Illinois Grade 7 Math

A constant increase in cost per added item does not by itself establish proportionality. A fixed fee can produce a linear rule whose total-to-quantity ratio changes. Comparing two clearly stated fictional pricing rules makes that distinction visible without relying on a graph’s shape alone.

Try the worked activity

Model A charges $2.50 for each printed poster and has no fixed fee. Model B charges the same $2.50 per poster plus one $6 setup fee per order. Assume these rules hold for the positive whole-number poster counts shown. Which total cost is proportional to poster count?

Follow the reasoning

For Model A, the totals for two, four, and six posters are $5, $10, and $15. Dividing each total by its poster count gives the same $2.50 per poster, so the relationship is proportional and can be written C = 2.50n. Model B gives totals of $11, $16, and $21. Their total-to-count ratios are $5.50, $4, and $3.50 per poster. These ratios differ because each spreads the same setup fee over a different quantity. Its rule C = 2.50n + 6 is not proportional.

Check the explanation

Both rules increase the total by $2.50 for one more poster, but only the first keeps the total-to-count ratio constant. A straight-line pattern alone is not sufficient; the proportional line must also pass through the origin. The model is stated for positive whole-number orders. Do not invent an actual business policy for zero posters or claim the fictional fees describe Illinois print shops. The coefficient 2.50 in Model B describes the additional-item cost, while the average total cost per poster also includes the setup fee.

Use the next session

1. Use each rule to calculate total cost for the same counts.
2. Divide the total by the count and compare the ratios.
3. Write the proportional equation and explain why the other rule differs.

Predict the totals for eight posters: $20 for A and $26 for B. Check that A still gives $2.50 per poster while B gives $3.25. Then compare the effects of doubling an order from two to four posters: A’s total doubles, but B’s does not because the fixed fee is charged once in each order. Use a graph to connect the constant ratio with a line through the origin, keeping the stated domain visible. These invented rules introduce proportionality and equations; broader learning also includes unit rates and multistep percent problems.

Selected references from Illinois’s adopted mathematics document

The Illinois mathematics standards directory links the Illinois Learning Standards document incorporating Common Core. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

Grade 7 is within Illinois’s IAR mathematics testing range. The linked learning resources have their own directions and scoring; a practice score does not establish an official state performance level. Read Illinois’s assessment overview.

Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.