Graphing Patterns on the Coordinate Plane: Plot Sequences and State the Rule

Algebra and Function Practice

When a numeric pattern is written as ordered pairs, graphing those pairs on a coordinate plane makes the pattern visible. Each point represents one term: the first coordinate is the term number, and the second coordinate is the term value. Connecting the points reveals whether the pattern grows steadily, shrinks, or changes in another way. In this lesson you will plot sequences, connect points in order, and write the pattern rule in words. You will also learn a convention for sketching points and a rubric that accepts valid alternative graph styles.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Reading ordered pairs as pattern terms

An ordered pair such as (3, 7) means the third term has value 7. The first coordinate is always the term number, and the second coordinate is the term value. A sequence of ordered pairs lists consecutive terms, so the first coordinates increase by 1 each time.

To plot a point, start at the origin (0, 0). Move right along the horizontal axis by the first coordinate, then move up along the vertical axis by the second coordinate. Mark the point and label it with its ordered pair if space allows. Use a consistent scale on each axis and state the scale near the axis.

1. Write the ordered pairs in a list and check that the first coordinates increase by 1.
2. Choose a scale that fits the largest first and second coordinates on your grid.
3. Plot each point by moving right from the origin, then up.
4. Label each point with its ordered pair or a small dot plus a letter.
5. Connect the points in order with a straight line or line segments.

Describing the rule in words

After plotting and connecting the points, compare consecutive second coordinates. If each term value increases by the same amount, the rule is add that amount to the previous term. If each term value decreases by the same amount, the rule is subtract that amount. If the change is not constant, describe the change you observe, such as doubling or alternating.

A complete rule statement names the starting value and the change. For example: start at 4 and add 3 to each term. You can also state the rule as a relationship between the term number and the term value, such as the term value is 3 times the term number plus 1. Both forms are valid when they correctly describe the plotted sequence.

1. Subtract each term value from the next term value to find the change.
2. Check whether the change is the same between every pair of consecutive terms.
3. Identify the first term value from the first ordered pair.
4. Write the rule as start value and change, or as a relationship between the two coordinates.
5. Test the rule on at least two ordered pairs to confirm it matches the graph.

Sketching convention and valid alternatives

For hand-drawn graphs, use this convention: draw the horizontal axis and vertical axis, mark equal spacing between grid lines, and write the scale. Plot points as small dots or crosses. Connect points with straight segments in the order of the term numbers. Label the axes with the quantity they represent, such as term number and term value.

Valid alternatives include using crosses instead of dots, drawing a single straight line through all points when the pattern is linear, or using a different but clearly labeled scale. Handwriting style varies, so point labels may be printed or cursive as long as they are readable. The rubric below accepts any representation that correctly shows the plotted points, the connection order, and a rule statement consistent with the graph.

1. Draw and label both axes with a stated scale.
2. Plot each ordered pair as a small dot or cross.
3. Connect points in order with straight segments or one straight line if appropriate.
4. Write the rule in words near the graph.
5. Check that the graph and rule agree for every plotted point.

Worked example

Plot the sequence (1, 4), (2, 7), (3, 10), (4, 13), (5, 16). Connect the points in order and write the rule in words.

Four practice exercises

Common mistakes

• Plotting the first coordinate as the vertical movement and the second as the horizontal movement. Always move right by the first coordinate, then up by the second coordinate. Re-plot one point and compare it with the ordered pair.
• Connecting points in an order that does not follow increasing term numbers. List the ordered pairs by term number first, then connect them in that listed order.
• Writing a rule that describes only the first change and not the whole sequence. Check the change between every pair of consecutive terms before writing the rule.
• Using an inconsistent scale on an axis, so equal grid spacing represents different amounts. State the scale and keep equal spacing for equal amounts on each axis.

Self-check rubric

• All ordered pairs are plotted with the first coordinate as horizontal movement and the second as vertical movement.
• Points are connected in order of increasing term number.
• The scale is stated and equal spacing represents equal amounts on each axis.
• The rule statement matches every plotted point and names the start value and change, or gives a correct relationship.
• The graph is readable, with labeled axes or a clear key, and point labels may be printed or cursive.

Questions and answers