Rate Slopes Answer Key
About This Worksheet
This worksheet helps seventh-grade students compare proportional relationships by looking at their rates on graphs. Each problem describes two proportional graphs using points on lines that pass through the origin. Students find the constant of proportionality for each relationship and decide which one has the greater rate. They then explain how the graph shows that difference. This activity helps students connect the numerical value of k with the steepness of a proportional line.
What Students Practice
Students calculate k by dividing the y-coordinate by the matching x-coordinate for each graph. After finding both constants, they compare the values and identify which relationship has the greater rate. They also explain that the graph with the larger constant rises more quickly and therefore appears steeper. This gives students practice with coordinate points, constants of proportionality, comparison, and graphical interpretation. The repeated side-by-side format makes the connection between rate and steepness much easier to see.
Why This Matters
Students often understand how to find a rate from numbers but do not immediately connect that rate with what a graph looks like. In a proportional relationship, a larger constant of proportionality creates a steeper line because y increases more for every one unit of x. This is an important bridge to the idea of slope in later algebra. Comparing two relationships this way also gives students a fair method for deciding which one changes faster. That reasoning appears in many real-world situations involving speed, cost, earnings, and growth.
Teaching Tips
Have students find both k-values before looking at which graph should be steeper. Then ask them to imagine that x increases by one unit in each relationship. The graph with the larger increase in y has the greater rate. Encourage students to use both the numerical constants and graph language in their explanation. A strong answer might say that one line is steeper because its y-value increases more for the same change in x.
Curriculum Standards
This worksheet strongly supports CCSS.MATH.CONTENT.7.RP.A.2.B by asking students to find and compare constants of proportionality from graph information. It also supports CCSS.MATH.CONTENT.7.RP.A.2.D because students explain what graphical features show about the relationship. The comparison work provides valuable preparation for understanding slope as a rate of change in later grades. It works well for enrichment, homework, independent practice, or review near the end of a Grade 7 proportional relationships unit.