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D Formulas and
D Formulas and Geometry
Geometry
Circles and Solids
Notes
Valerie wants to make a mold she can
later use to make candles. She decides to
use a cylinder-shaped mold. For the base
of the mold, she has cut a circle that has a
6-cm diameter.
6 cm diameter
13a If compasses are not
available, a pencil with a
short piece of string tied to
it could be used to draw
the circle. Remind
students that the diameter
is 6 cm, not the radius.
13b Students will need
strips about 20 cm long to
do this. They can cut this
from notebook paper.
13c To visually demonstrate the relationship
between diameter and
circumference, you could
use a piece of string the
length of a diameter and
show that it takes approximately 3 of these to reach
around the circle.
height
overlap
1 cm
base
13. a. Make an accurate drawing of a circle that is 6 cm in diameter.
Use a compass.
b. Use a strip of paper to find the size of the mantle of the mold.
Allow at least 1 cm overlap to glue the mantle together. What
are the measurements of the mantle without the overlap?
Valerie used this formula for the mantle of her mold:
circumference of a circle π diameter
c. Explain why this formula makes sense.
Fruit drinks come in cans of different sizes. Some cans are narrow
and tall; others are wide and short.
14. a. What shapes are juice cans usually?
b. Is it possible for cans in different shapes to contain the same
amount of liquid?
Reaching All Learners
Hands-On Learning
You might show different size cans to students and ask them to estimate
the volume. Why would a manufacturer make a narrow and tall tin can as
opposed to a wider can for paint? If you have an actual cylindrical carton
available, show students what the net would look like before starting with
problem 13. Toilet paper rolls are also useful.
English Language Learners
For problem 13b, you might want to explain what is meant by the mantle
of the mold.
38
Building Formulas
Solutions and Samples
Hints and Comments
13. a. Check the diameter of the circle students
made. Neatness of the drawing is important.
Materials
b. Note that the height of the mold is not
important.
Answers for the height will vary. Check the use
of measurement units.
Accept answers between 18 and 19 centimeters
for the width.
c. The formula gives Valerie the width of the
rectangle she needs for the mantle of her mold.
14. a. Juice cans are usually the shape of a cylinder.
b. Answers will vary. Most students will answer
that it is possible. Juice cans that look
completely different may contain the same
amount of liquid.
compass (one per student);
sheets of blank paper (one per student)
Overview
Students investigate the shape of cans and use a
formula to find the dimensions.
About the Mathematics
In this part of Section D, formulas for volume and
area, which students were introduced to in the unit
Reallotment, are reviewed and used in different
contexts. The net of a mold for candles and the net of
a juice can allow students to calculate the surface area
of a cylinder.
Planning
Students may work on problems 13 and 14 in pairs or
small groups.
Section D: Formulas and Geometry
38T
D Formulas and
Formulas and Geometry D
Geometry
This juice can is made up of two circles and a rectangle.
Notes
15a Before starting this
page, compare and
contrast the formulas for
area and circumference.
15b If the volume formula
is new to students, explain
the terms Base and Height.
15c Students need to find
out themselves that they
need the circumference of
a circle to find the area of
the mantle.
Or you may prefer to give
students the formula for
the circumference of a
circle before they start
working on problem 15.
15c Some students may
need to roll a sheet of
paper to model the
rectangle and see that one
dimension is the
circumference of the base
of the cylinder.
The can shown in the drawing has a height of 15 cm. The diameter of
the bottom is 7 cm.
15. a. Calculate the area of the bottom of the can.
b. Calculate the volume of the can. Remember that the formula
for the volume of any cylinder is:
Volume area of Base Height
c. What are the measurements of the rectangle that makes the
sides of the can?
d. The can is made of tin. How much tin (in cm2) is needed to
make this can?
This type of fruit juice is also available in cans that are twice as high.
16. a. Compare the amounts of fruit juice that each can contains.
b. How do the surface areas of the cans compare? Be prepared to
explain your answer without making calculations.
17. Suppose one can has double the diameter of another can.
a. Do you think the amount of liquid that fits in the larger can will
double? Give mathematical reasons to support your answer.
b. What can you tell about the surface area of the larger can
compared to that of the original can?
Reaching All Learners
Hands-On Learning
If students find it difficult to answer the questions on this page, it may be
helpful if you physically place one can on top of the other to see that the
volume doubles if the height doubles. In order to show what happens if
the diameter doubles, suggest that they draw circles to visualize the two
cylinders. First they trace the end of a can to represent the smaller can.
Then they use a compass to draw the larger circle.
Extension
After completing problem 15c, you might ask students why you should not
use the table made for problem 5a (on page 35) to calculate (28.3 50) ÷ 2
39.15. Discuss the difference between this type of table (which relates
one value to another by a formula) and a ratio table (which only includes
equivalent ratios).
39
Building Formulas
Solutions and Samples
Hints and Comments
15. a. The area of the bottom of the can is 38.5 cm2,
or 38.5 square centimeters.
The radius is 7 2 3.5 cm.
area π 3.5 3.5
Materials
b. The volume is 577.5 cm3 or 578 cm3 or 577.5
(578) cubic centimeters.
volume 38.5 15
c. The area of the rectangle is 330 cm2.
One side of the rectangle has the measure of
the height of the can, which is 15 centimeters.
One side of the rectangle has the measure of
the circumference of the can.
Use the formula for the circumference of a circle:
circumference π diameter
or
circumference 2 π radius
circumference π 7 ≈ 22 centimeters
area of the rectangle: 15 22 330 cm2
calculators (one per student)
Overview
Students calculate the surface area and volume of
cans. They investigate the influence of a change in
dimensions of the cans.
d. Tin needed for the can: 330 38.5 38.5 407 cm2. Note that the bottom and the top of
the can have the same area.
16. a. If the can becomes twice as high, the volume is
multiplied by 2. Explanations will vary.
Sample explanation:
The volume of a can can be computed with:
volume area of base height
Now multiply height by two:
volume area of base 2 height
Some students may need to use examples of
actual measurements.
b. The total area changes but less than two times.
Explanations will vary. Sample explanation:
The area of the top and the bottom of the can
does not change. The side of the rectangle that
is the circumference of the circle does not
change. The other side of the rectangle (the
height) is multiplied by two, so the area of the
rectangle is multiplied by two.
17. a. If the diameter of a can is doubled, the volume
is not doubled but multiplied by four.
Explanations will vary. Sample explanation:
If the diameter of a can is doubled, the radius is
also doubled. Since the base is a circle, the area
of the circle will be multiplied by four.
area1 π radius radius
area2 π (2 radius) (2 radius)
4 π radius radius
The height does not change, so the volume will
be multiplied by four.
Some students may need to use examples of
actual measurements.
b. The total area changes more than two times
but less than four times. Explanations will vary.
Sample explanation:
The surface area of top and bottom is multiplied
by four. The shortest side of the rectangle does
not change. The longest side is multiplied by
two, so the area of the rectangle is multiplied
by two.
Section D: Formulas and Geometry
39T