Learning goals: Connect a situation, table, graph, and equation; interpret units; build and critique a model.
Four ways to see the same relationship
A climbing gym charges a joining fee of $18 and $7 for each visit. Describe the total cost after \(v\) visits using words, a table, a graph, and an equation. The equation is \(C(v)=18+7v\). A table includes \((0,18),(1,25),(2,32)\). On a graph with visits horizontally and dollars vertically, the points rise seven dollars per visit. The intercept represents the joining fee, not a free visit.
Because visits are counted, the contextual domain consists of nonnegative integers. A continuous line is a useful extension of the model, but its point at half a visit is not an actual purchase. Distinguishing an algebraic domain from a contextual domain will matter throughout the course.
A disciplined modeling cycle
State the question, define variables and units, identify assumptions, choose a relationship, calculate, interpret, and check the model against the situation. Assumptions are part of the answer. The gym equation assumes no tax, discounts, or changes in price. If a promotion waives the fee, the intercept must change. A correct calculation using an unsuitable model is not a correct solution to the original problem.
Units are mathematical information
For constant motion, \(d=rt\). If rate is in miles per hour, time must be in hours to obtain miles. A trip lasting 90 minutes at 48 miles per hour covers \[d=48\frac{\text{mi}}{\text{h}}\cdot\frac{90}{60}\text{h}=72\text{ mi}.\] Multiplying 48 by 90 without converting would mix units. Conversion factors equal one: \(60\text{ min}/1\text{ h}\) changes units without changing the physical quantity.
Compare plans and explain a threshold
Plan A costs \(A(v)=18+7v\); Plan B costs \(B(v)=42+4v\). Set the costs equal: \(18+7v=42+4v\), so \(3v=24\) and \(v=8\). Both cost $74 at eight visits. The difference \(A(v)-B(v)=3v-24\) explains the entire comparison: Plan A is cheaper below eight visits, while Plan B is cheaper above eight. Checking one value on each side confirms the interpretation.
Investigate, critique, and extend
Suppose a table contains \((1,3),(2,5),(3,7)\). The linear rule \(y=2x+1\) fits, but finitely many points do not uniquely determine a rule unless a model family is specified. A graphing tool can suggest structure, reveal an arithmetic error, or test a conjecture. It cannot justify an unspoken assumption about how the situation behaves between or beyond measured points.
For a partner investigation, design two payment plans that agree at exactly six visits. Explain why your equations meet that requirement, identify when each is cheaper, and discuss whether a customer can make a fractional visit. Exchange plans and critique each other’s units, restrictions, and conclusions. A strong response combines computation with a clear account of what the result means.
Practice: attempt before reading the solutions
Show your reasoning, state restrictions, and check your result. For explanation questions, use complete mathematical sentences.
- A service charges $12 plus $3 per use. Write the cost for \(n\) uses and find the cost for nine uses.
- What does the coefficient 3 mean in \(C(n)=12+3n\)?
- Find the distance traveled in 40 minutes at 54 miles per hour.
- Compare \(A(n)=10+5n\) and \(B(n)=28+2n\). When are they equal?
- A rectangle has perimeter 30 cm. Express area using width \(w\) and state a physical domain.
- A model predicts 12.4 buses for a trip. What should a planner do?
- The table \((0,2),(1,6),(2,10)\) is assumed linear. Find its rule.
- A student uses a two-week plant-growth model to predict its height after 100 years. Critique the prediction.
Worked solutions and reasoning
- \(C(n)=12+3n\), with nonnegative integer \(n\). Then \(C(9)=12+27=39\) dollars.
- Each additional use increases cost by $3. Its units are dollars per use; the constant 12 is the fixed fee in dollars.
- Convert time to \(40/60=2/3\) hour. Then \(d=54(2/3)=36\) miles.
- \(10+5n=28+2n\) gives \(3n=18\), so \(n=6\). Both cost $40. B is cheaper for more than six uses.
- \(2w+2\ell=30\), so \(\ell=15-w\) and \(A(w)=w(15-w)\). A nondegenerate rectangle requires \(0\lt w\lt15\).
- If 12 buses cannot carry everyone and buses are identical, round up to 13. Interpret the context before rounding; fractional buses are not usable.
- Equal input steps increase the output by four, so the slope is four. The output at zero is two: \(y=4x+2\).
- This is extreme extrapolation. Growth rate, maturity, and survival may change; matching two weeks of data does not validate the model over 100 years.
Reflection
Choose one problem you initially found difficult. Explain the key idea, identify a possible mistake, and write a new problem that uses the same idea. Solve your new problem and check it independently.
Unit synthesis investigation: Two claims, two different arguments
A student claims that (A) the sum of any two consecutive integers is odd, and (B) any integer whose square is divisible by four must itself be divisible by four. Prove or disprove each statement. Then repair the false statement and justify your repair. Explain why checking only positive even integers divisible by four could hide the error.
Worked discussion — open after attempting
For A, write consecutive integers as \(n,n+1\). Their sum is \(2n+1\), which is odd by definition for every integer n. For B, use n=2: its square is four, but two is not divisible by four. A correct repair is “If an integer has a square divisible by four, the integer is even.” To justify it, suppose instead that the integer is odd, \(n=2k+1\). Then \(n^2=4k(k+1)+1\), whose remainder on division by four is one. It cannot be divisible by four, so n cannot be odd. Testing a narrow selection of even numbers would never test the allowed counterexample two or six. State the domain and justify all cases rather than treating examples as a proof.