FutureStepsSign in 12th grade learning.
Make the idea click.
Senior-year learning benefits from clear assumptions and independent checking. Choose the course you are taking, use the example as a model for reasoning, and practise explaining a solution so another person can follow it.
Try the worked exampleState the order and the assumptions
A discount followed by a fixed fee is different from applying the discount to both the item and the fee. Write the sequence before using a formula. In research or an argument, state the assumptions that make your conclusion reasonable.
Which operation happens first, and what quantity does the next operation act on?
Course ideas for 12th grade.
High-school learning is organized by course. The courses below are examples available for this grade selection, not a required schedule. Follow your school’s course placement and prerequisites.
Precalculus
Selected starter topics
Connect function domains, trigonometric models, directed motion and finite sums with their underlying reasoning.
- Explain why function order matters
- Connect radians to a circle’s radius
- Decide when a geometric sum has a limit
English IV
Selected starter topics
Evaluate language, subplots, drama, poetic patterns and allegory; improve explanations and arguments while synthesizing sources carefully.
- Evaluate whether a rebuttal answers the concern
- Inspect the evidence behind a source
- Connect a judgment to evidence and a criterion
U.S. Government
Selected starter topics
Connect founding principles and constitutional processes with rights, representation, civic participation and evidence about public choices.
- Distinguish national, state and shared powers
- Explain how a veto can be checked
- Separate proposing an amendment from ratifying it
Economics
Selected starter topics
Use production models, market schedules and economic indicators to explain choices and policy trade-offs.
- Explain why scarcity requires a choice
- Separate a change in demand from a price response
- Compare opportunity costs before specializing
Personal Financial Literacy and Economics
Selected starter topics
Connect wages and financial choices with budgets, education costs, borrowing, saving, insurance and fraud prevention.
- Distinguish gross pay from take-home pay
- Identify the trade-off between spending and saving
- Turn a savings target into a workable plan
These examples come from FutureSteps’ Texas starter collection. They introduce selected topics and support review; they are not a complete curriculum, school assessment or course credit.
Explain why function order matters
Model two operations with a composition and compare the reverse order.
A composition applies one function to the output of another. In f(g(x)), evaluate g first, then use its output as the input to f. The order matters because the second operation acts on a value already changed by the first.
Function composition is not generally commutative: f(g(x)) and g(f(x)) can differ. In a price model, applying a percentage discount before adding a fixed fee is different from discounting the fee as well. State the order before calculating.
A fictional store discounts an item by 20%, then adds a 10-dollar delivery fee. Let D(p) = 0.8p and F(p) = p + 10. Ignore tax.
- For an item priced at 50 dollars, apply the discount first: D(50) = 40 dollars.
- Then add delivery: F(D(50)) = F(40) = 50 dollars. This composition follows the stated store rule.
- Reverse the order to compare: D(F(50)) = D(60) = 48 dollars. This discounts delivery too, so it models a different rule.
Your turn.
Let D(p) = 0.9p and F(p) = p + 20. For a 100-dollar item, what is F(D(100)), with the discount applied before the fee?
- 108 dollars
- 110 dollars
- 120 dollars
Show the answer and explanation
110 dollars
D(100) = 90, then F(90) = 110 dollars. The 108-dollar result comes from reversing the order and discounting the fee as well.
Example from the FutureSteps Precalculus starter collection. Course reference: Texas TEKS · Precalculus. The example is app-authored; the reference does not imply endorsement or complete coverage.
A routine to try next.
- Translate the instructions into an ordered list of operations or claims.
- Calculate or argue from that order, keeping the assumptions visible.
- Change one assumption or reverse two steps and explain why the result changes.
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Questions before you begin.
Do these senior-year lessons provide course credit?
No. They are study and review examples from a starter collection. Your school determines course placement, assessment and credit. The economics examples teach concepts using fictional situations, not personal financial recommendations.
How does the in-app learning path build on these examples?
Eligible students choose a supported state, grade and courses. Lessons include explanations, worked examples and quick checks. Students can save Notes, mark their own progress and ask Study about a lesson when they have access. Progress labels reflect the student's selections, not a school grade or a mastery assessment.