Mathematische Sachaufgaben & Denkmodelle · Visual Fractions Area Models
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Typische Beispielaufgaben & Prüfungsfallen
Worked Problem
Aufgabe: Calculate 3/4 + 2/5. Show common denominator steps and simplified mixed fraction.
Schrittweise Herleitung:
- LCM(4, 5) = 20. 3/4 = 15/20, 2/5 = 8/20. Sum = 15/20 + 8/20 = 23/20 = 1 3/20.
Worked Problem
Aufgabe: Explain why 2/3 * 4/5 = 8/15 using an area grid model.
Schrittweise Herleitung:
- Unit square partitioned into 3 rows and 5 columns (15 tiles). Shading 2 rows and 4 columns creates an 8-tile overlap out of 15, yielding 8/15.
Arbeitsblatt für den Unterricht & Hausaufgaben
Q1. Variant 1: Ming eats 3/8 pizza, sister eats 1/4 pizza. What fraction eaten? What fraction remains?
Answer Key: 1/4 = 2/8. Total eaten = 3/8 + 2/8 = 5/8. Remaining = 1 - 5/8 = 3/8.
Q2. Variant 2: From a 6m rope, compare cutting 2/3 meter vs cutting 2/3 of its length. What remains?
Answer Key: Cutting 2/3 meter leaves 6 - 2/3 = 5 1/3 m. Cutting 2/3 of length leaves 6 * (1 - 2/3) = 2m.
Q3. Variant 3: Calculate 5/6 / 2/3 and explain why we invert and multiply.
Answer Key: 5/6 * 3/2 = 15/12 = 5/4 = 1 1/4. Dividing by 2/3 means dividing by 2 then multiplying by 3, which is multiplying by 3/2.
Häufige Fragen & typische Missverständnisse
Why can't we add numerators and denominators directly in fraction addition?
Because denominators define slice unit size! 1/2 and 1/3 are different sized blocks; you cannot sum counts until sliced to uniform size (common denominator).
What is fraction simplification (reduction)? Why does value stay constant?
Dividing numerator and denominator by their greatest common factor. Grouping multiple small tiles into fewer equivalent large blocks leaves total value invariant.
Distinguish proper, improper, and mixed fractions.
Proper: numerator < denominator (< 1). Improper: numerator ≥ denominator (≥ 1). Mixed: whole number + proper fraction (e.g. 1 2/3).
Why does dividing by a fraction equal multiplying by its reciprocal?
Multiply top and bottom of complex fraction [A] / [2/3] by 3/2: denominator becomes (2/3)*(3/2) = 1, leaving A * (3/2).