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Recognize proportional and non-proportional patterns informally

A proportional pattern relates two quantities through a constant multiplicative relationship: equivalent ratios or unit rates remain equal, and its table or graph pairs zero with zero and forms a line through the origin. Non-proportional patterns may have a changing ratio, a fixed starting amount, or an additive change without a constant ratio; distinguishing these in verbal descriptions, tables, graphs, and simple equations prepares learners for proportional reasoning and algebra without requiring formal slope calculations or generalized proofs.

Detailed Explanation: Recognize proportional and non-proportional patterns informally

A pattern is proportional when one quantity is always the same multiple of the other. You can check this by:

  1. Finding the unit rate: divide each output by its matching input.
  2. Seeing whether the unit rate stays the same.
  3. Checking that an input of 00 gives an output of 00.

Example: A taxi charges a starting fee of 3plusplus$2$ for each mile.

Miles, xxCost, yy
003$
115$
227$
339$

First, check what happens when the number of miles is 00. The cost is 3,not, not $0$. This means there is a fixed starting fee.

Next, compare the ratios for positive numbers of miles:

51=5,72=3.5,93=3\frac{5}{1}=5,\qquad \frac{7}{2}=3.5,\qquad \frac{9}{3}=3

The ratios are not equal, so there is no constant unit rate from miles to total cost.

Therefore, this is a non-proportional pattern. The cost increases by 2foreachmile,butitalsostartswithanextrafor each mile, but it also starts with an extra$3.Aproportionalpatternmusthaveaconstantratioandmustpair. A proportional pattern must have a constant ratio and must pair 0withwith0$.

Learn by doing: Recognize proportional and non-proportional patterns informally

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Ratios - Calculate Value from Item Ratio


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