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A practice area for the math concepts Dr. Kat teaches parents โ built so students can reinforce what their parents have taught them, with explanations on every answer.
Properties of Numbers: The Power of Nothing & Fractions
Foundations: zero as a 0D point, the nesting Number Families (N โ Z โ Q), what a fraction is, dimensionality, fraction-to-decimal conversion, types of fractions, signed fractions on the number line, and synthesis review.
Lesson Page 1, Section 1
Understand zero dimensions (0D) as a single point with no length, width, or height โ and why zero still belongs in the family of natural numbers.
Lesson Page 1, Section 2
Classify numbers into Natural (N), Integer (Z), and Rational (Q) sets and understand how the families nest inside each other.
Lesson Page 1, Section 3
Understand fractions as parts of a whole โ the numerator counts the parts we have; the denominator counts the equal parts that make up the whole.
Lesson Page 1, Section 4
Understand how the same number can be shown as a 0D point or on a 1D number line โ and why each view is useful.
Lesson Page 1, Section 5
Convert fractions into decimals using long division, and tell terminating decimals apart from repeating decimals.
Lesson Page 1, Section 6
Classify fractions as unit, proper, improper, or mixed by comparing the numerator to the denominator.
Lesson Page 1, Section 7
Recognize that fractions can be positive or negative, and apply the sign rules: an even count of negatives gives a positive; an odd count gives a negative.
Lesson Page 1, Wrap-Up
Revisit the full arc of Lesson Page 1 โ dimensions, number families, fractions, and decimals โ and check that the big ideas hold together as one picture.
Addition & Subtraction Across Domains
In this unit you'll discover that the rules for addition and subtraction stay the same no matter what kind of number you're working with โ but the details change with each domain. You'll start by matching and converting measurement units (feet and inches), then apply the exponent product rule to powers of 10. From there you'll move through plotting inequalities on a number line, combining like terms across the number domains N, Z, Q, R, and C, and reading summation notation (โ) as structured repeated addition. You'll also explore conjugates โ pairs of expressions that cancel their variable terms when added โ and finish by sharpening your decimal arithmetic: aligning decimal points to carry when adding and to borrow when subtracting. The big idea threading all ten lessons together is this: match the type first, then combine.
Lesson Page 2, Section 1
Recognize that feet and inches are related units, and understand why you can only add like units while using the conversion factor 1 ft = 12 in to connect the two scales.
Lesson Page 2, Section 1 (Standard Notation)
Apply the product rule for exponents โ when multiplying two powers that share the same base, add their exponents โ including cases with negative exponents.
Lesson Page 2, Section 2
Solve multi-part measurement addition problems by first converting all values to the same unit, then combining them.
Lesson Page 2, Section 3
Simplify compound inequalities by evaluating both sides, then correctly represent the solution on a number line using open circles for exclusive endpoints.
Lesson Page 2, Section 4
Combine like terms correctly when working in N, Z, Q (with absolute values), R (with ฯ and radicals), and C (with complex numbers) โ recognizing that unlike terms cannot be merged.
Lesson Page 2, Section 5
Simplify both sides of a comparison by combining like terms, then determine the correct inequality symbol (< , > , or =) between the two results.
Lesson Page 2, Section 6
Read summation (โ) notation, identify the starting index, ending index, and rule, then evaluate each term and sum them to find the total.
Lesson Page 2, Section 7
Identify the conjugate of an expression by changing the sign between its terms, and explain why adding an expression to its conjugate always eliminates the variable term.
Lesson Page 2, Section 8
Add decimal numbers accurately by aligning decimal points and carrying correctly when a column sums to 10 or more.
Lesson Page 2, Section 9
Subtract decimal numbers accurately by aligning decimal points and borrowing from the next column when the top digit is smaller than the bottom digit.
Processes with Functions & Limits
In this unit you'll build the tools that bridge arithmetic and algebra. You'll start by solving for an unknown โ finding exactly how much a temperature increased โ then move to adding fractions that carry physical units like ยฐF and ยฐC. From there you'll plot inclusive intervals on a number line, learning when to use a closed circle and when to leave it open. You'll sharpen your vocabulary by distinguishing expressions (no comparison sign) from equations and inequalities, and then explore function inverses โ discovering that addition and subtraction undo each other perfectly. The unit closes with two vocabulary-rich sections: classifying mathematical objects as sets, sequences, operations, functions, or relations; and expressing the same calculation three different ways โ as an equation, as a function, and as a function evaluated at a specific input (including a gentle introduction to imaginary-number addition).
Lesson Page 3, Section 1
Solve a one-step equation by subtracting the starting value from both sides to find how much a quantity increased.
Lesson Page 3, Section 2
Add fractions that carry units by finding a common denominator, adding the numerators, and keeping the unit on the result.
Lesson Page 3, Section 3
Identify inclusive inequalities (โฅ and โค) and graph them on a number line using a closed circle at the endpoint.
Lesson Page 3, Section 4
Identify mathematical expressions by checking whether they contain a comparison symbol (=, <, or >); expressions have none.
Lesson Page 3, Section 5
Find the inverse of a function by reversing the operation โ if the function adds, its inverse subtracts; if it subtracts, its inverse adds.
Lesson Page 3, Section 6
Classify a mathematical object as a set, sequence, operation, function, or relation by checking its defining property.
Lesson Page 3, Section 7
Express the same mathematical idea using an equation, a function, and a function evaluated at a specific input โ and recognize that all three forms produce the same result.
Arithmetic Sequences & Sums
In this unit you'll discover that adding things together follows the same core rules no matter what you're adding โ vectors, limits, powers of 10, scientific notation, or fractions. You'll start by learning that vectors are math arrows whose + and โ signs tell you direction, and you'll add 1D vectors like โจ2โฉ and โจ4โฉ using integer rules. From there you'll explore limits โ the idea of a value x getting closer and closer to a target โ and learn to evaluate lim x and lim xยฒ separately before combining them. One-sided limits introduce the notation x โ 3โป and x โ 3โบ to describe which direction you're approaching from. The second half of the unit builds your place-value toolkit: you'll rewrite powers of 10 to match before adding (4 ร 10ยฒ + 4 ร 10ยณ), add same-power scientific notation expressions, and adjust results into standard form (12 ร 10โธ โ 1.2 ร 10โน). You'll also break any number into its exponential, multiplicative, and additive place-value parts, and you'll add fractions with unlike denominators by multiplying by a/a โ the form of 1 that makes denominators match without changing a fraction's value. Across every topic, the unifying idea is the same: rewrite so things are consistent, then combine.
Lesson Page 4, Section 1
Understand that positive and negative signs in vector expressions represent direction (forward or backward), and that adding two vectors combines their movements.
Lesson Page 4, Section 2
Substitute the values uโ = โจ2โฉ and vโ = โจ4โฉ into vector expressions and use integer rules to compute the result.
Lesson Page 4, Section 3
Understand what a limit means, evaluate simple limits of the form lim x and lim xยฒ, and add the results together.
Lesson Page 4, Section 4
Distinguish between a left-hand limit (x โ 3โป) and a right-hand limit (x โ 3โบ), and understand how the direction of approach is shown in the notation.
Lesson Page 4, Section 5
Rewrite expressions with different powers of 10 so the exponents match, then add the coefficients to find the sum.
Lesson Page 4, Section 6
Add two numbers written in scientific notation when both have the same power of 10, by adding the coefficients and keeping the exponent.
Lesson Page 4, Section 7
Convert a result like 12 ร 10โธ into standard scientific notation (1.2 ร 10โน) by adjusting the coefficient to be between 1 and 10 and compensating with the exponent.
Lesson Page 4, Section 8
Break any number into its place value parts using three equivalent forms: exponential (powers of 10), multiplicative, and additive.
Lesson Page 4, Section 9
Find a common denominator for two or more unlike fractions and add them by multiplying each fraction by a form of 1 (like 20/20) that preserves its value.
More lesson pages coming as Dr. Kat's curriculum expands into Classical Quest.
Total across all four pages: 34 lessons ยท 211 questions
Dr. Kat is a classical-education math instructor who helps parents understand and teach their children's math curriculum. This practice section is built around her pedagogy โ the concepts, the progression, and the "Big Ideas" are hers. Classical Quest provides the practice experience that reinforces what she teaches.
Questions or feedback on specific lessons? Use the feedback button in the corner โ we read everything.