Whole-number arithmetic

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 3

ستتعلّم

• Multiply and divide multi-digit whole numbers, explaining each step of the algorithms through place value and the distributive law.
• Divide with remainder, writing a = qb + r with 0 ≤ r < b.
• Use the commutative, associative and distributive laws to calculate flexibly, for example 7 × 98 = 7 × 100 − 7 × 2.

تعريف

• a × b is the total number of objects in a equal groups of b, or the number of cells in an a-by-b array.
• Division with remainder: for whole numbers a and b with b > 0, there are unique whole numbers q and r with a = qb + r and 0 ≤ r < b.
• The distributive law: a × (b + c) = a × b + a × c.

مبرهنة

• Multiplication is commutative and associative and distributes over addition.
• The quotient and remainder in division with remainder are unique.
• Division by zero is undefined: for instance, no number q satisfies q × 0 = 5.

مفاهيم خاطئة شائعة

• Believing multiplication always makes a number bigger (false for 0 and 1, and later for fractions less than 1).
• Allowing a remainder as large as, or larger than, the divisor.
• Treating 'carrying' as a trick rather than as regrouping ten ones into one ten.

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اكتشافات اختيارية

• Rings, ideals and fields The laws that make arithmetic work (commutative, associative, distributive) later become the axioms of a ring.

مسارات التعلّم

• Conventional K–12 progression
• Adult beginner pathway

قراءات إضافية

• Liping Ma, Knowing and Teaching Elementary Mathematics (1999)
• Hung-Hsi Wu, Understanding Numbers in Elementary School Mathematics (2011)