Mathematics
Grade 10
15 min
Multiplication input/output tables
Multiplication input/output tables
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1
Introduction & Learning Objectives
Learning Objectives
Construct a multiplication input/output table to model the dilation of a circle.
Calculate the new radius, circumference, and area of a circle after applying a given scale factor.
Write the equation of a new circle after it has been dilated from its center.
Differentiate between the effect of a scale factor on a circle's linear dimensions (radius, circumference) versus its area.
Determine the scale factor required to transform a circle's area or circumference to a new given value.
Analyze the relationship between the input (scale factor) and various outputs (radius, area) using a table.
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Key Concepts & Vocabulary
TermDefinitionExample
Input/Output TableA table that organizes the relationship between an input value and a corresponding output value based on a specific rule. In this context, the input is the scale factor, and the output is a property of the new circle.If the rule is 'New Radius = Old Radius × Input', and the old radius is 5, an input of 2 gives an output of 10.
Scale Factor (k)The constant multiplicative value used to enlarge or reduce a geometric figure. A scale factor k > 1 results in an enlargement, while 0 < k < 1 results in a reduction.Applying a scale factor of 3 to a circle with a radius of 4 units results in a new circle with a radius of 12 units.
DilationA transformation that changes the size of a figure but not its shape. All lengths are multiplied by the...
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Core Formulas
Radius Scaling Rule
r_{new} = k \cdot r_{original}
To find the new radius after a dilation, multiply the original radius by the scale factor 'k'.
Circumference Scaling Rule
C_{new} = k \cdot C_{original}
Since C = 2Ï€r, the circumference scales linearly with the radius. Multiply the original circumference by the scale factor 'k' to find the new circumference.
Area Scaling Rule
A_{new} = k^2 \cdot A_{original}
Since A = πr², the area is proportional to the square of the radius. To find the new area, multiply the original area by the square of the scale factor, 'k²'.
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Challenging
The ratio of the areas of two circles is 9:4. The circumference of the larger circle is 18Ï€. Which input/output table entry correctly describes the smaller circle as a dilation of the larger one?
A.Input (k) = 2/3, Output (New Radius) = 6
B.Input (k) = 4/9, Output (New Radius) = 4
C.Input (k) = 3/2, Output (New Radius) = 13.5
D.Input (k) = 2/3, Output (New Radius) = 4
Challenging
A circle undergoes two successive dilations from its center. The first is by a scale factor of 4. The second is by a scale factor of 0.25. Which single entry in an input/output table represents the final state of the circle relative to its original state?
A.Input (k) = 4.25, Output (New Area) = 18.0625 * A_original
B.Input (k) = 1, Output (New Area) = A_original
C.Input (k) = 3.75, Output (New Area) = 14.0625 * A_original
D.Input (k) = 16, Output (New Area) = 256 * A_original
Challenging
A student creates the following input/output table for a circle with original radius 10. Identify the error. | Input (k) | Output (New Radius) | Output (New Area) | |---|---|---| | 2 | 20 | 200Ï€ | | 3 | 30 | 300Ï€ |
A.The New Radius column is incorrect; it should be k + 10.
B.The New Area column is incorrect; it was scaled by k instead of k².
C.The New Radius column is incorrect; it should be k² * 10.
D.There are no errors in the table.
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