Tables of values and the Cartesian plane
A rule, a table and a graph are three different ways of showing the same relationship. Earlier you read relationships off finished graphs; now you learn to build a graph yourself, starting from a rule, generating a table of values, and plotting those values on the Cartesian plane. Moving smoothly between these three forms is one of the most useful skills in algebra.
This year you learn to generate tables from a rule or a growing pattern and to plot the values as points. The plotted points reveal the shape of the relationship, and for the linear rules you meet this year, that shape is always a straight line.
From a rule to a table
A rule such as y equals 2x plus 1 is a recipe for turning any x into a y. To build a table, you choose some x values, often 0, 1, 2 and 3, and work out the y for each. When x is 3, the rule gives 2 times 3 plus 1, which is 7. Doing this for every x fills the table, and the table is simply the rule applied several times over. Looking down the y column, you can often spot a pattern: here each y is 2 more than the last, matching the 2 in the rule.
The relationship between adjacent values is worth noticing. Because the rule multiplies x by 2, every step of 1 in x produces a step of 2 in y. This steady step is what makes the pattern linear, and it is the same constant change that will show up as a straight line when you plot the points. Reading a table for these step-by-step patterns deepens your sense of how the rule behaves.
Plotting on the Cartesian plane
The Cartesian plane has two axes, a horizontal x-axis and a vertical y-axis, meeting at the origin. Every row of your table is a coordinate pair, written (x, y), that marks a single point. The pair (3, 7) means go 3 along the x-axis and 7 up the y-axis. Plotting each pair from the table places a series of points on the plane.
When the points come from a linear rule, they line up perfectly, and joining them produces a straight line, the graph of the rule. This completes the journey from rule to table to graph, and you can travel it in any direction: read a rule from a pattern of points, build a table to find missing values, or plot a table to picture a relationship. Holding all three views together, and moving between them with ease, is exactly the fluency this part of the curriculum is building toward.
Tables from a growing pattern
A table need not start from a written rule; it can come straight from a picture. A pattern of dots that grows figure by figure can be counted, and those counts fill a table just as a rule would. Reading a visually growing pattern into numbers is one of the two ways the curriculum asks you to generate a table.
What the constant does on the plane
Plotting two rules side by side shows what each number in a rule controls. The lines y equals 2x and y equals 2x plus 3 are equally steep, but the plus 3 lifts every point up by 3, so the constant decides where the line crosses the y-axis. Seeing this on the plane links the symbols of a rule to the picture it draws.
Reading a rule out of a table
The journey runs both ways: a table can hand you back its rule. If the y values climb by a steady amount each step, that step is the coefficient, and one pair pins down the constant. From x of 1, 2, 3, 4 giving y of 5, 7, 9, 11, the steady rise of 2 and the value 5 at x of 1 point straight to y equals 2x plus 3.
Teaching tip: a sheet of grid paper and a simple rule turn this into a satisfying hands-on task. Ask the student to choose the x values, fill the table, and plot the points themselves. The moment the scattered dots reveal a perfectly straight line is a small thrill that fixes the idea far better than watching it done.
A common slip is reversing the coordinates, plotting (x, y) as if it were (y, x). Reinforce that the first number is always the across value and the second is always the up value, and have the student say across then up each time until the order becomes automatic.