One thing after another

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 1

ستتعلّم

• Carry out chains of actions such as 'add 5, then double' and predict their results.
• Find examples where swapping the order of two actions changes the result, and examples where it does not.
• Undo a chain of actions by undoing its steps in reverse order, as with socks and shoes.
• Draw arrow pictures showing how objects are related or transformed, and follow paths of arrows.

تعريف

• Chaining two actions means doing the first and then doing the second to the result.
• An arrow picture shows objects as dots and actions or relationships as arrows from one dot to another.

مبرهنة

• 'Add 5, then double' and 'double, then add 5' usually give different results.
• To undo 'put on socks, then put on shoes', take off the shoes first and then the socks.
• Doing 'turn a quarter turn' four times in a row has the same effect as doing nothing.

مفاهيم خاطئة شائعة

• Assuming the order of two actions never matters.
• Undoing a chain by undoing its steps in the original order.

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اكتشافات اختيارية

• Composition Chaining actions becomes composition of functions g ∘ f, and undoing a chain becomes the rule (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹.
• Groups The socks-and-shoes rule (gh)⁻¹ = h⁻¹g⁻¹ holds in every group.
• ← Following and giving precise instructions
• ← Expressions, unknowns and rules
• ← Symmetry: slides, turns and flips