Combined isometries and invariants · Higher
| English | 中文 | Pinyin |
|---|---|---|
| invariant/ɪnˈveərɪənt/ | 不变量 | bù biàn liàng |
Reflecting a logo and then sliding it can differ from sliding it and then reflecting it. The order of instructions is part of the transformation.
- Reflecting a logo and then sliding it can differ from sliding it and then reflecting it. The order of instructions is part of the transformation.
- This lesson studies invariant 不变量: A property preserved by a specified transformation.
Choose the mathematical structure
- Apply each transformation to the current image in the stated order. Rotations, reflections and translations preserve lengths and angles, so their combinations also preserve them. Reflections reverse orientation; rotations and translations preserve it. Two reflections in parallel lines give a translation; in intersecting lines they give a rotation through twice the directed angle between the mirrors.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines invariant?
A property preserved by a specified transformation.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Start with P=(3,2). Reflect in the y-axis to get (-3,2), then translate by (2,1) to get (-1,3). Reversing the order gives (5,3) then (-5,3), a different point. Reflecting in x=0 followed by x=2 maps (3,2) to (-3,2) then (7,2), equivalent to translation by (4,0). Two reflections reverse orientation twice, restoring it. Reflecting in the x-axis then y-axis sends (3,2) to (-3,-2), a 180° rotation about the origin. Every pairwise length and angle is unchanged, but position generally changes.
Combined isometries and invariants
Apply each transformation to the current image in the stated order
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
For (3,2), reflect in y-axis then translate by (2,1). Find final x.
-3+2=-1.
Test a tempting shortcut
- Preserved length does not mean every point stays fixed. Do not commute transformations unless a checked argument permits it. A single reflection reverses orientation, while two reflections restore it.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Changing the order of transformations never changes the final image. This claim is false. Explain which definition or assumption it violates.
Reverse those operations on (3,2). Find final x.
Translate gives x=5; reflect gives -5.
Changing the order of transformations never changes the final image.
Preserved length does not mean every point stays fixed. Do not commute transformations unless a checked argument permits it. A single reflection reverses orientation, while two reflections restore it.
Interpret a new situation
- AQA G8 Higher requires changes and invariance under combinations of rigid transformations. Describe the resulting map completely and test it on more than one point before making a whole-shape claim.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Reflect in x=0 then x=2. Find the translation x-component.
x becomes -x, then 4-(-x)=x+4.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A property preserved by a specified transformation. Choose the relationship, show the method, check its assumptions and interpret the result.