Mathematics
Grade 10
15 min
Multiply fractions by whole numbers using number lines
Multiply fractions by whole numbers using number lines
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1
Introduction & Learning Objectives
Learning Objectives
Model the multiplication of a fraction by a whole number on a number line to represent the cumulative dimensions of 3D objects.
Apply number line visualization to calculate the total volume of multiple identical pyramids or cones.
Use a number line to determine the total length or height of a composite figure made of identical, fractionally-sized components.
Translate word problems involving stacking or sectioning 3D figures into a 'whole number × fraction' multiplication problem.
Prove the commutative property of multiplication (a × (b/c) = (b/c) × a) by constructing and interpreting a number line model in a geometric context.
Analyze and deconstruct complex 3D problems into repeated additions of a fractional quantity, suitable for number line re...
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Key Concepts & Vocabulary
TermDefinitionExample
Number Line ModelA visual representation of numbers as points on a line. For an expression like 'n × a/b', it represents 'n' consecutive jumps, each of size 'a/b', starting from zero.To show 3 × (2/5), you start at 0 and make 3 jumps, each of size 2/5, landing on 6/5.
Iterative AdditionThe principle that multiplication is a form of repeated addition. This is the core concept visualized by the number line jumps.The total volume of 4 stacked blocks, each with a volume of 3/4 m³, is (3/4) + (3/4) + (3/4) + (3/4), which is equivalent to 4 × (3/4) m³.
Unit FractionA fraction where the numerator is 1 (e.g., 1/c). In 3D contexts, this could be the fractional height of one layer or the fractional volume of one component relative to a whole.If a...
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Core Formulas
Multiplication as Repeated Jumps
n \times \frac{a}{b} = \underbrace{\frac{a}{b} + \frac{a}{b} + \dots + \frac{a}{b}}_{n \text{ times}}
To multiply a whole number 'n' by a fraction a/b on a number line, you start at zero and make 'n' consecutive jumps. The size of each jump is a/b.
The Multiplication Algorithm
n \times \frac{a}{b} = \frac{n \times a}{b}
This formula represents the final position after making the jumps on the number line. The total distance is the number of jumps ('n') times the numerator of the fraction ('a'), all over the original denominator ('b'), which defines the size of the fractional parts.
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Challenging
An artist is stacking 7 identical cubic blocks, each with a height of 2/3 meters, to build a column. Which statement accurately analyzes the total height on a number line and its implication for building a column of exactly 5 meters?
A.The total height is 14/3 m, or 4 2/3 m. The final jump lands 1/3 m short of 5 m, implying an 8th block would be too tall.
B.The total height is 14/21 m. This is less than 1 m, so many more blocks are needed.
C.The total height is 21/3 m, or 7 m. This is 2 m taller than the target.
D.The total height is 7 and 2/3 m. This is 2 and 2/3 m taller than the target.
Challenging
To prove the commutative property, 6 × (2/3) = (2/3) × 6, in the context of a rectangular prism's volume (V = lwh), how could a number line be used conceptually?
A.Model 6 jumps of size 2/3. This has no connection to a prism's volume.
B.Model the base area. Let a prism have a base of 6m by 2/3m. The area is 6 × (2/3) = 4. This can be shown as 6 jumps of 2/3. The same area results from a 2/3m by 6m base, proving commutativity geometrically.
C.Model 6 jumps of size 2/3 and 2/3 jumps of size 6. Since the second model is impossible to draw, the property cannot be proven this way.
D.Model the volume of 6 prisms, each with a volume of 2/3. Then model the volume of 2 prisms, each with a volume of 3. The results are different, so it's not commutative.
Challenging
A complex robotic arm is assembled from 9 identical cylindrical segments. Each segment has a length of 3/4 inches and a volume of 1/8 cubic inches. To find the total length of the arm, a technician draws a number line. Which number line representation is correct, and what is the final length?
A.9 jumps of size 1/8, resulting in a length of 9/8 inches.
B.single jump from 0 to 9, partitioned into 3/4, resulting in a length of 6.75 inches.
C.9 jumps of size 3/4, but the partitions are in eighths because of the volume, resulting in a length of 27/8 inches.
D.9 jumps of size 3/4, on a number line partitioned into fourths, resulting in a length of 27/4 inches.
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