Mathematics
Grade 10
15 min
Multiply fractions by whole numbers: input/output tables
Multiply fractions by whole numbers: input/output tables
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Introduction & Learning Objectives
Learning Objectives
Model the scaling of a single dimension of a 3D figure using an input/output table.
Define a linear function (the rule) for an input/output table that involves multiplying a fractional dimension by a whole number scaling factor.
Use input/output tables to generate a set of corresponding side lengths for a family of similar 3D figures.
Analyze the relationship between the whole number input (scaling factor) and the fractional output (new dimension) as a direct proportion.
Apply the concept of multiplying fractions by whole numbers to calculate the dimensions of scaled prisms, cylinders, and pyramids.
Prove that the ratio of corresponding side lengths generated from an input/output table is constant, confirming geometric similarity.
Ever wonder how 3D artis...
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Key Concepts & Vocabulary
TermDefinitionExample
Input/Output TableA table that organizes the relationship between two sets of values. An input value (x) is transformed by a rule to produce an output value (y). In this context, the input is a scaling factor, and the output is a new dimension.If the rule is y = (1/2)x, an input of 4 results in an output of 2.
Scaling Factor (k)A number by which the linear dimensions of a geometric figure are uniformly multiplied to create a similar figure. In this lesson, we focus on whole number scaling factors.Applying a scaling factor of 3 to a cube with a side length of 2 cm results in a similar cube with a side length of 6 cm.
Similar SolidsThree-dimensional figures that have the same shape but not necessarily the same size. The ratio of their corresponding linear dimensions is...
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Core Formulas
Multiplication of a Fraction by a Whole Number
\frac{a}{b} \cdot c = \frac{a \cdot c}{b}
To multiply a fraction by a whole number, multiply the numerator of the fraction by the whole number and keep the denominator the same. The whole number 'c' can be thought of as the fraction c/1.
Scaling Function for a Linear Dimension
d' = k \cdot d_{0}
The new dimension (d') of a scaled figure is found by multiplying the original dimension (d₀) by the whole number scaling factor (k). In an input/output table, d₀ is the constant in the rule, k is the input, and d' is the output.
Ratio of Corresponding Sides
\frac{d'_{1}}{d_{1}} = \frac{d'_{2}}{d_{2}} = k
For two solids to be similar, the ratio of any pair of corresponding linear dimensions must...
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Challenging
In an input/output table for scaling a pyramid's height, an input of k=3 produces an output of 11/2 units, and an input of k=5 produces an output of 55/6 units. What scaling factor 'k' is required to produce a new height of exactly 22 units?
A.11
B.12
C.6
D.4
Challenging
Given an input/output table defined by the rule d' = k * d₀, which of the following statements provides a formal proof that the ratio of any two scaled dimensions is equal to the ratio of their corresponding scaling factors?
A.Since d'₁ = k₁d₀ and d'₂ = k₂d₀, then d'₁ + d'₂ = d₀(k₁ + k₂).
B.Since d'₁ = k₁d₀ and d'₂ = k₂d₀, then d'₁d'₂ = (k₁k₂)d₀².
C.Since d'₁ = k₁d₀ and d'₂ = k₂d₀, the ratio d'₁/d'₂ = (k₁d₀)/(k₂d₀). The d₀ terms cancel, proving d'₁/d'₂ = k₁/k₂.
D.Since d'₁/k₁ = d₀ and d'₂/k₂ = d₀, then d'₁/k₁ = d'₂/k₂, which proves d'₁k₂ = d'₂k₁.
Challenging
A student incorrectly assumes the rule for a scaling table is additive and creates the rule y = x + 3/4. The correct rule should be multiplicative for an original dimension of 3/4. For which whole number input (scaling factor) do both the incorrect additive rule and the correct multiplicative rule produce the same output?
A.Because the ratio of Output/Input (y/x) is not constant, which is required for similarity.
B.Because the additive rule sometimes produces fractional outputs.
C.Because the additive rule only works for scaling factors greater than 10.
D.Because the additive rule is more complex to calculate than the multiplicative rule.
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