Mathematics
Grade 10
15 min
Multiply fractions and whole numbers: sorting
Multiply fractions and whole numbers: sorting
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1
Introduction & Learning Objectives
Learning Objectives
Apply formulas for the volume of prisms, pyramids, cones, and spheres.
Accurately calculate volumes of 3D figures where dimensions are given as whole numbers, fractions, or mixed numbers.
Fluently multiply fractions and whole numbers within the context of geometric formulas.
Compare the calculated volumes of multiple three-dimensional figures.
Sort a set of 3D figures in ascending or descending order based on their calculated volumes.
Justify the sorting order of figures using mathematical calculations as proof.
Imagine you're an architect comparing designs for a new skyscraper. How would you mathematically prove which of several 'pointy' glass pyramids has the greatest volume? 🏙️
This tutorial bridges a fundamental arithmetic skill—multip...
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Key Concepts & Vocabulary
TermDefinitionExample
VolumeThe measure of the amount of three-dimensional space an object occupies, expressed in cubic units (e.g., cmÂł, mÂł, ftÂł).A rectangular prism with side lengths 2m, 3m, and 4m has a volume of 2 Ă— 3 Ă— 4 = 24 mÂł.
Base Area (B)The area of the two-dimensional face that defines the base of a three-dimensional figure.For a cylinder with a radius of 5 inches, the Base Area is B = πr² = π(5)² = 25π in².
Improper FractionA fraction in which the numerator is greater than or equal to the denominator. Mixed numbers should be converted to improper fractions for easier multiplication.The mixed number 3 ½ is converted to the improper fraction 7/2.
Scalar Multiplication in GeometryThe process of multiplying the dimensions or properties of a figure by a constant number (a scalar),...
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Core Formulas
Volume of a Prism or Cylinder
V = B \times h
Used for figures with two parallel, congruent bases. Find the volume by multiplying the area of the base (B) by the figure's height (h).
Volume of a Pyramid or Cone
V = \frac{1}{3} B h
Used for figures with one base that rise to a single point (apex). The volume is one-third of the product of its base area (B) and its height (h). This formula requires multiplying by a fraction.
Volume of a Sphere
V = \frac{4}{3} \pi r^3
Used for perfectly round three-dimensional objects. The volume is calculated using its radius (r). This formula requires multiplying by the fraction 4/3.
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Challenging
You need to sort four figures in ascending order of volume. A: Cube with side 3. B: Sphere with radius 2. C: Rectangular prism with dimensions 2 x 3 x 4. D: Square pyramid with base side 4 and height 4 1/2.
A.D, C, A, B
B.C, D, A, B
C.C, A, D, B
D.D, A, C, B
Challenging
A prism has a base area B and height h. A pyramid has the same base area B. For the pyramid's volume to be exactly equal to the prism's volume, what must the pyramid's height be in relation to the prism's height h?
A.The pyramid's height must be 3h.
B.The pyramid's height must be (1/3)h.
C.The pyramid's height must be h.
D.The pyramid's height must be 9h.
Challenging
Three figures all have the same base radius 'r'. Figure A is a cylinder with height 'r'. Figure B is a cone with height '3r'. Figure C is a sphere with radius 'r'. Sort these figures in ascending order of volume.
A.A, B, C
B.B, A, C
C.= B, C
D.C, A = B
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