Plotting Straight Line Graphs from a Table of Coordinates

In (2, 6), the first number is the x-coordinate so we go 2 units across, and the second number is the y-coordinate so we go 6 units up. In (6, 2), we go 6 units across and only 2 units up. The two points are in completely different positions on the grid. This targets the “coordinates are interchangeable” misconception — the belief that the order of numbers in a coordinate pair does not matter.

The order of coordinates always matters: the first number controls horizontal position and the second controls vertical position. A useful memory aid is “along the corridor and up the stairs” — you always go across (x) before going up (y). Swapping the numbers gives a different point every time (unless both numbers happen to be equal).

Consider a table with x = 0, 1, 2, 3 and y = 8, 6, 4, 2. Plotting the coordinates (0, 8), (1, 6), (2, 4), (3, 2) and joining them gives a straight line that goes downhill — starting high on the left and finishing low on the right. The y-value decreases by 2 for every increase of 1 in x, which produces a negative gradient. This challenges the “straight lines always slope upwards” misconception.

Students sometimes assume straight line graphs must slope upwards because many early examples show this. But any relationship where y decreases as x increases produces a downward-sloping line. Horizontal lines (where y stays the same) are also possible.

The x-coordinate of 1.5 lies exactly halfway between the grid lines at 1 and 2 on the x-axis. We locate this midpoint, then go straight up to a height of 4 on the y-axis. The point sits neatly between the grid lines — it doesn’t need to land on an intersection. This addresses the “coordinates must be whole numbers” misconception.

Coordinate values do not have to be whole numbers. Any decimal or fraction can be plotted by finding the correct position between grid lines. Students often think only whole numbers are allowed because grid lines are drawn at whole-number intervals, but the spaces between grid lines represent all the values in between.

The x-coordinate tells us the horizontal position on the grid. Positive x-values go to the right of the y-axis, and negative x-values go to the left. Since −1 is negative, we go 1 unit to the left of the y-axis, then 3 units up. The point ends up in the top-left area of the grid (the second quadrant). This addresses the “negative signs can be ignored” misconception.

Students sometimes ignore the negative sign and plot (1, 3) instead, placing the point on the wrong side of the y-axis. The negative sign is not decoration — it reverses the direction from right to left.

Think of the y-axis like an elevator in a building, and the x-axis as a hallway. Positive y goes up to the top floors, negative y goes down to the basement. Positive x walks right down the hall, negative x walks left. You can’t reach the basement by pressing the up button!

The pattern in the table shows the y-value increasing by 2 every time the x-value increases by 1 (the constant difference is +2). Since a straight line always has a constant rate of change, we can use this pattern to predict future points. If we follow the pattern to x = 4, y will be 7 + 2 = 9. If we go one more step to x = 5, y will be 9 + 2 = 11. Therefore, (5, 11) lies perfectly on the line without us even needing to draw it!

This addresses the gap in understanding collinearity — teaching students that a straight line is not just a drawing, but a predictable mathematical rule.

Example: x = 0, 1, 2, 3 and y = 1, 3, 5, 7 (y increases by 2 each time the x increases by 1 — constant difference).

Another: x = 0, 1, 2, 3 and y = 10, 7, 4, 1 (y decreases by 3 each time — still a straight line, just sloping downward).

Creative: x = −1, 0, 1, 2 and y = −4, −1, 2, 5 (includes negative coordinates, y increases by 3 each time).

Trap: x = 0, 1, 2, 3 and y = 1, 2, 4, 8 — the y-values are doubling each time, so there IS a pattern, but the differences are +1, +2, +4 — not constant. These points form a curve, not a straight line. This exploits the “any pattern means a straight line” misconception.

Example: (−2, −3)

Another: (−5, −1)

Creative: (−0.5, −0.5) — non-integer coordinates can also be in the third quadrant.

Trap: (−4, 3) — the x-value is negative but the y-value is positive, so this point is actually in the second quadrant (top-left), not the third. This exploits the “any negative coordinate means third quadrant” misconception — students who only check for “a negative number somewhere” rather than “both values negative” will be caught out.

Example: (3, 0)

Another: (−2, 0)

Creative: (0, 0) — the origin lies on the x-axis (and on the y-axis too).

Trap: (0, 5) — this point is on the y-axis, not the x-axis. This exploits the “zero in the coordinates means on an axis” misconception — a student might think “there’s a zero in the coordinates, so it’s on an axis” without checking which coordinate is zero. For a point to be on the x-axis, the y-coordinate must be 0.

Example: x = 0, 1, 2, 3 and y = 1, 2, 4, 8 (doubling — differences are +1, +2, +4, not constant).

Another: x = 0, 1, 2, 3 and y = 0, 1, 4, 9 (square numbers — differences are +1, +3, +5).

Creative: x = 1, 2, 3, 4 and y = 12, 6, 4, 3 (halving/dividing pattern — these lie on a curve).

Trap: x = 0, 1, 2, 3 and y = 5, 8, 11, 14 — the y-values increase by 3 each time, which IS a constant difference, so these points DO form a straight line. This catches students with the “any clear pattern means not a straight line” misconception — the key test is whether the y-values change by the same amount each time.

Example: x = 0, 1, 2, 3 and y = 0.5, 1, 1.5, 2 (y increases by a constant 0.5 each time).

Another: x = 1, 2, 3, 4 and y = 2.5, 5, 7.5, 10 (y increases by 2.5).

Creative: x = 0, 1, 2, 3 and y = 1/3, 2/3, 1, 4/3 (using fractions, increasing by exactly 1/3 each step).

Trap: x = 0, 1, 2, 3 and y = 0.1, 0.2, 0.4, 0.8 — this incorporates decimals but uses a doubling pattern. Because the differences are not constant, it forms an exponential curve rather than a straight line.

This is always true. Equal steps in x combined with a constant change in y means the gradient (steepness) is the same between every pair of consecutive points. A constant gradient is exactly what defines a straight line. For example, x = 0, 1, 2, 3 with y = 3, 7, 11, 15 has a constant y-step of +4 for each x-step of +1, giving a straight line with gradient 4. Students who doubt this often hold the “constant difference is not enough to guarantee a straight line” misconception.

Students might doubt this for downward-sloping lines, but a constant negative change (e.g., y decreasing by 2 each time) also produces a straight line. Even a change of 0 is a constant change! If the y-values stay the same (e.g., y = 4, 4, 4, 4), they are changing by exactly 0 each time, which forms a perfect, flat, horizontal line.

True case: x = 0, 1, 2, 3 with y = 2, 4, 6, 8 — y increases as x increases, so the line slopes upward (positive gradient). False case: x = 0, 1, 2, 3 with y = 9, 6, 3, 0 — y decreases, so the line slopes downward (negative gradient). Also, x = 0, 1, 2, 3 with y = 5, 5, 5, 5 gives a horizontal line. This challenges the “straight lines always slope upwards” misconception — students often assume this because many textbook examples show positive gradients first.

True case: x = 0, 1, 2 with y = 0, 3, 6 — when x = 0, y = 0, so the line passes through the origin. False case: x = 0, 1, 2 with y = 4, 7, 10 — when x = 0, y = 4, so the line crosses the y-axis at (0, 4). This addresses the “all straight lines pass through the origin” misconception. A line only passes through the origin if y = 0 when x = 0. Many students assume all lines go through the origin because several common examples (like times tables) do.

This is never true. A horizontal line has the same y-value for every point, not the same x-value. For example, y = 4 has coordinates (0, 4), (1, 4), (2, 4), (3, 4). If the x-values were all the same, that would be a vertical line. This targets the “horizontal means same x-values” misconception — students mix up which value stays constant.

A helpful reminder is that “horizontal” relates to the horizon (flat, left to right), so the height (y) stays the same.

If x is always the same number (e.g., x = 3 with y = 0, 1, 2, 3), plotting these points means you go exactly 3 units right every time, but up by different amounts. Connecting them creates a vertical line straight up and down. This targets the misconception that tables must always have a changing x-value to be “valid” or to create a line.

The student has swapped the x and y coordinates. In the pair (2, 5), the first number (2) is the x-coordinate telling us the horizontal position, and the second number (5) is the y-coordinate telling us the vertical position. The student has read them in the wrong order, plotting (5, 2) instead of (2, 5).

The correct method is to always go across to the x-value first, then up (or down) to the y-value. “Along the corridor, up the stairs” is a common way to remember the order. This error places the point in a completely different location on the grid.

The student arrives at the correct answer — the line does pass through (0, 0) — but their reasoning reveals the “origin is where the graph starts” misconception. The origin is not “where the graph starts”; it is the fixed point on the grid where x = 0 and y = 0. The line passes through the origin because when x = 0, the corresponding y-value is also 0, giving the coordinate (0, 0).

This reasoning fails for other tables: for example, a line obeying y = 2x + 5 might have a table that also “starts” at x = 0 (x = 0, 1, 2, 3 with y = 5, 7, 9, 11). Even though the table starts at x = 0, the graph passes through (0, 5), not the origin. What matters is whether y = 0 when x = 0.

The student has ignored the negative sign on the x-coordinate. The “−” in front of the 3 means we go 3 units to the left of the y-axis, not to the right. The correct position is 3 left and 2 up, which places the point in the second quadrant (top-left area of the grid). The student’s plot of (3, 2) is in the first quadrant — the wrong side of the y-axis entirely.

Dropping negative signs is a very common error. The negative sign reverses the direction: negative x means left (not right), and negative y means down (not up). This student should check which quadrant their answer should be in before plotting.

The student has misused the axis scale, placing x-values at consecutive grid lines instead of at their true positions. By putting x = 2 at the second grid line (position 2) but treating it as position 1, and x = 4 at position 2 instead of position 4, the horizontal spacing is compressed. This distorts the straight line relationship into what appears to be a curve.

The correct approach is to check what each grid line represents before plotting. If each grid line = 1 unit, then x = 2 should be plotted at the 2nd grid line, x = 4 at the 4th grid line, and x = 6 at the 6th. The table does produce a straight line (y increases by 4 for each increase of 2 in x, giving a constant gradient of 2).