7.4(A) — Proportionality: Represent proportional relationships with tables, verbal descriptions, graphs, and equations in the form y = kx, where k is the constant of proportionality
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Generate a lesson like this 7th Grade · Math · pre-filled for youPresent a road trip scenario: "The Martinez family drives from Dallas to Houston at a constant speed. After 2 hours, they have traveled 140 miles. After 4 hours, 280 miles. After 6 hours, 420 miles." Students predict: How far after 3 hours? After 5 hours?
Students write their predictions on whiteboards. Share predictions — most will agree on the 3-hour mark. Ask: "How did you figure it out? Did you multiply? Divide?" Introduce the idea: this is a proportional relationship — as time doubles, distance doubles.
The road trip makes the abstract concrete. Ask follow-up: "If we drove for 10 hours at this rate, how far would we go? 100 hours?" Students will want to say "1,400 miles" for 20 hours — they're detecting the pattern. Now formalize it: "What rule did you use to get from hours to miles? Can you write it as an equation?"
Model the Dallas-to-Houston scenario using all 4 representations simultaneously: (1) Table: Hours | Miles: 0|0, 1|70, 2|140, 3|210, 4|280; (2) Equation: y = 70x; (3) Graph: coordinate plane, straight line through origin; (4) Verbal: "For every 1 hour driven, the car travels 70 miles."
Students complete a 4-quadrant organizer (one box each for table, equation, graph, verbal) using a new scenario: "A San Antonio taco truck sells breakfast tacos for $2.50 each." Students fill in the table, write the equation, sketch the graph, and write the verbal description.
The graph is the keystone representation — make sure students plot the point (0, 0) even though it's not in the data table. Ask: "What happens at 0 hours? You've ordered 0 tacos and paid $0 — does the graph start at the origin?" Students who miss the origin point have not fully understood proportionality. Connect to the road trip: "If you drive 0 hours, you travel 0 miles — the line always starts at (0,0)."
Partners receive 8 problem cards, each showing a proportional relationship in a different format: (1) Table, (2) Equation, (3) Graph, (4) Verbal description of rates. Partners identify k for each card and record it.
After completing all 8 cards, pairs check their answers against an answer key (hanging on the board). For any mismatches, pairs return to the card and find their error. Teacher circulates and notices which format caused the most errors.
Graphs often confuse students — they want to read k as the y-intercept or the x-value at a random point. Ask: "What is the y-value when x is 1?" That's always k in a proportional relationship (if it goes through the origin). The verbal descriptions are also tricky — if the description says "5 bags for $15," students must recognize k = 3 (dollars per bag). If students struggle, ask: "What operation connects the input to the output? Multiplication? Division? Addition?"
Present two real-world scenarios: "A Houston food truck charges $4 per taco. A Dallas food truck charges $3.50 per taco. Which is a better deal?" Students must compare the k values (4 vs. 3.5) and explain in context.
Students create a comparison table: Truck | Price per taco (k) | Cost for 6 tacos | Cost for 10 tacos. They answer: "If you want to buy 10 tacos, which truck is cheaper and by how much?" Write a complete sentence: "The Dallas truck is cheaper by $[X] when buying 10 tacos."
Watch for students who calculate the total cost for both trucks but don't reference k. Ask: "You found the total — which k is smaller? Does a smaller k mean a better deal or a worse deal?" The connection between k as unit rate and real-world decision-making is TEKS 7.4(A) applied to a real context. Students who can't articulate this yet need the guided question sequence, not the independent practice.
Students create their own proportional relationship scenario. They must: (1) Provide a verbal description, (2) Write the equation (y = kx), (3) Fill in a table with 4 data points including (0,0), (4) Sketch the graph with axes labeled.
Exit ticket on paper. Teacher collects and sorts into 3 piles: complete (all 4 representations correct), partial (3 of 4), incomplete (2 or fewer). The sorted exit tickets give you data for flexible grouping — same-day small-group re-teach for the incomplete pile.
Students who write "y = 5x" but leave out the verbal description need practice connecting representations. Students who write the table but graph a non-origin line need to revisit the definition. The sort takes 2 minutes after class and drives tomorrow's grouping. The student who gives you a proportional situation in the verbal description but writes the wrong equation probably hasn't generalized the k concept — small-group intervention targets that confusion.
Provide a partially completed four-quadrant organizer with the table pre-filled (4 rows) and k labeled for the first example. Students complete the equation, graph, and verbal description from the table. Allow graphing calculators for the partner practice. Reduce the exit ticket to the table and equation only.
Write a real-world problem where y = kx does NOT apply (e.g., y = 2x + 3, which is not proportional). Then answer: Why isn't this proportional? What would the graph look like? Would it pass through the origin? Then extend: Write a proportional relationship that gives the same answer at x = 5 but is not the same equation at x = 2. This problem requires deep understanding of k and proportionality.
Pre-teach: proportional, constant, rate, relationship, variable with visual examples and Spanish cognates (proporcional, constante, tasa). Provide a step-by-step graphic organizer for the four representations with arrows showing which part of the table becomes which part of the equation. Use real-world examples with Texas contexts (sports scores, food prices, distance to cities). Allow verbal explanation of the equation before written form.
Exit ticket: all 4 representations required. 4 = all 4 representations correct (equation through origin, table includes 0, graph passes through origin, verbal describes unit rate); 3 = 3 of 4 correct; 2 = 2 of 4 with correct equation; 1 = incomplete or incorrect. Students scoring 1-2 join small-group re-teach before the next class — the re-teach focuses on connecting the table to the equation (k = y/x when x is not 0). Students scoring 3 complete a self-correction on the missing representation using an answer key.
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