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Graphing Number Patterns from Rule Tables
Learn to complete rule tables, plot ordered pairs on a coordinate plane, and describe how a number pattern grows from the graph.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
A rule table links an input to an output using a simple operation, such as add three. Each row becomes an ordered pair that can be plotted on a coordinate plane. By graphing several pairs from the same rule, you can see the pattern's growth as a straight line of points. This lesson shows how to complete tables, plot points accurately, and answer questions about growth using the graph.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
From rule table to ordered pairs
A rule table has two columns, usually input and output. The rule tells you how to calculate each output from its input. For example, the rule add three means output equals input plus three. Each row gives one ordered pair written as input comma output.
To complete a table, apply the rule to every input. Keep the same order in each row so the pair is easy to read. If the input is two and the rule is add three, the output is five, giving the pair two comma five.
- Write the rule at the top of the table so every row uses the same operation.
- For each input, calculate the output by applying the rule once.
- Record the pair as input comma output, keeping the input first.
- Check one row by reversing the rule mentally to see if the input returns.
Plotting points on the coordinate plane
The coordinate plane has a horizontal axis and a vertical axis that meet at the origin. The first number in a pair tells how far to move along the horizontal axis. The second number tells how far to move along the vertical axis.
Start at the origin each time. Move right for a positive first number, then move up for a positive second number. Mark the point where those two moves meet. Label points only if the task asks for labels; otherwise a clear dot is enough.
- Draw and label both axes, marking equal spacing along each one.
- Start at the origin for the first pair and move along the horizontal axis by the first number.
- From that position, move parallel to the vertical axis by the second number.
- Mark the point with a small dot and repeat for each row of the table.
- Check that every plotted point matches a row in the table.
Reading growth from the graph
When the rule adds the same amount each time, the plotted points rise steadily. The vertical change between neighbouring points equals the number added by the rule. For the rule add three, each step to the right increases the output by three.
You can answer growth questions by reading the graph rather than recalculating. Find two neighbouring points, compare their vertical positions, and state the change. If the points do not rise steadily, recheck the table and the plotting.
- Choose two neighbouring plotted points from the same table.
- Compare their vertical positions to find the change in output.
- State the growth as the amount added for each increase of one in the input.
- Confirm the growth matches the rule in the table.
Worked example
A rule table uses the rule add four. The inputs are one, two, three, and four. Complete the table, plot the pairs on a coordinate plane, and state how the output grows as the input increases by one.
Worked answer
The outputs are five, six, seven, and eight, giving the pairs one comma five; two comma six; three comma seven; four comma eight. Plotted from the origin, the points rise steadily. Each time the input increases by one, the output increases by four, matching the rule add four.
Four practice exercises
Exercise 1
A rule table uses the rule add two. The inputs are zero, one, two, and three. Complete the table, plot the four pairs on a coordinate plane, and state the growth in the output when the input increases by one.
Show answer and check
The outputs are two, three, four, and five, giving the pairs zero comma two; one comma three; two comma four; three comma five. The points rise steadily, and the output grows by two for each increase of one in the input.
Why: Apply add two to each input: zero plus two is two, one plus two is three, two plus two is four, and three plus two is five. Plot each pair from the origin, moving right by the input and up by the output. Comparing neighbouring points shows a vertical change of two.
Check: Recalculate one row by subtracting two from its output to recover the input, then read one plotted point back from the graph and compare it with its table row.
Exercise 2
A rule table uses the rule add five. The inputs are one, two, three, and four. Complete the table, plot the pairs, and explain why the points rise by the same amount each time.
Show answer and check
The outputs are six, seven, eight, and nine, giving the pairs one comma six; two comma seven; three comma eight; four comma nine. The points rise by five each time because the rule adds five to every input.
Why: Each output equals its input plus five. Since the same amount is added to every input, each step to the right increases the output by exactly five, so the vertical change between neighbouring points is constant.
Check: Verify two neighbouring pairs by subtracting five from each output to recover the inputs, and confirm that the vertical gap between those plotted points equals five grid units.
Exercise 3
A rule table uses the rule subtract one. The inputs are three, four, five, and six. Complete the table, plot the pairs, and describe how the output changes as the input increases by one.
Show answer and check
The outputs are two, three, four, and five, giving the pairs three comma two; four comma three; five comma four; six comma five. As the input increases by one, the output increases by one, because each output is exactly one less than its input; when the input goes up by one, the output also goes up by one.
Why: Apply subtract one to each input: three minus one is two, four minus one is three, five minus one is four, and six minus one is five. The outputs rise by one between neighbouring rows because each output is one less than its input, and the inputs increase by one each row.
Check: Add one to each output to recover the input, then compare two neighbouring plotted points to confirm the vertical change is one unit.
Exercise 4
A rule table uses the rule add three. The inputs are two, four, six, and eight. Complete the table, plot the pairs, and state the growth in the output when the input increases by two.
Show answer and check
The outputs are five, seven, nine, and eleven, giving the pairs two comma five; four comma seven; six comma nine; eight comma eleven. When the input increases by two, the output increases by six.
Why: Apply add three to each input: two plus three is five, four plus three is seven, six plus three is nine, and eight plus three is eleven. Neighbouring inputs differ by two, and each output is three more than its input, so the output rises by six between neighbouring rows.
Check: Subtract three from each output to recover the inputs, then compare two neighbouring plotted points to confirm the vertical change is six units while the horizontal change is two units.
Common mistakes
- Plotting the output first and the input second. Always plot the input along the horizontal axis and the output along the vertical axis, matching the table order.
- Applying the rule twice to one input. Apply the rule exactly once per row; recalculate slowly and compare with a neighbouring row.
- Starting the second point from the first point instead of the origin. Return to the origin before plotting each new pair so every point is placed independently.
- Assuming the points must form a perfect straight line even when the table has a mistake. Check each row against the rule first; correct the table before judging the graph.
Self-check rubric
- The table is completed correctly for every input using the stated rule.
- Each ordered pair is written with the input first and the output second.
- Each point is plotted from the origin with the correct horizontal and vertical moves.
- The growth statement matches the rule and is supported by two neighbouring points.
- Valid alternative presentations, such as a neatly labelled grid with equal spacing, are accepted if the positions are correct.
Questions and answers
Do I have to connect the points with a line?
Only if the task asks. For a rule table, separate dots are enough to show the pattern, and connecting them is optional unless stated.
What if the rule subtracts instead of adds?
The same steps apply. The output is smaller than the input, but if the inputs increase, the outputs may still increase; the direction of the points depends on how the inputs change.
How can I check a plotted point quickly?
Read the point back from the graph: state its horizontal position, then its vertical position, and compare with the table row.
Why do the points rise by the same amount each time?
Because the rule adds the same number to every input, so each step to the right increases the output by that same number.