Fundamentals of Arithmetic, Place Value, and Geometry (Lecture Notes)

A comprehensive set of practice flashcards covering addition, subtraction, multiplication, division, place value, and basic geometry as described in the lecture notes.

What is addition as a binary operation on natural numbers?

Addition is the operation +: N × N → N that maps (a, b) to a + b, the sum of a and b.

What is the cardinal (Vereinigung) interpretation of addition?

Two disjoint sets are merged into a single total set; the size is a + b.

What is the ordinal (Fortschreiten) interpretation of addition on the number line?

From a, move b steps to the right to reach a + b.

What is the operatoric interpretation of addition?

Increase a by b (a + b).

What is the closure property of addition in the natural numbers?

For all a, b ∈ N, a + b ∈ N (the sum is still a natural number).

What is the commutative law for addition?

a + b = b + a; the order of the summands does not matter.

What is the associative law for addition?

(a + b) + c = a + (b + c); summands can be regrouped without changing the result.

What is the neutral element for addition in N?

0; a + 0 = a.

What is the monotonicity property of addition?

If a ≤ c and b ≤ d, then a + b ≤ c + d.

Is subtraction always defined in natural numbers? Why or why not?

No. Subtraction a − b is defined only if a ≥ b; otherwise the result is not in N.

What are the two handlungsformen for teaching/doing addition?

Time-sequential (adding quantities one after another) and spatial-simultaneous (quantities laid out together as a whole).

What is the 'Vereinigungsvorstellung' for addition?

Union interpretation: combining two disjoint sets to form a larger set (e.g., 3 red + 4 blue = 7).

What is the 'Ergänzungsvorstellung' for addition?

Completion interpretation: determining what is needed to reach a target amount (e.g., 5 + ? = 8 → missing 3).

What is the 'Fortschreitungs-/Handlungsvorstellung' for addition?

Progression: counting from a forward by b steps.

What is subtraction as the inverse operation of addition?

a − b = c ⇔ a = b + c; defined in N only if b ≤ a.

What are the three interpretations of subtraction listed in the notes?

Abzieh-/Entnahmevorstellung (removing part of a set), Ergänzungsvorstellung (how much is needed to reach), and Vergleichsvorstellung (difference between two numbers).

Is subtraction closed in N?

No. For example, 3 − 5 is not defined in N.

Is subtraction commutative?

No; generally a − b ≠ b − a.

Is subtraction associative?

No; (a − b) − c ≠ a − (b − c).

What is the neutral element on the right for subtraction?

a − 0 = a.

What is self-subtraction in subtraction?

a − a = 0.

What is the monotonicity property of subtraction?

For fixed b > 0, if a ≥ c then a − b ≥ c − b.

What are key didactic tips for teaching subtraction?

Use both Abziehen and Ergänzen perspectives, connect to addition, and later introduce negative numbers in Z to complete the system.

What are the two main handlungsformen for subtraction?

Time-sequential (Abziehen) and spatial-simultaneous (Vergleichen/Ergänzen).

What are the four fundamental representations of multiplication?

Repeated addition (time-sukzessiv), array/rectangle (spatial-simultane), Cartesian product (combinatorial), and scaling (proportionality).

What is the neutral element of multiplication?

1; a · 1 = a.

What is the absorbing element of multiplication?

0; a · 0 = 0.

What is the distributive law of multiplication over addition?

a · (b + c) = a · b + a · c.

What is the monotonicity property of multiplication for c > 0?

If a ≤ b, then a · c ≤ b · c.

What is the commutative law for multiplication?

a · b = b · a.

What is the associative law for multiplication?

(a · b) · c = a · (b · c).

What is the operator aspect of multiplication?

Multiplication as an operator in algebraic contexts (e.g., a · (b + c) = a · b + a · c).

What are the two common visual/structural representations used to teach multiplication?

Wiederholte Addition (time-sequential) and Anordnungs-/Rechteckvorstellung (array/rectangular layout).

What is cartesian product in the context of multiplication?

Counting the number of ordered pairs from two sets (e.g., 3 shirts × 4 pants = 12 outfits).

What is the meaning of Skalierung/Vervielfachung in multiplication?

Scaling: enlarging or reducing by a fixed factor (e.g., 4 m becomes 12 m if scaled 3×).

What is the operator-related expression for division in extended arithmetic?

Division can be seen as multiplication by the reciprocal: a : b = a × (1/b).

What are the two primary interpretations of division?

Partitive (Aufteilen: dividing a set into equal groups) and Quotitive (Wie oft passt hinein?: how many times b fits into a).

When is division defined in N?

In N, division a : b is defined only if b ≠ 0 and b divides a exactly (a is a multiple of b).

Is division closed in N?

No; for example, 5 : 2 is not a natural number.

Is division commutative or associative in general?

Division is neither commutative nor associative in general.

What is the right-neutral element for division?

a : 1 = a.

What is self-division in division?

a : a = 1 (for a ≠ 0).

What is division by zero?

Undefined; there is no c with b · c = a when b = 0 and a ≠ 0.

What is the monotonicity of division for fixed b > 0?

The function a ↦ a / b is monotonically increasing in a.

What are the four division interpretations and related strategies?

Partitive (Aufteilen), Quotitive (Wie oft passt hinein?), Mess-/Skalierung (length segmentation), and Operatoraspekt (division as multiplication by a reciprocal).

What is Kopfrechnen (mental arithmetic) and how does it differ from written methods?

Kopfrechnen is mental calculation without fixed notation; it relies on number sense and quick strategies.

What is Halbschriftlich and Schrifftlich arithmetic?

Halbschriftlich: strategies are visible with steps as a bridge to algorithm; Schriftlich: fixed, place-value oriented procedures.

What is Bündeln in place-value systems?

Bundling: grouping units (e.g., 10 ones → 1 ten) as the basis of the decimal system.

What is the formula for the place value (Stellenwert) of a digit in base 10?

Stellenwert = Ziper × 10^k, where k is the position index from the right (0 for units).

What is a Stellenwertsystem?

A numeral system where the value of a digit depends on its symbol (Zipernwert) and its position (Stellenwert), e.g., ∑ a_k · b^k for base b.

What is a real-world example of a base-10 bundle?

In the decimal system, 10 ones make 1 ten, 10 tens make 1 hundred, and 10 hundreds make 1 thousand.

What is the difference between a digit (Zipern) and a number (Zahl)?

A digit is a symbol (0–9 in base-10) used in a positional representation; its value depends on its position in the number.

What is the general form of a base-b numeral system?

For a number with digits an … a0 in base b: value = ∑{k=0}^n ak · b^k, with 0 ≤ a_k < b.

What is a basic property of numeral systems with regard to leading zeros?

Every number has a unique representation, up to leading zeros which are not allowed in standard form.

What basic geometric shapes are covered in the Grundlagen der Schulgeometrie for triangles?

Isosceles triangle (two equal sides), equilateral triangle (all sides equal, all angles 60°), right triangle (Thales' theorem and Pythagoras).

What does Thales' theorem state?

A point on a semicircle subtends a right angle; i.e., the angle in a semicircle is a right angle.

What does the Pythagorean theorem state?

In a right triangle with hypotenuse c and legs a and b: a^2 + b^2 = c^2.

What is Höhensatz?

Altitude theorem: h^2 = p × q, where p and q are segments into which the altitude divides the hypotenuse.

What is Kathetensatz?

Leg theorem: a^2 = c × p or b^2 = c × q, relating a leg to the hypotenuse and the projection p or q.

What are the basic types of quadrilaterals listed?

Square (4 equal sides, 4 right angles), Rectangle (opposite sides equal, 4 right angles), Parallelogram (opposite sides equal and parallel), Rhombus (4 equal sides, diagonals perpendicular), Trapezoid (two sides parallel), Kite/Drachen (two adjacent sides equal; diagonals perpendicular).

What are the basic solid shapes covered?

Cube (6 square faces), Cuboid (6 rectangles), Prism (two congruent bases), Pyramid (base plus triangular faces to a apex), Cylinder (two circular faces + curved surface), Cone (circular base + lateral surface to apex).