New Jersey Grade 7 Math

Two tables can increase at a constant rate while only one shows a proportional relationship. The selected New Jersey standard asks learners to recognize, represent, and interpret proportional quantities. The invented classroom hire offers here contrast a constant unit price with a separate starting charge.

Try this original activity

Offer A charges $4.50 per hour with no starting charge. Offer B charges the same hourly amount plus a $3 starting charge. Compare their costs at two, four, and six hours. Which total cost is proportional to time? Use ratios and the value at zero hours to explain.

Work through the reasoning

A has costs $9, $18, and $27. Dividing each cost by its time gives 9/2 = 18/4 = 27/6 = 4.50 dollars per hour. Its equation is C = 4.50t, and its extended mathematical model includes (0, 0). B has costs $12, $21, and $30. The ratios are 12/2 = 6, 21/4 = 5.25, and 30/6 = 5, so they do not stay equal. Its equation is C = 4.50t + 3; at t = 0 the model gives $3. Both costs rise by $9 when time rises by two hours, but only A maintains a constant cost-to-time ratio. A is proportional and B is not. The $3 starting charge changes the intercept without changing the hourly rate. At one hour A’s point (1, 4.50) directly shows its unit rate.

Check the common mistake

A straight line or a constant increase alone does not establish proportionality; the relationship must also pass through the origin in its mathematical model. Do not divide the change in cost by time and then mistake that slope for the total cost-to-time ratio in B. The zero-hour value is an extrapolation of the stated classroom pricing model, not a claim that a real business charges someone who makes no booking. The prices are invented and include no other charges.

Try a related case

Replace B’s starting charge with zero: its cost-to-time ratios then match A’s and its model passes through the origin. Alternatively keep the $3 charge and calculate costs at one and three hours, $7.50 and $16.50. Explain why the hourly increase remains constant while the total-cost ratio changes.

Selected 2023 New Jersey references for this activity

The 2023 New Jersey Student Learning Standards for Mathematics define grade-specific expectations. These selected references point to the activity’s focus; read the complete official Grade 7 page, its examples and clarifications, and the mathematical practices alongside the school’s teaching sequence.

Sources checked October 6, 2026. These focus labels are independent paraphrases of selected expectations, not a complete state checklist or certified alignment of the lesson library.

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.

The official 2026–2027 schedule lists NJSLA-A mathematics for Grades 3 through high school, including Grade 7. NJDOE describes the mathematics assessment as adaptive. Confirm the learner’s assigned pathway, tools, and school directions. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration specifications or a prediction of a state performance level. Read the official adaptive assessment overview.

Read the 2026–2027 statewide assessment schedule and confirm school directions. 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.