Students work with numerical patterns in a significant way by grade 3. This work includes numerical patterns found in t-charts/tables. A t-chart/table can be oriented two ways.
We will use the second orientation as the style.

Consider the t-chart shown on the right.

There are two kinds of related numerical patterns in the t-chart. There is the vertical (down) pattern and the horizontal (across) pattern.

The vertical pattern is easiest for students to figure out. It corresponds to a sequence of numbers that can also be described as the relationship between the outputs. In the case of the sample here, the sequence is 2, 4, 6, 8, 10. One way to describe the vertical pattern is that you add 2 each time. Another way to describe it is that you skip count by 2 each time.

The horizontal pattern is the relationship between the input and output numbers. This input/output rule allows you to figure out an output value for any given input value. You do not need to figure out all of the output values that come before it. In the case of the sample t-chart, one way to describe the rule is: double the input to get the output. Note that the rule must work for each pair of input/output values. It also must be constant - that is to say, the rule CANNOT change from row to row.

An input/output diagram can be used to show the relationship between input and output:

The input/output rule expresses a relationship between two things. For example, there is a relationship between how long a shower is on and the amount of water that is used. One can talk about the relationship qualitatively: "The longer the shower is on, the more water is used." One can also talk about the relationship quantitatively: "Every minute the shower is on, 23 litres of water are used." In order to talk quantitatively, one must measure things. In the case of the shower example, one must measure time and volume. Quantifying a relationship has advantages. In the case here, it would be useful for predicting how much water will be used if the shower were to be left on for two years. One does not actually have to waste water to answer the question. In other words, quantifying a relationship makes it possible to make predictions that can be useful for some human purpose such as designing automobile engines or determining the future sewage treatment needs of a city.

The input/output rule should first be developed by non-numerical contexts. The following are some possibilities:

Have students work with letters of the alphabet, where each letter is put through a shape changing machine. Discuss what kind of shape-changing rule to use. One possibility is: 'change sharp corners to curves'.

Have students provide a visual representation of the input/output rule (like the example below) to assist students in understanding its parts (input, output, rule).

Mixing colours could be another context. Have students work with a variety of paint colours. The input/output rule could be: 'mix red paint with another paint'. Have students predict and test the result of applying the rule.
For more details on numerical rules refer to:
Grade 4 Numerical Patterns
Try these:
Determine the input/output rule for each t-chart.