Algebra foundations: expressions, equations and roots
Ten lessons develop exact calculation, justified transformations and checking against original conditions. Written reasoning is self-reviewed with specific rubrics; it is not automatically graded. Further functions, systems and modeling units are still needed for full Algebra I/II coverage.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
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Algebra foundations: mixed unit assessment
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Read and evaluate an expression
Lesson details and objectives
Translate quantities into expressions and preserve signs, grouping and units.
- Distinguish an expression from an equation.
- Evaluate grouped expressions with signed and fractional inputs.
- Translate a fixed charge and a rate with consistent units.
5 practice tasks.
Explain why expressions are equivalent
Lesson details and objectives
Use distribution and structure without confusing a few matching values with an identity.
- Distribute through every term, including subtraction.
- Combine like terms using their coefficients.
- Distinguish identity, counterexample and domain-qualified equality.
5 practice tasks.
Solve equations with reversible steps
Lesson details and objectives
Preserve the solution set, classify exceptional cases and check the original equation.
- Solve linear equations containing groups and fractions.
- Distinguish unique solutions, identities and contradictions.
- Check a candidate in the original equation.
5 practice tasks.
Solve and interpret inequalities
Lesson details and objectives
Track order, endpoints and feasible values instead of treating an inequality as an equation.
- Reverse order when multiplying or dividing by a negative number.
- Represent interval intersections and unions.
- Apply integer and nonnegative constraints to contextual solutions.
5 practice tasks.
Rearrange formulas with conditions
Lesson details and objectives
Solve for a chosen symbol while keeping units and zero-coefficient cases visible.
- Isolate a chosen variable in a multi-symbol formula.
- State conditions required by division.
- Interpret exceptional cases and physical restrictions.
5 practice tasks.
Solve distance conditions with absolute value
Lesson details and objectives
Turn distance into two branches or an interval, including impossible negative distances.
- Interpret absolute value as nonnegative distance.
- Solve absolute-value equations and inequalities.
- Reject impossible right-hand sides before splitting cases.
5 practice tasks.
Solve rational equations without restoring excluded inputs
Lesson details and objectives
Record denominator restrictions before clearing fractions, and filter candidates afterward.
- Determine the original domain of a rational equation.
- Clear denominators on that domain.
- Check all candidates and distinguish cancellation from unrestricted equality.
5 practice tasks.
Check roots introduced by squaring
Lesson details and objectives
Respect the principal square root and use the original equation to remove extraneous candidates.
- Find real-domain and sign restrictions for square-root equations.
- Recognize that squaring is not generally reversible.
- Solve and check rational and irrational candidates appropriately.
5 practice tasks.
Factor structure and use zero products
Lesson details and objectives
Extract common factors, identify identities and distinguish factored expressions from equations.
- Factor common-factor, difference-of-squares and quadratic expressions.
- Verify a factorization by multiplication.
- Use the zero-product property only when the product equals zero.
5 practice tasks.
Classify and solve quadratic equations
Lesson details and objectives
Derive the formula by completing the square and distinguish repeated, irrational and complex roots.
- Use the quadratic formula with a nonzero quadratic coefficient.
- Interpret the discriminant over the real and complex numbers.
- Check roots and handle equations that are not actually quadratic.
5 practice tasks.