Mathematics
Grade 10
15 min
Multiply fractions by whole numbers using models
Multiply fractions by whole numbers using models
What you'll learn
- Draw a visual model (like a bar or circle) to represent a fraction multiplied by a whole number with at least 80% accuracy.
- Solve multiplication problems involving fractions and whole numbers using visual models and write the answer as a fraction or mixed number with 75% accuracy.
- Explain how a visual model represents the multiplication of a fraction by a whole number in their own words, demonstrating understanding of repeated addition.
- Identify and correct errors in visual models representing fraction multiplication by whole numbers, given example models with errors.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Model the multiplication of a fractional quantity by a whole number using the volume of 3D figures.
Calculate the total volume of a composite solid formed by stacking multiple, identical fractional parts of a base 3D model.
Interpret word problems involving repeated fractional volumes and translate them into a mathematical expression.
Prove the total volume of `n` objects, each with a volume of `(a/b)V`, is equal to `(n*a/b)V`.
Apply the concept of multiplying fractional volumes to solve problems related to density, capacity, and scaling of 3D figures.
Visually deconstruct a composite 3D figure into its repeated fractional components to determine its total volume.
Imagine a 3D printer building an object layer by layer. If you stop it after 5 identical lay...
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Key Concepts & Vocabulary
TermDefinitionExample
Unit ModelA single, complete three-dimensional figure (e.g., a cube, prism, or cylinder) whose volume represents 'one whole' in a calculation.A rectangular prism with dimensions 2cm x 3cm x 4cm has a unit volume of 24 cm³. This prism is our 'unit model'.
Fractional VolumeA specified portion (a/b) of the total volume of the Unit Model.If a cylinder with a total volume of 100π cm³ is filled 2/5 of the way with water, the fractional volume of the water is (2/5) * 100π = 40π cm³.
Scalar MultiplierThe whole number that represents how many identical fractional volumes are being combined or considered.If you have 6 identical bottles, each 1/3 full, the scalar multiplier is 6.
Composite VolumeThe total volume obtained by adding the volumes of multiple, of...
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Core Formulas
Total Fractional Volume Formula
V_{total} = n \times (\frac{a}{b} \times V_{unit})
Use this to find the total volume. Here, 'n' is the whole number (scalar multiplier), '(a/b)' is the fraction of the unit volume being used, and 'V_unit' is the volume of the single 3D model.
Volume of a Rectangular Prism
V = l \times w \times h
Calculates the unit volume when the 3D model is a rectangular prism with length 'l', width 'w', and height 'h'.
Volume of a Cylinder
V = \pi r^2 h
Calculates the unit volume when the 3D model is a cylinder with radius 'r' and height 'h'.
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Challenging
A 3D model of a building is a prism with a volume of 500 cm³. An architect creates a larger version by scaling every linear dimension by a factor of 3. From this new, larger version, 5 identical 'floor sections' are modeled, where each section represents 1/9 of the larger version's volume. What is the total volume of the 5 floor sections?
A.7,500 cm³
B.2,500 cm³
C.1,667 cm³
D.13,500 cm³
Challenging
A proof for V_total = n × (a/b × V_unit) can be justified by considering the volume as a discrete sum. Which mathematical concept best formalizes the idea of adding 'n' identical fractional quantities?
A.The Fundamental Theorem of Calculus
B.An arithmetic series or summation notation
C.The Pythagorean Theorem
D.The Law of Sines
Challenging
A machine produces small cylindrical gears from a large stock cylinder (V_stock = 10,000π cm³). Each gear has a volume equal to 1/200 of the stock cylinder. The machine produces 30 gears per hour. What is the total volume of gears produced in an 8-hour shift?
A.9,600π cm³
B.150π cm³
C.12,000π cm³
D.4,000π cm³
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What grade level is "Multiply fractions by whole numbers using models"?
Multiply fractions by whole numbers using models is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Multiply fractions by whole numbers using models?
You'll be able to: Draw a visual model (like a bar or circle) to represent a fraction multiplied by a whole number with at least 80% accuracy; Solve multiplication problems involving fractions and whole numbers using visual models and write the….
Is "Multiply fractions by whole numbers using models" free to practice?
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How many practice questions are included with Multiply fractions by whole numbers using models?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.