.
knows the properties of two-and three-dimensional figures.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Suggested Timeline
(Based on 45-minute classes)
Math Content
Day 1
Day 2
Day 3
Day 4
Learning Mode
Volume of a Right Rectangular Prism
Derive formula V=lwh using models
Small group;
Hands-on
Volume of a Cylinder
Derive formula V= r2h using models
Practice using V=lwh and V= r2h
Teacher Demo
Volume of a Right Rectangular Prism and a Cylinder
Apply V=lwh and V= r2h to real-world problems
SUGGESTED EXTENSION
Pair/individual;
Pencil-paper
Pair/small group;
Pencil-paper
Benchmark MA.B.1.3.3: The student understands and describes
how the change of a figure in such dimensions as length, width,
height, or radius affects its other measurements such as perimeter,
area, surface area, and volume.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Class _______ Date _____________Name__________________________________
Student Log for “Not Enough Cubes”
Your Mission: Follow the directions below. Use only the materials the teacher gives
your group. Have fun as you answer the question: How many whole cubes will it take to
fill the box and how did you figure it out?
Materials
One (1) open-top box
One (1) bag of centimeter cubes
Calculators
Pencils and paper
Overhead transparencies and pens
Step 1: Remove all cubes from the bag and place them in your group’s open-top box. Work together to
figure out how many whole cubes it will take to fill the box.
Step 2: Make sure that all group members agree upon how many cubes will fill the box. Then, write the
answer below and explain in writing below how your group figured out the answer.
______________________________________________________________________________________
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Step 3: Choose one group member to write your group’s answer and explanation on the blank
transparency. Choose another group member to put the completed transparency on the overhead projector
and explain it to the class—if your teacher calls on your group.
Step 4: After your teacher discusses formulas with the class, write down 2 formulas, which can be used to
find out how many cubes will fill a right rectangular prism.
Formula 1: _______________________ Formula 2: _________________________
Step 5: After your teacher shows you the cylinder demonstration, write down the formula for finding the
volume of a cylinder.
V = _____________________
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Class _______ Date _____________Name(s)__________________________________
__________________________________
__________________________________
SOLVE-IT: Real World Volume Problems
Set aside one activity sheet for recording your group’s final answers.
Raise your hands when your group answer sheet has been completed.
All answers must be correct to win.
All group members must be able to explain solutions to all problems.
1. Madison has two bread pans in her kitchen
in the shape of rectangular prisms. The first pan
is 11 in. long, 6 in. wide, and 6 in. high. The
second pan is 12 in. long, 5 in. wide, and 6 in.
high. How much greater is the volume, in cubic
inches, of the first pan than the second pan?
_____________
2. A swimming pool is 18 m long, 10 m wide
and 6 m deep. On Monday, the water level
measured 4.6m deep, with no people in it. On
Monday, what was the volume of the water, in
cubic meters?
3.
4. The layers of wedding cakes are often made
in several different sizes of pans that form
cylinders. If the second layer of a wedding cake
has a height of 8 inches and a radius of 6
inches, what is the volume, in cubic inches, of
the second layer of the cake? (Use 3.14 for .)
The volume of a rectangular cereal box is
2673 cm3. Its height is 27 cm. Its width is
one-third its height. What is its length, in
centimeters?
_____________________
_____________________
_____________________
*5. A box of salt, in the shape of a cylinder,
measures 10 cm in diameter and is 16 cm high.
What is the volume, in cm3, of the salt-filled
cylinder? (Use 3.14 for .)
___________________
* Read carefully.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Class _______ Date _____________Name__________________________________
Extended-Response Assessment Problem: Volume of Rectangular Prisms
Problem: An open-topped windowsill flower box has a length of 30 cm. Its width is 2/3
the measure of its length. Its height is ½ the measure of its width.
Part A: In the space below, draw the flower box described in the problem. Figure out all three dimensions
of the flower box and label its length, width, and height in centimeters. Show your work below.
Part B: If the height of the soil in the flower box is 4/5 of the height of the flower box, what is the volume
of the soil in cubic centimeters? Show your work below.
___________________
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Notes for the Teacher
Learning Activity: Not Enough Cubes!
Grades 7, 8
Topics: Deriving Volume Formulas of a Right Rectangular Prism and a Cylinder
Skills to Review: (a) area of rectangle, textbook pages: _________________________________;
(b)area of circle; textbook pages: ____________________________________.
Your Students’ Mission: Using centimeter cubes and a shallow box, students will figure
out how many whole cubes it will take to fill a given box. They will then derive and
justify the formulas for the volume of a right rectangular prism and the volume of a
cylinder (V=lwh and V= r2h).
Materials
Small identical, shallow, boxes with squared corners (1 per group and 1 for teacher)
Centimeter graph paper (10-20 pieces)
Double-stick tape (1)
Scissors (1)
Centimeter cubes (40-50 cubes per group)
Small zip-baggies
Clear plastic cylinder (1)
Copies of the Student Log (1 per student)
Calculators
Overhead projector and screen
Marker (1)
Blank overhead transparencies (2 per group)
Transparency pens (2 per group)
Advance Preparation
Search for small, shallow open-top plastic or cardboard boxes with square corners, one per group and
one for yourself. All boxes must be identical. Note: Small drawer organizers are sold that hold 21 X 14
X 5 centimeter cubes.
Run off 1 sheet of centimeter graph paper for each box. Cut the graph paper so that its area and the
number of cubes in the bottom layer of the box are the same (i.e. 21 X 14). Use double-stick tape to
stick the graph paper on the OUTSIDE of the bottom of each box. In marker, write “Area of Base (B)”
on the graph paper.
Similarly, cover the circular bottom of the plastic cylinder with graph paper. Write “Area of Base (B)”
on graph paper.
Count out centimeter cubes and place them in small zip-baggies.
Run copies of the Student Log (1 per student).
Make an answer key and a scoring rubric to let your students know what is expected of them.
Select a few problems from the textbook so students can practice using formulas.
Run copies of SOLVE-IT: Real World Volume Problems (1 per student and 1 extra for each group)
Run copies of the Practice Extended Response Assessment Problem. Prepare the scoring rubric you
will use and discuss with students.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
DAY 1
Volume of a Right Rectangular Prism
Derive formula V=lwh using models (Small group; Hands-on)
Delivering the Lesson
In preparing students for this activity, resist the urge to use the word “volume”. Only near the end
of the activity should you elicit the term “volume” from students.
Place a box in front of each group so that students DO NOT see the graph paper on the bottom of
the box.
Distribute bags of centimeter cubes, calculators, Student Logs, transparencies, and transparency
pens to each group.
Discuss with students the directions printed on their Student Logs. Do not offer suggestions for
solving the problem and start them working.
Discuss scoring rubric with students and post on wall.
Circulate as students work. Listen carefully and notice which students emerge as leaders—not
always “A” students.
Pick a group to present its work to the class, pointing out positive aspects of their written
explanation and its similarity to the FCAT short response and extended response solutions.
Answer Key for Student Log (question # 2)
Sample Response from Real Kids: “To find out how many cubes would fill the box, we had to first find
out how many cubes will cover the bottom layer of the box. You should’ve given us enough cubes to do
this, but since you didn’t we had to really think! We are pretty sure that 294 cubes will fit in the bottom
because the box is 21 cubes long and 14 cubes wide and 21 X 14 = 294. We stacked up cubes to the top of
the box to see how many layers there were. It was 5 cubes high, so we multiplied the number of layers
times the number of cubes in each layer and got 1470. 5 X 294 = 1470. Whew! “
Signed: Kayla, Jeremy, Vicki, Carlos the Great!
Deriving Formulas
After a group(s) has presented their work to the class, ask each group to put their cubes back in
baggies and then flip over the box. Next ask students to find the area of the base of the box by
using the graph paper taped to bottom (i.e., 294). They will count or multiply to get this number.
They should recognize this number as the number of cubes in the bottom of the box. Discuss the
concept of volume (use the word now). Lead students to tell you that the V= (Area of Base) X
height or V== (Area of Base) h or V = Bh (must use capital letter for area of base). Since the base
is a rectangle and the formula for the area of a rectangle is A=lw, by substitution, V= lwh.
Then, guide students to derive the following formulas for the volume of a right rectangular prism
and write formulas on their Student Logs.
Answer Key for Student Log (question # 4)
Formula 1: Volume = (Area of Base) X height or V = (Area of Base) h or V = Bh.
Formula 2: V= lwh
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Day 2
Volume of a Cylinder
Derive formula V= r2h using models (Teacher Demo)
Practice using V=lwh and V= r2h (Pair/individual; Pencil-paper)
Beginning the Lesson
Review yesterday’s Not Enough Cubes Activity with students. Remind them of their discovery
that the volume of a prism is the area of its base multiplied by its height (h) or
V = (Area of Base) h.
Hold up a clear plastic cylinder with graph paper taped on its circular base and the words “Area of
the Base (B)” written on the graph paper. Ask students how they would determine the number of
cubes that would fill the cylinder.
Discuss the differences between the shape of cubes and cylinders. Ideally, if they could count the
number of cubes that would fill the bottom layer of a cylinder, they could multiply the number by
the height of the cylinder (h). Like the right rectangular prism, Volume of Cylinder = (Area of
Base) h.
Guide students to see that, in this case, the area of the base is not a rectangle but a circle. Since
the area of a circle is A = r2 then by substitution, V= ( r2) h or V= r2h.
Applying the Formulas
As a class, practice applying the formulas. Be sure that a few problems state the diameter of the
circular base of a cylinder and several problems state the radius of the circle Explain that volume
is expressed in cubic units such as cm3, m 3, in3. Allow students to work in pairs on a few more
textbook problems.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Day 3
Volume of a Right Rectangular Prism and a Cylinder
Apply V=lwh and V= r2h to real-world problems (Pair/small group; Pencil-paper)
Distribute copies of “SOLVE-IT: Real World Volume Problems”. Allow students to work in pairs
or in cooperative groups. Encourage them to draw and label figures on notebook paper and show
work. [NOTE: You can easily convert these problems into short-response or extended-response
practice problems, if you prefer.]
Answer Key for “SOLVE-IT: Real World Volume Problems”: # 1. (a) Volume = 36 in3; (b) the
first pan; or the pan that is 11 in. X 6 in. X 6 in.; # 2. 828 m3; # 3. 11 cm; #4. 904.32 in3; * # 5. 1256
cm3. Solutions: # 1. Volume of first pan is 11 X 6 X 6 = 396 in3; volume of second pan is 12 X 5 X 6
= 360 in3; 396-360 = 36 in3 . # 2. 18 X 10 X 4.6= 828 m3; # 3. 27 X 9 X length = 2673, so 243 X
length = 2673 and 2673 243 = 18 cm; # 4. 6 X 6 X 3.14 X 8 = 904.32 in3; * # 5. Since diameter is 10
cm, radius is 10 2 = 5 cm and 5 X 5 X 3.14 X 16 =1256 cm3.
Distribute copies of the Short-Response Assessment Problem at the end of this lesson. Discuss
and post the scoring rubric you will be using and give students plenty of time to work.
Answer Key for the “Short-Response Assessment Problem for Volume of Rectangular Prisms”:
Part A:
Width: 2/3 X 30 cm = 20 cm.
Height: 1/2 X 20 cm = 10 cm.
Part B: 4800 cm3
4/5 of 10 = 8
30 X 20 X 8 = 4800 cm3
To the teacher: If the student’s answers to Part A, are incorrect, check to see if the
work in Part B is done correctly and score accordingly.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder
Revised 2/02/05
Class _______ Date _____________Name(s)__________________________________
__________________________________
__________________________________
__________________________________
SOLVE-IT: Real World Problems
Set aside one activity sheet for recording your group’s final answers.
Raise your hands when your group answer sheet has been completed.
All answers must be correct to win.
All group members must be able to explain solutions to all problems.
Anne C. Patterson 2004, Sept., 2004
Not Enough Cubes, Grades 7, 8, Volume of Prism/Cylinder