Grades 3–5Category TheoryPlanned
One thing after another
Doing one action and then another: chains of actions on shapes, numbers and objects, how the order can matter, how to undo a chain by undoing its steps in reverse order, and arrow pictures showing how one thing leads to another.
math.cat.chaining-actions
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 1
You will learn to
- 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.
Definition
- 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.
Theorem
- '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.
Common misconceptions
- Assuming the order of two actions never matters.
- Undoing a chain by undoing its steps in the original order.
Leads to
Optional discoveries
- 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