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.

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 table where the first ratio is 10/2=5, but others may vary, like assume full table x:2,4,6 y:10,18,26 with ratios 5,4.5,4.33 (not equivalent, not proportional), vs all 5 for proportional. In this question, the best conclusion is that you cannot conclude it is proportional unless all ratios y/x are equal for every row, emphasizing the need to check all pairs, not just one. A common error is thinking one matching ratio is enough to prove proportionality (like choice A or the student's claim), but all ratios must be constant for the relationship to be proportional. 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.

Identifying proportional relationships requires verifying all y/x ratios are equal (constant k), which means y=kx and graphs as a line through origin. If a table shows constant ratios y/x=4 for all pairs (like 8/2=4, 12/3=4, 16/4=4), then y=4x is proportional—this is the correct justification. The correct answer properly identifies that equal y/x ratios confirm proportionality with equation y=4x. Choice B incorrectly claims increasing values prevent proportionality (proportional relationships do increase when k>0), C wrongly suggests x/y must also be constant as a separate requirement (if y/x is constant, then x/y=1/k is automatically constant too), and D confuses constant differences with constant ratios (constant differences indicate linear but not necessarily proportional). Proper justification: (1) calculate y/x for all data pairs, (2) if all equal the same value k, then proportional y=kx, (3) this is equivalent to saying the graph passes through origin with slope k. Remember: proportionality is about constant RATIOS (multiplication), not constant DIFFERENCES (addition).