Geometry Items to Support Formative Assessment
Unit 4: Extending to Three Dimensions
G.GMD.A.1 Give an informal argument for the formulas for the volume of a cylinder,
pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit
arguments.
Item 1:
The volume of a cylinder is 102 cm3. Use what you know about the formula for the volume of
cylinders and cones to calculate the volume of a cone that has the same height and the same base
radius as the cylinder. Explain how you determined your answer.
Solution: 34 cm3. The volume of a cone is one-third the volume of a cylinder with the same
height and base radius, so I divided 102 by 3.
Item 2:
How is determining the volume of a cone similar to determining the volume of a pyramid? How
is it different?
Possible solution: You need to multiply by one-third to calculate the volume of both figures.
The difference is the area of the base since one is a rectangle and one is a circle.
Item 3:
The volume of a pyramid is 61 cm3. Use what you know about the formula for the volume of
pyramids and prisms to calculate the volume of a prism that has the same height and the same
base as the prism. Explain how you determined your answer.
Solution: 183cm3. The volume of a pyramid is one-third the volume of a prism with the same
height and base, so I multiplied 61 by 3.
Item 4:
Layla is working for a soft serve ice cream shop. The shop serves two different types of cones: a
cake cone and a sugar cone. She wants to know which type holds more ice cream in the cone.
Both cones have the same height and a diameter of 3 inches at the opening of the cone. The cake
cone consists of two cylinders, where the top cylinder is a third of the height of the entire cone.
Use what you know about the volume formulas to make an argument as to which cone will hold
the most ice cream.
Cake Cone:
Sugar Cone:
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Source: Google Images
Sample Answer:
Sugar Cone:
Cake Cone:
Volume of the top of cone:
The volume of the top of the cake cone is already the same volume as the sugar cone. Therefore,
the cake cone must hold more ice cream, since the cake cone still has a smaller cylinder for two
thirds of the remaining height.
G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
G.GMD.A.3/G.MG.A.3 Task:
Wally’s Winter Wonderland is offering a special workshop this holiday season. Patrons can
design their own snowman snow globe. The snow globes come in one standard size, a glass
sphere with a 12 cm diameter atop a wooden base. Participants will be able to design the
snowman that will be placed inside. They can either design a snowman whose body is made up
of 3 spheres or a 2-sphere snowman wearing a cylindrical top hat. In order for the glitter to
properly circulate throughout the snow globe, at least 50% of the volume of the globe must be
water. Design a snowman that can be placed in the standard snow globe.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Item 1:
Five cubes of ice with a base edge measuring 3 cm are melting in a cylindrical glass that has a
radius of 7 cm. How high will the liquid in the glass rise when all of the cubes have melted?
(Ignore the fact that ice takes up slightly more space than water).
Solution: about .87 cm
Item 2:
You are the restaurant manager of Renaissance Times. You recently ordered the three new types
of glasses for your guests, but you cannot remember which glass you had decided would be a
small, medium, or large. You do know all of the glasses have the same height, since they need to
fit in the kitchen dishwasher. Use the dimensions of each glass below to determine which glass
should be a small, which glass should be a medium, and which glass should be a large.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Answer:
Small Glass: Pyramid
Medium Glass: Cone
Large Glass: Cylinder
Item 3:
Ms. Greene, the manager of Paradise Lake, just ordered a duck and goose food dispenser like the
one shown below.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Source: https://www.gumball-machine.com/vending-machines/pet-food-vending-machines/coinoperated-duck-food-dispenser-1.html
She plans to use a paper cone to hold the pellets. The machine is set to dispense no more than
118 cm3 of food for each token. The cones Ms. Greene plans to use are 8cm in diameter. How
tall must the cone be to hold the food?
Solution: The height must be greater than 7 cm.
Item 4:
The Mapparium (shown below) in the Mary Baker Eddy library in Boston, MA is a three-story
stained-glass globe that depicts the 1935 perspective of the world.
http://www.marybakereddylibrary.org/exhibits/mapparium
More that 10 million visitors have crossed the thirty-foot glass bridge through the center of the
globe. What is the volume of the Mapparium?
Solution: about 14,137 cubic feet
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.