Mathematics
Grade 10
15 min
Multiply two unit fractions using models
Multiply two unit fractions using models
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1
Introduction & Learning Objectives
Learning Objectives
Model the product of two unit fractions as a fractional area on the face of a unit cube.
Represent the multiplication of unit fractions, 1/a × 1/b, as the area of a rectangle with vertices at (0,0,0), (1/a,0,0), (1/a,1/b,0), and (0,1/b,0) within a 3D coordinate system.
Extend the 2D model to three unit fractions to calculate a fractional volume within a unit cube.
Prove that the area of a fractional cross-section, represented by 1/a × 1/b, is geometrically equivalent to 1/(ab) by partitioning a unit square.
Analyze how scaling the dimensions of a prism's base by unit fractions affects its total surface area and volume.
Apply the concept to solve problems involving fractional cross-sectional areas of prisms and other 3D figures.
How can slicing a simp...
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Key Concepts & Vocabulary
TermDefinitionExample
Unit FractionA rational number written as a fraction where the numerator is 1 and the denominator is a positive integer.1/2, 1/5, and 1/12 are all unit fractions.
Unit CubeA cube whose edges are 1 unit in length. It serves as a fundamental model in 3D coordinate geometry, with a volume of 1 cubic unit and each face having an area of 1 square unit.A cube with vertices ranging from (0,0,0) to (1,1,1) in a Cartesian coordinate system.
Cross-SectionThe two-dimensional shape exposed when a three-dimensional solid is intersected by a plane.Slicing a rectangular prism parallel to its base reveals a rectangular cross-section identical in shape and size to the base.
Area Model of MultiplicationA visual representation where the product of two numbers is modeled as the area of...
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Core Formulas
Area of a Fractional Cross-Section on a Unit Face
A = \frac{1}{a} \times \frac{1}{b} = \frac{1}{ab}
This formula calculates the area (A) of a rectangular region on a unit square face of a 3D figure, where the region's dimensions are the unit fractions 1/a and 1/b. The model demonstrates that this area is one part of a grid containing 'a × b' total congruent parts.
Volume of a Fractional Prism within a Unit Cube
V = \frac{1}{a} \times \frac{1}{b} \times \frac{1}{c} = \frac{1}{abc}
This formula extends the area model into three dimensions. It calculates the volume (V) of a rectangular prism within a unit cube, where the prism's dimensions are the unit fractions 1/a, 1/b, and 1/c. The model shows this volume is one part of a larger unit cube partitioned into...
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Challenging
The area model provides a geometric proof that 1/a × 1/b = 1/(ab). The validity of this proof relies on which fundamental geometric axiom?
A.The Area Postulate, which states that the area of a rectangle is the product of its base and height.
B.The Parallel Postulate, which deals with parallel lines.
C.The Pythagorean Theorem, which relates the sides of a right triangle.
D.The definition of a circle, where all points are equidistant from the center.
Challenging
A rectangular area is modeled on a unit square with side lengths 1/a and 1/b, where a and b are distinct integers greater than 1. If the sum of the denominators, a + b, is 10, what is the smallest possible area for this rectangle?
A.1/16
B.1/21
C.1/25
D.1/24
Challenging
A small prism with dimensions 1/2, 1/3, and 1/4 is removed from one corner of a unit cube. What is the change in the total surface area of the cube?
A.The surface area decreases by 1/24.
B.The surface area increases by 1/12.
C.The surface area does not change.
D.The surface area decreases by 1/4.
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