Higher Level only. No prior matrix algebra is assumed beyond multiplication, which is why the order error is so common: the multiplication is new at the same moment the geometry is.
Step from view 2 to view 3 and back, twice. Nothing changes except which mirror line is dashed and where the tracked corner lands: (0, −2) one way, (0, 2) the other. The determinant line underneath reads −1 in both.
Then ask the class to tell the two transformations apart using only the determinant. They cannot. That is the lesson, and it is better reached by failing at the task than by being told.
View 4 is the other half of the sub-topic and it is quick: the unit square becomes 2 by 3, and 6 is both the area and the determinant. Leave it up while they do questions 1 and 2.
| Question | Answer |
|---|---|
| 1. det (2, 0; 0, 3) | 2×3 − 0×0 = 6. |
| 2. Area of the image of a triangle of area 4 | 4 × 6 = 24. |
| 3. det of a 30° rotation | cos²30° + sin²30° = 1, as for every rotation. |
| 4. (2, 0) rotated 90° anticlockwise, then reflected in the x-axis | (2,0) → (0,2) → (0,−2), so the y-coordinate is −2. |
| 5. What two equal determinants tell you | B. Area preserved, orientation flipped, and nothing about which transformation. |
Question 3 is worth doing without a calculator, because cos² + sin² = 1 is the identity from 3.8 and it is the whole answer. A student who types the matrix in and reads 1 off the screen has learned nothing from it; a student who sees the Pythagorean identity has learned that every rotation has determinant 1.
1 markEach transformation written as a matrix.
1 markThe product, in the right order.
1 markThe image, determinant or area factor as asked.
Markschemes give the order mark on the product itself, so a student who writes RM and then carries it through correctly loses the order mark and the answer mark both. It is the single most expensive habit in the sub-topic.
| They wrote | What happened |
|---|---|
| 5 | On question 1, added the diagonal rather than multiplying it. Usually a student who has met "trace" somewhere. |
| 36 | Squared the determinant. They have confused it with the enlargement case, where the factor k gives k², and have squared a determinant that was already the area factor. |
| 10 | On question 2, added 4 and 6. A scale factor multiplies, and the word "factor" is worth saying aloud. |
| 8 or 12 | On question 2, used only one of the two stretch factors. They have not noticed that both directions scale. |
| 0.866 | On question 3, gave cos 30°, the top-left entry. They have read an entry instead of computing ad − bc. |
| 2 on question 4 | The order error, and the one to hunt for. They reflected first. Reflecting (2,0) in the x-axis does nothing, so the rotation then gives (0,2). Everything about their working looks right. |
| 0 on question 4 | Gave the x-coordinate. Worth a word about reading the question, not about matrices. |
An answer of 2 on question 4 cannot be distinguished from correct work by looking at the sign or the size, which is why the send-(1,0)-through check in the student page matters more here than any arithmetic check.
"Why is the first transformation on the right?" Because of where the vector sits. The image is M(Rx), the vector is on the right, and whatever touches it first is next to it. Write M(Rx) = (MR)x on the board once and the convention stops being arbitrary. It is worth saying that this is also why function composition reads right to left: f(g(x)) does g first.
"Do I have to learn the matrices?" No, and teaching them as a list to memorise is the slower route. The first column is the image of (1, 0) and the second is the image of (0, 1). For a reflection in y = x, (1,0) goes to (0,1) and (0,1) goes to (1,0), so the matrix is (0, 1; 1, 0), derived in ten seconds with no recall.
"Can I do a translation with a matrix?" Not with a 2 by 2 one, because M0 = 0 for every matrix, so the origin can never move. A translation is a vector you add. If someone has met 3 by 3 homogeneous coordinates elsewhere, acknowledge it and say it is off this syllabus.
"What does a negative determinant mean physically?" The shape has been turned over, like a glove swapping hands. Label the triangle vertices P, Q, R in the figure and watch the order go from anticlockwise to clockwise. No amount of rotating can achieve it, which is why rotations have determinant +1 and reflections −1.
| Stage | What to do |
|---|---|
| Demonstrate | Store R and M, then put m*r and r*m on the screen one under the other. Two visibly different matrices. Then det(m*r) and det(r*m), both −1. Those four lines on one screen are the lesson, and they take about forty seconds once the matrices are stored. |
| Where they stick | Entering a matrix at all. On the Nspire it is ctrl × for the template and ctrl var to store; students who type the entries without storing have to retype them for every product. On the Casio, the dimension has to be set in MAT/VCT before any entry is accepted, and EXIT is what leaves the editor. Budget five minutes for this the first time and it never costs again. |
| The check | Send (1, 0) through by hand and compare it with the picture. It is two multiplications, it uses only the first column, and it catches the order error, which no determinant check can. |
Degrees or radians does not matter until a rotation by a named angle appears, at which point it matters completely. Set degrees before any question with a degree symbol in it.
| Step | What |
|---|---|
| 1 | Build three matrices from scratch using the images of (1,0) and (0,1). No list to copy. |
| 2 | Ask for the image of the flag under R, then under M. Straightforward, and it establishes the picture. |
| 3 | Now: rotate then reflect. Collect answers. Both (0,2) and (0,−2) will come back. |
| 4 | Figure views 2 and 3, stepped back and forth. Ask them to separate the two using the determinant. |
| 5 | M(Rx) = (MR)x on the board, once. |
| 6 | View 4: the determinant as an area factor, then questions 1 and 2. |
| 7 | The zero-determinant matrix (2, 4; 1, 2) as a closing thirty seconds, if there is time. |
Do not say "determinant −1 means a reflection". Every reflection has determinant −1 and the converse is false. The counterexample to have ready is (2, 0; 0, −½): its determinant is −1, so it preserves area and flips orientation, and it is not a reflection, because it stretches the x direction by 2 and squashes the y direction by a half. The determinant carries two pieces of information and no more; it is not an identification.
Do not say "matrix multiplication is just like numbers, but longer". The one way it is not like numbers is the one that matters here. Say the opposite: the order changes the answer, and that is the whole content of the composition work.