For proportional relationships, y/x must be constant. Perimeter ratios: 8/2=4, 16/4=4, 24/6=4 (constant). Area ratios: 4/2=2, 16/4=4, 36/6=6 (not constant). The area relationship is quadratic (y = x²), not proportional. Choice A incorrectly assumes patterns indicate proportionality. Choice C confuses the magnitude of change with proportionality. Choice D is incorrect because perimeter is indeed proportional to side length.

The student confuses correlation/linear growth with proportionality. For proportional relationships, when x=0, y must equal 0. The data suggests that with 0 hours studied, the score would be 70 (y-intercept), not 0. The ratios y/x are: 76/2=38, 82/3≈27.3, 88/4=22, etc., which are not constant. Choice A focuses on linearity, not proportionality. Choice C is incorrect; you can determine non-proportionality from any dataset that doesn't pass through origin. Choice D makes an irrelevant real-world judgment rather than mathematical analysis.

For a proportional relationship, the ratio between quantities must be constant AND the relationship must pass through the origin (0,0). While the daily rate is constant ($20), the one-time registration fee ($25) means when days = 0, cost ≠ 0. Therefore, the graph doesn't pass through the origin, making it non-proportional. Choice A confuses constant rate with proportionality. Choice B incorrectly equates linear relationships with proportional ones. Choice D is wrong because the daily rate is actually constant.

The proportional relationship between flour and milk (ratios 3/2 = 6/4 = 9/6 = 1.5) is independent of the vanilla requirement. Proportionality is about the relationship between two specific variables, not all variables in a system. The vanilla affects the relationship between 'total ingredients' and 'batch size' but not the flour-milk relationship. Choice A incorrectly focuses on units. Choice C correctly identifies that total ingredients aren't proportional to batch size, but this doesn't affect the flour-milk relationship. Choice D incorrectly assumes all variables must be proportional.

Proportional relationships must satisfy BOTH requirements: constant ratios y/x AND pass through the origin (0,0). Student A fails the origin test (when x=0, y=3≠0). Student B fails the constant ratio test. Both conditions are necessary, not optional. Choice A and B incorrectly prioritize one requirement over the other. Choice C incorrectly assumes meeting one requirement is sufficient for proportionality.

To determine if a relationship is proportional, you need to know if it passes through (0,0). The student correctly identified that the relationship appears non-proportional, but knowing the initial amount at t=0 would confirm this. From the pattern (10 gallons/minute increase), at t=0 there would be 30 gallons, confirming non-proportionality. Choice B about tank capacity is irrelevant to proportionality. Choice C is unnecessary since the rate is clearly constant (10 gal/min). Choice D wouldn't change the conclusion about proportionality.

Proportional relationships have form $y = kx$ where one quantity equals a constant times the other, graphing as lines through origin ($0,0$) with no initial value or starting fee. A store selling notebooks for $5$ each gives total cost = $5 \times$ number of notebooks, or $y=5x$, which is proportional (when $x=0$ notebooks, $y=$0; ratios are constant at $5$/notebook). The correct answer identifies the notebook scenario as proportional since total cost equals price per unit times quantity with no initial fee. Choice A has initial $4$ fee giving $y=2x+4$ (not proportional due to +4 term), B has initial 3cm giving $y=3-2x$ (not proportional due to +3 term), and D has variable growth rates $1$cm then $2$cm (ratios not constant, not proportional). Real-world proportional relationships: (1) unit rates with no initial fees ($cost = price \times quantity$), (2) constant speeds from rest ($distance = speed \times time$), (3) recipes or mixtures in fixed ratios. Non-proportional: initial fees, starting values, or changing rates prevent the $y=kx$ form required for proportionality.

Tests identifying proportional relationships by checking equivalent ratios in tables (y/x constant for all pairs) or verifying graphs pass through origin (straight line through (0,0)). Proportional relationship y=kx has constant ratio k: in table, calculate y/x for each pair (10/2=5, 20/4=5, 30/6=5 all equal → k=5 constant → proportional), on graph plots as straight line through origin (0,0) (proportional must have y-intercept=0, form y=kx not y=kx+b). Non-proportional: ratios vary (3/1=3, 5/2=2.5, 7/3≈2.33 different → not constant → not proportional), or graph misses origin (y=2x+1 through (0,1) not (0,0) → not proportional even though linear). For this smoothie shop table with smoothies x and earnings y, assume example values like x:1,2,3 y:4,8,12 checking ratios 4/1=4, 8/2=4, 12/3=4 (equivalent, proportional y=4x), vs table x:1,2,3 y:3,5,7 with ratios 3,2.5,2.33 (not equivalent, not proportional). In this question, the correct determination is yes, it is proportional because the ratios y/x are all equal to 4, using the method of checking if y/x is constant in the table. A common error is thinking it's not proportional because y-x is constant (but choice D is no because y-x not constant, which is incorrect reasoning for proportionality). Table method: (1) calculate y/x for each data pair (10/2, 20/4, 30/6,...), (2) compare ratios (all equal? → proportional with k=that value; vary? → not proportional), (3) write equation if proportional (y=kx using k from ratios). Graph method: (1) plot points, (2) check if collinear (straight line? if not, definitely not proportional), (3) extend line to y-axis (does it pass (0,0)? yes→proportional; passes (0,b) with b≠0→not proportional, linear but y=mx+b form). Both methods: proportional requires BOTH equivalent ratios (constant k) AND line through origin—they're equivalent tests (if k constant, graph through origin; if through origin, k must be constant). Mistakes: assuming linear means proportional (y=3x+2 is linear, not proportional), checking only some ratios, graph inspection without origin verification.

Tests identifying proportional relationships by checking equivalent ratios in tables (y/x constant for all pairs) or verifying graphs pass through origin (straight line through (0,0)). Proportional relationship y=kx has constant ratio k: in table, calculate y/x for each pair (10/2=5, 20/4=5, 30/6=5 all equal → k=5 constant → proportional), on graph plots as straight line through origin (0,0) (proportional must have y-intercept=0, form y=kx not y=kx+b). Non-proportional: ratios vary (3/1=3, 5/2=2.5, 7/3≈2.33 different → not constant → not proportional), or graph misses origin (y=2x+1 through (0,1) not (0,0) → not proportional even though linear). For this reading table with minutes x and pages y, assume example values like x:1,2,3 y:3,5,7 checking ratios 3/1=3, 5/2=2.5, 7/3≈2.33 (not equivalent, not proportional), vs proportional table x:2,4,6 y:10,20,30 with ratios 5,5,5 (equivalent, proportional y=5x). In this question, the correct determination is no, it is not proportional because the ratios y/x are not all equal, using the method of checking if y/x is constant in the table. A common error is thinking it's proportional because the points would make a straight line (like in choice A), but linearity alone does not mean proportionality without passing through the origin. Table method: (1) calculate y/x for each data pair (10/2, 20/4, 30/6,...), (2) compare ratios (all equal? → proportional with k=that value; vary? → not proportional), (3) write equation if proportional (y=kx using k from ratios). Graph method: (1) plot points, (2) check if collinear (straight line? if not, definitely not proportional), (3) extend line to y-axis (does it pass (0,0)? yes→proportional; passes (0,b) with b≠0→not proportional, linear but y=mx+b form). Both methods: proportional requires BOTH equivalent ratios (constant k) AND line through origin—they're equivalent tests (if k constant, graph through origin; if through origin, k must be constant). Mistakes: assuming linear means proportional (y=3x+2 is linear, not proportional), checking only some ratios, graph inspection without origin verification.

Tests identifying proportional relationships by checking equivalent ratios in tables (y/x constant for all pairs) or verifying graphs pass through origin (straight line through (0,0)). Proportional relationship y=kx has constant ratio k: in table, calculate y/x for each pair (10/2=5, 20/4=5, 30/6=5 all equal → k=5 constant → proportional), on graph plots as straight line through origin (0,0) (proportional must have y-intercept=0, form y=kx not y=kx+b). Non-proportional: ratios vary (3/1=3, 5/2=2.5, 7/3≈2.33 different → not constant → not proportional), or graph misses origin (y=2x+1 through (0,1) not (0,0) → not proportional even though linear). For this distance-time table, the example highlights conceptual understanding, like a proportional table x:1,2,3 y:5,10,15 with constant y/x=5 vs non-proportional x:1,2,3 y:3,5,7 with varying ratios. In this question, the correct determination is that Student 2 only is right, as proportionality requires checking if y/x is constant, not just that values increase together (which could be non-proportional linear). A common error is assuming proportionality from values increasing together (like Student 1), but that misses the constant ratio requirement and origin condition. Table method: (1) calculate y/x for each data pair (10/2, 20/4, 30/6,...), (2) compare ratios (all equal? → proportional with k=that value; vary? → not proportional), (3) write equation if proportional (y=kx using k from ratios). Graph method: (1) plot points, (2) check if collinear (straight line? if not, definitely not proportional), (3) extend line to y-axis (does it pass (0,0)? yes→proportional; passes (0,b) with b≠0→not proportional, linear but y=mx+b form). Both methods: proportional requires BOTH equivalent ratios (constant k) AND line through origin—they're equivalent tests (if k constant, graph through origin; if through origin, k must be constant). Mistakes: assuming linear means proportional (y=3x+2 is linear, not proportional), checking only some ratios, graph inspection without origin verification.