Multiplication input/output tables: find the rule

Learning Objectives Identify the constant of proportionality in tables relating different units of angle measure, such as degrees and radians. Formulate a multiplicative rule of the form y = kx to model the relationship between inputs and outputs in a table. Determine the rule for angular displacement given a table of time and angle data, identifying the constant as the angular velocity. Apply the derived rule to predict unknown output values (e.g., angular displacement) for given input values (e.g., time). Verify that a rule is consistently multiplicative by testing the ratio y/x for multiple data pairs. Connect the multiplicative constant 'k' to its physical or geometric meaning, such as a conversion factor or a rate of rotation. How does a satellite orbiting Ear...

TermDefinitionExample Input/Output TableA table that organizes pairs of related numbers. For each input value (x), there is a corresponding output value (y) determined by a specific rule.A table with inputs (Time in s) of 1, 2, 3 and outputs (Angle in rad) of π/2, π, 3π/2. Multiplicative RuleA rule that describes a direct proportion between two variables, expressed as y = kx. The output (y) is obtained by multiplying the input (x) by a fixed, non-zero number (k).If the rule is y = 4x, an input of 3 gives an output of 12. Constant of Proportionality (k)The constant multiplier in a multiplicative rule. It is the ratio of the output to the input (k = y/x) and remains the same for every pair in the relationship.In the relationship between degrees (x) and radians (y), the constant is k = π/180...

The Rule-Finding Formula k = \frac{y}{x} To find the constant of proportionality (k) in a multiplicative relationship, choose any corresponding input (x) and output (y) pair from the table and calculate their ratio. This constant is the 'rule'. General Multiplicative Rule y = kx Once the constant (k) is found, this formula represents the rule for the entire table. You can use it to find any output (y) for a given input (x), or vice-versa. Angular Displacement with Constant Velocity \theta(t) = \omega t A specific application of y = kx in physics. The angular displacement (θ) is the output, time (t) is the input, and the constant angular velocity (ω) is the multiplicative rule (k).