Mathematische Sachaufgaben & Denkmodelle · Arithmetic Logic Models
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Anschauliche mathematische Denkmodelle: Überschuss-Unterschied, Frosch im Brunnen und Flaschentausch
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Typische Beispielaufgaben & Prüfungsfallen
Worked Problem
Aufgabe: A snail in a 12m well climbs 4m by day and slips 2m by night. How many days to climb out?
Schrittweise Herleitung:
- Target before final day = 12 - 4 = 8m. Daily net = 4 - 2 = 2m. Days to 8m = 8 / 2 = 4 days. On day 5, it climbs 4m and reaches 12m. Total = 5 days.
Worked Problem
Aufgabe: 3 empty bottles trade for 1 full beer. You buy 24 beers. What is the maximum total beers you can drink?
Schrittweise Herleitung:
- Net exchange: 3 empties = 1 full (beer + empty) means 2 empties = 1 drink. Extra drinks = 24 / (3 - 1) = 12. Total drinks = 24 + 12 = 36 beers.
Arbeitsblatt für den Unterricht & Hausaufgaben
Q1. Variant 1: Ming has 18 pencils, Hua has 10. How many must Ming give Hua for equal? For Hua to have 3x Ming?
Answer Key: Equal: difference is 8; give 8/2 = 4. 3x: total is 28 (4 shares); Ming keeps 28/4 = 7; gives 18 - 7 = 11 pencils.
Q2. Variant 2: Candy sharing: 6 each lacks 12; 4 each leaves 8 surplus. How many kids and candies?
Answer Key: Kids = (Surplus + Deficit) / Rate Delta = (8 + 12) / (6 - 4) = 10 kids. Candies = 10 * 4 + 8 = 48 candies.
Q3. Variant 3: Monkey eats half + 1 peach daily. After 3 days, 1 peach remains. How many peaches originally?
Answer Key: Work backwards: Before day 3 = (1 + 1)*2 = 4. Before day 2 = (4 + 1)*2 = 10. Initial = (10 + 1)*2 = 22 peaches.
Häufige Fragen & typische Missverständnisse
Why is transferring items called "Give One, Shift Difference by Two"?
Moving 1 block from A to B reduces A by 1 and increases B by 1, altering their relative difference by exactly 2!
Why does well depth divided by daily net climb fail in the frog well problem?
Because net climb factors in the night slip. On the final day, the frog tops the rim during daytime and escapes; the subsequent night slip never happens!
If the shopkeeper refuses to lend an empty bottle, do you drink fewer sodas?
Yes, exactly 1 fewer! 2 remaining empties cannot redeem without borrowing; borrowing achieves the theoretical maximum.
How to summarize all surplus-deficit formulas in a single sentence?
Denominator is always the rate delta. Numerator rule: "Opposites Add, Sames Subtract" (Surplus+Deficit adds; Surplus-Surplus subtracts).