Graphing Ratios on the Coordinate Plane - Sixth Grade Mathematics
Graphing ratio tables on the coordinate plane visually proves proportional relationships, demonstrating that equivalent ratios always form a straight line passing through the origin.
Translating Ratio Tables into Ordered Pairs
Every row or column of a ratio table corresponds to an ordered pair (x, y) that can be plotted on a Cartesian coordinate plane.
Ratio Table: Earning Money Babysitting
Hours Worked (x) Earnings ($ y) Ordered Pair (x, y)
1 12 (1, 12)
2 24 (2, 24)
3 36 (3, 36)
4 48 (4, 48)
The Hallmarks of a Proportional Graph
When equivalent ratios are plotted on a coordinate plane, the resulting graph displays two non-negotiable geometric characteristics.
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| TWO CRITICAL CRITERIA OF A PROPORTIONAL GRAPH |
| |
| 1. The graph MUST be a STRAIGHT LINE (Linear). |
| The constant ratio produces a constant rate of change (slope). |
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| 2. The graph MUST PASS THROUGH THE ORIGIN: (0, 0). |
| Zero units of input must yield zero units of output! |
+-------------------------------------------------------------------------+
Visualizing the Proportional Line:
y (Earnings in $)
^
48| * (4, 48)
36| * (3, 36)
24| * (2, 24)
12| * (1, 12)
0+---|---|---|---|---> x (Hours Worked)
(0,0) 1 2 3 4
Notice: A straight ruler connects (0,0), (1,12), (2,24), (3,36), (4,48)!
Identifying Non-Proportional Graphs
If a graph fails either criterion, the relationship is NOT proportional: If a line is curved, the relationship is non-linear and not proportional. If a graph is a straight line but crosses the y-axis at (0, 5) instead of (0, 0), it is non-proportional because it includes a flat starting fee (for example, a taxi charging $5 just to enter the car).
Interpreting the Unit Rate as the Steepness (Slope)
The point (1, r) on a proportional coordinate graph holds special mathematical importance: r is the unit rate. In the babysitting graph above, the point (1, 12) indicates that for 1 hour of work, the earnings are $12. The greater the unit rate, the steeper the line rises. If a tutor earns $25 per hour, the point is (1, 25), and the line rises much more steeply than the $12 per hour line.
Using Graphs to Predict Values
Once a proportional line is established, you can use it to interpolate (find values within the data range) or extrapolate (predict values beyond the data range). To find the earnings for 2.5 hours, locate 2.5 on the horizontal x-axis, trace vertically up to the line, and read across to the y-axis to find $30.
Exercises
Exercise 1. A recipe uses 2 cups of sugar for every 3 cups of flour. Construct a table of four equivalent ordered pairs (x, y) where x is sugar and y is flour. State the unit rate of flour per cup of sugar.
Exercise 2. A graph shows a straight line passing through (2, 10), (4, 20), and (6, 30). Does this graph represent a proportional relationship? What is the value of y when x = 1?
Exercise 3. Explain why the linear equation y = 3x + 2 does not represent a proportional relationship by examining the coordinates when x = 0.
Solutions
Solution to Exercise 1. Let x = cups of sugar and y = cups of flour. The base ratio is (2, 3). Multiply by consecutive integers: (2 * 1, 3 * 1) = (2, 3); (2 * 2, 3 * 2) = (4, 6); (2 * 3, 3 * 3) = (6, 9); (2 * 4, 3 * 4) = (8, 12). Four ordered pairs are (2, 3), (4, 6), (6, 9), and (8, 12). The unit rate of flour per cup of sugar is y / x = 3 / 2 = 1.5 cups of flour per cup of sugar.
Solution to Exercise 2. Check the ratio y/x for all points: 10 / 2 = 5; 20 / 4 = 5; 30 / 6 = 5. Because the ratio y/x is constant (5) and the relationship is a straight line, it passes through (0, 0) since 5 * 0 = 0. Therefore, the graph represents a proportional relationship. When x = 1, y = 1 * 5 = 5 (the point is (1, 5)).
Solution to Exercise 3. In a proportional relationship, the graph must pass through the origin (0, 0). For the equation y = 3x + 2, substitute x = 0: y = 3(0) + 2 = 0 + 2 = 2. The graph passes through (0, 2), not (0, 0). Because zero input yields a non-zero output, the relationship is non-proportional.