[ { "lesson": 10, "step": 4, "reason": "Remove budget ambiguity and correct the total-cost truth value.", "before": { "q": "If a ticket costs $12 and you have x dollars, the equation for buying 3 tickets is:
3(12) = x
", "answer": false }, "after": { "q": "Each ticket costs $12. If x is the total cost of exactly 3 tickets, is 3(12) = x correct?", "answer": true } }, { "lesson": 11, "step": 3, "reason": "The keyed $11 is the pre-tip share, not the $15 total each pays.", "before": { "q": "Three friends split a bill equally and added a $12 tip. Total was $45. How much did each pay?" }, "after": { "q": "Three friends split the bill equally before adding a $12 tip. The total including the tip was $45. What was each friend’s share before the tip?" } }, { "lesson": 9, "step": 6, "reason": "Make the denominator unambiguous.", "before": { "choices": [ "r = C ÷ 2π", "r = 2πC", "r = C − 2π" ] }, "after": { "choices": [ "r = C ÷ (2π)", "r = 2πC", "r = C − 2π" ] } }, { "lesson": 19, "step": 9, "reason": "Remove conflicting truck capacity and distinguish equivalent full loads from whole trips.", "before": { "q": "A tank holds 6000 gallons. A truck holds 1 cubic foot. There are 7.5 gallons per cubic foot. How many truck loads to fill a 5240 ft³ pool?" }, "after": { "q": "A water truck carries 6000 gallons per load. Using 7.5 gallons per cubic foot, which expression gives the number of equivalent full loads needed for a 5240 ft³ pool?" } }, { "lesson": 26, "step": 9, "reason": "Solve x² − x − 1 = 0; old ordered pairs satisfied neither both equations.", "before": { "choices": [ "(2, 4) and (-1, 1)", "(1, 2) only", "(3, 9) only", "No solution" ], "answer": "(2, 4) and (-1, 1)" }, "after": { "choices": [ "((1 + √5)/2, (3 + √5)/2) and ((1 − √5)/2, (3 − √5)/2)", "(1, 2) only", "(3, 9) only", "No solution" ], "answer": "((1 + √5)/2, (3 + √5)/2) and ((1 − √5)/2, (3 − √5)/2)" } }, { "lesson": 26, "step": 10, "reason": "x² − 2x + 1 = (x − 1)² has one distinct solution, (1, 2).", "before": { "answer": "Two solutions" }, "after": { "answer": "One solution" } }, { "lesson": 27, "step": 1, "reason": "An x² term alone does not establish degree two.", "before": { "concept": "A quadratic equation has an x² term (x-squared). Linear equations only have x. Quadratics make curved graphs called parabolas.
Compare:
Linear: y = 2x + 3 (straight line)
Quadratic: y = x² + 2x + 3 (parabola curve)
" }, "after": { "concept": "A quadratic equation in one variable can be written as ax² + bx + c = 0 with a ≠ 0; its highest power is 2. Linear equations have highest power 1. Quadratics make curved graphs called parabolas.
Compare:
Linear: y = 2x + 3 (straight line)
Quadratic: y = x² + 2x + 3 (parabola curve)
" } }, { "lesson": 27, "step": 1, "reason": "Require degree two rather than merely an x² term.", "before": { "q": "What makes an equation quadratic?", "choices": [ "It has two x terms", "It has an x² term", "It equals zero", "It has a fraction" ], "answer": "It has an x² term" }, "after": { "q": "What makes a polynomial equation quadratic?", "choices": [ "Its highest variable power is 2 after simplifying", "It has an x term", "It has three solutions", "It contains a fraction" ], "answer": "Its highest variable power is 2 after simplifying" } }, { "lesson": 27, "step": 4, "reason": "Symmetry alone does not explain all curved graphs; ask the specific squaring fact.", "before": { "q": "Why does x² create a curved graph?" }, "after": { "q": "For y = x², why do x = 2 and x = -2 give the same y-value?" } }, { "lesson": 27, "step": 2, "reason": "Distinguish an expression from an equation.", "before": { "concept": "Standard form is ax² + bx + c. The letters a, b, and c are just numbers (called coefficients).
Example: 3x² + 5x - 2
a = 3 (number with x²)
b = 5 (number with x)
c = -2 (number alone)
" }, "after": { "concept": "A quadratic expression has standard form ax² + bx + c with a ≠ 0. A quadratic equation is written ax² + bx + c = 0. The letters a, b, and c are just numbers (called coefficients).
Example: 3x² + 5x - 2
a = 3 (number with x²)
b = 5 (number with x)
c = -2 (number alone)
" } }, { "lesson": 30, "step": 3, "reason": "Repeated roots are not two distinct solutions.", "before": { "concept": "After factoring, set each factor equal to zero and solve separately. You get TWO solutions!
Example: (x + 3)(x - 7) = 0
Factor 1: x + 3 = 0 → x = -3
Factor 2: x - 7 = 0 → x = 7
Solutions: x = -3 or x = 7
" }, "after": { "concept": "After factoring, set each factor equal to zero and solve separately. Distinct linear factors give two solutions; repeated factors give only one distinct solution.
Example: (x + 3)(x - 7) = 0
Factor 1: x + 3 = 0 → x = -3
Factor 2: x - 7 = 0 → x = 7
Solutions: x = -3 or x = 7
" } }, { "lesson": 31, "step": 2, "reason": "The middle coefficient is not twice the constant.", "before": { "concept": "This creates a perfect square because (x + n)² = x² + 2nx + n². The middle term is always TWICE the last number.
Check: (x + 4)²
(x + 4)(x + 4) = x² + 4x + 4x + 16
= x² + 8x + 16
Middle is 2 × 4 = 8 ✓
" }, "after": { "concept": "This creates a perfect square because (x + n)² = x² + 2nx + n². The coefficient of x is twice n; the constant term is n².
Check: (x + 4)²
(x + 4)(x + 4) = x² + 4x + 4x + 16
= x² + 8x + 16
Middle is 2 × 4 = 8 ✓
" } }, { "lesson": 32, "step": 1, "reason": "Do not promise two real roots for every quadratic.", "before": { "concept": "The Quadratic Formula solves ANY quadratic equation. For ax² + bx + c = 0, the solutions are x = (-b ± √(b² - 4ac)) / 2a.
Remember:
Numerator: -b plus/minus square root of (b² - 4ac)
Denominator: 2a
The ± gives you TWO solutions
" }, "after": { "concept": "The Quadratic Formula solves ANY quadratic equation. For ax² + bx + c = 0, the solutions are x = (-b ± √(b² - 4ac)) / 2a.
Remember:
Numerator: -b plus/minus square root of (b² - 4ac)
Denominator: 2a
For real coefficients with a ≠ 0, the discriminant determines whether there are two, one, or no real solutions
" } }, { "lesson": 42, "step": 4, "reason": "x and x+2 are consecutive even integers when x is even.", "before": { "q": "What is the product of consecutive integers x and (x + 2)?" }, "after": { "q": "What is the product of consecutive even integers x and (x + 2), where x is even?" } }, { "lesson": 42, "step": 2, "reason": "A constant dimension can leave degree two.", "before": { "concept": "Volume of a box is length × width × height. This creates cubic polynomials.
Box: length 2x, width x, height 3
Volume = 2x × x × 3
= 6x × x
= 6x²
" }, "after": { "concept": "Volume of a box is length × width × height. The degree depends on which dimensions vary; three variable dimensions can create a cubic polynomial.
Box: length 2x, width x, height 3
Volume = 2x × x × 3
= 6x × x
= 6x²
" } }, { "lesson": 45, "step": 2, "reason": "The zero-power rule excludes zero.", "before": { "q": "Why does any number to the zero power equal 1?" }, "after": { "q": "Why does any nonzero number to the zero power equal 1?" } }, { "lesson": 48, "step": 1, "reason": "State the real-domain condition.", "before": { "concept": "The product rule: √(ab) = √a × √b. You can split radicals when multiplying!
Example: √12
√12 = √(4 × 3)
= √4 × √3
= 2√3
" }, "after": { "concept": "For a ≥ 0 and b ≥ 0, the real square-root product rule is √(ab) = √a × √b. You can split radicals when multiplying!
Example: √12
√12 = √(4 × 3)
= √4 × √3
= 2√3
" } }, { "lesson": 48, "step": 4, "reason": "Every integer has square factor 1.", "before": { "concept": "A radical is simplified when no perfect square factors remain under the radical.
Examples:
√7 already simplified (7 has no perfect square factors)
√8 = 2√2 not simplified (8 = 4 × 2)
" }, "after": { "concept": "A radical is simplified when no perfect square factors greater than 1 remain under the radical.
Examples:
√7 already simplified (7 has no perfect square factors)
√8 = 2√2 not simplified (8 = 4 × 2)
" } }, { "lesson": 48, "step": 4, "reason": "The example incorrectly labels the simplified form as unsimplified.", "before": { "concept": "A radical is simplified when no perfect square factors greater than 1 remain under the radical.
Examples:
√7 already simplified (7 has no perfect square factors)
√8 = 2√2 not simplified (8 = 4 × 2)
" }, "after": { "concept": "A radical is simplified when no perfect square factors greater than 1 remain under the radical.
Examples:
√7 already simplified (7 has no perfect square factors)
√8 is not simplified (8 = 4 × 2); its simplified form is 2√2
" } }, { "lesson": 49, "step": 2, "reason": "Different starting radicands can become like radicals after simplification.", "before": { "concept": "Unlike radicals (different radicands) CANNOT be combined. They stay separate!
Cannot combine:
2√3 + 5√2 stays as 2√3 + 5√2
Can combine:
2√3 + 5√3 = 7√3
" }, "after": { "concept": "Simplify radicals first. If their radicands are still different, they cannot be combined as like radical terms.
Cannot combine:
2√3 + 5√2 stays as 2√3 + 5√2
Can combine:
2√3 + 5√3 = 7√3
" } }, { "lesson": 52, "step": 1, "reason": "Avoid claiming exactly four possible representations.", "before": { "concept": "Functions can be shown in four ways: equations, tables, graphs, and mappings. All show the same relationship!
Example: y = 2x
Equation: y = 2x
Table: x=1→y=2, x=2→y=4
Graph: straight line through origin
" }, "after": { "concept": "Four common ways to represent functions are equations, tables, graphs, and mappings. All show the same relationship!
Example: y = 2x
Equation: y = 2x
Table: x=1→y=2, x=2→y=4
Graph: straight line through origin
" } }, { "lesson": 52, "step": 1, "reason": "Scope the question to the representations taught.", "before": { "q": "How many ways can you represent a function?" }, "after": { "q": "How many common representations are listed here: equations, tables, graphs, and mappings?" } }, { "lesson": 52, "step": 3, "reason": "Repeated inputs are only a problem when outputs differ.", "before": { "concept": "In a function table, each input (x) appears only once. If an input appears twice with different outputs, it's not a function.
Function table:
x=1→y=3, x=2→y=5, x=3→y=7 ✓
NOT a function:
x=1→y=3, x=1→y=5 (same input, different outputs)
" }, "after": { "concept": "In a function table, each input (x) has exactly one output. Repeating an identical input-output pair is allowed. If an input appears twice with different outputs, it's not a function.
Function table:
x=1→y=3, x=2→y=5, x=3→y=7 ✓
NOT a function:
x=1→y=3, x=1→y=5 (same input, different outputs)
" } }, { "lesson": 52, "step": 3, "reason": "Explain conflicting outputs, not repeated inputs alone.", "before": { "choices": [ "Yes, all different outputs", "No, x=2 appears twice", "Yes, inputs can repeat", "Need the equation" ], "answer": "No, x=2 appears twice" }, "after": { "choices": [ "Yes, all different outputs", "No, x=2 has two different outputs", "Yes, inputs can repeat", "Need the equation" ], "answer": "No, x=2 has two different outputs" } }, { "lesson": 52, "step": 9, "reason": "Explain conflicting outputs, not repeated inputs alone.", "before": { "choices": [ "Yes, all y-values different", "No, x=2 appears twice", "Yes, x-values can repeat", "Not enough data" ], "answer": "No, x=2 appears twice" }, "after": { "choices": [ "Yes, all y-values different", "No, x=2 has two different outputs", "Yes, x-values can repeat", "Not enough data" ], "answer": "No, x=2 has two different outputs" } }, { "lesson": 53, "step": 4, "reason": "A line does not start at its y-intercept.", "before": { "q": "For f(x) = x + 4, where does the graph start?" }, "after": { "q": "For f(x) = x + 4, where does the graph cross the y-axis?" } }, { "lesson": 54, "step": 8, "reason": "Retain the excluded zero input when cancelling x.", "before": { "q": "If f(x) = x² and g(x) = x, what is (f / g)(x)?", "choices": [ "x", "x³", "1", "2x" ], "answer": "x" }, "after": { "q": "If f(x) = x² and g(x) = x, what is (f / g)(x)? State the restriction on x.", "choices": [ "x, x ≠ 0", "x³", "1", "2x" ], "answer": "x, x ≠ 0" } }, { "lesson": 54, "step": 12, "reason": "Retain the excluded zero input when cancelling x.", "before": { "q": "If f(x) = 6x² and g(x) = 2x, what is (f / g)(x)?", "choices": [ "3x", "3x²", "12x³", "3" ], "answer": "3x" }, "after": { "q": "If f(x) = 6x² and g(x) = 2x, what is (f / g)(x)? State the restriction on x.", "choices": [ "3x, x ≠ 0", "3x²", "12x³", "3" ], "answer": "3x, x ≠ 0" } }, { "lesson": 56, "step": 1, "reason": "Not every function has an inverse function on its original domain.", "before": { "concept": "An inverse function \"undoes\" what the original function does. If f takes you from 2 to 7, then f⁻¹ takes you back from 7 to 2.
Example: f(x) = x + 3
f(2) = 5 (add 3)
f⁻¹(5) = 2 (subtract 3 to undo)
" }, "after": { "concept": "A one-to-one function has an inverse function that \"undoes\" it. Other functions need a suitable domain restriction first. If f takes you from 2 to 7, then f⁻¹ takes you back from 7 to 2.
Example: f(x) = x + 3
f(2) = 5 (add 3)
f⁻¹(5) = 2 (subtract 3 to undo)
" } }, { "lesson": 56, "step": 12, "reason": "Remove a second equivalent correct choice (-x+5 equals 5-x).", "before": { "choices": [ "x - 5", "-x + 5", "5 - x", "-x - 5" ] }, "after": { "choices": [ "x - 5", "x + 5", "5 - x", "-x - 5" ] } }, { "lesson": 58, "step": 4, "reason": "A ticket count can be zero unless a purchase is required.", "before": { "q": "For T(m) = cost of m movie tickets at $12 each, what's a reasonable domain?", "choices": [ "All real numbers", "Positive integers only", "Negative numbers", "Fractions only" ], "answer": "Positive integers only" }, "after": { "q": "For T(m) = 12m, the cost of m movie tickets including buying zero tickets, what is a reasonable domain?", "choices": [ "All real numbers", "Nonnegative integers (0, 1, 2, ...)", "Negative numbers", "Fractions only" ], "answer": "Nonnegative integers (0, 1, 2, ...)" } }, { "lesson": 9, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": null }, "after": { "concept": "

Assume each variable used as a divisor is nonzero.

" } }, { "lesson": 41, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": "To divide a polynomial by a monomial, divide EACH term separately.
Example: (6x³ + 12x²) ÷ 3x
6x³ ÷ 3x = 2x²
12x² ÷ 3x = 4x
Answer: 2x² + 4x
" }, "after": { "concept": "To divide a polynomial by a monomial, divide EACH term separately.
Example: (6x³ + 12x²) ÷ 3x
6x³ ÷ 3x = 2x²
12x² ÷ 3x = 4x
Answer: 2x² + 4x

Assume every divisor is nonzero; in questions dividing by a power of x, x ≠ 0.

" } }, { "lesson": 43, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": "When multiplying powers with the same base, ADD the exponents. The base stays the same!
Product Rule: x^a × x^b = x^(a+b)
x³ × x⁵ = x^(3+5) = x⁸
2⁴ × 2³ = 2^(4+3) = 2⁷
" }, "after": { "concept": "When multiplying powers with the same base, ADD the exponents. The base stays the same!
Product Rule: x^a × x^b = x^(a+b)
x³ × x⁵ = x^(3+5) = x⁸
2⁴ × 2³ = 2^(4+3) = 2⁷

Assume bases in denominators are nonzero; in the exponent-identification question assume x > 0 and x ≠ 1.

" } }, { "lesson": 44, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": "When raising a power to another power, MULTIPLY the exponents.
Power of a Power: (x^a)^b = x^(a×b)
(x³)⁴ = x^(3×4) = x¹²
(2²)³ = 2^(2×3) = 2⁶
" }, "after": { "concept": "When raising a power to another power, MULTIPLY the exponents.
Power of a Power: (x^a)^b = x^(a×b)
(x³)⁴ = x^(3×4) = x¹²
(2²)³ = 2^(2×3) = 2⁶

Assume denominators are nonzero.

" } }, { "lesson": 45, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": "ANY nonzero number raised to the power of zero equals 1. This is always true!
Zero Exponent Rule: x⁰ = 1
5⁰ = 1
100⁰ = 1
x⁰ = 1 (if x ≠ 0)
" }, "after": { "concept": "ANY nonzero number raised to the power of zero equals 1. This is always true!
Zero Exponent Rule: x⁰ = 1
5⁰ = 1
100⁰ = 1
x⁰ = 1 (if x ≠ 0)

Assume variable bases are nonzero wherever zero or negative powers appear.

" } }, { "lesson": 47, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": "A radical asks \"what number, when multiplied by itself, gives this?\" The square root √ is the most common radical.
Examples:
√9 = 3 (because 3 × 3 = 9)
√16 = 4 (because 4 × 4 = 16)
√25 = 5
" }, "after": { "concept": "A radical asks \"what number, when multiplied by itself, gives this?\" The square root √ is the most common radical.
Examples:
√9 = 3 (because 3 × 3 = 9)
√16 = 4 (because 4 × 4 = 16)
√25 = 5

Work over the real numbers; radicands of even roots must be nonnegative.

" } }, { "lesson": 50, "step": 1, "reason": "Make domain assumptions visible.", "before": { "concept": "A fractional exponent with 1 in the numerator means a root! The denominator tells you which root.
Pattern: x^(1/n) = ⁿ√x
x^(1/2) = √x (square root)
x^(1/3) = ³√x (cube root)
8^(1/3) = ³√8 = 2
" }, "after": { "concept": "A fractional exponent with 1 in the numerator means a root! The denominator tells you which root.
Pattern: x^(1/n) = ⁿ√x
x^(1/2) = √x (square root)
x^(1/3) = ³√x (cube root)
8^(1/3) = ³√8 = 2

For the variable-base exponent rules in this lesson, assume x > 0.

" } } ]